Pinning the Order: The Nature of Quantum Criticality in the Hubbard Modelon Honeycomb Lattice

Fakher F. Assaad1,3 and Igor F. Herbut2,3

1Institut fur Theoretische Physik und Astrophysik, Universitat Wurzburg, Am Hubland, D-97074 Wurzburg, Germany2Department of Physics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada

3Max-Planck-Institut fur Physik Komplexer Systeme, Nothnitzer Strasse 38, 01187 Dresden, Germany(Received 1 May 2013; published 26 August 2013)

In numerical simulations, spontaneously broken symmetry is often detected by computing two-point

correlation functions of the appropriate local order parameter. This approach, however, computes the

square of the local order parameter, and so when it is small, very large system sizes at high precisions are

required to obtain reliable results. Alternatively, one can pin the order by introducing a local symmetry-

breaking field and then measure the induced local order parameter infinitely far from the pinning center.

The method is tested here at length for the Hubbard model on honeycomb lattice, within the realm of the

projective auxiliary-field quantum Monte Carlo algorithm. With our enhanced resolution, we find a direct

and continuous quantum phase transition between the semimetallic and the insulating antiferromagnetic

states with increase of the interaction. The single-particle gap, measured in units of Hubbard U, tracks the

staggered magnetization. An excellent data collapse is obtained by finite-size scaling, with the values of

the critical exponents in accord with the Gross-Neveu universality class of the transition.

DOI: 10.1103/PhysRevX.3.031010 Subject Areas: Computational Physics, Mesoscopics, Strongly

Correlated Materials

I. INTRODUCTION

Detecting spontaneous symmetry-broken phases in nu-merical simulations often relies on the measure of correla-tion function. For instance, the magnetically ordered phaseis characterized by long-ranged spin-spin correlations,whereas the superconducting state exhibits long-rangedpair correlations in the appropriate symmetry channel. Afundamental caveat with such an approach is that onemeasures the square of the order parameter. If the latterquantity is small, the quantity one attempts to obtain byextrapolating numerical data to the thermodynamic limit isquadratically smaller. As a consequence, very large systemsizes at high precision are required in addition to an appro-priate finite-size extrapolation formula.

The aim of this article is twofold. We will document asimple and very efficient alternative method to detect mag-netically ordered phases in SU(2)-invariant Hubbard-typemodels in the realm of projective quantum Monte Carlomethods. With the enhanced resolution, we will revisit thesemimetal-to-insulator transition on graphene’s honey-comb lattice, which has recently been under considerabledebate [1,2].

Honeycomb lattice is a bipartite, nonfrustrated lattice,which at half-filling and small Hubbard repulsion U hoststhe semimetallic state of electrons, as in graphene. Whenthe repulsion is increased, one eventually expects a phase

transition into an insulating state with antiferromagneticorder [3–5] which, because of gapless Dirac fermionicexcitations being present on the semimetallic side, shouldbelong to a particular Gross-Neveu universality class [5,6].Starting from the strong coupling limit and noting that

the insulator-to-metal transition occurs at values of theHubbard interaction lesser than the bandwidth allows forthe proliferation of higher-order ring-exchange terms in aneffective spin model aimed at describing the magneticinsulating state in the vicinity of the transition [7]. Thispoint of view opens the possibility that the melting of themagnetic order is unrelated to the metal-insulator transi-tion. Recent quantum Monte Carlo calculations [1] sug-gested that there is an intermediate spin-liquid phase with asingle-particle gap but nomagnetic ordering, separating thesemimetal and magnetic insulator. Similar results havebeen put forward for the related�-flux model on the squarelattice [8]. The results of Ref. [1] have been challenged byrecent studies. Entropy calculations do not favor ground-state degeneracy, as expected for the Z2 spin liquid [9].Moreover, Ref. [2] shows that extrapolating from signifi-cantly larger system sizes would suggest almost completedisappearance of the spin liquid from the phase diagram.The latter conclusion is reinforced here, where we findexcellent data collapse and identical finite-size scaling ofboth the single-particle gap and staggered magnetization,with the distinct values of critical exponents, in accord withthe Gross-Neveu universality class [5,6].From the technical point of view, our approach is very

similar in spirit to an approach considered in Ref. [10]. Byintroducing a local magnetic field at, say, the origin, weexplicitly break the SU(2) spin symmetry. In the presence

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 3.0 License. Further distri-bution of this work must maintain attribution to the author(s) andthe published article’s title, journal citation, and DOI.

PHYSICAL REVIEW X 3, 031010 (2013)

2160-3308=13=3(3)=031010(8) 031010-1 Published by the American Physical Society

of long-range order and in the thermodynamic limit, anyfield will pin the order along the direction of the externalfield. Thereby, order can be detected by computing directlythe magnetization infinitely far from the pinning field. Theupside of such an approach is that one measures directlythe order parameter rather than its square. This procedureamounts to evaluating a single-particle quantity, which isoften much more stable than correlation functions. Thedownsides are threefold. One explicitly breaks SU(2)spin symmetry such that spin sectors mix and it becomescomputationally more expensive to reach the ground state.Since the computational cost scales linearly with the pro-jection parameter, this problem is tractable. The seconddifficulty lies in the ordering of limits. To obtain resultsthat are independent on the magnitude of the pinning field,it is important to first take the thermodynamic limit andthen the limit of infinite distance from the pinning field. Ina practical implementation, this ordering of limits has, as aconsequence, some leftover dependence of the magnetiza-tion on the magnitude of the pinning field. This feature isparticularly visible when the pinning field is small. Thefinal drawback is that it is not always possible to introducea pinning field without generating a negative sign problem.For instance, in the Kane-Mele-Hubbard model [11–13],the spin order lies in the x-y plane. Adding a magnetic fieldalong this quantization axis introduces a sign problem. Onthe other hand, the method is applicable to SUðNÞ sym-metric Hubbard-Heisenberg models [14].

The organization and main results of the article are thefollowing. We focus on the Hubbard model on honeycomblattice at the filling one-half, for which the presence of anintermediate spin-liquid phase has been controversial [1,2].After introducing and testing the approach in the nextsection, we provide a phase diagram of the Hubbard modelin Sec. III. The data point to the fact that the staggeredmoment follows rather precisely the single-particle gap,when the gap is measured in the natural units of theHubbard U, suggesting a direct quantum phase transitionbetween the semimetallic and the insulating antiferromag-netic phases. Furthermore, an excellent finite-size scalingof the data for both the staggered magnetization and thesingle-particle gap is found by assuming the values of thecritical exponents � ¼ 0:79 and � ¼ 0:88. These valuesare the ones found in the first-order expansion for theGross-Neveu-Yukawa field theory of this quantum phasetransition [6], around its upper critical (spatial) dimensionof three. Altogether, the data strongly support the existenceof a single quantum critical point separating the semime-tallic and the insulating antiferromagnetic phases of theHubbard model, with the quantum criticality belonging tothe Gross-Neveu universality class [5].

II. MODEL AND METHOD

As in Ref. [1], we will consider the half-filled Hubbardmodel on the honeycomb lattice

HtU ¼ �tX

hi;ji;�cyi;�cj;� þU

Xi

ðni;" � 1=2Þðni;# � 1=2Þ:

(1)

The hopping is restricted to nearest neighbors, so that thebipartite nature of the lattice allows us to avoid the negativesign problem.Generically, to detect antiferromagnetic ordering, we

compute spin-spin correlations:

m ¼ limL!1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

N

XNi¼1

eiQ�ihS0 � SiiHtU

vuut : (2)

Here, N ¼ 2L2 corresponds to the number of orbitals,and L is the linear length of the lattice. A finite value ofm signalizes long-range order and is equivalent to sponta-neous symmetry breaking. In particular, includinga magnetic-field term with an appropriate Fouriercomponent

Hh ¼ hXi

eiQ�iSzi (3)

gives

m ¼ limh!0

limL!1

1

L2

Xi

eiQ�ihSzi iHtUþHh: (4)

The ordering of limits is crucial. One first has to take thethermodynamic limit to allow for the collapse ofAnderson’s tower of states and then the limit of vanishingmagnetic field h. Such an approach was, for instance, usedin Ref. [15].It is more convenient to consider a local field since, as

we will see below, this trick lifts the burden of taking thelimit h ! 0 numerically. The local pinning field is given bythe term

Hloc ¼ h0Sz0 (5)

in the Hamiltonian. Using the representation �i;0 ¼1L2

Pqe

iq�i of the Kronecker symbol shows that each

Fourier component comes with an amplitude h0=L2, so

that taking the thermodynamic limit is equivalent to takingthe amplitude of the relevant Fourier component to zero.With the local field construction, the appropriate orderingof limits for an L� L lattice reads

m ¼ limi!1

limL!1 eiQ�ihSzi iHtUþHloc

: (6)

That is, one first has to take the thermodynamic limit—again to guarantee the collapse of the tower of states in thepresence of long-range order—and only then can one takethe distance from the pinning center to infinity [16]. Inother words, the distance from the pinning center sets anenergy scale that has to be larger than the finite-size spingap. As an efficient estimator for the evaluation of theordered moment, we thus propose

FAKHER F. ASSAAD AND IGOR F. HERBUT PHYS. REV. X 3, 031010 (2013)

031010-2

m ¼ limL!1

1

L2

Xi

eiQ�ihSzi iHtUþHloc: (7)

We have tested the above approach for the Hubbardmodel on the honeycomb lattice. Ground-state calculationswere carried out with the projective auxiliary-field quan-tum Monte Carlo (QMC) algorithm, which is based on theequation

hOiH ¼ lim�!1

h�Tje��H=2Oe��H=2j�Tih�Tje��Hj�Ti

: (8)

Here, � is a projection parameter, and the trial wavefunction is required to be nonorthogonal to the groundstate. For H ¼ HtU þHloc, the inclusion of the magneticfield does not generate a negative sign problem. We havechosen the trial wave function to be the ground state of thenoninteracting Hamiltonian HT ¼ Ht þHloc in the S

z ¼ 0sector. The implementation of the algorithm followsclosely Refs. [1,17]. The major difference is the use of asymmetric Trotter breakup, which ensures the hermiticityof the imaginary time propagator for any value of the timediscretization��. It also leads to smaller systematic errors.

Figure 1(a) plots the local moment at U=t ¼ 5 usingdifferent methods. In this case, magnetic ordering is robust,such that various approaches can be compared. The data setat h0 ¼ 0 corresponds to the correlation functions ofEq. (2). For this set of runs, we use a spin-singlet trialwave function, and the projection parameter �t ¼ 40 suf-fices to guarantee convergence to the ground state. Thisquick convergence stems from the fact that the trial wavefunction is orthogonal to the low-lying spin excitations[18]. The runs at finite values of the pinning field corre-spond to the quantity of Eq. (7). In the presence of a finitepinning field, SU(2) spin symmetry is broken and the trialwave function overlaps with all spin sectors. Consequently,a large value of the projection parameter �t ¼ 320 isrequired to guarantee convergence to the ground statewithin the quoted accuracy. Note that the CPU time scalesonly linearly with the projection parameter, so that suchlarge projection parameters are still numerically tractable.It is also worth pointing out that the observable of Eq. (7)corresponds to a single-particle quantity and shows verylittle fluctuation. As is evident in Fig. 1(a), finite-sizeeffects are strongly dependent on the specific choice ofthe pinning field. Nevertheless, convergence to valuesconsistent with the generic approach based on Eq. (2) isobtained for relatively large values of the pinning field. Ifthe pinning field is chosen too small, larger lattices areapparently required to ensure that the finite-size spin gap,set by v=L with v the spin-wave velocity, is smaller thanthe energy scale set by the pinning field. This expectation isconfirmed by the data, which show a systematic upturn as afunction of the system size for smaller values of thepinning field. Such a nonmonotonic finite-size behaviorcomplicates a finite-size scaling analysis. For this reason,

we propose to use a relatively large value of the pinningfield [19].At U=t ¼ 4, the local moment is smaller and hard to

detect. The h0 ¼ 0 data set of Fig. 1(b) compares ourresults for the correlation function of Eq. (2) to those ofRef. [2]. As is apparent, the agreement up to our largestlattice size L ¼ 18 is remarkable. Without the largestlattice sizes L ¼ 24 and L ¼ 36, extrapolation to the ther-modynamic limit is hard because of the downward turnpresent in the finite-size results. The data sets stemmingfrom the pinning-field approach provide an alternativeperspective and, on the whole, confirm the result ofRef. [2]. For the considered field range, there is consider-able scatter in the finite-size results, but nevertheless theextrapolation to the thermodynamic limit seems to be fieldindependent, as expected from the above considerations.We conclude this section by mentioning that we have

tested the approach for the noninteracting case and the

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.05 0.1 0.15 0.2

m

1/L

U/t=5

(a)h0 = 0.05h0 = 0.1 h0 = 0.4 h0 = 1.0 h0 = 5.0 h0 = 0.0

0

0.05

0.1

0.15

0.2

0.25

0 0.05 0.1 0.15 0.2

m

1/L

U/t=4

(b)h0=0.2h0=0.3h0=0.4h0=1.0h0=5.0h0=0.0

Sorella et al.

FIG. 1. Comparison of the pinning-field and correlation-function approaches to determine the staggered moment at(a)U=t ¼ 5 and (b)U=t ¼ 4. The data sets at h0 ¼ 0 correspondto the correlations functions, and values of �t ¼ 40 are sufficientto converge to the ground state. For nonvanishing pinning fields,projection parameters �t ¼ 320 are required to guarantee con-vergence. We use ��t ¼ 0:1, which for the symmetric Trotterdecomposition yields converged results within our numericalaccuracy. At U=t ¼ 4, comparison with the results of Ref. [2]shows excellent agreement. Lines corresponds to least-squaresfits to the form aþ b=Lþ c=L2.

PINNING THE ORDER: THE NATURE OF QUANTUM . . . PHYS. REV. X 3, 031010 (2013)

031010-3

method successfully demonstrates the absence of long-range magnetic order. Hence, both for critical states aswell as for magnetically ordered phases, the pinning-fieldapproach does provide an efficient tool. Further testing ofthis approach for Heisenberg bilayers is presently inprogress [20].

III. PHASE DIAGRAM OF THE HUBBARDMODEL ON HONEYCOMB LATTICE

We have used the above approach to revisit the magneticphase diagram of the Hubbard model on the honeycomblattice. At weak couplings, the model is known to have astable semimetallic state. In the strong coupling limit andbecause of the absence of frustration, an antiferromagneticMott insulator is present. The nature of the transition be-tween these two states has been studied in the past [3–5] andis presently controversial [1,2].

A. Single-particle gap

To pin down the coupling strength beyond which thesingle-particle gap opens, we have repeated calcula-tions for the time-displaced single-particle imaginary

time Green function at the nodal point: GðK; �Þ ¼P�hcyK;�ð�ÞcK;�ð� ¼ 0Þi. As is evident in Fig. 2 and with

the symmetric Trotter decomposition, the Trotter system-atic error is negligible within our accuracy. Fitting the datato an exponential form allows us to extract the single-particle gap, which we plot as a function of system sizein Fig. 2. Assuming a polynomial form for the extrapola-tion to the thermodynamic limit, we find a small but finitesingle-particle gap for U=t � 3:7. This finding is particu-larly interesting when compared to the results of Sorellaet al. [2] that at U=t ¼ 3:8 point to the absence of long-range magnetic order. This result can be taken as anindication for a possible intermediate phase. However,the analysis that we present below suggests a differentinterpretation.

B. Magnetization from pinning fields

We have used the pinning-field approach to computethe staggered moment as a function of U=t. The results ath0 ¼ 5t are reported in Fig. 3. The extrapolation to thethermodynamic limit is carried out using a polynomialscaling up to second order in 1=L. Figure 4 plots the so-obtained staggered moment for two choices of the pinningfield as well as the single-particle gap. Several commentsare in order.(i) Within our accuracy, and maybe most importantly,

with the polynomial fit used in extrapolating the datato the thermodynamic limit, it appears that thesingle-particle gap opens right when magnetic order-ing sets in. The only mismatch is at U=t ¼ 3:7,where we do not detect magnetic ordering but wedo detect a small single-particle gap.

0.0001

0.001

0.01

0.1

1

10

0 5 10 15 20

G(K

,τ

τ

)

τt

U/t=3.8

L=3, ∆ τ∆

τ∆τ∆τ∆τ∆

τ∆

=0.1, 2/15 L=6, =0.1, 2/15

(a)

L=18, =0.1L=15, =0.1L=12, =0.1L=9, =0.1

L=9, =2/15

0

0.1

0.2

0.3

0.4

0.5

0 0.05 0.1 0.15 0.2

∆ sp

1/L

(b)

U/t=3.7U/t=3.8U/t=3.9U/t=4.0U/t=4.1

FIG. 2. Single-particle gap. (a) The raw data at U=t ¼ 3:8. Asis apparent, the systematic error stemming from the finite Trotterstep is negligible within our accuracy. The data support a largeimaginary time range consistent with a single exponential decay.Lines are least-squares fits of the tail of the imaginary timeGreen function to the form Ze��sp�. Here, Z corresponds to thesingle-particle residue and �sp to the single-particle gap. (b) Size

dependence and extrapolation of the single-particle gap.

0

0.02

0.04

0.06

0.08

0.1

0.12

0 0.05 0.1 0.15 0.2

m

1/L

h0 = 5.0

U/t = 5.0U/t = 4.6U/t = 4.3U/t = 4.2U/t = 4.1U/t = 4.0U/t = 3.9U/t = 3.8U/t = 3.7

FIG. 3. Magnetic moment at pinning field h0 ¼ 5t as a func-tion of U=t. Here, we have used �t ¼ 320 and ��t ¼ 0:1. Thesolid lines are least-squares fits to the form aþ b=Lþ c=L2.

FAKHER F. ASSAAD AND IGOR F. HERBUT PHYS. REV. X 3, 031010 (2013)

031010-4

(ii) The QMC data in Fig. 4 show that over a wideparameter range, the single-particle gap, measuredin units of the Hubbard U, tracks the staggeredmagnetization. We take this fact as a strong indica-tion that the magnetization provides the only rele-vant scale in the problem, determining directly thesingle-particle gap. We will see below that thisconclusion, based here on a simple, polynomialextrapolation of the finite-size data, is also obtained,if a more refined data analysis is performed.

(iii) The data in Fig. 4 exhibit an unusual inflectionpoint at approximately U=t ¼ 4:1. Such an inflec-tion point is clearly absent at the mean-field level(see the inset of Fig. 4). We will discuss the impli-cations of this inflection point in the next section.Let us finally note that in previous calculations [1],we were unable to resolve staggered momentslesser than m ’ 0:03. We thereby missed this in-flection point in the polynomially extrapolatedmagnetization curve and concluded the presenceof an intermediate phase [21].

C. Finite-size scaling

As mentioned above, one of the particularities of thedata presented in Fig. 4 is the occurrence of an inflec-tion point atU=t ¼ 4:1. It is a natural question to ask if thisrather peculiar feature may be an artifact of using a simplepolynomial fitting procedure that one would indeedexpect to fail close to criticality. The polynomial procedurecould result in an overestimation of the magnetization inthe vicinity of the critical point between the semimetallicand the insulating phases of the Hubbard model. As weexplain next, arguments in favor of this conjecture areprovided by the large-N treatment of the Gross-Neveumodel [5] and the � expansion around three spatial dimen-

sions in the equivalent Gross-Neveu-Yukawa field theory,formulated in Ref. [6]. Given the order parameter exponent�, as well as the correlation length exponent �, the stag-gered magnetization scales as

m ’ jU�Ucj� ’ ��=�: (9)

Using the standard scaling laws [22], the exponent �=�may conveniently be expressed in terms of the anomalousdimension for the order parameter as

�

�� 1

2ð½dþ z� � 2þ Þ; (10)

where dþ z is the effective dimensionality of the system.If we assume that the Lorentz invariance is emergent at thecritical point, as it indeed is close to the upper criticaldimension dup ¼ 3 of the Gross-Neveu-Yukawa theory

[6], and maybe even more generally [23], the dynamicalcritical exponent is z ¼ 1. If, then, the anomalous dimen-sion of the order parameter is such that < 3� d, we findthat the combination of the exponents �=� < 1 and ourpolynomial fitting procedure in the previous section couldvery well overestimate the value of the staggered magne-tization. In fact, both the large-N approach [5] and theexpansion around the upper critical dimension [6] suggestsuch an overestimation. Within the first order of the ex-pansion in the parameter � ¼ 3� d, for example, ¼4�=5, so that

�

�¼ 1� �

10þOð�2Þ: (11)

In two dimensions, then, �=� ’ 0:9.To look for the signs of the Gross-Neveu criticality in the

Hubbard model, we carry out a finite-size scaling analysisbased on the usual scaling form

m ¼ L��=�F½L1=�ðU�UcÞ�: (12)

Figure 5(a) plots mL�=� versus U for the magnetizationdata at the fixed field h0 ¼ 5. [We will omit at this pointthe second scaling variable h0L

y�d, since the scaling di-mension y� d ¼ ð�� Þ=2 � 1. This second argumentof the scaling function, present in principle, is thereforeeffectively constant at a fixed h0, and its inclusion does notvisibly affect the quality of scaling. For further discussionof this point, see the Appendix.] As a guide, we have usedthe first-order �-expansion value of �=� ¼ 0:9. Fivecurves then all cross at a single point Uc=t ’ 3:8, therebyproviding a first nontrivial indication of the critical point.This value of Uc is slightly larger than that obtained withthe polynomial fit. The �-expansion value of the correla-tion length exponent reads [6]

� ¼ 12 þ 21

55�þOð�2Þ: (13)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

3.5 4 4.5 5 5.5 6

m

U/t

m, h0=1.0 m, h0=5.0

∆sp/U

1.5 2 2.5 3 0

0.1

0.2

0.3

m

U/t

Mean field

FIG. 4. Staggered moment extrapolated to the thermodynamiclimit (see Fig. 3) for two values of the pinning field. We haveequally plotted the single-particle gap in units of U. The insetplots the staggered magnetization as obtained from a mean-fieldspin-density-wave ansatz.

PINNING THE ORDER: THE NATURE OF QUANTUM . . . PHYS. REV. X 3, 031010 (2013)

031010-5

With this value of �, again at � ¼ 1, we obtain an excellentdata collapse, as is shown in Fig. 5(b).

In accord with the Gross-Neveu-Yukawa theory, thenumerical data of Fig. 4 support the interpretation thatthe magnetization is the only scale in the problem. Tofurther check this interpretation, we have scaled thesingle-particle gap to the form

�sp

U¼ L��=� ~F½L1=�ðU�UcÞ�; (14)

again using the �-expansion exponents in twodimensions. Figure 6(a) shows that the crossing point of

the�sp

U L�=� curves again occurs at Uc=t ’ 3:8, and

Fig. 6(b) shows the collapse. It is quite remarkable thatwithin our precision, the two scaling functions are equal:~F ¼ F.Hence, the scaling analysis of our QMC within the

Gross-Neveu scenario is consistent with a single continu-ous quantum phase transition between the semimetal andthe antiferromagnetic insulators, and suggests that the first-order expansion around the upper critical dimension in the

Gross-Neveu-Yukawa theory may already yield ratheraccurate values of the critical exponents.

IV. DISCUSSION AND CONCLUSIONS

We have introduced an alternative method to computethe staggered magnetization. By introducing a local mag-netic field, we pin the quantization axis of the ordered state.The staggered magnetic moment then corresponds to thelocal magnetization infinitely far away from the pinningcenter. The approach has the major advantage that wecompute directly the magnetization as opposed to itssquare when measuring correlation functions. We cantherefore expect improved resolution when the local mo-ment is small. One advantage of the approach is an internalcross-check that requires the staggered moment to beindependent on the numerical value of the pinning field.We have been able to reach this internal cross-check onlyin the case of large pinning fields, which is consistent withthe approach proposed by Ref. [10], where the pinningfield is set to infinity on the boundary of the lattice. If thepinning field is too small, the approach suffers from large

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5

(∆sp

/U )

L

U/t

(a)L=6L=9

L=12L=15L=18

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

-2 0 2 4 6 8 10

m L

, (∆

sp/U

) L

L1 / (U - Uc)/Uc

(b)L=6, mL=9, m

L=12, mL=15, mL=18, m

L=6L=9

L=12L=15L=18

ν

ν

β/

νβ

/ν

β/

FIG. 6. Data collapse for the single-particle gap. The expo-nents are taken for the � expansion of Ref. [6]. (a) The crossingpoint pins down the value of Uc. (b) The data collapse againusing Uc=t ¼ 3:78. For comparison, we have included the datafor the magnetization.

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5

m L

β / ν

β / ν

ν

U/t

(a)L=6L=9

L=12L=15L=18

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

-2 0 2 4 6 8 10

m L

L1 / (U - Uc)/Uc

(b)L=6L=9

L=12L=15L=18

FIG. 5. Data collapse for the magnetization presented in Fig. 3.The exponents are taken for the � expansion of Ref. [6]. (a) Thecrossing point pins down the value of Uc. (b) The data collapse,using Uc=t ¼ 3:78.

FAKHER F. ASSAAD AND IGOR F. HERBUT PHYS. REV. X 3, 031010 (2013)

031010-6

and nonmonotonic size effects, since the energy scale setby the local magnetic field is unable to overcome the finite-size spin gap. Under these circumstances, the extrapolatedvalue of the magnetization has the tendency of underesti-mating the order. In this article, we have only considered avery specific form of the pinning field. The number ofdifferent choices of pinning fields provides a playgroundfor optimization of the approach and for minimization ofthe size effects.

The application to the Hubbard model on the honey-comb lattice sheds new light on the phase diagram of thiswell-known problem. The enhanced precision in compari-son to Refs. [1,2] reveals that the staggered moment hasthe same functional form as the single-particle gap (asmeasured in units of U). Remarkably, an excellent datacollapse onto a single universal curve is found in the finite-size scaling of both quantities, with the values of thecritical exponents characteristic of the Gross-Neveu criti-cality between the semimetallic and the magnetic insulat-ing phases.

ACKNOWLEDGMENTS

F. F. A. is very indebted to T. Lang, Z. Y. Meng, A.Muramatsu, and S. Wessel for many valuable comments.We equally acknowledge discussions with A. Chernyshev,L. Fritz, M. Hohenadler, and S. Sorella. This work wasinitiated at the KITP during the workshop on ‘‘FrustratedMagnetism and Quantum Spin Liquids From Theory andModels to Experiments’’ coordinated by K. Kanoda, P. Lee,A. Vishwanath, and S. White and finalized at the MPI-PKS(Dresden). Funding from the DFG under GrantsNo. AS120/9-1 and No. AS120/10-1 (ForschergruppeFOR 1807) is acknowledged. I. F. H. has been supportedby the NSERC of Canada. We thank the JulichSupercomputing Centre and the Leibniz-Rechenzentrumin Munich for generous allocation of CPU time.

APPENDIX

Strictly speaking, the finite-size scaling form for themagnetization in the presence of the external field is

m ¼ L��=�G

�L

; hy

�¼ L��=�G

�L

;h0Ld

y

�; (A1)

where GðX; YÞ is the scaling function of two variables, and is the (diverging) correlation length. Here, we have usedthe fact that the relevant Fourier component of the local

pinning field scales as h0Ld . The dimension y of the uniform

external field is [22]

y ¼ dþ zþ 2�

2: (A2)

Away from criticality, is bounded, and in the thermo-dynamic limit, the magnetization scales to its field-independent value. The data in Fig. 1 confirm this feature

and show that the extrapolated value of the magnetizationis h0 independent, provided that for the considered latticesizes, h0 is chosen to be large enough. It is also interestingto point out that for values of h0 in the range 1t < h0 < 5t,the finite-size value of the magnetization is next to inde-pendent on the value of the pinning field.At criticality, we can replace by L to obtain

m ¼ L��=�Gð1; h0Ly�dÞ: (A3)

With the Lorentz symmetry at the critical point, the valueof the dynamical critical exponent is pinned to z ¼ 1, andthe scaling dimension of the local field h0 is

y� d ¼ 3� d�

2: (A4)

In the � expansion, then,

y� d ¼ �

10þOð�2Þ; (A5)

and rather small, presumably even for � ¼ 1 (d ¼ 2).For a fixed value of h0, therefore, the secondargument of the scaling function G is almost constant.To be more precise, we can use the asymptotic form

Gð1;YÞ/Y1=� with 1� ¼ �

�y [22], such that in the large-L

limit, m / h1=�0 L��=�þ½ðy�dÞ=��. Within the � expansion,ðy�dÞ� ’0:05, which results in a very small correction for

considered lattice sizes.Hence, on the whole, the scaling function G depends

rather weakly on the second argument, and for practicalpurposes, it suffices to neglect it.

[1] Z. Y. Meng, T. C. Lang, S. Wessel, F. F. Assaad, and A.Muramatsu, Quantum Spin Liquid Emerging in Two-Dimensional Correlated Dirac Fermions, Nature(London) 464, 847 (2010).

[2] S. Sorella, Y. Otsuka, and S. Yunoki, Absence of a SpinLiquid Phase in the Hubbard Model on the HoneycombLattice, Sci. Rep. 2, 992 (2012).

[3] S. Sorella and E. Tosatti, Semi-Metal-Insulator Transitionof the Hubbard Model in the Honeycomb Lattice,Europhys. Lett. 19, 699 (1992).

[4] T. Paiva, R. T. Scalettar, W. Zheng, R. R. P. Singh, and J.Oitmaa, Ground-State and Finite-Temperature Signaturesof Quantum Phase Transitions in the Half-Filled HubbardModel on a Honeycomb Lattice, Phys. Rev. B 72, 085123(2005).

[5] I. F. Herbut, Interactions and Phase Transitions onGraphene’s Honeycomb Lattice, Phys. Rev. Lett. 97,146401 (2006).

[6] I. F. Herbut, V. Juricic, and O. Vafek, Relativistic MottCriticality in Graphene, Phys. Rev. B 80, 075432 (2009).

[7] H.-Y. Yang, A. F. Albuquerque, S. Capponi, A.M. Lauchli,and K. P. Schmidt, Effective Spin Couplings in the MottInsulator of the Honeycomb Lattice Hubbard Model,New J. Phys. 14, 115027 (2012).

PINNING THE ORDER: THE NATURE OF QUANTUM . . . PHYS. REV. X 3, 031010 (2013)

031010-7

[8] C.-C. Chang and R. T. Scalettar, Quantum DisorderedPhase near the Mott Transition in the Staggered-FluxHubbard Model on a Square Lattice, Phys. Rev. Lett.109, 026404 (2012).

[9] B. K. Clark, Searching for Topological Degeneracy inthe Hubbard Model with Quantum Monte Carlo,arXiv:1305.0278.

[10] S. R. White and A. L. Chernyshev, Neel Order in Squareand Triangular Lattice Heisenberg Models, Phys. Rev.Lett. 99, 127004 (2007).

[11] M. Hohenadler, T. C. Lang, and F. F. Assaad, CorrelationEffects in Quantum Spin-Hall Insulators: A QuantumMonte Carlo Study, Phys. Rev. Lett. 106, 100403(2011).

[12] M. Hohenadler, Z. Y. Meng, T. C. Lang, S. Wessel, A.Muramatsu, and F. F. Assaad, Quantum Phase Transitionsin the Kane-Mele-Hubbard Model, Phys. Rev. B 85,115132 (2012).

[13] S. Rachel and K. Le Hur, Topological Insulators and MottPhysics from the Hubbard Interaction, Phys. Rev. B 82,075106 (2010).

[14] F. F. Assaad, Phase Diagram of the Half-Filled Two-Dimensional SU(N) Hubbard-Heisenberg Model: AQuantum Monte Carlo Study, Phys. Rev. B 71, 075103(2005).

[15] M.V. Ulybyshev, P. V. Buividovich, M. I. Katsnelson, andM. I. Polikarpov, Monte Carlo Study of the Semimetal-Insulator Phase Transition in Monolayer Graphene withRealistic Interelectron Interaction Potential, Phys. Rev.Lett. 111, 056801 (2013).

[16] Clearly, one cannot exchange the limits in Eq. (6).However, one can measure the magnetization at the largestdistance available on the lattice, thereby tying together the

thermodynamic and infinite distances from the pinningcenter limits. This procedure can lead to spurious results.

[17] F. F. Assaad and H.G. Evertz, in Computational Many-Particle Physics, Lecture Notes in Physics Vol. 739, editedby H. Fehske, R. Schneider, and A. Weiße (Springer-Verlag, Berlin, 2008), p. 277–356.

[18] S. Capponi and F. F. Assaad, Spin and Charge Dynamicsof the Ferromagnetic and Antiferromagnetic Two-Dimensional Half-Filled Kondo Lattice Model, Phys.Rev. B 63, 155114 (2001).

[19] Here, large means comparable to the bandwidth. If thepinning field is much larger than the bandwidth, chargefluctuations on the pinning site will be blocked by anenergy scale set by h0. Thereby, in the limit h0 ! 1, thepinning site effectively drops out of the Hamiltonian andno symmetry breaking occurs. The very slight drop in themagnetization at h0 ¼ 5 and on small lattices in Fig. 1(a)could be a precursor of this effect.

[20] S. Wessel (private communication).[21] In Ref. [1], a small but finite spin gap with maximal value

at U=t ¼ 4 was reported. The spin gap is determined withthe same method as the single-particle gap, by fitting thetail of the imaginary time-displaced spin-spin correlationfunctions to a single exponential. Enhancing the imagi-nary time range for the fit produces a slight decrease of thespin gap for the larger lattices sizes. This slight decreaserenders the extrapolation to the thermodynamic limit in-conclusive.

[22] I. Herbut, A Modern Approach to Critical Phenomena(Cambridge University Press, Cambridge, England, 2007).

[23] I. F. Herbut, V. Juricic, and B. Roy, Theory of InteractingElectrons on the Honeycomb Lattice, Phys. Rev. B 79,085116 (2009).

FAKHER F. ASSAAD AND IGOR F. HERBUT PHYS. REV. X 3, 031010 (2013)

031010-8