A Mott Glass to Superfluid Transition for Random Bosons in Two Dimensions

S. Iyer, D. Pekker, and G. RefaelDepartment of Physics, California Institute of Technology,

MC 149-33, 1200 E. California Blvd., Pasadena, CA 91125(Dated: October 18, 2011)

We study the zero temperature superfluid-insulator transition for a two-dimensional model ofinteracting, lattice bosons in the presence of quenched disorder and particle-hole symmetry. Wefollow the approach of a recent series of papers by Altman, Kafri, Polkovnikov, and Refael, in whichthe strong disorder renormalization group is used to study disordered bosons in one dimension.Adapting this method to two dimensions, we study several different species of disorder and uncoveruniversal features of the superfluid-insulator transition. In particular, we locate an unstable finitedisorder fixed point that governs the transition between the superfluid and a gapless, glassy insulator.We present numerical evidence that this glassy phase is the incompressible Mott glass and that thetransition from this phase to the superfluid is driven by percolation-type process. Finally, we provideestimates of the critical exponents governing this transition.

I. INTRODUCTION

In the 1980s, seminal experiments on helium adsorbedin Vycor first attracted the attention of theorists to therandom boson problem1,2. The onset of superfluidity inthis disordered system showed, in some respects, strik-ing similarities to ideal Bose gas behavior. Thus, in thedecade before the realization of Bose-Einstein conden-sation in cold atomic gases, disordered bosonic systemswere actually proposed as possible realizations of this elu-sive phenomenon. While studies of disordered bosons didnot ultimately lead to the observation of Bose-Einsteincondensation, the random boson problem continued tostimulate theoretical activity because of its considerablerichness. The interplay of interactions and disorder leadsto a variety of phases in bosonic systems; the descriptionof these phases and the transitions between them is anongoing challenge3,4.

The theoretical investigation of random bosons in onedimension was pioneered by Giamarchi and Schulz, whodescribed the transition to superfluidity in the presenceof perturbatively weak disorder5. Subsequently, Fisher,Weichman, Grinstein, and Fisher expanded upon thework of Giamarchi and Schulz by establishing the nowcanonical zero temperature phase diagram of the Bose-Hubbard model with chemical potential disorder. Thesuperfluid and Mott insulating phases of the clean modelare separated by another insulating phase, the Bose glass.In this glassy phase, there exist rare-regions in which theenergetic gap for adding another boson to the systemvanishes. However, any additional bosons are localizedby the disordered environment, rendering the phase agapless, compressible insulator. Fisher et al. argued thatthere should be no direct superfluid-Mott insulator tran-sition in the presence of quenched, uncorrelated disorder,or in other words, that the Bose glass always intervenesbetween the phases of the clean model4. The general pic-ture formulated by these authors has been vindicated byover two decades of subsequent theoretical and numericalwork6. On the experimental front, following the observa-tion of the Mott insulator to superfluid transition in cold

atomic gases by Greiner et al.7, there has been progresstowards the realization of a Bose glass, including sugges-tive but inconclusive evidence that the phase has alreadybeen observed8–11.

Meanwhile, evidence has accumulated that disorderedbosonic systems may exhibit more exotic glassy phases inthe presence of particle-hole symmetry. One such phase,the so-called Mott glass, differs from the Bose glass inthat it is incompressible. This phase was originally pro-posed for systems of disordered fermions by Giamarchi,Le Doussal, and Orignac, but these authors predictedthat it can exist in bosonic systems as well12. Subse-quently, Altman, Kafri, Polkovnikov, and Refael studieda variant of the Bose-Hubbard model in which chemicalpotential disorder is omitted in favor of strong disorderin the on-site interaction and hopping. In the limit oflarge, commensurate boson filling, this model is equiv-alent to a chain of quantum rotors that can describe aJosephson junction array. Relative to the large filling,this rotor model exhibits an exact particle-hole symmetrywhich has important consequences for the phase diagram.Through a strong disorder renormalization group analy-sis, Altman et al. found that this symmetry results in theappearance of a Mott glass phase in this model13. Thesame authors later considered a generalization of this ro-tor model which allows for random offsets in the filling,effectively reintroducing a chemical potential. Genericdisorder in the random offsets violates the particle-holesymmetry, and in this case, the rotor model exhibits theusual Bose glass phase14,15. This breakdown of the Mottglass demonstrates the link between exotic phases andsymmetries of the disordered Hamiltonian. Irrespectiveof the identity of the glassy phase, Altman et al. alsofound that the superfluid-insulator transition at strongdisorder lives in a different universality class from theweak disorder transition of Giamarchi and Schulz13,14.

In this paper, we study the two-dimensional analogueof the rotor model considered by Altman et al. in orderto investigate the particle-hole symmetric random bo-son problem in d > 1. One of our goals is to look forthe Mott glass, but more generally, we aim to charac-

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terize the phases and the superfluid-insulator transitionof our model. Like Altman et al. the tool we use todo this is the strong disorder real space renormalizationgroup, first formulated by Ma and Dasgupta16 to studythe 1D Heisenberg antiferromagnet about thirty yearsago. A numerical application of the RG by Bhatt andLee followed shortly thereafter17, and the method waslater expanded upon and applied to more general spinmodels by Daniel Fisher18. Strong disorder renormaliza-tion has proven to be a powerful tool in the analysis ofseveral disordered systems, especially in one dimension.In higher dimensions, application of the RG has beenrarer, because in addition to the generic intractabilityof analytical approaches19, there are few known transi-tions that exhibit so-called infinite randomness, a prop-erty that guarantees that the RG becomes asymptoti-cally exact near criticality. The random boson model isnot expected to exhibit infinite randomness and, indeed,the numerical data we present in this paper is consistentwith this expectation. Hence, in addition to physicalquestions about the phases and phase transitions of themodel, our work also aims to address a methodologicalquestion: might the strong disorder RG give useful infor-mation about a model, even when confronted with thetwin difficulties of higher dimensionality and the absenceof infinite randomness?

Summary of the Results

Our main results are as follows: we present numericalevidence for the existence of an unstable finite disorderfixed point of the RG flow, near which the distributionsof Josephson couplings Jjk and charging energies Uj inthe rotor model flow to universal forms. A schematicpicture of this unstable fixed point and the flows in itsvicinity is given in Figure 1.

To the left of the diagram, flows propagate towards aregime in which the ratio of J , the mean of the Joseph-son couplings, to U , the mean of the charging energies,vanishes; meanwhile, the ratio of ∆J , the width of theJosephson coupling distribution, to J grows very large.These flows terminate in one of two insulating phases.The first is a conventional Mott insulator, in which itis energetically unfavorable for the particle number tofluctuate from the large filling at any site. The other isa glassy phase, in which there exist rare-regions of su-perfluid ordering. As the thermodynamic limit is ap-proached, arbitrarily large rare-regions appear, drivingthe gap for charging the system to zero. However, thedensity of the largest clusters decays exponentially intheir size, and the size of the largest cluster in a typi-cal sample does not scale extensively in the size of thesystem. Moreover, the largest clusters are so rare thatthey cannot generate a finite compressibility. Thus, thephase is a Mott glass.

This insulating phase gives way to global superfluiditywhen the rare-regions of superfluid ordering percolate,

producing a macroscopic cluster of superfluid ordering.The appearance of the macroscopic cluster is associatedwith flows that propagate towards the lower right of Fig-ure 1, indicating that the unstable fixed point governsthe glass-superfluid transition. Our numerical implemen-tation of the strong disorder RG allows us to extract es-timates for the critical exponents that characterize thistransition. We are thus able to construct a compellingpicture of the superfluid-insulator transition: a picturethat must, however, be checked by other methods be-cause of the perils of employing the strong disorder RGmethod in the vicinity of a finite disorder fixed point.

Organization of the Paper

We begin our analysis of the disordered rotor modelin Section II by introducing the model, noting its rela-tionship to the standard disordered Bose-Hubbard modeland its special symmetries. We also discuss the cleanlimit and the disordered problem in one dimension. Sec-tion III is devoted to a description of the strong disor-der renormalization group, as applied to the disorderedrotor model. In discussing the method, we emphasizethe adaptations needed to use the technique in dimen-sions greater than one. We then present data collectedfrom our numerical implementation of the strong disor-der renormalization procedure in Section IV and sub-sequently explore what the data can tell us about thezero temperature phases and quantum phase transitionsof our random boson model in Section V. In Section VI,we summarize the results, make connections to experi-ments, and give an outlook.

II. THE MODEL

In motivating the model that we study in this paper,we begin by writing down a disordered Bose-HubbardHamiltonian that includes randomness in the interactionand hopping along with the usual chemical potential dis-order:

Hbh = −∑〈jk〉

tjk(a†j ak + a†kaj) +∑j

Uj a†j aj(a

†j aj − 1)

−∑j

µj a†j aj (1)

Here, the creation and annihilation operators satisfybosonic commutation relations:

[aj , a†k] = δjk (2)

and the hopping is between all nearest-neighbor sites onan L × L square lattice with periodic boundary condi-tions. An alternative representation of this model is givenby the number and phase operators:

aj = eiφj√Nj (3)

3

!"#$%&'()*"+#

!"#,'-.+/'01#

FIG. 1: A schematization of the universal features of the pro-posed flow diagram. The x-axis gives the ratio of the meanof the renormalized Josephson coupling distribution to themean of the renormalized charging energy distribution. They-axis gives the ratio of the standard deviation of the Joseph-son coupling distribution to its mean. In this context, theJosephson coupling distribution only includes the dominant2N couplings in the renormalized J distribution. See the textof Section ?? for the reasoning behind the exclusion of weakerJosephson couplings from statistics.

[φj , Nk] = iδjk (4)

In terms of these operators:

Hbh = −∑〈jk〉

tjk

(√Nje

−iφjeiφk√Nk

+

√Nke

−iφkeiφj√Nj

)+∑j

Uj(Nj − 1)Nj −∑j

µjNj (5)

To obtain a large, commensurate boson filling N0, thechemical potential is tuned such that the on-site inter-action and chemical potential terms are minimized forNj = N0. This consideration fixes µj = (2N0 − 1)Uj .Then, if we expand the number operators around thislarge filling as Nj = N0 + nj , the Hamiltonian becomes:

Hbh = −∑〈jk〉

tjkN0

(√1 +

njN0

e−iφjeiφk√

1 +nkN0

+

√1 +

nkN0

e−iφkeiφj√

1 +njN0

)+∑j

Uj n2j + (const.) (6)

The operators nj now correspond to the particle num-ber deviations from the large filling N0. As such, nj cantake on any integer value from −N0 to∞, but we assumethat N0 is so large that we can let nj run from −∞ to

∞. The same approximation allows us to drop sublead-ing (in 1

N0) terms in the hopping. We finally define the

couplings Jjk = 2tjkN0 to arrive at the quantum rotorHamiltonian:

H = −∑〈jk〉

Jjk cos(φj − φk) +∑j

Uj n2j (7)

This model, constructed as the large filling limit of aBose-Hubbard Hamiltonian, can also be viewed as a de-scription of a two-dimensional array of superconductingislands connected by Josephson junctions13,14. Moreover,Vosk and Altman have recently demonstrated that theone-dimensional model is relevant to cold atomic gasesof rubidium-8720.

When the Josephson couplings Jjk and charging en-ergies Uj are uniform, the rotor model (7) exhibits aquantum phase transition between superfluid and Mottinsulating phases at zero temperature. This transition isin the universality class of the three-dimensional classicalXY model3,4,21, and one recent study determines that thetransition occurs at J

U ≈ 0.34522. The critical exponentgoverning the divergence of the correlation length at theclean transition is ν ≈ 0.66323. This exponent violatesthe Harris criterion:

νd ≥ 2 (8)

when d = 2. Violation of the Harris criterion genericallyindicates that disorder will either change the universal-ity class of the clean transition or completely smear itaway. In the former case, a Griffiths phase will sepa-rate the phases of the clean system24,25. The nature ofthis intervening phase depends upon the specific model inquestion, but it is generically insulating. In 1D at T = 0,Altman et al. found an incompressible Mott glass phaseand a quantum phase transition of Kosterlitz-Thoulesstype between this glassy phase and the superfluid13. TheRG fixed point that controls this transition actually oc-curs at a point in the flow diagram where all Uj = 0,meaning that the transition can be tuned by only vary-ing the disorder in the Josephson couplings at arbitrarilyweak interaction strength.

In our work, we introduce disorder by choosing theinitial distributions of charging energies and Josephsoncouplings, Pi(U) and Pi(J) to have one of the followingforms:

1. Gaussian distributions truncated at three standard de-viations:

Pi(x) ∝ exp

[− (x− x0)2

2σ2x

](9)

for x ∈ (x0 − 3σx, x0 + 3σx).

2. Power law distributions with upper and lower cutoffs:

Pi(x) =η + 1

xη+1max − xη+1

min

xη (10)

for x ∈ (xmin, xmax).

4

3. Flat distributions with upper and lower cutoffs:

Pi(x) =1

xmax − xmin(11)

for x ∈ (xmin, xmax). Of course, this is just a powerlaw with exponent η = 0.

4. “Bimodal” distributions consisting of two flat peakscentered at x` and xh:

Pi(x) =1

2δx(12)

for x ∈ (x` − δx2 , x` + δx

2 ) and x ∈ (xh − δx2 , xh + δx

2 ).

All of these distributions have positive upper and lowercutoffs (xmin and xmax respectively) and have zero weightfor x outside of these bounds. This restriction avoidsthe complications of frustration in the phase degrees offreedom and the pathologies of the particle sinks that re-sult from on-site charging spectrums that are unboundedfrom below.

Even in the presence of disorder, the Hamiltonian (7)respects two important symmetries. First, there is theglobal U(1) phase rotation symmetry:

φj → φj + ϕ (13)

This means that the Hamiltonian conserves total particlenumber:

ntot =∑j

nj (14)

The model is also globally particle-hole symmetric:

nj → −njφj → −φj (15)

The particle-hole symmetry exists because the chemicalpotential coupling to the true particle number Nj hasbeen tuned precisely to the value that enforces the large,commensurate filling. If this chemical potential is al-lowed to deviate from this value, then it would manifestin a chemical potential coupling to the particle numberfluctuation nj , or equivalently, in offsets to the large fill-ing. Such terms are absent from our rotor Hamiltonian(7), but let us momentarily consider the on-site chargingspectrum (as a function of nj) in the general situationwhere a filling offset δnj may be present:

Ej(nj) = Uj(nj − δnj)2 (16)

The integer value of nj that minimizes this energychanges at half-integer δnj . For sufficiently strong disor-der in δnj , the introduction of an arbitrarily small globalchemical potential shift will bring a finite fraction of sitesj arbitrarily close to these density changing points. Thus,a finite density of particles will be added to the system,making it compressible. Nevertheless, these particles will

be localized by the disordered environment, leaving thesystem globally insulating. This is the mechanism behindthe formation of the Bose glass4. The situation changesat two special particle-hole symmetric points. One oc-curs when δnj is restricted to be integer or half-integer.Then, at the half-integer sites, there is a degeneracy inthe charging spectrum:

Ej

(δnj −

1

2

)= Ej

(δnj +

1

2

)(17)

In the 1D model, Altman et al. showed that this degen-eracy gives rise to a random singlet glass14, but we willnot explore this situation in this paper. Instead, we willfocus on the other particle-hole symmetric point whereδnj = 0 for all sites j. Now, if the Uj are distributed suchthat Umin > 0, the on-site charging spectrum always hasa unique minimum that is protected by a gap. The usualmechanism for Bose glass formation is evaded, and thisallows for the possibility of realizing more exotic glassyphases. Thus, the particle-hole symmetry is a crucialfeature of our model that can influence the nature of theintervening glassy phase.

III. STRONG DISORDER RENORMALIZATIONOF THE 2D DISORDERED ROTOR MODEL

As mentioned previously, the tool that we use to studythe disordered rotor model is the strong disorder realspace renormalization method. Here we briefly reviewthe method and discuss its application to the model athand.

At first glance, problems involving strong quencheddisorder may appear to be substantially more compli-cated than their clean counterparts. However, one way tomotivate the strong disorder renormalization procedureis to consider that, in some cases, strong disorder can ac-tually serve as an advantage. In particular, disorder canmake the problem of finding the quantum ground state ofa model more local. Having identified the strongest of allthe disordered couplings in the Hamiltonian, we can thenuse the assumption of strong disorder to argue that othercouplings in the vicinity (in real space) of this dominantcoupling are likely to be much weaker. This means thatthe ground state can locally be approximated by satisfy-ing the dominant coupling. Other terms in the Hamil-tonian can then be incorporated as corrections. Quiteoften, these other terms are treated by means of per-turbation theory, but this need not always be the case.These corrections manifest as new couplings in the model,and thus, the procedure yields a new effective Hamilto-nian. Since part of the ground state is specified in thisstep, some degrees of freedom of the system are deci-mated away. By repeating the procedure, we can itera-tively specify the entire ground state16,17. Moreover, wecan examine the way in which the probability distribu-tions of the disordered couplings flow as the renormal-ization proceeds. One possibility is that the the model

5

looks more and more disordered at larger length scalesnear criticality. This is the case for random transversefield Ising models in one and two dimensions19,26,27. Insuch cases, the model flows towards infinite randomness,and the strong disorder renormalization group becomesasymptotically exact near criticality18. Of course, it isalso possible to flow towards finite or weak disorder, andin these cases, the reliability of the RG is less clear. Wediscuss this issue extensively, as it pertains to the disor-dered rotor model, in Appendix D.

A. The Basic RG Steps

We now concretize these ideas by application to therotor Hamiltonian (7). In our model of random bosons in2D, there are two types of disordered couplings, namelycharging energies and Josephson couplings. In each stepof the renormalization, we identify the maximum of allof these couplings, which defines the RG scale:

Ω = max [Uj, Jjk] (18)

How we then proceed depends upon which type of cou-pling is dominant.

Site Decimation

Consider the case where the charging energy on siteX is dominant. We define a local Hamiltonian in whichthis charging energy term is chosen to be the unperturbedpiece. All Josephson couplings entering the correspond-ing site are considered to be perturbations:

HX = UX n2X −

∑k

JXk cos (φX − φk) (19)

Satisfying the dominant coupling means setting nX = 0to leading order. This defines a degenerate manifold of lo-cal ground states: |0, nk〉. In these kets, the first termcorresponds to zero number fluctuation on site X andthe second specifies the number fluctuations on all sitesconnected to X by a Josephson coupling. The degener-ate space is infinitely large, corresponding to all possiblechoices of nk. However, all matrix elements of the per-turbative Josephson couplings in this ground state mani-fold vanish. The leading corrections then come from sec-ond order degenerate perturbation theory, in which wecalculate corrections coming from excited states:

|0, nk〉′ ≈ |0, nk〉 (20)

+∑m∈k

JXm2UX

(|1, nm − 1〉+ |−1, nm + 1〉)

In the terms giving the perturbative corrections, we as-sume that the number fluctuations on all neighboringsites except m remain unmodified from their values innk. We next consider the matrix elements of these

!"

#"

!"

#"

FIG. 2: The site decimation RG step: The site marked withthe x has the dominant charging energy and is decimatedaway, generating bonds between neighboring sites j and kwith the effective coupling given in equation (21). The newlocal structure of the lattice is shown to the right.

states in HX . Up to a constant term, these matrix el-ements are identically those that would result from aneffective Josephson coupling:

Jjk =JjXJXkUX

(21)

between each two sites that were coupled to site X beforethe decimation step13. This process of site decimation isillustrated in Figure 2.

Link Decimation

Now, suppose that a Josephson coupling is the dom-inant energy scale. In this case, the local Hamiltonianis:

Hjk = Uj n2j + Ukn

2k − Jjk cos (φj − φk) (22)

The local approximation to be made here is that, to low-est order, the phases on these adjacent sites should belocked together. In other words, the degree of freedomto be specified is the relative phase φj − φk. This moti-vates the introduction of new operators:

nC = nj + nk

φC =Ukφj + Uj φkUj + Uk

nR =Uj nj − UknkUj + Uk

φR = φj − φk (23)

These operators satisfy the commutation relations:

[φC , nC ] = i

[φR, nR] = i (24)

6

!" #" $"

FIG. 3: The link decimation RG step: The crossed link hasthe dominant Josephson coupling. The two sites it joins aremerged into one cluster, resulting the effective lattice struc-ture shown to the right. The cluster C has an effective charg-ing energy given in equation (26).

with all other commutators vanishing. Thus, the trans-formation preserves the algebra of number and phase op-erators. A subtlety arises for the relative coordinate op-

erators nR and φR because, as defined above, nR neednot be an integer and φR ∈ [−2π, 2π) as opposed toφR ∈ [0, 2π). To deal with this difficulty, we may makethe additional approximation of treating φR as a non-compact variable. This makes nR continuous instead ofdiscrete. Then, in terms of the new cluster and relativecoordinate operators, the local Hamiltonian (22) reads:

Hjk =UjUkUj + Uk

n2C + (Uj + Uk)n2

R − Jjk cos (φR) (25)

To lowest order, we set φR = 0. This decimation of therelative phase leaves the cluster phase φC unspecified, sotwo phase degrees of freedom have been reduced to one.The first term in Hjk shows that the inverse chargingenergies add like the capacitances of capacitors in parallelto give the charging energy for the cluster:

UC =1

1Uj

+ 1Uk

=UjUkUj + Uk

(26)

Figure 3 depicts this process of link decimation13.Higher order corrections to this picture, arising from

harmonic vibrations of the phases that make up the clus-ter, can be obtained by considering the part of the lo-cal Hamiltonian involving the relative coordinate. Theseterms act approximately like a simple harmonic oscilla-tor Hamiltonian on the basis of nR eigenstates. Thus,the ground state for the relative coordinate can be ap-proximated by a simple harmonic oscillator ground state:

|ψR〉 ≈γ

18

π14

∫ ∞−∞

dnR exp

[−γ

12

2n2R

]|nR〉 (27)

with:

γ =2(Uj + Uk)

Jjk(28)

This approximation is used in the numerics to computeDebye-Waller factors that modify Josephson couplings

entering the newly formed cluster. Quantum fluctua-tions of φR weaken the phase coherence of the cluster,and consequently, suppress these Josephson couplings.Mathematically, the Debye-Waller factors arise because,in writing down the local, two site Hamiltonian (22), we

have neglected that φR also appears in the other linkspenetrating the two sites j and k. Consider a Josephsoncoupling from a third site m to the site j. This corre-sponds to a term in the full Hamiltonian (7):

cos (φm − φj) = cos(φm − φC − µ1φR

)= cos

(φm − φC

)cos(µ1φR

)+ sin

(φm − φC

)sin(µ1φR

)≈ cos

(φm − φC

)〈cos

(µ1φR

)〉+

sin(φm − φC

)〈sin

(µ1φR

)〉 (29)

with:

µj =Uj

Uj + Uk(30)

The angle brackets in the final line of equation (29) referto averages taken in the relative coordinate ground state(27). The expectation value of the sine vanishes, and theexpectation value of the cosine yields the Debye-Wallerfactor:

cDW,j ≈sin2(πµj)

π2

∞∑q=−∞

(q2 + µ2j )

(q2 − µ2j )

2exp

(−γ

12

4q2

)(31)

In the numerics, we truncate the calculation of this sumat a specified order, |qmax| = 20, and multiply the Joseph-son coupling Jmj by the result to find the new Joseph-

son coupling JmC penetrating the cluster. Note that theDebye-Waller factor for links penetrating the site k is, ingeneral, not equal to cDW,j , but its calculation is com-pletely analogous.

B. Adaptations for 2D

Sum Rule

As shown by Altman, Kafri, Polkovnikov, and Refael,the two renormalization steps outlined above form thebasis of the strong disorder renormalization group for thedisordered rotor model in 1D13. In higher dimensions,the geometry of the lattice changes as the renormaliza-tion proceeds, and this presents complications. For ex-ample, in Figure 4, decimation of a site X produces aneffective Josephson coupling between two sites j and kthat are already linked to one another by a preexistingJosephson coupling. In our numerics, we sum the pre-existing and new coupling to form the effective coupling

7

!" #" !" #"

FIG. 4: Site decimation with the sum rule: The site with thedominant charging energy is coupled to two sites that are al-ready coupled to one another. These two sites are indicatedwith a solid black border. After site decimation, the effec-tive Josephson coupling between those two sites is the sum ofthe old coupling and the effective coupling generated throughdecimation of the site with the cross (see equation (32)).

!"

#"$" $"%"

FIG. 5: Link decimation with the sum rule: The two sites con-nected by the dominant Josephson coupling are both coupledto a third site, drawn with a solid border. The Josephson cou-pling between the third site and the cluster formed throughsite decimation is the sum of the original couplings betweenthat site and the two sites forming the cluster (see equation(33)), up to corrections coming from Debye-Waller factors.

between the remaining sites:

Jjk = Jjk +JjXJXkUX

(32)

A similar situation can arise during link decimation. InFigure 5, a cluster is formed by two sites j and k, each ofwhich is connected to a third site m. Up to correctionscoming from Debye-Waller factors, the effective Joseph-son coupling joining site m to the cluster is:

JmC = Jmj + Jmk (33)

Some authors replace the sum rule in equations (32)and (33) with a maximum rule 19,26,27. The motivationbehind the maximum rule is that, in the strong disorderlimit:

max

[Jjk,

JjXJXkUX

]≈ Jjk +

JjXJXkUX

(34)

This should be a good approximation in an infinite dis-order context. For our model however, we find that thesum rule increases the class of distributions which findthe unstable fixed point depicted schematically in Figure1. For further discussion of the difference between thesum and maximum rules, please consult Appendix A.

Thresholding

In dimensions greater than one, there is a tendency forthe numerics to slow down considerably if the renormal-

ization procedure involves a lot of site decimation. Again,the source of the problem is the evolution of lattice underthe RG. If a site X is decimated, then effective links aregenerated between each pair of sites that were previouslycoupled to X. Thus, site decimation generates many newcouplings, increasing the coordination number of the ef-fective lattice. At the same time, the site decimationstep takes computer time quadratic in the coordinationnumber of the site being decimated. To apply the pro-cedure to large lattices, it is necessary to find a way tocircumvent this difficulty.

At the beginning of the RG, we specify a thresholdingparameter, which we call α. During a site decimation, ifa new Josephson coupling is created between sites j andk such that:

Jjk =JjXJXkUX

< αUX = αΩ (35)

then the coupling is thrown away. For convenience inimplementation, the new bond is ignored only if it doesnot sum with a preexisting Josephson coupling. If α ischosen to be very small, then ignoring the coupling willhopefully not affect the future course of RG. However, tobe more careful, it is better to perform an extrapolationin the threshold α to see if the numerics converge. Usingthis thresholding procedure, we are able to reach latticesup to size 300× 300, if we additionally require averagingover a reasonably large number of disorder samples. Inthis paper, unless otherwise stated, we always use 103

samples for any given choice of distributions.

Distribution Flows

Typically, in an application of the strong disorderrenormalization method, it is interesting to monitor theflow of the distributions of the various couplings as theRG proceeds. This is straightforward for a 1D chain, butin higher dimensions, there is yet again a complicationfrom the evolving lattice structure. As the renormal-ization proceeds, it is possible to generate very highlyconnected lattices. Many of the effective Josephson cou-plings will, however, be exceedingly small. Incorporatingthese anomalously small couplings into statistics can bemisleading. Despite the large number of weak bonds,there may exist a number of strong bonds sufficient toproduce superfluid clusters. In fact, including the weakbonds in statistics is analogous to polluting the statisticswith the inactive next-nearest neighbor Josephson cou-plings in the original lattice. It is more appropriate tofollow Motrunich et al. and focus on the largest O(N)

Josephson couplings, where N is the number of sites re-maining in the effective lattice27. In the remainder of thepaper, the “Josephson coupling distribution” will there-fore refer solely to the dominant 2N effective Josephsoncouplings at any stage in the RG, and all statistics willbe done only on these 2N couplings.

8

0 0.2 0.4 0.6 0.80

2

4

J/U

J/J

0 20 40 60 80 1000

10

20

30

= 1 x 10 5

= 5 x 10 5

FIG. 6: The projection, in the ∆J/J vs. J/U plane, of theflows of the coupling distributions at different stages of theRG. The initial distributions Pi(U) and Pi(J) are both trun-cated Gaussians, and J0 (the center of the initial J distribu-tion) is used as the tuning parameter. Each flow correspondsto a different choice of the tuning parameter. The flows startat the bottom of the figure and go up and to the left or up andto the right. A smaller value of the thresholding parameter isused near criticality as indicated by the legend. All runs weredone on L = 100 lattices.

IV. NUMERICAL APPLICATION OF THESTRONG DISORDER RG

In this section, we present numerical data collectedfrom our implementation of the strong disorder renor-malization group. First, we explore the strong disorderRG flows of the distributions of charging energies andJosephson couplings. This investigation points to the ex-istence of an unstable fixed point of the RG flow. We findthat the presence of this fixed point is robust to many dif-ferent changes in the choices of the initial distributions.Next, we examine the distributions generated by the RGnear this fixed point and find that universal physics arisesin its vicinity. Subsequently, we proceed away from thefixed point to study properties of the phases of the dis-ordered rotor model. We find phases that we tentativelyidentify as Mott insulating, glassy, and superfluid, andfurthermore, we find that the unstable fixed point gov-erns the putative glass-superfluid transition. We deferdetailed interpretation of the data and analysis of thetransition to Section V.

A. Flow Diagrams and the Finite Disorder FixedPoint

In Figures 6-8, we plot flows of quantities that charac-terize the Josephson coupling and charging energy distri-butions. We emphasize again that, in the context of ourstudy of distribution flows, the “Josephson coupling dis-tribution” actually only includes the greatest 2N Joseph-son couplings, where N is the number of sites remain-ing in the effective lattice. After M decimation steps ofthe RG, we stop the procedure and look at the remain-ing charging energies and these dominant Josephson cou-plings. We then use these values to update estimates forthe mean and standard deviation corresponding to that

0 0.2 0.4 0.6 0.8 10

2

4

6

8

J/U

J/J

0 20 40 60 80 1000

10

20

30

= 5 x 10 6

= 2.5 x 10 5

FIG. 7: Same as Figure 6, except Pi(U) is Gaussian andPi(J) ∝ −1.6, with cutoffs chosen to make the latter distri-bution very broad. The parameter U0 is used to tune throughthe transition. The flows begin near the center of the figure.To the left of the figure, flows initially propagate towards thebottom left but eventually turn around and propagate to-wards the top left. To the right of the figure, flows initiallypropagate towards the bottom left but eventually turn aroundand propagate towards the bottom right. All runs were doneon L = 300 lattices.

0 0.2 0.4 0.60

2

4

6

J/U

J/J

0 20 40 60 80 1000

10

20

30

= 1 x 10 5

= 5 x 10 5

FIG. 8: Same as Figure 6, except Pi(U) ∝ U5 and Pi(J) ∝J−3. The tuning parameter is Jmin, the lower cutoff of Pi(J).All runs were done on L = 150 lattices.

step in the RG. For each realization of the randomness(i.e. each sample), we do this for many different choices ofM , and we repeat the process for 103 realizations of therandomness. Ultimately, this procedure gives a “flow”that characterizes the disorder-averaged evolution of thedistributions at different stages of the renormalization.

The x-axes of Figures 6-8 give the ratio of the meansof the two distributions. Meanwhile, the y-axes give theratio of the standard deviation of the Josephson couplingdistribution to the mean of the distribution. The plotsactually show 2D projections of flows that occur in thespace of all possible distributions. At the very least, theseplots imply the existence of a third axis, namely ∆U

U,

which may carry important information. Nevertheless,these highly simplified 2D pictures are surprisingly ef-fective in describing the fate of the model with differentchoices of parameters. In interpreting Figures 6-8, thereader will likely find it helpful to glance back to Figure1, which shows a schematization of the flows.

Figure 6 specifically corresponds to flows for initial dis-tributions Pi(U) and Pi(J) that are Gaussian. The cen-ter of the Josephson coupling distribution is used as thetuning parameter. To the left of the plot are two flows

9

that propagate to the top left of the diagram, towards

small JU

and large ∆JJ

. Since these flows propagate to-

wards high U , it is tempting to identify them as flowingtowards an insulating regime. Meanwhile, to the rightof the plot, there are seven flows that propagate towardshigh J , and it is tempting to identify these as propagat-ing towards a superfluid regime. At the interface betweenthese two behaviors, the flows “slow down” and travel ashorter distance in the plane. This behavior is suggestiveof a separatrix flow that terminates at an unstable fixedpoint, as shown in Figure 1.

Our next goal will be to show that the behavior indi-cating the presence of this unstable fixed point is robustto changes in the choice of the initial distributions. InFigure 7, Pi(U) is a Gaussian, and Pi(J) ∝ J−1.6. Thecenter of the charging energy distribution, U0, is used asthe tuning parameter. The numerical choices place theflows initially above and to the right of the location ofthe unstable fixed point in the previous figure. From thepoint of view of the strong disorder RG procedure, thischoice of initial distributions is advantageous, becausethe flows begin in a regime of high disorder in J , wherethe procedure is more accurate. Later in the paper, afterpresenting evidence of the universal physics that emergesin the disordered rotor model, we will focus on this choiceof distributions exclusively. Therefore, we have collectedadditional details about these distributions in AppendixC. Note that the leftmost flows in Figure 7 initially prop-agate towards the lower left hand corner of the figure;then, they they turn upward, continuing onward to lowerJU

but now also towards high ∆JJ

. Hence, they share thesame qualitative fate as the leftmost flows in Figure 6. Tothe right of Figure 7, the flows initially also propagate to-wards the lower left; however, these flows ultimately turn

around and propagate towards high JU

. The separatrixthat divides these two classes of flows appears to termi-nate in the same critical region that was seen in Figure6.

In Figure 8, we make yet another choice of initial dis-tributions. Now, Pi(J) ∝ J−3 and Pi(U) ∝ U5. Theresulting flow diagram again suggests the presence of anunstable fixed point in the same critical region. It wouldbe misleading, however, to suggest that every flow dia-gram generated by the RG will have the nice propertiesof Figures 6-8. We provide a counterexample in Figure9, in which Pi(U) is bimodal and Pi(J) ∝ J−1.6. Panel(a) shows the extremely complicated behavior of someof the flows. These features are reflections of the struc-tural details of the bimodal distribution. We will seeshortly that, at least in the vicinity of criticality, the RGworks to wash away these details and construct univer-sal distributions. After these universal distributions aresomewhat well approximated, the flows should be morewell behaved, but in Figure 9, we see a non-universal eraof the flows, where the complexities of the initial distri-butions can manifest in complicated flows. To bring outthis point more clearly, we have removed data for theearly stages of the RG in panel (b). Now the “late RG-

(a)

0 0.5 1 1.50

2

4

J/U

J/J

0 20 40 60 80 1000

10

20

30

= 1 x 10 5

= 5 x 10 5

(b)

0 0.5 1 1.50

1

2

3

J/U

J/J

FIG. 9: In these numerical flow diagrams, Pi(U) is bimodaland Pi(J) ∝ J−1.6, with cutoffs chosen to make the latterdistribution very broad. In panel (a), we show a few sam-ple flows that start near the top of the figure and initiallypropagate towards the lower left hand corner. The complexfeatures of these flows reflect the structural details of the bi-modal U distribution. In panel (b), we exhibit the flows atlate RG times, when the procedure has had an opportunityto renormalize away the microscopic details of the initial dis-tributions. Then, the flows are, at least qualitatively, moresimilar to those seen in Figure 7. All runs were done onL = 200 lattices.

time” behavior of the flows falls more nicely in line withwhat is seen in Figure 7.

B. Universal Distributions

Near the unstable finite disorder fixed point of the RGflow, we expect universal physics to emerge. Certain as-pects of the critical behavior should be independent ofmicroscopic details, including the structure of the initialdistributions. The universality of the fixed point shouldbecome evident in the forms of the renormalized distribu-tions generated through the RG: whatever the initial dis-tributions may be, they should evolve towards universalforms, provided that they put the system near criticality.

We first focus on determining the universal form ofthe fixed point Josephson coupling distribution. Figure10 shows data for the four different choices of the ini-tial distributions that we explore in this paper. In thefirst three panels, Pi(U) and Pi(J) have the same qual-itative form, and in panel (d), we have Pi(J) ∝ J−1.6

and Pi(U) ∝ U1.6. We tune the parameters characteriz-ing the distributions such that the flows propagate nearthe unstable fixed point, run the numerics on 100 × 100lattices, and plot the initial distributions alongside therenormalized distributions when 100 sites remain in the

10

effective lattice. For the renormalized distributions, weagain only include the dominant 2N = 200 Josephsoncouplings for each sample. The renormalized distribu-tions suggest that the RG indeed washes away the detailsof the initial choices, leaving a power law in each case.The universality of this power law is more striking in Fig-ure 11, where we plot the renormalized distributions forthe four cases together. In this plot, we scale J for eachof the four cases by the average RG scale Ω when only100 sites remain. This scaling causes the distributions tonearly collapse onto one another, revealing the universalform:

Puniv(J) ∝ J−ϕ (36)

We will momentarily defer providing a numerical esti-mate of ϕ, in anticipation of presenting higher qualitydata, taken from runs on larger lattices, below.

The charging energy distributions produced by the RGin the four cases shown in Figure 10, while showing hintsof universality, do not demonstrate it to the same strik-ing degree as the Josephson coupling distributions. Thereason for this is the following: in three out of the fourcases, the initial distributions have Jmax < Umax. Severalof the initial distributions we study in this paper satisfythis property, because in dimensions greater than one, in-teresting choices of distributions typically have most barecharging energies greater than most bare Josephson cou-plings. Otherwise, they almost certainly yield superfluidbehavior. Consequently, for three out of the four casesin Figure 10, the RG begins with only site decimations.These site decimations dramatically modify the Joseph-son coupling distribution, but the charging energy distri-bution is, to a large extent, only truncated from aboveby the renormalization scale. Later on in the procedure,after many sites have been decimated away, the RG en-ters a regime where the charging energy and Josephsoncoupling scales compete. Only then do link decimationsbegin to occur, and only then can the charging energydistribution begin to evolve in a nontrivial way. How-ever, by this point, there is far less RG time remainingfor the fixed point distribution to emerge.

There are two ways to circumvent this difficulty. Onestrategy is to note that this problem of insufficient RGtime would not arise if we had access to arbitrarily largelattices. We could follow the renormalization as long asnecessary to construct the universal distributions. Thus,we can try to explore larger lattices up to the limits setby our computational capabilities. On the other hand,another solution is to work with very wide initial distri-butions of Josephson couplings. These are distributionswhich have large ∆J

J. As such, they correspond to flows

that begin above the unstable fixed point in our ∆JJ

vs.JU

flow diagrams. Using such distributions, it is possibleto engineer situations where most bare charging energiesexceed most bare Josephson couplings but where, dueto the presence of a small fraction of anomalously largeJosephson couplings, Jmax > Umax at the beginning of

(a)

4 2 0 215

10

5

0

ln(J)

ln(P(J))

0 20 40 60 80 1000

5

10

15

20

initialrenormalized

(b)

6 4 2 0 2 415

10

5

0

ln(J)

ln(P(J))

(c)

4 2 0 2 415

10

5

0

ln(J)

ln(P(J))

(d)

3 2 1 0 1 2 315

10

5

0

ln(J)

ln(P(J))

FIG. 10: Log-log plots of initial and renormalized Josephsoncoupling distributions for near-critical flows. All runs weredone on L = 100 lattices with α = 5× 10−6. Each plot showsthe initial distribution and the distribution when the effec-tive lattice has 1% of the original number of sites. The initialdistributions have four different forms, but the distributionsafter renormalization show a universal power law. Note thatthe plots of initial distributions in these plots were not con-structed from actual data (i. e. actual numerical sampling ofthe distributions) but were instead constructed by hand.

11

6 4 2 0 215

10

5

0

ln(J/ )

ln(P(J))

1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

GaussianPower LawBimodalFlat

FIG. 11: The distributions from Figure 10, with the Joseph-son coupling strength scaled by the mean RG scale Ω at thecorresponding stage of the Rg. The near collapse of the distri-butions reveals the universal power law form of the Josephsoncoupling distribution near the unstable fixed point. A refinedversion of this plot, showing data for larger lattices (but alsodifferent choices of initial distributions), can be found in panel(b) of Figure 13.

the RG. If the parameters are chosen appropriately, therenormalization procedure will begin with a few link dec-imations, until the charging energy and Josephson cou-pling scales meet. After this point, the RG will featurean interplay of site and link decimations. Thus, boththe Josephson coupling and charging energy distributionswill evolve nontrivially.

To target the fixed point charging energy distribution,we apply both of the strategies. We proceed to 200×200lattices and compute renormalized distributions when theeffective lattice has 200 sites remaining. Additionally, wework with very wide initial Josephson coupling distribu-tions. In order to achieve large initial ∆J

J, we restrict our

attention to power law distributions of Josephson cou-plings of the form Pi(J) ∝ J−1.6. We vary the choice ofthe initial charging energy distribution and tune the pa-rameters near criticality. The results are shown in Figure12. Now, the RG does generate a universal form for thecharging energy distribution, and in Figure 13, we scalethe renormalized distributions by the corresponding RGscales to expose the universality more clearly. Figure13 suggests that the functional form of the fixed pointcharging energy distribution may be:

Puniv(U) ∝ 1

Uβexp

[−gUU

](37)

We have been unable to extract a good estimate of β.Panel (a) of Figure 13 presupposes β ≈ 1, and the lin-earity of the plots suggests that this may be close to theactual value. Taking β = 1, we can estimate:

gUΩ≈ 0.67± 0.03 (38)

where Umax is the largest charging energy in the renor-malized distribution. We should note that qualitativelysimilar charging energy distributions to those seen in Fig-ure 12 still emerge near criticality if we relax the re-strictions of initially power law J distributions and ini-tially high ∆J

J. This is true of the distributions studied

in Figure 10, even in the flat and bimodal cases whereJmax < Umin initially and cluster formation can only formdue to the use of the sum rule. As argued above, the addi-tional restrictions we impose on Pi(J) in Figure 12 simplyallow the RG to construct the fixed point distributionsmore cleanly.

In the lower panel of Figure 13, we additionally presentdata for the Josephson coupling distributions at the samestage of the RG. From this data, we can estimate:

ϕ ≈ 1.16± 0.04 (39)

Before proceeding, we should note that the form of thefixed point J distribution allows us to construct an ar-gument for the validity of the RG near criticality. Weexpand upon this argument greatly in Appendix D, butwe will sketch the basic premise here. Essentially, weshould consider the implications of the fixed point distri-butions for the reliability of each of the RG steps. Thevalidity of site decimation rests on the reliability of theperturbative treatment of the Josephson couplings en-tering the site with the dominant charging energy. Theform of the critical Josephson coupling distribution im-mediately guarantees that most Josephson couplings aremuch weaker than the RG scale. For the distributionspictured in Figure 13, the ratio of the median J to the

RG scale isJtyp

Ω ≈ 0.10 ± 0.01. Hence, the site deci-mation is usually very safe. In the case of link decima-tion, a similar argument allows us to ignore, to leadingorder, other Josephson couplings penetrating the sitesjoined by the dominant coupling. However, the struc-ture of the critical charging energy distribution, shownin Figure 13, actually suggests that there can be a largenumber of charging energies of the same order as the RGscale; in particular, the ratio of the median U to the RG

scale isUtyp

Ω ≈ 0.66 ± 0.01. Consequently, the questionof the reliability of link decimation reduces to the fol-lowing: in a single junction (or two-site) problem, howreliable is cluster formation when both charging energiesare weaker than the Josephson coupling but potentiallyof the same order-of-magnitude? We address this ques-tion in Appendix D and find that the link decimationstep also seems to be reasonably safe.

C. Physical Properties and Finite Size Scaling

To determine a preliminary classification of the phasesof the model, we now measure four physical properties.First, we measure smax, the size of the largest clusterformed by link decimations during the renormalizationprocedure. This corresponds to the largest domain ofsuperfluid ordering in the system. We also measure s2,the size of the second largest cluster. We will see that thebehavior of this quantity differs dramatically from thatof smax in the superfluid phase, and therefore, both areinteresting quantities to measure.

We also calculate the charging gap for the system,∆min. We can estimate this quantity by simply mea-

12

(a)

4 2 0 2 414121086

ln(U)

ln(P(U))

0 20 40 60 80 1000

5

10

15

20

initialrenormalized

(b)

4 2 0 2 415

10

5

ln(U)

ln(P(U))

(c)

6 4 2 0 215

10

5

ln(U)

ln(P(U))

(d)

6 4 2 0 215

10

5

ln(U)

ln(P(U))

FIG. 12: Log-log plots of initial and renormalized chargingenergy distributions for near-critical flows. All runs were doneon L = 200 lattices with α = 5 × 10−6. Each plot showsthe initial distribution and the distribution when the effectivelattice has 0.5% of the original number of sites. The initialcharging energy distributions have four different forms, butthe distributions after renormalization show a universal powerlaw. Note that the plots of initial distributions in these plotswere not constructed from actual data (i. e. actual numericalsampling of the distributions) but were instead constructedby hand.

(a)

0 5 10 15 20 2520

15

10

5

/U

ln((U

/)P(U))

1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

GaussianPower LawBimodalFlat

(b)

6 4 2 0 215

10

5

0

ln(J/ )

ln(P(J))

FIG. 13: In panel (a), the renormalized distributions fromFigure 12 are plotted together, with the charging energiesU scaled by the corresponding RG scale Ω. In panel (b), weprovide a similar plot for the renormalized Josephson couplingdistributions produced by the runs in Figure 12.

suring the charging energy of the final site remaining inthe RG.

Finally, we measure a susceptibility towards superfluidordering. Consider adding a perturbation of the followingform the to rotor model Hamiltonian:

H ′ = −h∑j

cos(φj) (40)

In the RG, we can evaluate the linear response of the sys-tem to this perturbation and measure the susceptibility:

χ =1

L2

∑j

∂〈cos φj〉∂h

∣∣∣∣∣h=0

(41)

The terms of this sum are computed during site decima-tion. Perturbation theory gives a single site susceptibilityof 1

UX, where X is the site being decimated. Neglecting

corrections from harmonic fluctuations, we can find thecontribution of a cluster to the susceptibility by multi-plying the perturbative result by s2, where s is the sizeof the cluster. One factor of s arises from the fact thatthe cluster represents s terms in the original sum (41),and the other follows from the fact that the effective fieldcoupling to the cluster phase is enhanced s times when sphases rotate together. When harmonic fluctuations aretaken into account, both of these factors of s should bereplaced by a renormalized factor which we denote as b.This b-factor accounts for the fact that quantum fluctu-ations weaken the phase coherence of the cluster. For abare site, b = 1, and when two sites j and k merge, therenormalized b-factor for the cluster C is:

bC = bjcDW,j + bkcDW,k (42)

13

where cDW,j and cDW,k are the Debye-Waller factors (31).Hence, the total contribution of the cluster to the suscep-tibility, before the normalization by 1

L2 , is:

χC =b2CUC

(43)

where bC and UC are the b-factor and charging energy ofthe cluster respectively. Further details of this calcula-tion can be found in Appendix B.

For each lattice size, we obtain an estimate for the fourquantities smax, s2, ∆min, and χ. Then, we examine howthese estimates vary as we raise L. For certain types ofdistributions, computational limitations force us to workon relatively small lattices. This occurs, for example,when both Pi(U) and Pi(J) are bimodal, and we studythis case in Figure 14.

In panel (a) of Figure 14, there is no cluster formationwhatsoever. Hence, smax = s2 = 1. This results in a gap∆min that remains asymptotically constant as it cannotbe lower than the lower bound of the initial chargingenergy distribution. The susceptibility χ also remainsasymptotically constant.

Next, in panel (b), we find a regime in which link dec-imations do occur and clusters do form. Moreover, smax

and s2 grow with system size, with what appears to bea power law for the largest lattice sizes that we explore.This growth is, however, subquadratic in L, meaning thatsmax grows subextensively with lattice size. Meanwhile,∆min decays with a power slower than L−2, and the sus-ceptibility χ remains constant with L.

In panel (c), all quantities show power law behavior outto L = 100, including the susceptibility which appears togrow with a very slow power. The growth of smax is stillslower than L2, so the largest cluster is still subextensive.

Finally, in panel (d), we find a regime in which smax

grows as L2, reflecting the formation of macroscopic clus-ters that scale extensively with the size of the lattice. Thegap ∆min decays as L−2, and the susceptibility shows anapproximately L4 growth for large L. Perhaps surpris-ingly, although s2 grows with system size, it does so moreslowly than it does in panel (c).

We now turn to a class of distributions for which wecan reach larger lattice sizes. In particular, we return tothe data set described in Appendix C, in which Pi(U)is Gaussian and Pi(J) ∝ J−1.6. Data for this choice ofdistributions is shown in Figure 15.

Panel (a) of this figure qualitatively reproduces thefeatures of panel (a) of Figure 14. Panel (b) of the newfigure, on the other hand, differs from panel (b) of Figure14 in an important way. For large L, the power lawbehaviors of smax, s2 and ∆min are lost, and all threequantities vary more slowly. In Figure 14, this effect mayhave been hidden by the use of smaller system sizes.

If the parameters are tuned such that the correspond-ing flow propagates very close to the unstable fixed point,then we do find a regime in which all quantities shownearly power law behavior out to L = 300. This regimeis depicted in panel (c) of Figure 15.

(a)

2 3 4 54

2

0

2x 10 3

ln(L)

ln(p

rop(

L)/p

rop(

10)) Mott Insulator

2000 4000 6000 8000 100002

1

0

1x 10 3

ln(L)

ln(p

rop(

L)/p

rop(

25)) Mott Insulator

smaxs2

min

(b)

2 3 4 50.5

0

0.5

ln(L)

ln(prop(L)/prop(10)) Glass

(c)

2 3 4 55

0

5

ln(L)ln(prop(L)/prop(10)) Near Critical

(d)

2 3 4 55

0

5

10

ln(L)

ln(prop(L)/prop(10)) Superfluid

FIG. 14: Four characteristic behaviors of smax, s2, ∆min, andχ as a function of system size. Here, Pi(U) and Pi(J) areboth bimodal, and Jh, the center of the higher peak of theJosephson coupling distribution, is used as the tuning param-eter. All quantities have been normalized by their value forL = 10, and data is shown for L = 10 to L = 100. In panel(a), the quantities reflect the purely local physics of the Mottinsulator. In panel (b), smax and s2 grow subextensively withsystem size, with what appears to be a power law. The gapalso decays with a slow power, and the susceptibility remainsconstant. In panel (c), all quantities show power law behaviorout to L = 100. The reference line shows L2 growth. Panel(d) reflects the macroscopic clusters of the superfluid phase.The cluster size smax is parallel to the L2 reference line, andthe susceptibility χ is parallel to the L4 reference line for largeL.

Tuning past this point, we enter a regime in whichmacroscopic clusters form. Panel (d) of Figure 15 showsthe behavior of the four quantities in this regime, and wesee that most of the essential features of the correspond-

14

ing panel of Figure 14 are reproduced. An importantfeature of the plot in panel (d) is that we can clearly seethat the behavior of s2 in this regime is closer to thatobserved in panel (b) than in the intervening panel (c).

The plots in Figures 14 and 15 are suggestively labeledwith their corresponding phase identifications. We willprovisionally use these labels for convenience in referringto these regimes, in advance of presenting arguments forthese classifications in Section V. In the flow diagramsthat we presented earlier, choices of initial distributionsthat correspond to the Mott insulating and glassy behav-iors shown in panels (a) and (b) of Figures 14 and 15 flow

to the stable insulating region where JU→ 0. The super-

fluid behavior in panel (d) of the figures corresponds to

flows towards the high JU

regime of the flow diagrams.The pure power law behavior of panel (c) emerges forflows that propagate very close to the proposed unstablefixed point. This suggests that this fixed point may con-trol the glass to superfluid transition of the disorderedrotor model.

For now, we assume that this is the case and investi-gate more closely the behavior of physical quantities inthe vicinity of this proposed transition. Having providedevidence of the universality that emerges near the criticalpoint, here and in the remainder of this paper, we willfocus exclusively on the choice of distributions detailedin Appendix C. In Figure 16, we show that plots of phys-ical quantities vs. tuning parameter, taken for differentL, can be collapsed onto universal curves. We will usethis scaling collapse in Section V to determine the criti-cal exponents governing the putative transition betweenglassy and superfluid phases.

D. Cluster Densities and b-factors

The physical property that distinguishes the Mott glassfrom the Bose glass is the compressibility. Later, we willshow that, in order to calculate the compressibility, it isinsufficient to consider just the minimum charging gap.Within each sample, the RG may form several clusters,each of which implies a local gap for adding and removingbosons. We will need to monitor all of these local gapsto find the compressibility.

More specifically, in this section, we will calculate thedensity (number per unit area) of clusters of a given sizeand of clusters with gaps in a given range of energy. Wecall these densities ρ(s) and ρ(∆) respectively. The latterquantity gives a “density-of-states” for addition of singlebosons or holes to the system. For a single sample cor-responding to a specific choice of initial distributions, wemonitor the size and local charging gap for all the clustersformed during the renormalization, excluding bare sitesthat are not clustered by the RG. We pool data for 103

samples, choose a discretization of energy to determinea histogram bin size, compute a histogram of gaps anda histogram of cluster sizes, and finally normalize thesehistograms by the total simulated surface area: L2 times

(a)

3 4 5 62

1

0

1x 10 3

ln(L)

ln(p

rop(

L)/p

rop(

25)) Mott Insulator

2000 4000 6000 8000 100002

1

0

1x 10 3

ln(L)

ln(p

rop(

L)/p

rop(

25)) Mott Insulator

smaxs2

min

(b)

3 4 5 65

0

5

ln(L)

ln(prop(L)/prop(25)) Glass

(c)

3 4 5 65

0

5

ln(L)ln(prop(L)/prop(25)) Near Critical

(d)

3 4 5 65

0

5

10

ln(L)

ln(prop(L)/prop(25)) Superfluid

FIG. 15: Four characteristic behaviors of smax, s2, ∆min, andχ as a function of system size. The initial distributions arethose described in Appendix C. All quantities have been nor-malized by their value for L = 25, and data is shown forL = 25 to L = 300. In the four panels, U0 = 400, 9.2, 9, and8 respectively. Panel (a) reflects the purely local physics of theMott insulator. Panel (b) shows a glassy regime characterizedby rare-regions superfluid clusters that grow subextensively insystem size. The reference line shows the power law that smax

obeys near criticality. This nearly critical regime is shown inin panel (c). The reference line here shows L2 growth. Fi-nally, panel (d) shows the superfluid regime, in which the plotof smax vs. L is parallel to the L2 reference line for large L.The plot of the susceptibility χ vs. L is expected to be parallelto the L4 reference line for very large L but does not quitereach this behavior for the system sizes that we have used.

the number of samples.

Our study of these densities will bring into focus thecrucial difference between two types of clusters formedby the RG: rare-regions clusters and the macroscopic

15

(a)

500 0 500 15005

0

5

(U0 U0,c)L1/

ln(smax/Ld f)

0150 1501504

2

0

2

4

(b)

1000 0 1000 30006

4

2

0

(U0 U0,c)L1/

ln(s2/Ld

f)

0200 100 100 2002005

4

3

2

1

FIG. 16: Panel (a) shows the scaling collapse of smax as afunction of tuning parameter, and panels (b) shows a similarcollapse of s2. Each line corresponds to a different value of thelattice size. We show data for L = 25, 50, 75, 100, 150, 200,and 300. The insets show magnified views of the vicinity ofthe critical point for the four largest lattice sizes. To see thevalues of the exponents ν and df used in each panel, consultequations (71)-(75).

clusters that characterize the superfluid phase. We will,therefore, also take the opportunity to examine how theb-factors, which quantify the effect of harmonic fluctua-tions on the susceptibility, vary as a function of s for thetwo types of clusters.

The Charging Gap Density ρ(∆)

Note that ρ(s) and ρ(∆) are not particularly inter-esting for choices of distributions and parameters thatyield the Mott insulating behavior from Figures 14 and15. The profile of ρ(∆) will be identical to the profileof the initial charging energy distribution, and becausewe choose this distribution to be bounded from below bysome positive Umin, it can be shown that this always cor-responds to an incompressible phase. Hence, we beginby focusing on the glassy regime.

Figure 17 shows the gap density for a choice of pa-rameters in the glassy phase. As the size of the latticeis raised, the density profile remains essentially invari-ant at large ∆, but smaller gaps, corresponding to largerclusters, begin to appear. However, these smaller gapssimply fill out a decay to 0 as ∆→ 0.

Now, we turn our attention to the putative superfluidphase. The gap density in this phase is shown in Figure18. In panel (a), there is an invariant piece to the dis-tribution, but at very low ∆, a second peak appears aswell. Panel (b) of Figure 18 shows a magnified view ofthis low ∆ peak. This peak propagates towards ∆ = 0 as

4 2 0 220

15

10

5

ln( )

ln(())

0 50 100 150 200 2500

2

4

6x 10 4

()

L = 75L = 150L = 300

FIG. 17: The number per unity area of clusters with local gapnear ∆ for a choice of parameters in the glassy phase. Theinitial distributions are those described in Appendix C, withPi(U) Gaussian and Pi(J) ∝ J−1.6. The value of the tuningparameter is U0 = 8.8, and this puts the system just barelyon the glassy side of the transition at U0 ≈ 9. Data is shownfor L = 75, 150, and 300 lattices. The density decays fasterthan a power law at small ∆.

(a)

0 2 4 6 80

2

4

6x 10 4

()

0 50 100 150 200 2500

2

4

6x 10 4

()

L = 75L = 150L = 300

(b)

0 0.05 0.1 0.150

2

4x 10 5

()

FIG. 18: In panel (a), the density (number per unit area) ofclusters with a given gap ∆ for a choice of parameters in thesuperfluid phase. The initial distributions are those describedin Appendix C, with Pi(U) Gaussian and Pi(J) ∝ J−1.6. Thevalue of the tuning parameter is U0 = 9.2, and this puts thesystem just barely on the superfluid side of the transition atU0 ≈ 9. Data is shown for L = 75, 150, and 300 lattices. Thedensity profiles exhibit two peaks. The broad peak that isvisible in panel (a) remains invariant with L. To expose thesecond peak, we provide a magnified view of the low ∆ partof the density in panel (b). This peak simultaneously shrinksand propagates towards ∆ = 0 as the system size is raised.

the system size is raised. Accompanying the propagationis a shrinking: the integrated weight of the low ∆ peakshrinks as L−2. The consequences of these two effectsneed to be taken into account carefully to calculate thecompressibility.

16

(a)

0 2 4 6 820

10

0

ln(s)

ln((s))

(b)

0 2 4 6 8 10 1220

10

0

ln(s)

ln((s))

FIG. 19: Sweeps of ρ(s) as the system is tuned through thesuperfluid-insulator transition. The initial distributions arethose described in Appendix C, with Pi(U) Gaussian andPi(J) ∝ J−1.6. The tuning parameter is U0, the center ofPi(U). Panel (a) shows the sweep from deep in the glassyphase (U0 = 20) to the transition (U0 = 9). The closest dataset to the transition is indicated by the black dotted line. Thisline is reproduced in panel (b), which shows the sweep fromthe transition into the superfluid phase (up to U0 = 6.5). Inthe superfluid phase, the density plot is characterized by apeak at large s and a remnant decay at low s.

The Cluster Size Density ρ(s)

We now consider how ρ(s), the density of clusters ofsize s, varies as we sweep through the glassy regime andinto the superfluid. Panel (a) of Figure 19 shows theapproach to the transition from the glassy side. Veryclose to the transition at U0 ≈ 9, ρ(s) exhibits a strikingpower law decay. Proceeding into the proposed glassyphase, the power law decay persists at small s. However,this behavior is cut off by some scale s, beyond whichρ(s) decays very rapidly, essentially exponentially.

In panel (b), we proceed in the other direction fromthe putative transition, into the superfluid regime. Apeak, corresponding to the macroscopic clusters, appearsat large s. As in Figure 18, the macroscopic cluster ineach sample is dressed by rare-regions clusters, and theseclusters are represented by the remnant decay at smalls. This decay closely resembles the decay well inside theglassy regime. In summary, ρ(s) exhibits a pure powerlaw decay in the vicinity of the proposed glass-superfluidtransition; tuning away from criticality in either direc-tion, and excluding the macroscopic clusters of the su-perfluid phase, the power law form of ρ(s) only survivesup to a scale s. For s > s, clusters become exponentiallyrare.

A type of scaling collapse can be performed for theρ(s) curves from Figure 19, and this collapse is shown in

50 0 50 10010

5

0

(U U0,c)s

ln(s

(s

))

FIG. 20: Scaling collapse of the ρ(s) curves from Figure 19.Small cluster sizes (s < 20) need to be discarded, becausethey are non-universal. Large cluster sizes (s > 50) needto be discarded, because they are noisy. Then, the remainingρ(s) curves, taken for different values of the tuning parameter,collapse onto a universal curve.

Figure 20. We will see below that this collapse gives acomplementary set of critical exponents which are relatedby scaling to those that can be extracted from Figure 16.

Susceptibility b-factors

The data presented above for the cluster densitiesρ(s) and ρ(∆) highlights the difference between the rare-regions and macroscopic clusters generated by the RG. Astudy of how the b-factors for the clusters vary as a func-tion of s can bring out another difference between thetwo types of clusters. Recall that the b-factor was intro-duced in equation (43) to quantity the effect of harmonicfluctuations on the susceptibility of a superfluid cluster.As such, understanding the behavior of the b-factors willalso be essential in explaining the behavior of χ in thevarious phases of our model.

In Figure 21, we plot the average value of b for a clusterof size s and plot it against s. Again, we work withL = 300 lattices and the choice of distributions describedin Appendix C. Panel (a) shows data for the glassy phaseand for the non-macroscopic clusters of the superfluidphase. We see that b varies with s as a power law:

b(s) ∼ sζ (44)

and that the power is consistent for seven different choicesof the tuning parameter U0. We will provide an estimateof ζ in Section V. Panel (b) of Figure 21 shows data forthe macroscopic clusters when U0 = 8.8. Now, b(s) ∝ s.This behavior can be anticipated from a simple pictureof macroscopic cluster growth, which we will discuss inSection V.

V. THE PHASES AND QUANTUM PHASETRANSITIONS OF THE DISORDERED ROTOR

MODEL

Having collected representative numerical data in theprevious section, we now assess what the data tells us

17

(a)

0 2 4 6 8 100

2

4

6

ln(s)

ln(b(s))

0 1000 2000 3000 4000 50000

2

4

6

ln(s)

ln(b

(s))

149.69.298.88.47

(b)

1000 2000 3000 40000

200

400

600

s

b(s)

FIG. 21: The behavior of the mean b-factor for clusters ofsize s as a function of s. In panel (a), we show data for theglassy regime and for the non-macroscopic clusters of the su-perfluid regime. The initial distributions are those describedin Appendix C, and data is shown for seven values of thetuning parameter U0 on L = 300 lattices. The log-log plotshows power law behavior of b(s) vs. s. In panel (b), we showdata for the macroscopic clusters when U0 = 8.8. The plotindicates that b(s) ∼ s for macroscopic clusters.

about the phases and phase transitions of our model. Ourfirst task will be to confirm the association of phases withthe behaviors of physical properties that we observed inFigures 14 and 15. To this end, we will have to pre-emptively introduce one of the main conclusions of thediscussion below: that our data points to a superfluid-insulator transition driven by a percolation-type process.The rare-regions clusters of the glassy phase find one an-other, and their phases cohere, producing a macroscopiccluster of superfluid ordering and driving the transitionto long range order and global superfluidity.

Motivated by this intuitive picture of the transition,the logic of the discussion below will be the following:in the proposed glassy and superfluid regimes of Figure15, the cluster size density ρ(s) exhibits the universalfeatures that we would expect from non-percolating andpercolating phases of standard models of percolation. Wecan use these correspondences to extrapolate the behav-iors seen in Figure 15 to the thermodynamic limit, inthe process showing that these phases will indeed havethe properties expected of glassy and superfluid phases.Furthermore, we can analyze the critical point and ex-tract the critical exponents that govern the superfluid-insulator transition. After characterizing this transition,we will finally return to the question of the identity ofthe glassy phase and determine if a Mott glass is presentin the disordered rotor model (7).

Before proceeding, we should clarify that, although ourRG produces clusters with size distributions similar to

models of percolation, our transition is not the standardpercolation transition. Indeed, the exponents that we re-cover are significantly different from the percolation ex-ponents on a 2D square lattice. However, the analogyto percolation allows us to easily identify the relationsbetween the various exponents and the scaling functionsfor the observables.

A. Phases of the Model

Mott Insulator

We briefly digress to describe the phase of our modelin which the percolation picture is irrelevant, simply be-cause there are no regions of superfluid ordering. In aMott insulating phase, the system wants to pin the num-ber fluctuation to zero on each site to avoid the energeticcosts of charging. Hence, smax and s2 simply stay pinnedat one as L is raised. Meanwhile, ∆min should asymptoteto a constant, reflecting the fact that the Mott insulatoris gapped. A phase without any cluster formation can bedescribed by completely local physics. This means thatthe susceptibility can be approximated by disorder aver-aging a single site problem. In other words, χ should alsostay constant as the system size is increased. Thus, ina Mott insulating phase, we expect exactly the behaviorseen in panel (a) of Figures 14 and 15.

Glass

In non-percolating phases of standard models of per-colation, the cluster size density is expected to go as:

ρ(s) = cs−τf(ss

)(45)

where c is a constant. The function f(x) is expected tobe approximately constant for x < 1 and to rapidly decayfor x > 128. Equation (45) is consistent with what wehave observed in our proposed glassy phase in panel (a) ofFigure 19 and is also consistent with the expectation that,in a Griffiths phase, the frequency of occurrence of largerare-regions decays exponentially in their size25. In ournumerics, f(x) seems to follow a pure exponential decayf(x) ∼ e−x, but the conclusions we present below would

be qualitatively unchanged if, for example, f(x) ∼ e−xλ .With the form of equation (45) in hand, we can now

formulate a simple argument for the asymptotic behaviorof smax. In particular, the order-of-magnitude of smax isset by the condition28:

L2L2∑

s=smax

ρL(s) ≈ 1 (46)

If the left hand side of equation (46) is less than 1, thenit is unlikely that even one cluster of size greater than

18

or equal to smax will be present in a sample of size L2.In equation (46), ρL(s) is the finite size approximant tothe function ρ(s) that appears in equation (45). The up-per limit of the sum in equation (46) is taken as L2 be-cause larger clusters simply cannot appear in the finitesize sample. With enough sampling of the random dis-tributions, it should, in principle, be able to accuratelyrepresent the approximant ρL(s) out to s = L2. Thedata indicates that ρL(s) will simply reproduce ρ(s) outto this value of s, so in the calculations below, we canreplace the approximant ρL(s) by ρ(s). This will not bepossible in the superfluid phase.

For large L, where we also expect smax > s, we canuse equation (46) to find smax by computing

L2L2∑

s=smax

ρ(s) ≈ L2

∫ L2

smax

dsρ(s)

= cL2

∫ L2

smax

ds exp[−ss− τ ln(s)

]≈ cL2

∫ L2

smax

ds exp[−ss

]= csL2

(e−

smaxs − e−L

2

s

)(47)

Setting this expression equal to 1 and inverting for smax,we find that, asymptotically in L:

smax ∼ lnL (48)

For large clusters, the link decimation rule for addition ofcharging energies (26) implies that U ∼ s−1, and there-fore:

∆min ∼1

lnL(49)

An entirely analogous condition to equation (46) canbe formulated for s2. We simply replace the right handside of equation (46) with two, indicating that we wantto find the value of s such that there are likely to betwo clusters of that size or greater in a typical sample.However, the remainder of the calculation is qualitativelyunaffected by this change. Therefore, s2 should also growlogarithmically in this regime:

s2 ∼ lnL (50)

Finally, we turn to the susceptibility. This can be com-puted as follows:

χ =1

L2

∑C

b2CUC

∼ 1

L2

∑C

sCb2C

∼ 1

L2

smax∑s=1

ρ(s)L2(b(s))2s

=

smax∑s=1

ρ(s)(b(s))2s (51)

In this calculation, the sum over C is a sum over clus-ters, with sC being the size of the cluster. Then, the sumover s is, as before, a sum over cluster sizes, and b(s) isthe average value of the b-factor for a cluster of size s.Figure 21 shows that, in the glassy regime, b varies as apower of s and that this power remains the same all theway up to criticality. While we do not have a completeunderstanding of this behavior, we can still understandthe asymptotic behavior of χ by reasoning that b can, atmost, grow linearly in s. This follows from an interpre-tation of the b-factor as the effective number of rotorsthat order with the field in a cluster of size s. Since ρ(s)decays exponentially for large s while b(s) grows at mostas a power, the sum (51) converges, and χ should beasymptotically constant:

limL→∞

χ = χ∞ (52)

All of these behaviors are consistent with what hasbeen observed numerically in panel (b) of Figure 15.Moreover, since logarithmic behavior can be difficult todiscriminate from a slow power law at low L, they arealso consistent with panel (b) of Figure 14. Thus, underthe numerically justified assumption that this regime cor-responds to a non-percolating phase, we can deduce that,as L increases, arbitrarily large rare-regions of superfluidordering will appear, driving the gap to zero. However,the typical size of these regions grows extremely slowly(i. e. logarithmically) with system size. The fraction ofsites occupied by the largest cluster in a typical samplevanishes as L→∞, so there is no long range order. Thebehavior of this phase for large L corresponds to whatwe should expect for a glassy phase.

Critical Region

At the critical point of a percolation transition, thecluster size scale in equation (45) is expected to divergeas:

s ∼ |g − gc|−1σ (53)

where g is the tuning parameter for the transition. Thisdivergence is related to the divergence of of a correlationlength which indicates the typical linear extent of thelargest clusters:

ξ ∼ |g − gc|−ν (54)

By equation (53), ρ(s) is a power law at criticality:

ρ(s) = cs−τ (55)

This means that the critical point is characterized by ascale invariant, fractal structure of clusters28. In turn,this implies that ξ and s are related by a fractal dimen-sion:

s = ξdf (56)

19

Equations (53), (54), and (56) together imply:

σ =1

νdf(57)

We will use this scaling law in our analysis of the transi-tion below28.

For the present purposes, note that equation (55) isonce again consistent with what we have observed nu-merically in Figure 19. Now, the calculation for smax

becomes:

L2L2∑

s=smax

ρ(s) ≈ cL2

∫ L2

smax

dss−τ = 1 (58)

which, when inverted, yields:

ln smax =2

τ − 1lnL− 1

τ − 1ln

(τ − 1

c

)+ ln

(1 +

c

(τ − 1)L2(τ−2)

)(59)

Asymptotically, as long as τ > 2, this simply correspondsto a power law growth of smax:

smax ∼ L2

τ−1 (60)

On the other hand, since df is the exponent that connectslength and cluster size scales at the transition, equation(60) yields another scaling relation:

df =2

τ − 1(61)

Equations (57) and (61) are the usual scaling laws con-necting exponents at a percolation transiiton28.

In accordance with equation 60, the gap should closeas:

∆min ∼ L−2

τ−1 = L−df (62)

Furthermore, as in the glassy regime, the qualitative be-havior of the second largest cluster size s2 should be iden-tical to that of smax:

s2 ∼ L2

τ−1 (63)

Turning to the susceptibility, we find that it no longerconverges to a finite value. At criticality, power lawbehavior of b(s) follows naturally from scale invariance.When ρ(s) ∼ s−τ and b(s) ∼ sζ :

χ =

smax∑s=1

ρ(s)(b(s))2s

∼∫ smax

1

dss1+2ζ−τ

∼ s2+2ζ−τmax

∼ L4+4ζ−2ττ−1

= Ldf (1+2ζ)−2 (64)

From Figure 21, we can estimate the exponent ζ:

ζ ≈ 0.68± 0.01 (65)

With another choice of initial distributions (Pi(U) bi-modal and Pi(J) ∝ J−1.6), we have found a similar valuefor ζ. If df (1 + 2ζ) > 2, then χ asymptotically divergesas a power law, as seen in Figure 15. We will providean estimate of df shortly in equation (72). For now, wenote that the power observed for χ vs. L in Figure 15 isconsistent with this estimate of df and the estimate forζ that is given above. In Figure 14, the relatively smallsystem sizes likely put us out of the scaling regime for χ,and this is probably responsible for the extremely slowgrowth of χ with L.

Superfluid

The percolating phase is characterized by the presenceof a macroscopic cluster that grows with the size of thelattice, so trivially:

smax ∝ L2 (66)

and therefore:

∆min ∝ L−2 (67)

It is possible to construct an argument along the lines ofcondition (46) for the behavior of smax in equation (66),but in this case, it is important not to substitute theinfinite lattice density ρ(s) for the finite size approximantρL(s). The subtlety lies in the high s peak observed in thedensity plots in panel (b) of Figure 19. Consistent withtheir low ∆ counterparts in the plots of the gap density,the location of these peaks propagates as L2 towards highs as L is raised. Simultaneously, the integrated weight ofthe peaks shrinks as L−2, reflecting the fact that there isonly one macroscopic cluster in each sample. Balancingthe shrinking and propagation, it is possible to see that,in order to achieve the condition (46), smax must scale asshown above in equation (66).

The reasoning above has important implications forthe behavior of s2. Since the weight of the high s peakof ρL(s) shrinks as L−2, the second largest cluster mustbe drawn from the remnant low s decay. The physicalpicture behind this low s decay is the following: supposewe remove the sites belonging to the macroscopic clusterfrom the original lattice. What remains is a depleted lat-tice from which some of the lowest charging energies andhighest Josephson couplings have been removed. Conse-quently, the depleted lattice is globally insulating. Nev-ertheless, rare-regions of superfluid ordering can form ex-actly as in the glassy phase. It follows that s2 will growwith L as in the glassy phase:

s2 ∼ lnL (68)

This is responsible for the peak in s2 at criticality.

20

Mott Insulator Glass Critical Superfluid

smax const. ln(L) Ldf L2

s2 const. ln(L) Ldf ln(L)

∆min const. 1ln(L)

L−df L−2

χ const. const. Ldf (1+2ζ)−2 L4

TABLE I: Large lattice size (L) behavior of physical proper-ties in the three phases and at the critical point of the glass-superfluid transition. Note that our estimate of df from datafor s2, given in equation (75), differs quantitatively from ourestimate from smax, given in equation (72). We suspect thatthe estimate from the s2 data is influenced by finite size ef-fects, and we therefore take the estimate (72) to be our bestapproximation to df .

To calculate the susceptibility, we first take into ac-count the contribution of the macroscopic cluster. Thebehavior of b(s) for a macroscopic cluster can be in-ferred from a simplified picture of a large cluster merg-ing with single neighbors. As the macroscopic clustergrows, its charging energy becomes smaller, driving theDebye-Waller factor for the cluster to one. At the sametime, the clusters Josephson couplings to other sites growthrough link addition processes. This means that theDebye-Waller factor for another site that is merging withthe macroscopic cluster also approaches unity. Therefore,the b-factor addition rule (42) approximately becomes

bC = bC + 1, and bC ∼ sC follows. Then:

χmac =1

L2× (b(smax))2

∆min

∼ 1

L2× s2

max

∆min

∼ L4 (69)

The rare-regions clusters dressing the macroscopic clusteradd a subleading contribution to the susceptibility, whichwe call χrr. The reasoning of equation (52) indicatesthat this contribution should be asymptotically constant.Thus, the quartic behavior of equation (69) is the correctasymptotic behavior. Finite size corrections from χrr willmodify this behavior, however, and this is probably whywe do not quite see χ reach the L4 behavior in panel (d)of Figure 15.

We have now provided arguments for the behaviorsobserved in each panel of that figure, and we summarizethis information in Table I.

B. Quantum Phase Transitons

Glass-Superfluid Transition

Earlier, we remarked in passing that systems exhibit-ing the behaviors that we have now identified as Mottinsulating and glassy eventually propagate towards the

putative insulating region to the top left of the numeri-cal flow diagrams. Correspondingly, the systems exhibit-ing superfluid behavior propagate towards the putativesuperfluid region to the bottom right. We can now ver-ify our tentative identifications of these stable regionsof the diagram. We can also determine that the unsta-ble fixed point controls the transition between the glassand the superfluid, the superfluid-insulator transition ofour disordered rotor model. This allows us to draw theschematic picture of the flow diagram that we presentedin Figure 1.

We will now focus on the critical region and extractcritical exponents governing the glass-superfluid transi-tion. Estimating these exponents requires formulatingscaling hypotheses for the behavior of physical quantitiesin the critical region. In the case of smax

28:

smax = Ldfh1(L

ξ) = Ldf h1(L

1ν (g − gc)) (70)

Exactly at criticality, smax ∼ Ldf asymptotically, so plot-ting smax

Ldfvs. (g−gc) generates a crossing of the curves for

different lattice sizes. Slightly away from criticality, thepower law behavior should persist if L < ξ. For L > ξ,the system recognizes that it is not critical and we shouldsee a crossover to logarithmic growth on the glassy sideand L2 growth on the superfluid side. Hence, Lξ is the im-

portant ratio near criticality, and this motivates scalingform (70). The scaling form, in turn, implies that we can

produce scaling collapse by plotting smax

Ldfvs. (g− gc)L

1ν .

This is what we have done in panel (a) of Figure 16. Thisscaling collapse leads to the estimates:

ν ≈ 1.13± 0.02 (71)

df ≈ 1.13± 0.04 (72)

These estimates and error bars are obtained through thefollowing procedure: first, we automate the collapse pro-cedure. Next, we partition the 1000 disorder samples foreach value of U0 into bins of size 25. Then, for each ofthe bins, we focus on values of U0 near the transitionand optimize the collapse of the curves for two values ofL (L = 150 and L = 300). Through this process, we findthe best values of ν and df for each bin, thus obtaining40 estimates of each exponent. From these estimates, wecan find a mean and an error bar. We thank Bryan Clarkfor suggesting this method of putting error bars on ourestimates. Note that we have repeated this method toperform scaling collapse of smax for a different choice ofinitial distributions: bimodal Pi(U) and Pi(J) ∝ J−1.6.We have recovered estimates for ν and df which are con-sistent with (71) and (72), providing further evidence ofthe universality of the transition.

A completely analogous scaling hypothesis can bemade for s2:

s2 = Ldfh2(L

ξ) = Ldf h2(L

1ν (g − gc)) (73)

21

Then, the exponents needed to produce the collapse inpanel (c) of Figure 16 are:

ν ≈ 1.08± 0.03 (74)

df ≈ 1.29± 0.04 (75)

While the estimate for ν is consistent with the estimate(71), the estimate for df is considerably higher than thevalue that we found previously. We have not been able tofind a satisfactory explanation of this behavior. Perhapswe have not explored sufficiently large lattice sizes to puts2 truly in the scaling regime. We will proceed by usingour first estimates for the exponents, equations (71) and(72).

A scaling hypothesis can also be formulated for thedensity ρ(s). From equation (45), we see that, for fixedlattice size L, sτρ(s) should depend only on the com-

bination ss ∼ s(g − gc)

1σ . Hence, by plotting sτρ(s) vs.

s(g−gc)1σ and tuning until the curves for different choices

of g collapse, we ought to be able to extract estimates forτ and σ. On the other hand, τ and σ are related to ν anddf through the scaling relations (61) and (57), so fromthe estimates (71) and (72), we can infer:

τ ≈ 2.76± 0.06 (76)

σ ≈ 0.78± 0.02 (77)

In Figure 20, we attempt to produce collapse of ρ(s) forL = 300 lattices using these estimates of τ and σ. Toproduce this plot, we need to discard data points for smallcluster sizes (s < 20), because they are non-universal,and for large cluster sizes (s > 50), because they arenoisy. Once we do this, the collapse works fairly well,indicating that we have found a consistent set of criticalexponents obeying the necessary scaling relations.

Comments on the Insulator-Insulator Transition

Before proceeding further, we should note that our nu-merics do not accurately capture the boundary betweenthe Mott insulator and the Mott glass. Several authorshave argued that we should expect glassy behavior tooccur whenever the ratio of the largest bare Josephsoncoupling to the lowest bare charging energy Jmax

Uminexceeds

the value of the clean transition, because this conditionallows for the presence of regions in which the systemlocally looks like it is in the superfluid phase6,15. How-ever, in the strong disorder RG treatment, some distri-butions that satisfy this criterion still produce Mott in-sulating behavior out to the largest lattice sizes that weinvestigate. Since the glassy phase occurs due to rare-regions or Griffiths effects, a finite size system will onlylook glassy if it is large enough for the rare-regions toappear. This suggests that some choices of parameters

which yield Mott insulating behavior on finite size latticesmay actually yield glassy behavior in the thermodynamiclimit. Of course, this difficulty necessarily afflicts all nu-merical methods (except those that rely on mean-fieldtype approximations29), since they are confined to oper-ate on finite size systems.

We will not comment on this transition further, butwe take this opportunity to refer the reader to papers byKruger et al. and Pollet et al. , which present two view-points on the Mott insulator to Bose glass transition6,30.

C. Identifying the Glass

Finally, we return to the question of the identity ofthe glassy phase. Is the phase a Bose glass or a Mottglass? A definitive diagnosis requires a measurement ofthe compressibility:

κ =1

L2

∑j

∂〈nj〉∂µ

∣∣∣∣µ=0

(78)

The compressibility is more subtle to measure than thequantities that we have already discussed. Any finitesize system is gapped and therefore incompressible. Onthe other hand, in the thermodynamic limit, the gap canvanish and the compressibility need not be zero.

How can we measure the compressibility of the glassyphase in the RG? In Figures 17 and 18, we presenteddata for the density (number per unit area) of clusterswith a given gap ∆. With this density profile in hand, wecan calculate the density of particles introduced to thesystem by a small chemical potential shift µ:

ρex =

∫ µ

0

d∆ρ(∆)n(∆)

=

∫ µ

0

d∆ρ(∆)b µ2∆− 1

2c

≈∫ µ

0

d∆ρ(∆)

(µ

2∆− 1

2

)(79)

Here, n(∆) is the number of particles added to a clus-ter with gap ∆ if the chemical potential is µ. If ρ(∆)stays finite as ∆ → 0, the integral is divergent, and thesystem is infinitely compressible. Suppose alternativelythat ρ(∆) vanishes as ∆β for small ∆. Then:∫ µ

0

d∆∆β

(µ

2∆− 1

2

)=

µβ+1

2β(β + 1)(80)

The derivative of the integral vanishes at µ = 0 for β >0. Thus, the system is incompressible. Comparing tothe data shown in Figures 17, we see that there is nonumerical evidence for a finite glass compressibility; thegap density appears to vanish even faster than a powerlaw as ∆ → 0. This is consistent with the behavior ofρ(s) in equation (45), since ∆ is expected to scale as s−1.Hence, the numerics imply that the Mott glass intervenes

22

between the Mott insulator and the superfluid in thismodel.

At first glance, the preceding argument may be trou-bling. Due to the shrinking of the low ∆ peak in panel(b) of Figure 18, the gap density ρ(∆) also appears tovanish as ∆ → 0 in the superfluid phase. The caveat isthat it is necessary to more carefully evaluate the com-peting effects of the shrinking and the propagation. Thelow ∆ peak in Figure 18 represents the macroscopic su-perfluid clusters that form in the superfluid phase. Theseclusters do not appear in proportion to the surface areaof the sample, as is the case for rare-regions clusters; in-stead, one such macroscopic cluster appears in each sam-ple. Therefore, the density of macroscopic clusters willgo as 1

L2 , and this is responsible for the shrinking of thelow ∆ peak. The propagation of the peak, meanwhile,reflects the fact that the gap closes as L−2. For a fixedchoice of µ, the number of bosons that will be added tothese macroscopic clusters scales as:

µ

2∆− 1

2∝ µL2 (81)

for large L. Then, the density of particles introduced tothe system is:

ρex ∝1

L2× µL2 = µ (82)

This directly implies that the compressibility (78) is aconstant in the thermodynamic limit, so we recover theexpected result that the superfluid phase is compressible.

VI. CONCLUSION

While the interplay of disorder and interactions inbosonic systems has attracted considerable interest fornearly three decades, the random boson problem remainsa fertile source of intriguing physics. In this paper,we have investigated a particular model of disorderedbosons, the two-dimensional rotor (or Josephson junc-tion) model. Our strong disorder RG analysis suggeststhe presence of an unstable finite disorder fixed pointof the RG flow, near which the coupling distributionsflow to universal forms. Furthermore, the strong disorderrenormalization procedure indicates the presence of threephases of the model: the Mott insulating and superfluidphases of the clean model are separated in the phase dia-gram by an intervening glassy phase. The unstable fixedpoint governs the transition between the superfluid andthis glassy phase, and the transition is driven by a kindof percolation. The RG procedure also provides evidencethat this glassy phase is, in fact, the incompressible Mottglass.

Our work is a numerical extension into two dimen-sions of the one-dimensional study by Altman, Kafri,Polkovnikov, and Refael13. The 2D fixed point, how-ever, differs from the 1D fixed point in an important

way. The 1D fixed point occurs at vanishing interac-tion strength (all charging energies Uj = 0). Thus, itcorresponds to a completely classical model and revealsthat the superfluid-insulator transition can be tuned byvarying the width of the Josephson coupling distributionat arbitrarily small interaction strength. The 2D fixedpoint is, in contrast, fully quantum. Indeed, in the crit-ical distributions generated by the strong disorder RG,the charging energy distribution is peaked near the RGscale while the Josephson coupling distribution is peakedwell below.

On the other hand, the fixed point that we have identi-fied in this paper is similar to its 1D counterpart in thatit does not exhibit infinite randomness. Finite disorderfixed points are not optimal settings for strong disorderrenormalization analyses, because the procedure does notbecome asymptotically exact near criticality and is, inthis sense, and uncontrolled approximation. We haveproceeded with such an analysis anyway. In doing so,have found a robust fixed point controlling the superfluid-insulator transition and phases exhibiting sensible phys-ical properties. While this may be surprising given theperils of the method, we have attempted to argue forthe appropriateness of the method, as an approximation,through an analysis of the RG steps in light of the formsof the fixed point distributions Puniv(U) and Puniv(J).We certainly acknowledge that there are other subtletiesdue to the lack of infinite randomness; for example, thenotion of a superfluid cluster is not completely sharp,and consequently, percolation of superfluid clusters canonly be an approximate picture of the transition31. Nev-ertheless, the structure of the fixed point Josephson dis-tribution (36) suggests that the picture may be a goodapproximation, and we take this opportunity to remindthe reader that we extensively discuss the reliability ofthe RG, in light of the properties of the fixed point, inAppendix D. Moreover, the self-consistency of our nu-merical results, especially the striking universality androbustness of the unstable fixed point, gives us confidencethat our strong disorder RG analysis provides useful in-formation about the system. With the potential caveatsin mind, we therefore turn to exploring connections withother theoretical, numerical, and experimental work.

The Mott glass phase of the two-dimensional model isthe straightforward analogue of the phase found in onedimension by Altman et al. The charging gap, the energyneeded to add or remove a boson from the system, van-ishes due to the presence of arbitrarily large rare-regionsthat superfluid order. However, there is no true longrange order because these rare-regions grow subexten-sively with system size. If a small chemical potentialshift is turned on, it becomes energetically preferable toadd bosons somewhere in the system, specifically in thelargest of the rare-region superfluid clusters. Neverthe-less, these clusters do not occur with sufficient number toproduce a finite density of bosons, and the glass remainsincompressible. The Mott glass that was identified byRoscilde and Haas in a related spin-one model relies on

23

the same mechanism32. However, the original proposalof Giamarchi, Le Doussal, and Orignac is fundamentallydistinct12. In their Mott glass, the charging gap remainsfinite, guaranteeing a vanishing of the compressibility;however, gaplessness is achieved through the closing ofa mobility gap for transport of particles between nearbyinsulating and superfluid patches. Sengupta and Haashave argued that particle-hole symmetry, a crucial ingre-dient in the formation of our Mott glass, is not necessaryfor the realization of the phase through this alternativemechanism33.

In the superfluid phase, true long range order emergesbecause the largest cluster scales with the size of thelattice. In this sense, this cluster is macroscopic. De-spite this, the macroscopic cluster may only occupy avery small fraction of the total number of lattice sites.Because the clustering procedure can merge sites thatare not nearest neighbors in the bare lattice, the frac-tion of insulating sites may even exceed the standard 2Dsquare lattice percolation threshold. Thus, the superfluidphase of our model is a depleted superfluid. Here too, ourwork may connect with the work of Roscilde and Haas,in which disorder is introduced into the spin-one modelprecisely through site depletion32. Site depleted super-fluids have also been recently studied by Fernandes andSchmalian34.

We have remarked that several experimental groupsare currently working on studying disordered bosonic sys-tems in ultracold atoms8–11. Dirty superconductors pro-vide another experimental context that may be relevantto the physics of this paper. The question of the appli-cability of bosonic pictures to the 2D superconductor-insulator quantum phase transition is long-standing. Re-cently, this problem has been addressed numerically byBouadim et al. , who used quantum Monte Carlo simu-lations to study a fermionic model of the transition andfound that bosonic physics emerges at criticality35. Thisissue has also been addressed experimentally by Crane etal. , who studied indium oxide thin films and measureda superfluid stiffness in the insulating state. This indi-cates that Cooper pairs may survive into the insulatingregion and leaves open the possibility that the transitionis driven by percolation of superconducting regions36.The relationship of percolation to the superconductor-insulator transition has also been explored in experi-ments on granular superconductors by Frydman et al.37.Finally, Roscilde and Haas have proposed the possibil-ity of realizing a Mott glass to superfluid transition innickel based spin-one antiferromagnets, where particle-hole symmetry may be easier to realize in the guiseof Z2 symmetry32. Very recently, Yu et al. have fol-lowed up on this proposal with an experimental inves-tigation of Bose and Mott glass phases in bromine-dopeddichloro-tetrakis-thiourea-nickel38. This system is three-dimensional, so the character of the transition would nat-urally differ from what we have calculated in this paper.

One immediate extension of our study would be toconsider adding random filling offsets to the disordered

rotor model, as Altman et al. did in a follow-up to theirwork on the 1D model. Such a model would likely containa Bose glass phase, and it would be interesting to see ifthe Bose glass to superfluid transition lives in the sameuniversality class as the transition explored in this paper.In one dimension, Altman et al. found that the propertiesof the transition are independent of the nature of theglassy phase, but perhaps this is not so in d > 114.

Another interesting extension may be to study the ro-tor model defined on random networks. Suppose we donot begin with a square lattice but rather with a gener-alized network of mean connectivity z = 4. At its criticalpoint, would such a model flow to the same fixed point asthe model defined on a square lattice? The fact that thestrong disorder RG modifies the initial lattice structureinto a more general network suggests that this may be thecase for at least some types of random networks. Next,suppose we vary the mean connectivity from z = 4. Isthere a range of connectivities for which random networkmodels access our fixed point?

Perhaps it would be better to precede such investiga-tions with a better characterization of the fixed pointitself. In one dimension, Altman et al. were able to writedown master equations for the RG flow, solve them tofind fixed point distributions, and then verify numeri-cally that these distributions are stable13. In two dimen-sions, a direct analytical approach is more difficult, soperhaps the approach can be inverted. Our work pro-vides suggestive numerical evidence regarding the formsof the universal distributions that characterize the criti-cal point of the disordered rotor model (7). A Lyapunovanalysis of these distributions, in which the RG is usedas a tool to identify irrelevant directions in the space ofpossible distributions, could be a useful step in clarifyingthe critical forms still further. Then, analytically veri-fying these forms as attractor solutions of the RG flowmay be an easier task than analytically identifying themwould have been in the absence of any numerical guid-ance.

Recent work by Vosk and Altman suggests yet an-other direction for understanding the results of the RGand connecting them to experiment. These authors havederived the 1D version of the rotor model as an effec-tive description of continuum bosons. In doing so, theyhave established a connection between the strong disor-der RG treatment of Altman, Kafri, Polkovnikov, andRefael and cold atom experiments on rubidium-87. Re-markably, the distributions that Vosk and Altman deriveare of the same form as the fixed point distributions foundby the strong disorder RG13,20. If such a treatment canbe extended to the 2D case, that would be very valu-able, both as a clarification of the critical behavior andas an indication of the relevance of the results to currentexperiments.

24

Acknowledgments

It is our pleasure to acknowledge useful discussionswith Anton Akhmerov, Ravin Bhatt, Francois Crepin,David Huse, Joel Moore, Nikolai Prokof’ev, Srini-vas Raghu, Boris Svistunov, Nandini Trivedi, and AriTurner. We particularly thank Olexei Motrunich forcomputer time and Bryan Clark for pointing out howto estimate errors on the critical exponents that we ob-tain from scaling collapse. SI would like to acknowl-edge Robert Dondero for helpful suggestions for resolvingproblems with the RG code. SI would also like to expressgratitude to the organizers of the 2010 Boulder Schoolfor Condensed Matter and Materials Physics, the 2011Cargese School on Disordered Systems, and the 2011Princeton Summer School for Condensed Matter Physics.SI and DP both thank the 2010 International Centre forTheoretical Sciences school in Mysore. This material isbased upon work supported, in part, by the National Sci-ence Foundation under Grant No. 1066293 and the hospi-tality of the Aspen Center for Physics. DP acknowledgesfinancial support by the Lee A. DuBridge Fellowship, andGR acknowledges support from the Packard Foundation.

Appendix A: Sum Rule vs. Maximum Rule

In this appendix, we present a short argument for whyit may be preferable to use the sum rule (see equations(32) and (33)) instead of the maximum rule (see equa-tion (34)). In dimensions greater than one, it should beeasier to form ordered (e. g. superfluid) phases. Indeed,the transition for the clean rotor model occurs when J

U is

substantially smaller than one22. Suppose we begin withthe clean model at its critical point and then disorder itby increasing some Josephson couplings and decreasingsome charging energies. Suppose further that we do thissuch that that the greatest Josephson coupling Jmax isstill less than the weakest charging energy Umin. Then,using the maximum rule, the strong disorder renormal-ization procedure will predict no cluster formation at all.In other words, it will predict the ground state to be aMott insulator, and this result seems inconsistent withthe location of the clean transition. As mentioned previ-ously, several authors have argued that, if Jmax

Uminexceeds

the ratio JU at the clean transition, then we should ex-

pect glassy behavior, because there can be rare-regions inwhich the system looks locally superfluid6. With the sumrule, the procedure has a mechanism to circumvent thisinconsistency. The Josephson coupling scale can actuallygrow through decimation and compete with the chargingenergy scale. Thus, there can be cluster formation, andthe procedure can find ground states that are glassy orsuperfluid, even when all Josephson couplings of the baremodel are less than the minimum bare charging energy.This indeed occurs in the numerics, as we have notedwhile presenting the numerical data above.

The notion of the Josephson coupling scale increasingthrough the RG may be a source of concern to some read-ers. The increase actually corresponds to the generationof multiple effective couplings between two sites throughdifferent paths in the lattice. This still happens whenthe maximum rule is used, but it is hidden through thediscarding of certain paths. If the coupling through eachpath is treated as an independent Josephson coupling,then the Josephson coupling scale does decrease as therenormalization proceeds. However, when it is time todetermine the next decimation step, it is necessary toconsider all of the couplings between any two sites. Forthis reason, it makes sense to sum all the couplings intoone effective coupling between the sites.

Appendix B: Measuring Physical Properties in theRG

Here, we work out two examples of how estimates ofphysical properties can be extracted from the RG proce-dure.

1. Particle Number Variance

First, consider the quantity:∑j

〈n2j 〉 (B1)

This particle number variance gives the mean squarednumber fluctuation away from the large filling, summedover all sites in the lattice. When normalized by 1

L2 , wefind numerically that this quantity stays constant as thesystem size is increased for all choices of distributionsand parameters. As such, this quantity is completelyuninteresting for discriminating between phases of themodel, but we do calculate it for comparison to exactdiagonalization in Appendix D.

The calculation of the particle number variance (B1)is most straightforward when clusters do not form, solet us first consider the case where some site X is notclustered with any other site during the renormalization.At some stage in the procedure, the site is decimatedaway. The number fluctuation on site X is locked tozero at leading order with corrections incorporated insecond order perturbation theory. An approximation tothe ground state value of 〈n2

X〉 can be obtained from theperturbative expansion of the state. In particular:

〈n2X〉 ≈

1

2

∑k

J2Xk

(UX + Uk)2(B2)

When clusters do form, the calculation is trickier, butit can be performed by carefully keeping track of how theoperator that we are targeting is written in terms of thecluster and relative number operators introduced in link

25

decimation. To illustrate this, suppose we are trying tomeasure the operator:

〈aj n2j + akn

2k〉 = aj〈n2

j 〉+ ak〈n2k〉 (B3)

The factors aj and ak are just numbers. In (B1), allthese factors are one, but we will motivate the inclusionof more general a factors here shortly. If sites j and k aremerged into a cluster, then we switch to the operators nCand nR to find:

aj n2j + akn

2k =

ajU2k + akU

2j

(Uj + Uk)2n2C +

ajUk − akUjUj + Uk

nRnC

+(aj + ak)n2R (B4)

During link decimation, the relative coordinate is spec-ified, so the expectation value of the final term can befound immediately from the harmonic approximation:

(aj + ak)〈n2R〉 ≈ (aj + ak)

1

2γ12

(B5)

Furthermore, the harmonic theory also predicts that theexpectation value of the term linear in nR will vanish.The calculation of the term proportional to n2

C must bedeferred to later in the renormalization procedure. Thus,we keep the operator n2

C in the portion of the sum (B1)that remains to be evaluated, where it appears just likethe n2 for a bare site, but multiplied by a renormalizedaC coefficient. This was the motivation for including thea factors; though the bare values of these factors are allequal, different values can be generated through clusterformation. If the cluster is merged with another clus-ter in a future link decimation, we repeat the procedureabove. When the cluster is finally decimated in a sitedecimation, the cluster’s contributions to the sum (B1)are calculated through equation (B2) and then multipliedby the appropriate a factor.

The procedure outlined above can run into a difficultythat we can anticipate by thinking about the two-siteproblem. Suppose two sites, labelled 1 and 2, are con-nected by a Josephson coupling J12. Furthermore, sup-pose U1 > U2, but both charging energies are greaterthan J12. Then, we would decimate site 1 first and ob-tain an estimate for the site’s contribution to the particlenumber variance (B1) through equation (B2). Next, wewould decimate site 2 and find that it does not contributeat all to (B1) because there are no remaining links. How-ever, the contribution of site 2 should, in fact, be equalto that of site 1, so our estimate is off by a factor oftwo. We can verify this by adopting cluster and relativecoordinates (23) and then calculating (B1) by doing per-turbation theory for the relative coordinate. To partiallyresolve this difficulty, we can keep track of which sites areunclustered by the RG process. At the end, we can returnto the original lattice and calculate the contributions ofthese unclustered sites to (B1) using the bare couplings.For sites that are clustered by the RG, we reason that

the main contribution to (B1) comes from internal fluc-tuations of the cluster, so this correction may not be soimportant.

It is possible to raise another objection to our calcu-lation of (B1). The perturbation theory that leads tothe result in (B2) incorporates the charging energies onsites neighboring site X. However, the perturbation the-ory leading to the RG rule (21) does not. How do weresolve this contradiction? When we calculate effectiveJosephson couplings in the RG, what we want to calcu-late is the effective rate of tunneling, once a boson hasleft one neighboring site and before it arrives at another,through the link-site-link system. For this purpose, it isappropriate to treat the sites neighboring site X as fic-titious charging energy-free islands. On the other hand,when calculating observable quantities like (B1), it is im-portant to account for the fact that the ability move aparticle or hole from site X to a neighboring site alsodepends on how hard it is to charge the neighboring site.

2. Susceptibility

In our studies of physical properties of the phases of thedisordered rotor model, we considered the susceptibilityχ. We defined this quantity in equation (41) as the linearresponse of the system to a uniform ordering field (40)

coupling to cos(φj).To calculate χ, we consider how to calculate the ex-

pectation value: ∑j

bj〈cos(φj)〉 (B6)

in the presence of an infinitesmal ordering field h. Aswith the a-factors in the calculation of the particle num-ber variance, all the bare bj = 1. When clusters form,the effective b-factor for the cluster can differ from unity.

If a bare site is decimated, then perturbation theory inh gives:

〈cos(φX)〉 =h

UX(B7)

Since bX = 1 for a bare site, (B7) is the contribution ofsite X to the quantity (B6).

When a link decimation joins two sites into a cluster,the corresponding terms in the sum (B6) merge as:

bj cos(φj) + bk cos(φk) = bj cos(φC + µj φR)

+bk cos(φC − µkφR)

≈ bj cos(φC)〈cos(µj φR)〉 (B8)

−bj sin(φC)〈sin(µj φR)〉+bk cos(φC)〈cos(µkφR)〉+bk sin(φC)〈sin(µkφR)〉

= (bjcDW,j + bkcDW,k) cos(φC)

26

In this calculation, µj and µk are the ratios introducedin equation (30), and cDW,j and cDW,k are precisely theDebye-Waller factors given in equation (31). We can readoff the renormalized b-factor for the cluster from the cal-culation above, and the resulting expression is given inequation (42).

Next, it is important to note that the ordering fieldterms in the Hamiltonian (40) transform in the same

way. In other words, the term coupling to cos(φC) inthe Hamiltonian should appear multiplied by bC aftera merger. Physically, this corresponds to the fact that,when the cluster phase rotates, sC bare phases rotatesemi-coherently. Complete coherence is lost due to quan-tum fluctuations, which are accounted for by Debye-Waller factors. The factor bC can be thought of as theeffective number of rotors that are coherently orderingwith the field. The energetic cost of φC straying fromthe direction of the ordering field is therefore amplifiedby this factor. When the cluster is finally decimated, theperturbation is amplified by this amount. Furthermore,

cos(φC) appears in the sum (B6) multiplied by bC , so thetotal contribution of the cluster to the sum is:

bC〈cos(φC)〉 =hb2CUC

(B9)

From equations (B7) and (B9), the calculation of thelinear response to an infinitesmal h (i. e. the susceptibilityχ) follows immediately.

Appendix C: The Pi(U) Gaussian, Pi(J) Power LawData Set

In the work reported above, we have explored thesuperfluid-insulator transition of the disordered rotormodel using several different species of disorder (seeequations (9)-(12)). In doing so, we have exposed uni-versal features of the critical behavior. After providingnumerical evidence of this universality however, we focuson data for one particular choice of initial distributions.In this appendix, we describe this choice of distributionsin greater detail.

The choice of distributions in question first appears inFigure 7. The initial distribution of charging energiesPi(U) is taken to be a Gaussian with center at U0 andwidth σU = 2. Hence, the form of the distribution is:

Pi(U) ∝ exp

[− (U − U0)2

8

](C1)

Recall that the Gaussian distribution is truncated at3σU = 6, so the distribution only has weight in the in-terval U ∈ (U0 − 6, U0 + 6). We leave U0 unspecified forthe moment, because it is the parameter that we use totune through the transition.

The initial Josephson coupling distribution Pi(J) is apower law of the form J−1.6. We choose the cutoffs sothat the distribution has weight for J ∈ (0.5, 100). This is

5 10 15 200.95

1

1.05

1.1

U0

(smax)

FIG. 22: A test of the convergence of smax in the thresholdingparameter α. The variable υ is the ratio of the estimate ofsmax for αh, the less conservative value of the thresholdingparameter, to α`, the more conservative value. We plot υagainst the tuning parameter U0. Smaller values of αh andα` are used in the vicinity of the transition at U0 = 9. Seethe text of Appendix C for details.

a very wide power law distribution, so the correspondingflows begin well above the unstable finite disorder fixedpoint in the numerical flow diagrams. Explicitly:

Pi(J) ≈ 0.413J−1.6 (C2)

For this choice of distributions, we have we have ac-quired data for U0 = 400, 20, 18, 16, 14, 12, 10, 9.8, 9.6,9.4, 9.2, 9, 8.8, 8.6, 8.4, 8.2, 8, 7.5, 7, and 6.5. We havealways acquired data for L = 25, 50, 75, 100, 150, 200,and 300. In all cases, we have pooled data for 103 disor-der samples. All the data is consistent in indicating thatU0 = 9 is closest to criticality, and therefore:

U0,c ≈ 9.0± 0.2 (C3)

Close to criticality, it is better to use a lower value ofthe thresholding parameter α. For 10 ≥ U0 ≥ 8, wehave run the RG with αh = 10−5 and α` = 5 × 10−6

to test for convergence. Further away from criticality,we have instead used αh = 5 × 10−5 and α` = 2.5 ×10−5. Figure 22 shows a test of the convergence of themaximum cluster size smax in the thresholding parameterα. We plot:

υ =smax(αh)

smax(α`)(C4)

vs. the tuning parameter U0. The ratio (C4) is essentiallyalways within two error bars of unity. We take this as anindication that physical properties have converged.

Appendix D: Further Comments on the Use of theStrong Disorder RG in a Finite Disorder Context

This appendix is devoted to exploring, in further de-tail, the validity of the RG procedure. We first expandupon the argument, introduced earlier in the paper, forthe reliability of the RG near criticality. Then, we moveaway from criticality and assess the performance of the

27

RG in the various phases of the disordered rotor model(7). Next, we focus on each of the RG steps, considercircumstances in which they may fail to capture impor-tant physics, and formulate tests to ensure that the RGis trustworthy in these situations. Finally, we present acomparison of the RG to exact diagonalization of smallsystems.

1. Review of the Argument for the RG atCriticality

Our confidence in the RG procedure near criticalityrests on the form of the critical Josephson coupling distri-bution, reported in equations (36) and (39). Infinite ran-domness develops when P (J) ∝ J−1, and our numericalevidence suggests that the critical distribution of the dis-ordered rotor model decays even more strongly19. Never-theless, the critical distribution does not exhibit infiniterandomness, because as seen in Figure 11, it is cut offfrom below. Recall that the lower cutoff of the “Joseph-son coupling distribution” is set by our choice to retain,in statistics, only the dominant 2N Josephson couplings,where N is the number of sites remaining in the effec-tive lattice. Then, the appropriate way to interpret thedistributions in Figure 10 is the following: penetratingany given site in the effective lattice, there are likely tobe on the order of four Josephson couplings drawn fromthe depicted distribution. There may be other links pen-etrating the site, but these will be even weaker. We canestimate the typical strength of the four dominant linksby comparing the median of the critical Josephson cou-pling distribution to the maximum. Doing this for thefour distributions depicted in panel (b) of Figure 13, we

find that the ratioJtyp

Ω is 0.10± 0.01. Hence, the typical

link is quantitatively weak compared to JU ≈ 0.345 at the

clean transition22.The considerations above form a strong argument for

the validity of the site decimation RG step. Here, weseek out the dominant effective charging energy in theeffective lattice, and treat the links penetrating the sitein perturbation theory. This perturbation theory is likelya very good approximation, because the Josephson cou-plings penetrating the site in question are usually ex-tremely weak.

Now, we turn to the link decimation step, in which weseek out the largest Josephson coupling in the lattice andmerge the corresponding sites into a cluster. Although,the Josephson coupling being decimated is the largest en-ergy scale in the system, there is a high probability thatall the other links penetrating the two sites being joinedwill be very small. However, the critical distribution ofcharging energies is not peaked at low U . The structureof the distributions plotted in Figure 12 suggests thatit is quite likely that one or both sites being merged willhave a charging energy of the same order as the RG scale,thus violating the strong disorder hypothesis. Our treat-ment of the quantum fluctuations of the relative phase

of the cluster is based on the harmonic approximation(27). Is this approximation appropriate when the charg-ing energies of the two-site problem are comparable inmagnitude to the Josephson coupling? Alternatively, dothe quantum fluctuations grow so large that the clus-tering becomes meaningless? We address this questionas follows: using the fact that the remaining links areweak, we isolate the two site problem and solve it exactly,treating the remaining links via second order perturba-tion theory. Comparing the results of the RG with theexact solution, we find that, even in this worst case sce-nario, the RG procedures reasonably accurate effectivecouplings. The evidence for this claim is given in section3 below.

2. Reliability of the RG in the Phases

How reliable is the RG when we move away from criti-cality and into the phases of the disordered rotor model?To gain some insight into this question, we can consultFigure 23, which expands upon Figures 11 and 13 byplotting renormalized J and U distributions away fromcriticality.

(a)

8 6 4 2 0 215

10

5

0

ln(J/ )

ln(P(J))

0 200 400 600 800 10000

10

20

30

glassnear criticalsuperfluid

(b)

8 6 4 2 0 215

10

5

ln(U/ )

ln(P(U))

FIG. 23: In panel (a), a sweep of renormalized J distributionthrough the glass and into the superfluid. The initial distribu-tions are those described in Appendix C: Pi(U) is Gaussianand Pi(J) ∝ J−1.6. All data is taken for L = 300 lattices,and the renormalized distribution is computed when 300 sitesremain in the effective lattice. The values of U0 shown are18, 12, 9.6, 9 (critical), 8.4, and 7.5. Panel (b) shows the cor-responding sweep of the renormalized U distribution at thesame stage of the RG.

Proceeding into the glassy regime, the arguments pre-sented above for the validity of the RG near criticalitygenerally become stronger. The primary reason for this

28

is that the renormalized J distributions become progres-sively broader than at criticality. In the flow diagrams(Figures 6-9), this is reflected in the apparent divergenceof ∆J

J. Consequently, the assumption of isolating local

degrees of freedom becomes better as we get deeper intothe glass. One complication is that the renormalized Udistribution becomes more strongly peaked near the RGscale. This may pose trouble for the link decimation stepand makes it especially important to consider the reli-ability of this step when a strong Jjk couples two siteswith charging energies Uj ≈ Uk ≈ Jjk. As mentionedpreviously, we will study this worst-case scenario in part3 of this appendix and find that the RG still works rea-sonably well.

Now, we turn to the superfluid phase. As we proceedaway from criticality, the renormalized charging energydistribution seems to become approximately flat. In com-parison to criticality, the renormalized U distribution alsobecomes broader, indicating that we need not be as wor-ried about encountering a strong J coupling sites withcomparably strong U values. However, the J distribu-tion also seems to become flat deep in the superfluid,and this is problematic. For example, during link dec-imation, it may not be a reasonable approximation toisolate the two-site problem centered on the strongestJosephson coupling. As we proceed into the superfluid,it is necessary to be more dubious of the RG results;nearer to criticality however, the arguments used at thecritical point are probably approximately valid.

The plots in Figure 23 attempt to elucidate system-atics in the behavior of the renormalized distributionsin the insulating and superfluid regimes, but we shouldmention that this figure should be interpreted with somecare. For the choice of distributions in Appendix C, flowsterminating in the insulating or superfluid regions of theflow diagram nevertheless propagate in the vicinity of theunstable fixed point along the way. This can be seen, forinstance, in Figure 7. For certain choices of initial dis-tributions, there can be flows towards the insulating orsuperfluid regimes which never propagate anywhere nearthe unstable fixed point. Consequently, the RG neverhas an opportunity to wash away the details of the ini-tial distributions and allow the universal properties of thefixed point to emerge. Hence, the renormalized distribu-tions generated along such flow trajectories are unlikelyto exhibit the clean systematic properties seen in Figure23.

3. Analysis of Potential Problems with the RG

Next, we address some potential difficulties with thearguments presented above for the reliability of the renor-malization procedure in the critical region. These diffi-culties are rooted in the lack of strong randomness in thecritical charging energy distribution.

Consider the lattice geometry shown in Figure 24. Sup-pose that all links except the Josephson coupling between

sites 1 and 2 are perturbatively weak. Now, suppose fur-ther that J12, U1, and U2 are comparable in magnitude,but J12 is the largest of the three. We would like to for-mulate a test of whether the RG appropriately handlesthis situation. Within the RG, a link decimation wouldmerge sites 1 and 2 into a cluster. All links penetrat-ing sites 1 and 2 would be modified by their correspond-ing Debye-Waller factors (31) cDW,1 and cDW,2. Then,because all the remaining links are assumed to be veryweak, the cluster of sites 1 and 2 will be decimated, pro-ducing an effective coupling between sites 3 and 5:

J35,RG = cDW,1cDW,2J13J25(U1 + U2)

U1U2(D1)

Another approach to calculating this effective couplingwould be the following: take the two-site problem of sites1 and 2 and exactly diagonalize it. Then, to leadingorder, sites 1 and 2 should be locked into the two-siteground state, with perturbative corrections coming fromthe Josephson couplings J13, J14, and J25. This pertur-bation theory leads to an effective coupling through thesites 1 and 2. This alternative procedure is perhaps moreappropriate, because it does not presuppose the harmonicapproximation. In Figure 25, we assess how much of anerror we make by adopting the harmonic approximation.Holding J fixed, we sweep U = U1 = U2 through J ,comparing the RG with the alternative method outlinedabove. We see that the usual RG performs reasonablywell, implying that the harmonic approximation is safe.

Finally, we consider another potentially dangerous sce-nario. We return to the lattice shown in Figure 24. Now,we assume that J12 is greater than all other Josephsoncouplings, but it too is much weaker than the chargingenergies U1 and U2. In particular, J12

U2= 0.05. Then,

we sweep U1 such that it passes through a regime where|U1−U2| < J12. The danger here is that the RG may ig-nore resonance effects associated with this region. Withinthe usual RG, sites 1 and 2 will be decimated in turn togive:

J35,RG ≈J13J12J25

U1U2(D2)

J34,RG ≈J13J14

U1(D3)

where we ignore subleading corrections coming from po-tential applications of the sum rule, depending on theorder of decimation of sites 1 and 2. To consider poten-tial resonance effects, we can also implement the samehybrid exact diagonalization and RG procedure that weused above. In Figure 26, we compare the two methodsand find excellent agreement.

4. Strong Disorder RG vs. Exact Diagonalization

As a final test of the RG procedure, we now comparethe RG to exact diagonalization of small systems. We

29

!"

#"

$" %" &"

FIG. 24: The graph structure for the tests reported in Figures25 and 26. The links J13, J14, and J25 are assumed to beperturbatively weak. The charging energies U1 and U2 andthe Josephson coupling J12 are varied to explore potentiallytroublesome scenarios.

0 20 40 60 80 100 1200.5

1

1.5

2

J/U

J eff,E

D+R

G/J

eff,p

ure

RG

FIG. 25: In this test, J = J12 is assumed to be the strongestcoupling in the system, but U = U1 = U2 may be of thesame order. An effective coupling between sites 3 and 5 iscalculated using two methods. One is the usual RG schemeused in this paper. Another is a hybrid exact diagonalizationand RG scheme: The two-site problem of sites 1 and 2 isexactly diagonalized. Then, the resulting cluster is decimatedaway, and perturbation theory is used to calculate an effectivecoupling between sites 3 and 5. The two candidate valuesfor the effective coupling J35 are compared in the plot, as afunction of J

U.

0 0.5 1 1.5 20

5

10

U1/U2

J eff

0 500 1000 1500 20000

5

10

U1/U2

J eff

J35,ED+RGJ34,ED+RGJ35,pure RGJ34,pure RG

FIG. 26: In this test, J12 = 0.05 and U2 = 1. Hence, J12 isrelatively quite weak. However, we vary U1 such that it passesthrough a regime where |U1 − U2| < J12, where there maybe a danger of resonance effects. We calculate the effectivecouplings J34 and J35 using two schemes. One is the usualRG scheme used in this paper. Another is a hybrid exactdiagonalization and RG scheme: the two-site problem of sites1 and 2 is exactly diagonalized. Then, the resulting cluster isdecimated away, and perturbation theory is used to calculatean effective coupling between sites 3 and 4 and between sites 3and 5. The effective couplings predicted by the two methodsare compared in the plot, as a function of U1

U2. No resonance

effects are observed.

0 1 2 30

2

4

6

particle number variance from ED

parti

cle

num

ber

varia

nce

from

RG

0 100 200 300 4000

10

20

min from ED

min

from

RG

GGGPFP

FIG. 27: A sample-by-sample comparison of the particle num-ber variance predictions from exact diagonalization and fromthe renormalization procedure. The disordered couplings aredrawn from three different choices of the distributions, with100 samples per distribution type. See the text of AppendixD for details on the distribution choices GG, GP, and FP.Also pictured is the coincidence line y = x, along which thepoints would ideally fall for full quantitative agreement.

truncate the possible number fluctuation on each site tonj = −1, 0, 1, interpret these three values as possible z-axis spin projections of a spin-one object, and in doingso, arrive at a “spin-one” model:

H = −∑〈jk〉

Jjk2

(S+j S−k + S−j S

+k ) +

∑j

UjS2zj (D4)

The Hilbert space of this spin-one model grows with the

size of the lattice as 3L2

. The particle number conser-vation of the rotor model manifests here as total spinconservation along the z-axis. This means that we canpartition the Hilbert space into different total spin sec-tors and diagonalize the sectors separately. For mostground state expectation values, we just need to diago-nalize the total spin zero sector, and to calculate a charg-ing gap, the only additional diagonalization needed is forthe total spin one sector. Despite these simplifications,computational limitations restrict us to studying 3 × 3lattices using CLAPACK. Testing the RG against exactdiagonalization cannot directly tell us about the reliabil-ity of the RG at criticality, because exact diagonalizationis limited to very small system sizes. However, a compar-ison with exact diagonalization can tell us how well theRG captures information about small patches of a largerlattice, and this information is potentially quite valuablefor building confidence in the RG.

Another complication arises precisely due to theHilbert space truncation: the spin-one model may notalways approximate the full rotor model well. This isespecially true when there is cluster formation, becausethen the rotor model has more of an opportunity to accessparticle number fluctuations that exceed 1 in magnitude.In other words, the strong disorder renormalization groupand the exact diagonalization of the spin-one model con-stitute different approximations to the behavior of ourrandom boson model. We cannot expect the two ap-proximations to show full quantitative agreement, butwe proceed with the comparison, despite its limitations,

30

0 5 10 15 200

10

20

min from ED

min

from

RG

0 100 200 300 4000

10

20

min from ED

min

from

RG

GGGPFP

FIG. 28: A sample-by-sample comparison of the charging gappredictions from exact diagonalization and from the renormal-ization procedure. The disordered couplings are drawn fromthree different choices of the distributions, with 100 samplesper distribution type. See the text of Appendix D for detailson the distribution choices GG, GP, and FP. Also picturedis the coincidence line y = x, along which the points wouldideally fall for full quantitative agreement.

with the hope of at least seeing qualitative correspon-dence between the two methods.

Our general approach to comparing the RG with exactdiagonalization will be to measure physical quantities, onindividual 3× 3 samples, using both methods and assessif there is a correlation between the predictions. The firstquantity that we compare is the particle number variance(B1). The interested reader may consult Appendix B tosee how this quantity is calculated during the renormal-ization procedure. We also compare the charging gap∆min, the minimum energy needed to add a particle orhole to the system. This quantity is typically estimatedduring site decimation. The logic behind site decimationis that the charging energy for some site X is greaterthan all other scales in the problem, so the site can bedisconnected from the rest of the lattice to leading or-der. Then, the charging energy gives an estimate for thelocal charging gap on that site. During the RG, we findmany such charging gaps from the various site decima-tions. The minimum among all of these gives an estimatefor the charging gap for the whole system. This minimumis always given by the charging energy of the final remain-ing site, so an estimate of ∆min can be simply obtainedby renormalizing down to a single site problem and mea-suring the charging energy of the remaining site. For the

purposes of comparison to exact diagonalization however,we find that we obtain better quantitative agreement be-tween the two methods if we renormalize down to aneffective two-site problem and then perform exact diag-onalization on that system. Exactly diagonalizing thentot = 0 and ntot = 1 sectors of this two-site problemthen yields a charging gap for the system. In this exactdiagonlization, we need not truncate the on-site numberfluctuation to nX = −1, 0, 1. Instead, in the numerics,we typically truncate to nX = −100 . . . 100.

In Figures 27 and 28, we present comparisons for threedifferent data sets. In the first data set, we use take Pi(U)and Pi(J) to be Gaussian. We fix U0 = 10 and σU = 3.Then, we randomly sample J0 in the interval (0, 5) andσJ in the interval (0, J03 ). The aim is to approximatesome of the environments that the RG encounters in runssuch as those reported in Figure 6. The second data setuses the distributions described in Appendix C: Pi(U)is Gaussian and Pi(J) ∝ J−1.6. We randomly sampleU0 ∈ (6.5, 20). Here, the motivation is to look at thetypes of environments that the RG encounters when itapproaches the unstable fixed point from above. In thefinal data set, we try to mimic 3×3 patches that the RGmight encounter near criticality. To this end, the initialJ distribution is fixed to a power law P (J) ∝ J−1.16

(see equation (39)) and the cutoffs are chosen so that theratio of Jmin to Jmax is approximately that observed inpanel (b) of Figure 13. The distribution Pi(U) is flat withUmax = Jmax and with Umin randomly sampled such thatthe ratio of Umin to Umax lies in (e−2, e−1). Figures 27and 28 identify these three data sets with the labels GG,GP, and FP respectively.

Both figures show that the predictions of the RG areclearly correlated with the predictions from exact diago-nalization, although the level of quantitative correspon-dence varies. For the particle number variance, quanti-tative agreement is lost at higher values of the variance,essentially corresponding to cases in which there is clus-tering. One potential source of error could be the Hilbertspace truncation of the exact diagonalization, althoughthe structure of the harmonic ground state (27) makesit unlikely that this could account for the entire discrep-ancy. Nevertheless, these comparisons suggest that theRG is retaining useful information about the system.

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