Dual theory of the superfluid-Bose-glass transition in the disordered Bose-Hubbard model in one and...

PHYSICAL REVIEW B 1 JUNE 1998-IVOLUME 57, NUMBER 21

Dual theory of the superfluid-Bose-glass transition in the disordered Bose-Hubbard modelin one and two dimensions

Igor F. Herbut*Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver,

British Columbia, Canada V6T 1Z1~Received 19 November 1997!

I study the zero-temperature phase transition between the superfluid and the insulating ground states of theBose-Hubbard model in a random chemical potential and at large integer average number of particles per site.Duality transformation maps the pure Bose-Hubbard model onto the sine-Gordon theory in one dimension~1D!, and onto the three dimensional Higgs electrodynamics in two dimensions~2D!. In 1D the randomchemical potential in the dual theory couples to the space derivative of the dual field, and appears as a randommagnetic field along the imaginary time direction in 2D. I show that the transition from the superfluid stateboth in 1D and 2D is always controlled by the random critical point. This arises due to a coupling constant inthe dual theory with replicas that becomes generated at large distances by the random chemical potential, andrepresents a relevant perturbation at the pure superfluid-Mott insulator fixed point. At large distances the dualtheory in 1D becomes equivalent to Haldane’s macroscopic representation of a disordered quantum fluid,where the generated term is identified with the random backscattering. In 2D the generated coupling corre-sponds to the random mass of the complex field that represents vortex loops. I calculate the critical exponentsat the superfluid-Bose-glass fixed point in 2D to ben51.38 andz51.93, and the universal conductivity at thetransition sc50.25e

*2 /h, using the one-loop field-theoretic renormalization group in fixed dimension.

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I. INTRODUCTION

In recent years, a new paradigm in physics of condenmatter has emerged under the name of superconduinsulator~SI! transition. Among the quantum many-particsystems that are believed to exhibit this type of transitbetween their ground states at zero temperature are4He inrandom media,1 thin superconducting films,2 and Josephsonjunction arrays.3 At T50, as some parameter in the Hamtonian is varied, such a system is expected to show eizero or infinite dc linear resistance, with the transition cauby purely quantum fluctuations that qualitatively alter tmany-body ground state. Experimentally, tsuperconductor-insulator transition is manifested as a chafrom continuously increasing to sharply decreasing retance of the system as temperature is lowered, in accordwith the notion that atT50 the ground state is either ainsulator or a superconductor.

It has been argued4,5 that as a model for superconductoinsulator transitions it suffices to consider a Hamiltonianinteracting bosons in random external potential. The baassumption behind the idea is that the transition correspoto the onset of phase coherence of the already preforCooper pairs, and not to the formation of the pairs theselves. The ground state of the bosons can then be esuperfluid~SF! or insulating; the insulating state may aridue to repulsive interactions, in which case it is an incopressible Mott insulator~MI ! with a gap, or from a combi-nation of interaction and localization effects, which maysult in the formation of the gapless Bose glass phase~BG!.Particularly interesting is the SI transition in two dimensio~2D!. At finite temperature 2D are the lower critical dime

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sion for superconductivity with the superconducting trantion being of the Kosterlitz-Thouless type.6 Also, for nonin-teracting particles 2D are the lower critical dimension for tAnderson localization, with all single-particle states weaklocalized.7 Arguably, the main source of interest in 2superconductor-insulator transitions comes fromsuggestion8 that right at the quantum (T50) critical point dcconductivity should be finite and universal, determined oby the universality class of the transition. This exotic posbility of normal but universal diffusion results from a delcate balance between localization and superfluid fluctuatioand is unique for 2D.

Although the experimental support for universality of thconductivity at the transition is still rather weak,2 the calcu-lation of this and other critical quantities for different univesality classes of dirty-boson systems in 2D is a fundameand unsolved problem. Without disorder, the transition insystem of interacting bosons on a lattice is always betwthe Mott insulator and superfluid phases. With integer avage number of particles per site, the MI-SF transition isthe universality class of the classical 3DXY model.9,5 Thecritical exponents in this case are well known to bez51,n'0.667, and the universal conductivity has been calculaby a variety of methods, including 1/N and the Monte Carlocalculations,10 and thee expansion.11 With disorder present,the critical behavior at the superconductor-insulator trantion in 2D is much less understood, due to apparent nonistence of the upper critical dimension where the thewould have a critical point at a weak coupling.5,12 In fact,even the very nature of the transition at commensurate bodensities has been a matter of debate: simple argume5

suggest that at weak disorder the transition into the su

13 729 © 1998 The American Physical Society

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13 730 57IGOR F. HERBUT

fluid phase proceeds from the Bose-glass phase, while mof the numerical studies see a direct MI-SF transition.13,14 Inthe absence of a controlled analytical approach to the plem with disorder, most of our knowledge of the criticbehavior at the BG-SF transition originates from the numcal studies.15–17 On the analytical side, large-N expansion,18

real-space methods,19 and strong-coupling expansion20 haveall been brought to bear. While these calculations give vaable information on the phase diagram of the system,critical exponents are difficult to extract and less reliable.particular, the universal conductivity at the BG-SF transitihas hitherto been calculated only numerically.

In the present paper I study the critical behavior atT50superconductor-insulator transition in 1D and 2D systemsinteracting lattice bosons in Gaussian random chemicaltential and at commensurate densities within dual descripof the problem21 that focuses on topological defects in thmany-body ground state. This approach, in which one loat the destruction of the long-range order by proliferationthe defects, has proven invaluable for several problems inpast, the Kosterlitz-Thouless transition being a primeample. Let me first describe the main results for the simp1D system. By duality the disordered Bose-Hubbard momaps onto the sine-Gordon theory in which the randchemical potential couples linearly to the space derivativethe dual field. The elimination of the fast modes of the dfield performed perturbatively in the strength of the lattipotential always leads to exponential suppression ofsmall periodic potential by disorder, so that at some smacutoff the lattice potential in the theory scales to zero. Mimportantly, while renormalizing to zero the periodic potetial together with the random chemical potential generatesadditional disorder term, so that at long length scales the dtheory is without the periodic potential term, but contaitwo different disorder terms. This long length scale effecttheory is equivalent to Haldane’s macroscoprepresentation22,5 of the disordered quantum fluid in 1Dwhere the generated term arises from the backscatterinthe random potential. Once it helped generate the imporbackscattering term in the action, the random chemicaltential that corresponds to forward scattering in the lolength scale theory becomes redundant and can be remfrom the problem by an appropriate change of variabThus the 1D disordered Bose-Hubbard model at large lenscales is described by Haldane’s macroscopic theory for dbosons with no trace of the lattice potential. The transitfrom the superfluid state is even at infinitesimal disorder ctrolled by the random fixed point in 1D,23,5 which I interpretas a sign that the transition is between the superfluid andBose-glass phases. I end the analysis of the 1D problemthe discussion of the shape of the BG-SF phase boundathe interaction-disorder plane.

The lesson from the 1D example is that the transitfrom the superfluid state in the disordered system is alwinto the Bose glass, and that it is controlled by the fixed pothat can be located easily in the space of coupling constof the dual theory. This suggests the same line of attacthe 2D version of the problem. A short account of the resuin 2D has already been published.24 The dual theory for 2Ddirty bosons atT50 and with short-range repulsion is cosiderably more complicated than its 1D counterpart, and

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the form of 211D classical, anisotropic Higgs electrodynamics ~Ginzburg-Landau superconductor! in random magneticfield, directed and correlated along one dimension. The cplex Higgs field describes closed vortex loops in spaceimaginary time, which interact via long-range forces meated by the gauge field, whose Maxwell term in the actdescribes the sound excitations in the parent superfluid sCondensation of the vortex field implies destruction of tsuperfluid by the appearance of infinitely large vortex loowhich is also reflected in the induced Higgs mass ofgauge field. At the pure critical point that controls the MI-Stransition, weak anisotropy that arises from the underlyquantum nature of the problem is argued to be irrelevaMore interestingly, weak disorder in the form of a randocorrelated magnetic field turns out to be precisely marginand I assume it is marginally irrelevant. Similarly, as in t1D case, the random chemical potential generates additidisorder that can be understood as the random ‘‘mass’’ ofvortex field, which is strongly relevant at the pure MI-Sfixed point. The flow at the critical surface eventually endsthe attractive random point, which is again interpreted asBG-SF critical point.

The critical exponents and the universal resistivity atBG-SF critical point in 2D are determined to the lowest oder in fixed-point values of the coupling constants in the dtheory, using the field-theoretic renormalization groupfixed dimension.25 The calculations are performed directly211 dimensions since the duality between the Bose-HubbHamiltonian and the Higgs electrodynamics holds onlythe specific dimension, and I also want to determine the uversal conductivity at the critical point, which is again finiexclusively in 211D. The idea of this method is to expresthe universal quantities like the critical exponents or the uversal conductivity as perturbative series in renormalizinstead of bare, coupling constants, so that they assume fivalues at the transition, determined by the critical pointthe theory. The lowest-order resultsz51.93 andn51.38 ob-tained this way compare reasonably well with the numericalculations,17 even though I do not have a truly small prameter. The conductivity is obtained using the Kubo fomula for the response of the disorder field to an exterpotential. By duality of charges and vortices this way odirectly calculates the boson resistivity, instead of the cductivity, as a series in critical coupling constants. Invertithe final result I obtain the universal conductivitysc

50.25(e*2 /h) at the quantum SF-BG critical point, in roug

agreement with the numerical results,16,17 and apparentlysmaller than the value suggested by experiments. Subtleinvolved in interpretation of the experimental measuremeof sc are also discussed.

The paper is organized in the following manner. In tnext section I define the dirty-boson lattice model and uduality transformations to arrive at the lattice theory in desity representation in general dimension. In Sec. III I stuthe dual theory in 1D and discuss the renormalization snario and the phase diagram. In Sec. IV the dual represetion in 2D is derived. The one-loop renormalization of thcontinuum version of the dual theory is performed andcritical exponents and the conductivity at the BG-SF trantion are calculated. In the last section the summary anddiscussion of the main conclusions is given. Technical

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57 13 731DUAL THEORY OF THE SUPERFLUID-BOSE-GLASS . . .

tails are presented in the Appendixes.

II. HAMILTONIAN AND DUALITY TRANSFORMATION

I study the Hamiltonian for a system of charge-e* bosonswith on-site repulsionU.0, written in number-phase representation~Josephson-junction array Hamiltonian!:5,17,21

H5U

2(i

ni22(

ihi ni2t(

i ,n

cos~f i 1 n2f i !, ~1!

where indexi labels the sites of a 1D~2D! lattice, n5 x,y isa unit vector,ni represents a deviation from a largeintegeraverage number of bosons per site,hi is a Gaussian randomchemical potential withhihj5whd i , j and hi50, and f i isthe phase variable canonically conjugate to the numbebosons,@f i ,n j #5 id i , j . The Hamiltonian~1! is in the sameuniversality class as the Bose-Hubbard model.26,17I calculatethe partition function of the system

Z5Tr exp$2bH%, ~2!

b51/T, in the basis of states that diagonalize the partinumbers: 15($ni %

u$ni%&^$ni%u, $ni%5$n1 , . . . ,nNL%, 2`

,ni,` and integer, whereNL is the total number of latticesites. Following the standard procedure for constructionthe path-integral representation of the partition function27

one arrives at the expression

Z5 ($ni ,a%

E2p

p

)i 51

NL

)a51

N

df i ,a exp2S eU

2 (i ,a

ni ,a2

2e(i ,a

hini ,a2te (i ,n,a

cos~f i 1 n,a2f i ,a!

1 i(i ,a

f i ,a~ni ,a112ni ,a!D , ~3!

with the boundary conditions only on the number of bosoni ,05ni ,N , e5b/N, and the limite→0 is assumed. At thispoint there are two ways to proceed: utilizing the Poisssummation formula one may perform the sums over numbof bosons and arrive at the phase representation, from wit follows that without disorder the MI-SF transition atT50is in the universality class of theXY model.9,5 Disorder inthis formulation enters via an imaginary boundary term,ducing significantly the utility of the phase representationthat case. Nevertheless, the phase representation leadsrally to the field theory for the superfluid order parametstudied extensively in the literature.12 Alternatively, one canattempt to integrate the phases out to be left with the pation function in number representation. To that end I useVillain representation of the cosine term in the last equati

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i ,n,a

mi ,a,n2

2et

1 i (i ,n,a

mi ,a,n~f i 1 n,a2f i ,a!D , ~4!

wheremW i ,a are one-component~two-component! integer val-ued currents in 1D~2D! at the sites of the two-~three-!dimensional lattice labeled by indices (i ,a). The integrationover the phases now can be performed exactly, and its eis to produce the constraint on the currents and numberbosons:

¹ rW•mW i ,a1¹t ni ,a50, ~5!

where¹ rW and¹t are the lattice gradients~finite differences!in space and imaginary time directions, respectively. Theequation may be interpreted as a continuity equation, andtwo- ~three-! component vectorMW i ,a5(mW i ,a ,ni ,a) representsthe conserved bosonic current in 1D~2D!. Equation~5! en-sures that only closed current loops in space and imagintime contribute to the statistical sum. The partition functinow assumes the form

Z5 ($MW i ,a%

d~¹•MW !exp2S eU

2 (i ,a

ni ,a2 2e(

i ,ahini ,a

11

2et(i ,a mW i ,a2 D . ~6!

Note that the action in the exponent is completely real. Astands, the last form of the partition function is still difficuto study analytically due to discreetness of the variables,is more amenable to numerical methods.17 It is possible how-ever to further transform the partition function into a motelling expression,28–30as, in this context, was pointed out bFisher and Lee.21 The form of the transformation differs in1D and 2D, so I will address the two cases separately.

III. ONE DIMENSION

A. Dual theory

In 1D the constraint¹•MW 50 may be resolved by introducing a unique integer scalar variableN, so that MW 5(2¹tN,¹xN). The theory then takes the form

Z5 limy→0

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dAi ,a exp2S eU

2 (i ,a

~¹xAi ,a!2

11

2et(i ,a ~¹tAi ,a!22e(i ,a

hi~¹xAi ,a!

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y(i ,a cos~2pAi ,a! D , ~7!

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where the limity→0 serves to force the real-valued field$Ai ,a% to take only integer values. Hereafter I will takeT50. Assuming that softening the constraint ony will notchange the universality class of the transition29,30 in the con-tinuum limit a→0, e→0 the dual theory becomes

S5KE dx dt$@]tu~x,t!#21c2@]xu~x,t!#2%

2E dx dt h~x!]xu~x,t!

2vE dx dt cos2u~x,t!, ~8!

where pA(x,t)5u(x,t), c25Uta2, K51/(2tp2a), and vmay be understood as a~finite! strength of the lattice potential with the lattice constanta. Averaging over the Gaussiarandom potential by introducing replicas in the standway,31 one finally arrives at the form

S5K (a51

N E dx dt$@]tua~x,t!#21c2@]xua~x,t!#2%

2wh (a,b51

N E dx dt dt8]xua~x,t!]xub~x,t8!

2v (a51

N E dx dt cos2ua~x,t!, ~9!

where Greek indices now enumerate replicas, and the lN→0 at the end of the calculation is assumed. Ifwh50,dual to the Bose-Hubbard model~1! in 1D is the familiarsine-Gordon theory that has the transition in theXY univer-sality class asK or v are varied.6 At large K the configura-tions with u close to constant dominate the statistical suwhich implies a sharp particle number at each site and thfore by uncertainty relation, disordered phases. Noticeby simple power counting the disorder couplingwh is rel-evant at the pure fixed point~located atK* c* 5 1

4 , v* 50,see below!, so one may expect that with disorder the natuof the transition in the system should be changed. I wdemonstrate that indeed that is the case, although, inteingly, the couplingwh will ultimately be irrelevant for thecritical behavior. It is however crucial for bringing about thcorrect form of the effective theory at large distances,which I turn next.

B. Renormalization

I now integrate out the components of the fieldu in Eq.~9! with the momentaL/s,uku,L, L;1/a, ands'1, andwith any frequencyv, perturbatively in strength of the perodic potentialv. The calculation is standard, with the onnontrivial step being a matrix inversion needed to obtainpropagator for the dual fieldu, which in the presence odisorder is nontrivial in replica indices. The interested reais refereed to Appendix A for the details, and here I will onmention that this can be done exactly in the limitN→0. Tothe lowest order inv the recursion relations are

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wh

2K2c4D v1O~v3!, ~10!

dK

d ln~s!5

dc

d ln~s!5O~v2!, ~11!

dwh

d ln~s!5wh1O~v2!, ~12!

and I introduce the dimensionless combinationswh /L→wh , v/L2→v and pK→K. At wh50 one may choosethe units of lengths and energy so thatc51, so Eqs.~10!–~12! reduce to the celebrated Kosterlitz recursion relatio6

with K playing the role of temperature for the equivaleclassical 2DXY model. The crucial new feature brought bdisorder is the presence of thewh term in the recursion relation ~10! with a negative sign. Physically this is exactly whone expects: at large distances the random chemical potewashes away the lattice potential. In fact, sincewh is a rel-evant perturbation at the pure fixed point~at v* 50, K*5 1

4 ), we see that it will make a weak enough periodic ptential exponentially irrelevant at large distances for anyK.If this would be the end of the story we would come tosomewhat paradoxical conclusion that disorder always tua Mott insulator into a superfluid. This would follow fromthe observation that oncev50 in Eq. ~9! one can eliminatethe wh term altogether by a shift in the fieldsua(x,t),23 nomatter how largewh might have become at that scale. Thwould be incorrect, however, for the following reason: to tsecond order inv a new disorder term in the action~9! be-comesgeneratedduring the mode elimination

2D~s! (a,b51

N E dx dt dt8cos2@ua~x,t!2ub~x,t8!#,

~13!

with

D~s!5v2wh ln~s!

4K2c4L2, ~14!

wherev andwh are now left dimensionful. The reader wirecognize the term~13! as precisely the one appearingHaldane’s course-grained representation for the problem22,5

and which arises there from the components of the randpotential with the wave vectork;2pr0 and may be under-stood as the backscattering term. Thewh term would corre-spond to purely forward scattering in the large length sctheory in 1D. Once the coupling constantD has becomegenerated and smallv has scaled to zero we may freely shthe dual variables to completely eliminate thewh term fromthe problem. At a smaller cutoffL8 the effective theorytherefore acquires the familiar form

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SL85K~L8!(a

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~15!

with the effect of integration over the fast modes absorbinto the effective values of the coupling constants. Foweak initialv the value ofD(L8) at the scaleL8 at whichvvanishes will be small, so one may monitor the change ofeffective couplings in Eq.~15! during further mode elimina-tion perturbatively inD. The recursion relations for this casare well known:23

dD

d ln~s!5S 32

1

K DD, ~16!

dK

d ln~s!5

4D

K, ~17!

where again I putc51. The transition from the superfluistate is thus always ultimately controlled by the random fixpoint atD* 50 andK* 5 1

3 , at which the correlation lengthexponentn5` and the dynamical exponent isz51. This isin accord with Ref. 5 where the same conclusion wreached using only slightly different reasoning. Similar argments have also recently been advanced in Ref. 32. Thedom fixed point has a natural interpretation as the BGcritical point, although the nature of the insulating phase cnot be directly inferred from the recursion relations at smD. Although the exponents have the same values as aMI-SF fixed point in 1D, it is a different critical point: forinstance, the superfluid two-point correlation function attransition decays algebraically at large distances withpower 1

3 , instead of14 at the MI-SF transition.23

C. The phase diagram

Although the precise phase diagram for the 1D BoHubbard model will depend on the microscopic details, ocan still understand some of its general features on the bof the above renormalization scenario. Let me first assuthat the periodic potential is weak, thatc51, and think of thephases in theK2wh plane. In these unitsK5AU/t/2p2. Atzero disorder, forK,Kc'

14 the system is in the superflui

phase, and forK.Kc it is a Mott insulator. Now take thepoint with K5Kc , but wh.0 and small. According to therenormalization scenario discussed above, at some lalength scale the periodic potential disappears, and if it wweak initially we end up with an effective theory~15! at thesmaller cutoffL8 with small D(L8) and K(L8)' 1

4 . Therecursion relations~16!–~17! then imply that at even largelength scalesD will renormalize to zero, so the system isthe superfluid phase. The same remains true at smallwh aslong asK,Kc(wh), 1

3 , whereKc(wh) is the initial value ofK for which K(L8)5 1

3 . SinceK increases under renormaization @i.e., the O(v2) term in Eq. ~11! is positive#, Kc,Kc(wh), 1

3 for a smallwh . I conclude that~1! there is an

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intervening Bose-glass phase between the superfluid andMott insulator at weak disorder, and~2! the BG-SF phaseboundary must curve upward not to intersect theK5 1

3 line~see Fig. 1!.

What happens at large disorderwh cannot be concludedon the basis of the perturbative renormalization group, sithis would eventually correspond to largeD in Eqs. ~16!–~17!, which is outside of their domain of validity. I still expect that at anyK at arbitrarily large disorder the systemmust be in the localized phase, so the BG-SF phase bounshould turn left, as suggested by the dashed line in FigWhat happens at smallerK is an interesting question, sincone is approaching a singular limit of noninteracting pticles (K50). Since in this limit the system should belocalized phase for any disorder, I expect the phase bouneventually to turn downwards, so that there may be a retrance of the localized phase asK decreases at fixed disorde

The reader could object to the conclusion that at any dorder the transition from the superfluid state is always intBose glass may be an artifact of the assumption thatdisorder distribution is Gaussian and hence unbounded.can easily show however that averaging over a boundedtribution ~square shaped, for instance! again gives thewhterm in Eq.~9! as the most relevant one, together with termof higher order inu. Thus I still expect the dual theory alarge length scales to have the same form as in Eq.~15!,which leads to the BG-SF transition. What will change is tha bounded distribution should lead to a finite region with tMott insulator phase, as drawn in Fig. 1, instead of onlyhalf line atwh50 andK.Kc , as in the Gaussian case. Thtransition between the Mott insulator and the Bose glshould be first order, since it corresponds to a local collaof the gap.20

Finally, there is a question of what happens if at the mcroscopic scalev@wh . At K5Kc, 1

4 , without disorder oneis right on the MI-SF separatrix in theK2v renormalization-group flow diagram. At smallwh andK5Kc it seems plau-sible that small disorder will makev renormalize downwardfasterthan it would without it, so that the system would scatowards smallv and someK, 1

4 . If this is the case, all of the

FIG. 1. The phase diagram for the commensurate 1D BoHubbard model in random chemical potential.K5AU/t/2p2 andwh measures the strength of disorder. The dashed line represemore speculative part of the BG-SF phase boundary. The BGtransition is expected to be first order. For further discussion,the text.

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previous conclusions would remain the same. Thus, evethis case, it is natural to expect that the transition is againthe BG-SF type.

IV. TWO DIMENSIONS

A. Dual theory

In 211-dimensional lattice theory in Eq. 6 the constraon MW may be resolved by introducing another set of intethree-component vectors byMW 5¹3NW , where again the lat-tice version of the 3D curl operator is assumed. Then

Z5 limy→0

($ lW i ,a%

E2`

`

)i ,a

dAW i ,a expS i2p (i ,a

lW i ,a•AW i ,a

1y

2 (i ,a

lW i ,a2 2S@¹3AW # D , ~18!

where

S5eU

2 (i ,a

~¹3AW !t22e(

i ,ahi~¹3AW !t1

1

2et(i ,a ~¹3AW !rW2 ,

~19!

and I introduce an auxiliary set of integer three-componvectors$ lW i ,a%, the summation over which in the limity→0forces the real valued gauge fields$AW i ,a% to take strictly in-teger values. To assure the gauge invariance of the abexpression the constraint¹• lW50 must be enforced. This ithe crucial difference from the 1D case, where one cohave also introduced the two-component vectors$ lW i ,a%, butdue to the lack of gauge freedom in 111 dimensions, thistime without the constraint. So in 1D one may sum ov

$ lW ia% freely and end up with the partition function in Eq.~7!.In 211D however we need new degrees of freedom toforce the constraint. In the fixed gauge¹•AW 50 the partitionfunction can finally be written as

Z5 limy→0

E )n

@dundAW n# expS 1

y(n,m

cos~un1m2un

22pAW n,m!2S@¹3AW # D , ~20!

as can be checked by going backwards from Eq.~20! in theVillain approximation and by performing the integrationover angles$un%, which precisely enforces the constrai¹• lW50. Indexn now labels the sites of athree-dimensional

lattice, m5 x,y,t, and2p,un,p. With e finite, one rec-ognizes the last form of the partition function atT50 as the3D ‘‘frozen’’ 28 anisotropic lattice superconductor in randomagnetic field along one~time! axis.

The expression~20! is the sought dual representationthe original problem in 2D. The angles$u i% represent disor-der variables,33 since they randomize the configurationsthe gauge-fieldAW . In fact, without the coupling to the disorder variables, the action in Eq.~20! would be quadratic in thegauge field, which should be identified with the phonon ecitation mode in the parent superfluid state of the origi

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t

ve

d

r

-

-l

bosons. The long-range order foru variables destroys superfluidity by gapping this sound mode in the pure case. In tsense the system described by Eq.~20! is dual to the originalHamiltonian~1!: the superfluid phase of bosons corresponto the disordered phase ofu variables, while the insulatingphase~MI or BG! of bosons corresponds to the orderphase ofu variables. Assuming that by leaving small bfinite y in Eq. ~20! the universality class of the transition aT50 is not changed,29,24one may write the soft-spin versioof the lattice gauge theory in Eq.~20! as34

S5E d2rW dzS u@¹2 iqAW ~rW,z!#C~rW,z!u21m2uC~rW,z!u2

1l

2uC~rW,z!u41

1

2@¹3AW ~rW,z!#21

g

2@¹3AW ~rW,z!# z

2

1h~rW !@¹3AW ~rW,z!# zD , ~21!

which is a Ginzburg-Landau theory for a strongly typesuperconductor~assumingl@q2) in transverse gauge¹•AW50, in the random magnetic field along thez axis. The non-zero dimensionless parameterg indicates an inherent anisoropy of the quantum problem. The transition is tunedvarying the massm, and the condensation of the compledisorder fieldC signals the destruction of superfluidity.

B. Renormalization

In the field theory~21! disorder appears as the randomagnetic field along thez direction. This term now plays arole similar to thewh term in 1D. Note that the randommagnetic field depends only on two~spatial! coordinates andis completely correlated in time, as is typical for quantuproblems with static disorder.35. The quantity of interest isthe disorder-averaged zero-temperature free energyF

52 lnZ. Using a replica trick to average over disorder, oarrives at the action for interacting replicas of the system

S85 (a51

N E d2rW dzS u~¹2 iqAW a!Cau21m2uCau2

1l

2uCau41

1

2~¹3AW a!21

g

2~¹3AW a! z

2

2wh

2(b51

N E dz8@¹3AW a~rW,z!# z@¹3AW b~rW,z8!# zD .

~22!

Similarly as in the dual theory in 1D in Eq.~9!, the effect ofaveraging over disorder is to introduce an off-diagonal~inreplica indices! contribution to the gauge-field propagator,determined by the last term in Eq.~22!.

The point of transition in the dual theory is reachedtuning the renormalized mass of the disorder fieldm to zero.Simple power counting tells us that atm50 one needs toconsider four dimensionless coupling constants: the dchargeq25q2/p, the quartic term couplingl5l/p, the an-isotropy parameterg, and the width of the random-field dis

.ano

. 2rtthe

rtig

enein

aom

e

Ih

tieet

rea

d

ce

cne

eld


tribution wh5wh /p, wherep is an arbitrary infrared scaleHowever, a little analysis shows that this is not enough,to complete the theory one needs to include one more cpling constant. Consider, for example, the diagram in Figsince the gauge-field propagator has an off-diagonal pareplica indices, this diagram generates the quartic termcouples different replicas even though this term is absinitially. This is analogous to the way the standard quaterm l would get generated by the fluctuations of the gaufield in the pure theory. We already saw that a similar geration of a new disorder term happens in 1D theory as wTo complete the theory one therefore must add the followterm to the action in Eq.~22!:

S952w

2 (a,b51

n E d2rW dz dz8uCa~rW,z!u2uCb~rW,z8!u2,

~23!

with w.0. Note that this term has precisely the form thwould arise from averaging over an independently randmass in Eq.~21!, correlated along thez direction and withthe Gaussian distribution with a widthw. Thus the fulltheory S5S81S9 contains five coupling constants at thcritical surfacem50: the fifth dimensionless coupling isw5w/pp2.

To address the quantum (T50) critical behavior at the Stransition, I consider the renormalized theory right at ttransition pointm50:

Sr5 (a51

N E d2rW dzS ZCuZm~¹m2 iqAW a,m!Cau21l8

2uCau4

2w8

2(b51

N E dz8uCa~rW,z!u2uCb~rW,z8!u2

1ZA

2~¹3AW a!21

gZg

2~¹3AW a! z

22

Zhwh

2

3 (b51

N E dz8@¹3AW a~rW,z!# z@¹3AW b~rW,z8!# zD . ~24!

The lowest-order contributions to the renormalized quantiare determined by the diagrams in Fig. 3. The infrared divgences are regulated by evaluating quartic vertices atusual symmetric point

kW i•kW j5~4d i , j21!p2

4, ~25!

FIG. 2. Diagram of the orderwh2q4 that generates the quarti

interaction w between different replicas in 2D. The dashed lirepresents the gauge-field propagator.

du-:inatntce-ll.g

t

e

sr-he

with an additional conditionk1,z5k2,z52k3,z52k4,z , nec-essary because the disorder is correlated along thez axis, sothat individualkz components, and not only their sums, aconserved in thew vertex. The polarization is evaluated atfinite momentumc•p, wherec is a constant to be specifieshortly. The result of the one-loop renormalizations is

ZC5121

4q2, ~26!

Zz5111

8w, ~27!

ZA511c

16q2, ~28!

l85l22A211

8l22

1

2A2q4

12lwFA2 ln~11A2!11

A6ln~A21A3!G , ~29!

w85w12w2S 1

A2ln~11A2!1

1

A6ln~A21A3!D

21

2A2lw, ~30!

Zg5Zh5Zx5Zy51. ~31!

The nontrivial numerical factors in Eqs.~29!–~30! arise fromthe integration performed directly in 3D and from the choiof the renormalization point~25!.

FIG. 3. The lowest-order contributions to the self-energy~a!, l8~b!, w8 ~c!, andZA ~d!. The dashed line represents the gauge-fipropagator, the wavy line stands for the interaction vertexw anddots denote the interaction vertexl.

p

r.oinru

ouline

carg

rg

oth

a

ahee,y

alin1

rt

nsbeint

llyan-

em

e-gr I

-

usly

e,

theue

sti-

l


Let me first discuss the renormalization of the anisotroparameterg and the random-field disorder variablewh . Therenormalized couplingsg r5gZg /ZA and wh,r5whZh /ZAobey the equations

dg r

dt52g r S c

16qr

21O~ qr4! D1O~g r

2!, ~32!

dwh,r

dt5wh,r~12hA!1O~wh,r

2 !, ~33!

wheret52 ln(p) and

hA5d ln~ZA!

dt~34!

is the anomalous dimension of the gauge-field propagatosmall anisotropy appears to be irrelevant at any fixed pwith a finite dual charge. I checked that this remains twhen the termO(qr

4) is included in Eq.~32!. Note that an-isotropy in the gauge-theory cannot simply be rescaledas would be possible in theXY model. Although the duatheory~21! should be strongly and not weakly anisotropicthe continuum limita→0, e→0, I will assume that there arno stable anisotropic (gÞ0) fixed points in the theory. Thiswould be in agreement with what is known for the classiscalar electrodynamics close to four dimensions for a lanumber of complex field components.36 I therefore set an-isotropyg50 hereafter.

The scaling equation for the renormalized dual chaqr

25q2/ZA may be written exactly as

dqr2

dt5qr

2~12hA!, ~35!

as follows from the definition of the anomalous dimensionthe gauge-field propagator. Thus at any fixed point intheory with a finite dual charge,hA51.37 In particular, thisimplies exact vanishing of the linear term in Eq.~33! at theMI-SF fixed point of the pure system. The fact that the scing dimension of the random-field disorderwh is simply 12hA , follows from the observation that the transverse pof the polarization is diagonal in the replica indices to tlowest order inwh . The term depicted in Fig. 4, for exampldoes not contribute toZh since it turns out to be completellongitudinal.

The fate of the random-field disorderwh is determined bythe higher-order terms in Eq.~33!. Marginality of wh followsfrom the quantum nature of disorder: if the disorder wcorrelated along slightly less than one dimension, its scadimension would have been negative. Guided by the

FIG. 4. A purely longitudinal contribution to the off-diagona~in replica indices! polarization.

y

At

e

t,

le

e

fe

l-

rt

sg

D

problem, I will make a simplifying assumption that apafrom generating the quartic termw, disorder couplingwh isirrelevant at the MI-SF fixed point. If this assumption turout to be incorrect, the calculation presented here wouldpertinent to the crossover regime towards a new critical powith whÞ0. However, sincewh can at worst be marginallyrelevant at the pure fixed point, and therefore can initiagrow much slower thanw, we are assured of a region forsmall initial wh where ignoring its possible growth is a sesible approximation.

The scenario in 2D is thus similar in spirit to what walready encountered in 1D: initially disorder in the probleis represented by the random magnetic fieldwh , which Iassume is ultimately unimportant for the BG-SF critical bhavior. However,wh generates another disorderlike couplinw that is relevant and which grows. In the rest of the papewill then set wh50, and follow the renormalization of theremaining three coupling constants.

The one-loopb functions for the remaining coupling constantsqr

2 , wr5w8/ZC2 , andl r5l8/ZC

2 are

dqr2

dt5qr

22c

16qr

4 , ~36!

dl r

dt5l rF11

1

2qr

212S A2 ln~11A2!11

A6ln~A2

1A3!D wr G22A211

8l r

221

2A2qr

4 , ~37!

dwr

dt5wrS 21

1

2qr

221

2A2l r D 12S 1

A2ln~11A2!

11

A6ln~A21A3!D wr

2 . ~38!

For wr50 these equations reduce to those studied previoin the context of critical behavior of superconductors.37 Inthat case, for a choicec.5.17 there are four fixed points inthe theory: Gaussian (qr

25l r50) and 3D XY (qr250,

l r ,xy52.09), both unstable in direction of the dual chargtricritical (qr

2516/c, l r5l2), unstable in thel direction,

and the MI-SF critical point (qr2516/c, l r5l1), believed to

be of ‘‘inverted’’ XY type29,30 ~see the inset on Fig. 5!. TheGaussian and the tricritical fixed points are connected bystraight separatrix that determines the tricritical initial valof the Ginzburg-Landau parameterk5Al/2q25kc in thesuperconducting problem. The tricritical value ofk has beenestimated both analytically38 and numerically,39 and we maytune the renormalization point to match that number:kc

50.42/A2 requiresc520. It is worth noting that forw50our one-loop approximation gives encouragingly good emates of the critical exponents both at the unstableXY (n50.63) and at the stable, charged fixed point (n50.61, h520.20).40

se

ldm(a

thDory,

ucantr-r

thseon

hr

fin

g

es,

theob-

g

c--erlo

e-an-he

on

asim-de-

er

he


A small disorderw is relevant at the MI-SF fixed point, afollows from the lastb function, and more generally may bexpected on the basis of the Harris criterion.41 Bothperturbative37 and nonperturbative,38,29,30calculations of thecorrelation length exponent at the pure critical point yievaluesn,1, suggesting relevance of the correlated randomass disorder. Starting from the strongly type-II regionl@q2), the flows are attracted by the random critical pointqc

250.8, wc53.71, andlc529.72, which I identify as theBG-SF critical point~Fig. 5!. The BG-SF critical point existsfor any choice of the parametrization byc, unlike thecharged fixed points in the pure theory. This is similar todisordered classical scalar electrodynamics close to 442

where randomness restores the critical point in the theThe flow towards the BG-SF critical point is oscillatorwhich should lead to oscillatory corrections to scaling.

C. Critical exponents

Having obtained the approximate renormalization-groflow and the fixed points, I proceed to calculate the critiexponents. First, note that the correlation length exponenand the dynamical exponentz may be assigned a purely themodynamical meaning, since the BG-SF transition tempeture Tc;dzn and the zero-temperature compressibilityk;dn(d2z). d is a parameter measuring the distance fromBG-SF transition atT50. One may therefore calculate theexponents from any representation of the partition functiThe correlation length critical exponentn is naturally definedaway from the critical surfacem50, while the calculationhere is performed precisely at the critical surface. Nevertless, I may still obtainn from the knowledge of yet anotherenormalization factorZC2 ~Ref. 25! that accounts for therenormalization of theuCu2 term in the theory~22!, evalu-ated at the critical surfacem50. I find

ZC2511l

42

w

2, ~39!

to the lowest order in the coupling constants@Fig. 3~a!#. Inanalogy with the thermal critical phenomena, one may dethe exponentg as

FIG. 5. Renormalization-group flow and the fixed points in t

q5qc plane. The inset shows the flow in thew50 plane~Ref. 37!.

-

t

e,y.

pl

a-

e

.

e-

e

g512 limp→0

d ln ZC2

d ln p2, ~40!

so that the exponentn then follows from the standard scalinrelation

n5g

22hd

. ~41!

hd is the anomalous dimension of the vortex fieldC,

hd52 limp→0

d lnZC

d lnp. ~42!

Using the lowest-order Eqs.~26! and ~39!, expressing thebare coupling constants in terms of the renormalized onand taking the infrared limitp→0, the correlation lengthexponent becomes a series in the fixed-point values ofrenormalized coupling constants. To the lowest order Itain

n51

2S 11lc2qc

224wc

8D 51.38. ~43!

Similarly, the dynamical critical exponentz is defined by

z21511 limp→0

d ln Zz

d ln p. ~44!

From Eq. ~27!, to the lowest order in the critical couplinconstants I find

z511wc

451.93. ~45!

At the BG-SF critical pointn.1, as expected.41 The dy-namical exponent is very close to two, which was conjetured to be exact5 in 2D, based on the assumption that compressibility right at the BG-SF transition stays finite. Thvalues are also in good agreement with the Monte Caresults of Ref. 17 (n50.960.1 andz5260.1) and the real-space study of Ref. 19 (n51.4 andz51.7).

In contradistinction to the critical exponentsn andz, thestandard exponenth characterizes the behavior of the corrlation function for the superfluid order parameter at the trsition, and cannot be straightforwardly obtained from tdual theory.

D. Universal conductivity

Since the calculation is performed right at the transitipoint ~where the mass of the dual fieldm50), one maycalculate the universal dc conductivity in a similar waythe critical exponents. Before proceeding however, twoportant points need to be clarified. First, the dual theoryscribes the dirty-boson system atT50, and therefore theuniversal conductivity I will calculate corresponds toT→0,v→0 order of limits. The reader should note that the ord

chni

thiofoe

alenngy

ths

anthednoi

hoic

ece

or

mrnd

e-uc

iv

m-nc-u-

itwe

t of

rst


of limits in typical experiments is actually reversed, whiraises subtle issues about the actual dissipation mechatested by the experiments43 to which I will return in theconcluding section. Nevertheless, here I computeT50 criti-cal conductivity as a fundamental characteristic ofBG-SF fixed point. The second point concerns my intentto obtain the conductivity of bosons from the dual theorythe vortex field. This question is intimately linked with thmechanism of dissipation in the quantum (T50) problem.As already mentioned, the dual theory describes the dispearance of superfluidity due to destruction of the gapmode, which, as argued, can happen only through condetion of vortices. The physical picture is that on the insulatiside of the transition vortex loops in 3D classical electrodnamics blow up, completely disordering the phase ofsuperfluid order parameter. The ground state of the Boglass insulator is therefore a superfluid of free vorticesantivortices, which represent the 2D space projection of211D vortex loops, while the original bosons are localizby random potential. In contrast, in the superfluid groustate bosons are phase coherent while vortices and antivces are tightly bound into pairs. Right at the transitionseems natural to assume that both bosons and vortices sbe mobile, which then leads to a finite metallconductivity.10 Consider the current of bosons

I b5e* Nb ~46!

from one end of the sample to another; motion of vorticperpendicular to the direction of the boson current induthe voltage by the Josephson relation

Vb5hNv

e*, ~47!

whereNv is the vortex flux. The boson resistance is theref

Rb5Nv

Nb

h

e*2

. ~48!

On the other hand, vortices ‘‘see’’ bosons exactly the saway as bosons see them;44 writing the above relations focurrent and ‘‘voltage’’ of vortices instead of bosons we fithat

Rv5Vv

I v

5Nb

Nv

h

ev2

, ~49!

whereev is an arbitrary ‘‘charge’’ of a vortex. Combiningthe last two equations I find

sb215

sv

ev2/h

h

e*2

, ~50!

where sb and sv are boson and vortex conductivities, rspectively, and I used the fact that conductivity and condtance are the same in two dimensions.

The last expression enables one to calculate the resistof the bosonssb

21 by using the Kubo formula for vortex

sm

enr

p-sssa-

-ee-de

drti-tuld

ss

e

e

-

ity

conductivitysv . The derivation of the Kubo formula in thetheory with replicas is presented in Appendix B. The siplest way to evaluate the integrals over the correlation futions in Eq.~B8! is to use the scheme of dimensional reglarization. Recalling that the couplingsl and q2 havedimensions of frequency, it follows that in the static limv→0 to the lowest order in these coupling constantsneed to calculate the integrals in Eq.~B8! only at kW50, sothat they will give no contribution tosv to this order. To thelowest ordersv can then be written as@kW5(0W ,v)#

sv5~sv,01sv,1!ev

2

\, ~51!

where

sv,05 limv→0

2

vS E d3qW

~2p!3

1

q222E d3qW

~2p!2

qx2

q2~qW 1kW !2D 51

16

~52!

in dimensional regularization, in agreement with the resulChaet al.10 sv,1 is given by the diagrams in Fig. 6:

sv,1528

vE d3qW

~2p!3

S~qW !

q4~qW 1kW !2

24w

v E d3qW d2pW'

~2p!5

qxpx

q2~p'2 1qz

2!~qW 1kW !2@p'2 1~qz1v!2#

~53!

in the static limitv→0. The second term insv,1 vanishesdue toqx integration. The self-energy appearing in the fiterm is to be calculated to the lowest order inw at the tran-sition pointm50. For small momenta it can be written as

S~qW !5w

2plnuqzu, ~54!

as follows from the result for the dynamical exponentz.Inserting the last expression into Eq.~53! and introducingFeynman parameters to integrate overqx andqy leads to

sv,15I

4p3

w

v2. ~55!

The integral in the numerator is

FIG. 6. Contributions of the orderw to the vortex conductivitysv .

sc

ng

to-thu

he

io

diane

ollye-linpuicssetht

am

fotz

he

e-heanthe

Dle.

nt,ly.n-al-Fa

hatn-e

t inr

g-eld,tiones

t inldran-

tui-

uid.the

ramix

en

am

heghtthattos-reentere-thats of

he

ialndst ofsi-

uldit

port

inte


I 5E2`

`

dqzE0

11dt

~ t21! lnuqzu

qz212tqz1t

55.89. ~56!

Recognizing the combinationw/pv2 as the dimensionlescouplingw, the lowest-order result for the dc vortex condutivity becomes

sv5S p

81

I

2plimv→0

wD ev2

h, ~57!

and therefore formally divergent due to the infrared~static!limit. Rewriting sv as a series in the renormalized coupliwr however, to the lowest order

sv5ZC22S p

81

I

2plimv→0

wr D ev2

h, ~58!

where I introduced the wave-function renormalization facZC to account for the fact thatwr describes correlation functions of the rescaled vortex field. To the lowest orderboson dc resistivity is then determined by the critical copling constants as

sb215S p

81

p

16qc

21I

2pwcD h

e*2

. ~59!

Numerical values of the critical couplings, as found from tone-loopb functions of Eqs.~36!–~38!, yield the estimate ofthe universal conductivity of bosons at the BG-SF transitin 2D:

sb50.25e*2

h. ~60!

V. SUMMARY AND DISCUSSION

In the present paper it was shown that the system ofordered interacting bosons at a commensurate density hsuperfluid-insulator transition at zero temperature goverby a random fixed point, which was interpreted as a signthe BG-SF transition. This critical point can be naturaidentified within the dual theory for the transition that dscribes topological defects in the ground state. The coupconstant that represents the disorder relevant at theMI-SF fixed point is not initially present in the microscopdual theory, but becomes generated at larger length scalethe microscopic disorder and the lattice potential. I discusthe phase diagram in 1D, and calculated approximatelycritical exponentsn andz and the universal conductivity athe BG-SF transition in 2D.

I argued that only in the dual theory the BG-SF criticpoint may become apparent. It is not obvious to which terwould the generated disorder in Eq.~13! correspond in thetheory written in terms of the superfluid order parameter,example. The situation is reminiscent of the KosterliThouless transition in the 2DXY model: while in principlethe information on the transition is contained in t

-

r

e-

n

s-s adf

gre

byde

ls

r-

Ginzburg-Landau-Wilson functional, its extraction is hoplessly difficult; on the other hand, by focusing directly on tbehavior of vortices the problem of the critical behavior cbe solved exactly. It appears the same is accomplished bydual formulation of the dirty-boson problem, at least in 1where the random critical point is perturbatively accessibIn 2D the dual theory has a strong-coupling critical poiand one can obtain the critical quantities only approximate

It is interesting how the relevant disorder coupling is geerated by the random chemical potential and the lattice,though they are both ultimately irrelevant for the BG-Scritical behavior. In 1D the recursion relations imply thatweak lattice potential always renormalizes to zero, but twhile doing so it conspires with the random chemical potetial to induce another disorder term in the theory. At somlarger length scale the initial disorder becomes redundan1D, and one is left with an effective theory in the familiaHaldane’s form. In a similar way, in 2D the random manetic field generates the random mass for the vortex fiwhich immediately becomes the most relevant perturbaat the pure fixed point and whose flow ultimately determinthe critical behavior. I assumed, but could not prove, tha2D the initial disorder in form of the random magnetic fiealso becomes irrelevant once it generated the importantdom mass.

The proposed phase diagram in 1D has one counterintive feature: there is a region forK.Kc where byincreasingdisorder one turns a Bose-glass insulator into a superflThis comes as a consequence of renormalization intheory and I believe it is a genuine effect. The phase diagin Fig. 1 is in disagreement with the density-matrrenormalization-group study of Paiet al.,13 who claimed adirect MI-SF transition at small disorder. This work has becriticized recently by Prokof’ev and Svistunov45 who per-formed a Monte Carlo calculation to find the phase diagrin qualitative agreement with the one discussed here.

The performed renormalization-group calculation of tcritical exponents in 2D, although approximate, has a sliadvantage when compared with other ad hoc schemes init leaves room for a systematic improvement by goinghigher orders in perturbation theory. Although this is posible in principle, it is rather complicated since there athree couplings to consider. The relatively good agreembetween my results and those of other approaches is thfore somewhat fortuitous. The reader should also notethe values of the critical exponents are close to the resultother studies17,19 that are performed atincommensurateden-sities of bosons. In 1D it is easy to show that nothing in tconclusions is changed by allowinghÞ0, since by shiftingthe dual field one may always placeh under the cosine termin Eq. ~9!, so it disappears together with the lattice potentat large length scales. In 2D incommensurability correspoto a nonzero magnetic field along one direction, the effecwhich at the random fixed point is presently not clear. Phycally one would expect that the incommensurability shobecome irrelevant at the BG-SF critical point, althoughwould be desirable to have a concrete calculation in supof this claim.46

The universal conductivity at the BG-SF critical pointEq. ~60! turns out to be somewhat larger than in the MonCarlo calculations of Wallinet al.,17 and smaller than in Ba-

n

an

-cel

ret

ith-

P-reth

inthnd

e

idi--af

an

r

nd

reco

by

ef

ged

theovered

rier

ion


trouni et al.16 It is smaller than at the MI-SF transition,10,11

which seems plausible. It is also smaller than what is fouin most experiments, wheresc;(2e)2/h, and it has recentlybeen proposed that this is not a coincidence. DamleSachdev43 pointed out that at the critical points becomes anontrivial universal function ofv/T, and that the experimentally relevant limit corresponds to the behavior of this funtion at small values of its argument. In this regime the revant dissipation mechanism is the random scatteringthermally created quasiparticles, rather than the cohetransport of the field-induced defects in the ground stawhich would correspond to the opposite limitv/T→`. Thedirect comparison of the conductivity calculated here wthe present experimental data2 is therefore probably not appropriate. I may still note that the number in Eq.~60! isintriguingly close to the recent measurement in a thinfilm:47 sc'0.25(2e)2/h. The experiment that would measure the high-frequency conductivity at low temperatuwould be closer to probing the quantum regime for whichresult in Eq.~60! is directly relevant.

Finally, the kinship between the BG-SF critical points1D and 2D suggests that one could try to approachstrong-coupling critical point in 2D via an expansion arou1D where the BG-SF fixed point happens to be locatedzero disorder. Although the apparent dissimilarity betwethe forms of dual theories~9! in 1D and~22! in 2D at firstsight makes this idea unlikely to work, all that is neededanalyticity of the critical exponents as functions of themension for 1,D,2, for such an expansion not to be impossible in principle. Some steps in this direction haveready been taken in Ref. 48, and this will be a subject ofuture publication.49

ACKNOWLEDGMENTS

This work has been supported by NSERC of Canadathe Izaak Walton Killam foundation.

APPENDIX A: DUAL FIELD PROPAGATOR IN 1D

From the dual theory~9! the inverse of the propagator fothe fieldu is

G215K~v21c2k2! I 2whk2d~v!M , ~A1!

whereG is a N3N matrix, I is a unit matrix, andM is alsoa N3N matrix with all elements equal to one. The secoterm is due to disorder, and thed function in frequency origi-nates in the quantum nature of the problem: disorder isresented by a potential random in space, but completelyrelated in imaginary time. By noticing thatM25NM wemay write

G51

K~v21c2k2!I S 11 (

n51

` S whd~v!

Kc2 D n

Nn21M D .

~A2!

In the relevant limit of the vanishing number of replicasN→0, the propagator therefore takes a simple form:

d

d

--ofnte,

b

se

e

atn

s

l-a

d

p-r-

G51

K~v21c2k2!I 1

whd~v!

K2c4k2M . ~A3!

APPENDIX B: KUBO FORMULA FOR DISORDEREDDUAL THEORY

We want to find the response of the system describedEq. ~22! to the external vector potentialaW that couples to thevortex field via minimal coupling. I work in units where thcharge of the vortex fieldev that determines the strength othe coupling to the external probe~not to be confused withthe running dual chargeq) and\ are both set to unity. Withthe external probe, the kinetic part of the action in Eq.~22!becomes

S85 (a51

N E d2rW dzu~¹2 iqAW a1 iaW !Cau2, ~B1!

with the rest of the action unchanged. The disorder-averacurrent of the disorder field is then given by

^ jW~rW,z!&52dF@aW #

daW ~rW,z!U

aW 50

. ~B2!

Angular brackets denote the quantum average overground state and the overline stands for the averagedisorder configurations. The vortex conductivity is definby linear response to the vortex ‘‘electric field’’ as

^ jW~rW,z!&52E d2rW8 dz8sv~rW2rW8,z2z8!]aW ~rW8,z8!

i ]z8.

~B3!

Integrating by parts the last equation and going to Fouspace, I find

sv~v!521

vE d2rW dz exp~ ivz!

d2F@a#

dax~rW,z!dax~0W ,0!U

aW 50

,

~B4!

where I choseaW to be along thex direction andv is imagi-nary ~Matsubara! frequency. Using the replica trick to writeF and performing the differentiations in Eq.~B4!, I obtain

sv~v!51

vlimN→0

1

NS 2(a

N E C exp~2S!uCa~0W ,0!u2

2ECE d2rW dz exp~ ivz!

3exp~2S! j x~rW,z! j x~0W ,0!D , ~B5!

where the current is given by the standard expresssummed over replicas

mr-es,

a

ceion


jW5(a

N

~ iCa* ¹Ca2 iCa¹Ca* 12quCau2AW a!. ~B6!

Since

^A&5

EC

exp~2S!A

EC

exp~2S!

5EC

exp~2S!A, ~B7!

in the limit N→0, inserting the above expression for thcurrent, the last equation can be rewritten in the final for

sv~kW !52

v

ev2

\limN→0

S E d3qW

~2p!3^C1* ~qW !C1~qW !&

22

NV(

a,b

N E d3qW d3pW

~2p!6qxpx^Ca* ~qW !Ca~qW 1kW !

u

o

a

.

h,

e:

3Cb* ~pW 1kW !Cb~pW !&

22

Nq2V2(

a,b

Nd3qW 1d3qW 2d3pW 1d3pW 2

~2p!12

3^Ca* ~qW 1!Ca~qW 2!Cb* ~pW 1!Cb~pW 2!Aa,x

3~kW1qW 12qW 2!Ab,x~pW 12pW 22kW !& D , ~B8!

with kW5(0W ,v), V is the 3D volume, and I restored the corect units. Apart from the appearance of the replica indicthe first two terms are the same as in Chaet al.10 Simplepower-counting in the last expression implies thatsv doesnot scale with frequency, and at criticality is given byuniversal constant, in its natural unitsev

2/\. This conclusionremains true in the fully renormalized theory as well, sinthe current operator cannot acquire anomalous dimensdue to gauge invariance.

*Address after September 1998: Department of Physics, DalhoUniversity, Halifax, Nova Scotia, Canada B3H 3J5.

1P. A. Crowell, F. W. van Keuls, and J. D. Reppy, Phys. Rev. Le75, 1106~1995!; Phys. Rev. B55, 12 620~1997!.

2Y. Liu and A. M. Goldman, Mod. Phys. Lett. B8, 277 ~1994!,and references therein; J. M. Valles, Jr., R. C. Dynes, and JGarno, Phys. Rev. Lett.69, 3567 ~1992!; A. Yazdani and A.Kapitulnik, ibid. 74, 3037~1995!.

3H. S. J. van der Zant, W. J. Elion, L. J. Geerligs, and J. E. MoPhys. Rev. B54, 10 081~1996!.

4M. Ma and P. A. Lee, Phys. Rev. B32, 5658~1985!; M. Ma, B.I. Halperin, and P. A. Lee,ibid. 34, 3136~1986!.

5M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. FishPhys. Rev. B40, 546 ~1989!.

6J. M. Kosterlitz and D. J. Thouless, J. Phys. C6, 1181~1973!; J.M. Kosterlitz, ibid. 7, 1046~1974!.

7E. Abrahams, P. W. Anderson, D. Licciardello, and T. V. Rmakrishnan, Phys. Rev. Lett.42, 673 ~1979!.

8M. P. A. Fisher, G. Grinstein, and S. M. Girvin, Phys. Rev. Le64, 587 ~1990!.

9S. Doniach, Phys. Rev. B24, 5063~1981!.10M. Cha, M. P. A. Fisher, S. M. Girvin, M. Wallin, and A. P

Young, Phys. Rev. B44, 6883~1991!.11R. Fazio and D. Zappala, Phys. Rev. B53, R8883~1996!.12P. B. Weichman and K. Kim, Phys. Rev. B40, 813 ~1989!; R.

Mukhopadhyay and P. B. Weichman, Phys. Rev. Lett.76, 2977~1996!.

13W. Krauth, N. Trivedi, and D. Ceperley, Phys. Rev. Lett.67,2307~1991!; R. V. Pai, R. Pandit, H. R. Krishnamurthy, and SRamasesha,ibid. 76, 2937 ~1996!; J. Kisker and H. Rieger,Phys. Rev. B55, R11 981~1997!.

14See also, F. Pazmandi, G. T. Zimanyi, and R. Scalettar, PRev. Lett. 75, 1356 ~1995!; F. Pazmandi and G. T. ZimanyiPhys. Rev. B57, 5044~1998!.

15K. J. Runge, Phys. Rev. B45, 13 136~1992!.16G. G. Batrouni, B. Larson, R. T. Scalettar, J. Tobochnik, and

Wang, Phys. Rev. B48, 9628~1994!.17M. Wallin, E. Sorensen, S. M. Girvin, and A. P. Young, Phy

Rev. B49, 12 115~1994!.18Y. Tu and P. B. Weichman, Phys. Rev. Lett.73, 6 ~1994!; Y. B.

sie

tt.

. P.

ij,

er,

-

tt.

.

ys.

J.

s.

Kim and X. G. Wen, Phys. Rev. B49, 4043~1994!.19L. Zhang and M. Ma, Phys. Rev. B45, 4855~1992!; K. Singh and

D. Rokhsar,ibid. 46, 3002~1992!.20J. Freericks and H. Monien, Phys. Rev. B53, 2691~1996!.21M. P. A. Fisher and D. H. Lee, Phys. Rev. B39, 2756~1989!.22F. D. M. Haldane, Phys. Rev. Lett.47, 1840~1981!.23T. Giamarchi and H. J. Shulz, Phys. Rev. B37, 325 ~1988!.24I. F. Herbut, Phys. Rev. Lett.79, 3502~1997!.25G. Parisi, J. Stat. Phys.23, 49 ~1980!; J. Zinn-Justin,Quantum

Field Theory and Critical Phenomena~Oxford University Press,Oxford, 1993!, Chap. 27.

26M. P. A. Fisher and G. Grinstein, Phys. Rev. Lett.60, 208~1988!.27See, for example, J. Negele and H. Orland,Quantum Many-

Particle Systems~Addison-Wesley, Reading, MA, 1988!.28M. Peskin, Ann. Phys.~N.Y.! 113, 122~1978!; P. R. Thomas and

M. Stone, Nucl. Phys. B144, 513 ~1978!.29C. Dasgupta and B. I. Halperin, Phys. Rev. Lett.47, 1556~1981!.30I. F. Herbut, J. Phys. A30, 423 ~1997!.31S. Edwards and P. W. Anderson, J. Phys. F5, 965 ~1975!.32B. Svistunov, Phys. Rev. B54, 16 131~1996!.33H. Kleinert, Gauge Fields in Condensed Matter~World Scien-

tific, Singapore, 1989!, Vol 1.34More formally, the theory in Eq.~21! can be derived by introduc-

ing a Hubbard-Stratonovich fieldC to decouple the kinetic en-ergy term in Eq.~20!, and then integrating over the dual anglesu. The action~21! is the result of expansion in powers ofC and^C&;^eiu&.

35S. N. Dorogovtsev, Phys. Lett. A76, 169 ~1980!; D. Boyanovskiand J. Cardy, Phys. Rev. B26, 154 ~1982!; I. D. Lawrie and V.V. Prudnikov, J. Phys. C17, 1655~1984!.

36T. C. Lubensky and J.-H. Chen, Phys. Rev. B17, 366 ~1978!.37I. F. Herbut and Z. Tesˇanovic, Phys. Rev. Lett.76, 4588~1996!;

78, 980 ~1997!; I. D. Lawrie, ibid. 78, 979 ~1997!.38B. Bergerhoff, F. Freire, D. F. Litim, S. Lola, and C. Wetterich,

Phys. Rev. B53, 5734~1996!.39J. Bartholomew, Phys. Rev. B28, 5378~1983!.40The reader may be wondering about the role of the parameterc

on which the critical exponents, otherwise universal quantities,seem to depend. The point is that the coefficients of the pertur-bative series for theb functions are nonuniversal, which implies

tioivese

-

,

r ofm-


the same for the exponents to any finite order of the perturbatheory in fixed dimension. This is an artifact of the perturbatcalculation without any physical meaning. Our trick is to uthis to our advantage and choose the scheme~parametrized by‘‘ c’ ’) that yields the result onkc in agreement with the previous numerical calculations.

41A. B. Harris, J. Phys. C7, 1671~1974!; J. T. Chayes, L. ChayesD. S. Fisher, and T. Spencer, Phys. Rev. Lett.57, 2999~1986!.

42D. Boyanovsky and J. L. Cardy, Phys. Rev. B25, 7058~1982!.43K. Damle and S. Sachdev, Phys. Rev. B56, 8714~1997!.

n44F. D. M. Haldane, Phys. Rev. Lett.51, 605 ~1983!.45N. V. Prokof’ev and B. V. Svistunov, cond-mat/9706169~unpub-

lished!.46See the second paper under Ref. 12 for an argument in favo

this proposition within the theory for the superfluid order paraeter.

47K. Kagawa, K. Inagaki, and S. Tanda, Phys. Rev. B53, R2979~1996!.

48I. F. Herbut, Phys. Rev. B57, 1303~1998!.49 I. F. Herbut, cond-mat/9802141~unpublished!.

Dual theory of the superfluid-Bose-glass transition in the disordered Bose-Hubbard model in one and...

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