Faulkner, M. F., Bramwell, S. T., & Holdsworth, P. C. W. (2015).Topological-sector fluctuations and ergodicity breaking at the Berezinskii-Kosterlitz-Thouless transition. Physical Review B: Condensed Matter andMaterials Physics, 91(15), [155412].https://doi.org/10.1103/PhysRevB.91.155412

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Topological-sector fluctuations and ergodicity breaking at theBerezinskii-Kosterlitz-Thouless transition

Michael F. Faulkner,1, 2, ∗ Steven T. Bramwell,1 and Peter C. W. Holdsworth2

1London Centre for Nanotechnology and Department of Physics and Astronomy,University College London, 17-19 Gordon Street, London WC1H 0AH, United Kingdom

2Laboratoire de Physique, Universite de Lyon, Ecole NormaleSuperieure de Lyon, 46 allee d’Italie, 69364 Lyon Cedex 07, France

The Berezinskii-Kosterlitz-Thouless (BKT) phase transition drives the unbinding of topologicaldefects in many two-dimensional systems. In the two-dimensional Coulomb gas, it corresponds to aninsulator-conductor transition driven by charge deconfinement. We investigate the global topologicalproperties of this transition, both analytically and by numerical simulation, using a lattice-fielddescription of the two-dimensional Coulomb gas on a torus. The BKT transition is shown to be anergodicity breaking between the topological sectors of the electric field, which implies a definition oftopological order in terms of broken ergodicity. The breakdown of local topological order at the BKTtransition leads to the excitation of global topological defects in the electric field, corresponding todifferent topological sectors. The quantized nature of these classical excitations, and their strictsuppression by ergodicity breaking in the low-temperature phase, afford striking global signaturesof topological-sector fluctuations at the BKT transition. We discuss how these signatures could bedetected in experiments on, for example, magnetic films and cold-atom systems.

I. INTRODUCTION

Topological physics [1] emerges in many condensed-matter systems, including superfluids and super-conductors [2–4], topological insulators [5], exciton-polariton condensates [6], and magnetic textures suchas skyrmions [7]. Among two-dimensional systems, aprototypical application of topology concerns the quan-tum Hall effect in the two-dimensional electron gas [8–10], while many other examples relate to the physics oftopological defects identified by Berezinskii, Kosterlitzand Thouless (BKT) [11–13]. These include Josephsonjunction arrays [14–17], films composed of Bose-Einsteincondensates [18, 19], superfluid films [20], liquid-crystaland polymer films [21], superinsulators [22, 23], and mag-netic films and layers [24–27]. In such systems, the BKTphase transition drives the thermal dissociation of boundpairs of local topological-defects [11–13, 28]. The idea ofa topological defect (defined in the footnote [29]) is in-deed one of the most basic and important applications oftopology in condensed-matter physics [30].

An important discovery of BKT and later authors [11–13, 28] is that the defect-mediated transition of the planerotator (or 2D-XY) model and its analogues can bemapped to the insulator-conductor transition of a two-dimensional Coulomb gas [31]. The long-range Coulombinteractions emerge from a purely local Hamiltonian, sothat the mapping at the microscopic level, although com-plete [32], is far from transparent. However, as Maggsand co-workers [33–38] have shown in three dimensions,a Coulomb fluid can be transformed into a local prob-lem by using an electric-field representation and intro-ducing a freely fluctuating auxiliary gauge field. Fol-

∗ michael.faulkner@bristol.ac.uk

lowing this work, it is straightforward to show that theXY Hamiltonians that admit a BKT transition map onto this generalized electrostatic problem in two dimen-sions. A practical consequence of the phase-space ex-tension to a fluctuating auxiliary gauge field is the de-velopment of purely local algorithms for the simulationof Coulomb fluids in both three [33–38] and two dimen-sions [39], which circumvent the technical difficulties as-sociated with long-range interactions. In particular, thelogarithmic potential that governs charge-charge interac-tions in the two-dimensional Coulomb gas is dealt withlocally, allowing a new approach to the efficient simula-tion of two-dimensional Coulombic systems.

In this paper, we exploit these developments to formu-late and simulate a lattice-field description of the two-dimensional Coulomb gas for the purpose of investigatingthe topological properties of the BKT transition. TheBKT transition is topological in the sense that it sep-arates a topologically ordered phase from a disorderedone. Topological order in this context means that thelocal topological defects (charges in the two-dimensionalCoulomb gas) are confined. Vallat and Beck [32] con-sidered the two-dimensional XY model on a torus, andshowed how a winding field can be associated with theglobal topology of the system. In the high-temperaturephase, where charge is deconfined, non-zero values ofthis winding field define global topological defects thatare distinct from the local topological defects driving theBKT transition. Here we show that the lattice-field de-scription naturally lends itself to classifying and investi-gating this property. In this paper, we treat only the two-dimensional Coulomb gas, but in a further publication wewill extend our analysis to the case of two-dimensionalXY models on the torus.

Our key observation is that the topology of the torus onwhich the Coulomb gas is placed generates a multiplicityof states in the lattice electric-field representation that

2

are equivalent for charge configurations but not energeti-cally degenerate. Given an arbitrary charge distribution,one is at liberty to add an integer multiple of some con-stant to each component of the harmonic mode of theelectric field while leaving the charge distribution un-changed. This global topology associated with the BKTtransition describes the winding of charges around thetorus. In the high-temperature phase, charge deconfine-ment allows for fluctuations in the winding componentof the harmonic mode, which can be classified as differ-ent topological sectors. Below the transition, however,the binding of charge pairs causes the winding compo-nent to be zero. Topological-sector fluctuations in theelectric field therefore mark the appearance of the high-temperature, topologically disordered phase at the BKTtransition.

The present study of topological-sector fluctuations inthe two-dimensional Coulomb gas may be compared toprevious studies on the three-dimensional Coulomb phaseof spin-ice materials and models [40–43]. In spin ice, theonset of topological-sector fluctuations is shown to sig-nal a Curie law crossover [40] for the zero-field suscep-tibility and a Kasteleyn transition in the presence of anapplied field [41, 42]. Our study of the BKT transitionreveals aspects of topological-sector fluctuations that arenot found in either of these established cases. For exam-ple, by our analysis, the two-dimensional Coulomb gasshould be considered to present an ergodicity-breakingtransition to a topologically ordered phase in the absenceof an applied field, whereas spin ice has no equivalentphase.

The paper is organized as follows. In Section II,we introduce the lattice-field representation of the two-dimensional Coulomb gas on a torus and use this to definethe partition function and the topological sectors of theelectric field. We use numerical simulations to demon-strate that topological-sector fluctuations appear in thehigh-temperature (conducting) phase but not in the low-temperature (insulating) phase. In Section III, we showthat the reason for the strict suppression of topological-sector fluctuations in the low-temperature phase is er-godicity breaking at the transition. A finite-size scalinganalysis, given in Section IV, confirms that in the ther-modynamic limit, ergodicity is broken precisely at TBKT.Conclusions and comparisons with experimental systemsare discussed in Section V.

II. TOPOLOGICAL-SECTOR FLUCTUATIONS

Using the unit system defined in Appendix A, we for-mulate the two-dimensional Coulomb gas using discretevector calculus on a square lattice with periodic bound-ary conditions (PBCs) applied. The PBCs enforce thetoroidal topology but not the curvature of a true torus.All functions are defined to be the discrete counterpartsof smooth vector fields [44], and any lattice vector field

F is defined component-wise via [44]

F(x) := Fx

(x +

a

2ex

)ex + Fy

(x +

a

2ey

)ey, (1)

where x is any lattice site and ex/y is the unit vector

in the x/y direction. The operators ∇ and ∇ are theforwards and backwards finite-difference operators on alattice, respectively, and the lattice Laplacian is definedby ∇2 := ∇ · ∇ [44]. The most general electric field Emay be Helmholtz decomposed into the sum of a Poisson(divergence-full) component −∇φ, a rotational compo-

nent E and a harmonic component E:

E(x) = −∇φ(x) + E(x) + E. (2)

This electric field is the most general solution to Gauss’law on a lattice:

∇ ·E(x) = ρ(x)/ε0. (3)

Here, ρ(x) := qm(x)/a2 is the charge density at eachlattice site x, q is the elementary charge, the integer mdenotes the charge species, a is the lattice spacing and ε0is the electric permittivity of free space (see Appendix A).

Using the field E adds an auxiliary field E to the usualsolution of electrostatics, as in the electrostatic modelof Maggs and Rossetto (MR) [33]. This allows us tosimulate the physics of Coulombic interactions on a lat-tice via local electric-field updates, avoiding the need totreat computationally intensive long-range interactions,as outlined in Appendix B.

The validity of introducing the auxiliary field is seen inthe context of the separability of the partition functioninto its Coulombic and auxiliary components: the auxil-iary field contributes to the internal energy of the system,but it is statistically independent of the Coulombic ele-ment. In Appendix B, we give a full description of thealgorithm and a derivation of the partition function forthe Coulomb gas of multi-valued charges.

The internal energy of the electric fields correspondingto a given charge and auxiliary-field configuration is givenby

U0 =ε0a

2

2

∑x∈D|E(x)|2, (4)

where D is the set of all lattice points. To represent theCoulomb gas in the grand canonical ensemble, we add acore-energy term UCore, given by

UCore :=a4

2

∑x∈D

εc [m(x)] ρ(x)2, (5)

where εc(m) is the core-energy constant of each chargemq, and εc(m) = εc(−m) since charges are excited tothe vacuum in neutral pairs. The grand-canonical en-ergy of the system U = U0 + UCore may be expanded bycombining Eqs. (2), (4) and (5) to give a sum of terms

3

arising from the different field components, which add tothe core energy:

U = USelf + UInt + URot + UHarm + UCore. (6)

Here, respectively, URot := ε0a2∑

x∈D |E(x)|2/2 and

UHarm := ε0L2|E|2/2 are the auxiliary-field and

harmonic-mode components of the grand-canonical en-ergy, and USelf and UInt are the self-energy and Coulom-bic charge-charge interaction components. As outlinedin detail in Appendix B, the latter two components maybe expressed in terms of the lattice Green’s function G,according to USelf := a4G(0)

∑x∈D ρ(x)2/2ε0, UInt :=

a4∑

xi 6=xj∈D ρ(xi)G(xi,xj)ρ(xj)/2ε0, where G(0) :=

G(x,x). Note that, while UInt can be negative, the sumUSelf + UInt is necessarily ≥ 0, as it arises from the termin |∇φ|2.

Using the above results, we may define the chemicalpotential for the introduction of a charge mq:

µm := −[G(0)

ε0+ εc(m)

]m2q2

2. (7)

In the following, we specialize to a Coulomb gas of n pairsof elementary charges of chemical potential µ := µ1, bysetting εc(m = 0,±1) = 0 and εc(m 6= 0,±1) =∞ [45].

The harmonic mode E is a uniform field found by av-eraging the total electric field E(x) over x. In a simplyconnected space, the average field may be convenientlyrelated to the average polarization P arising from an ef-fective surface charge distribution by E = −P/ε0. Fora charge-neutral system in a simply connected space,P :=

∑x∈D xρ(x)/N is invariant with respect to the ori-

gin shift x 7→ x+x0, and is therefore origin-independent.The situation is more complicated on a toroidal surfaceas E can also depend on a harmonic-field component thatcorresponds to a charge winding around the torus, and itis necessary to adopt a convention to define distances be-tween points (the concepts ‘close together’ and ‘far apart’are ambiguous on a torus). In Appendix C, we show indetail how it is possible to define origin-independent po-larization Ep and winding Ew components of the har-monic mode such that

E = Ep + Ew. (8)

Here,

Ew =q

Lε0w, (9)

where w is an integer-valued winding field chosen suchthat

Ep,x/y ∈(− q

2Lε0,

q

2Lε0

], (10)

and L is the lattice length.This decomposition of the harmonic field has the fol-

lowing interpretation. A charge pair may unbind and

wind around the torus in opposing directions before as-suming its original configuration. When a single chargewinds around the torus in the x/y direction, the x/ycomponent of the harmonic mode of the electric fieldEx/y increases by ±q/Lε0. As shown in Appendix C,the lowest-energy harmonic mode that describes an ar-bitrary charge distribution is therefore an element of theset (−q/2Lε0, q/2Lε0] and is defined as the polarizationcomponent in Eq. (10) by applying modular arithmeticto E. The remainder is the winding component. Themodulo operation removes any need for a ‘distances’ con-vention to define the polarization component, as well asany origin dependence of the field components (see Ap-pendix C for further details).

With these results we may use the integer-valued wind-ing field w to define the topological sector of the systemas the number of times charges wind around the torusin the x and y directions, with all non-trivial topologicalsectors given by w 6= 0. The topological sector of the sys-tem changes any time a charge pair unbinds and windsaround the torus and hence thermal fluctuations of thetopological sector are closely related to the unbinding ofcharge pairs, as elucidated further below.

The statistical mechanics of the topological-sector fluc-tuations may now be formulated by considering how thepolarization and winding components of the harmonicmode enter the lattice partition function. As shown inAppendix B, the partition function splits into two sta-tistically independent components. One component isthe Coulombic partition function ZCoul and contains allinformation about the charge-charge correlations, whilethe other is the auxiliary-field partition function ZRot

and contains all information about the auxiliary field:the auxiliary field can freely fluctuate without affectingcharge-charge correlations (see Appendix B). The parti-tion function is written as

Z = ZCoulZRot, (11)

where ZCoul is given by

ZCoul =∑{ρ(x)}

exp

−βa4

2ε0

∑xi 6=xj

ρ(xi)G(xi,xj)ρ(xj)

×∑

w∈Z2

exp

(−L

2βε02|Ep +

q

Lε0w|2)

× δ(∑

x∈Dρ(x)

)eβµn. (12)

Here, β := 1/kBT is the inverse temperature and the sum∑{ρ(x)} :=

∑{a2ρ(x)∈{0,±q}}.

The first exponential of Eq. (12) describes the anhar-monic charge-charge interactions, the second describesthe polarization and winding state of the system, andthe third describes the sum of the self-energies associatedwith each elementary charge. The sum over the windingfield w is necessitated by the degeneracy of the harmonic

4

FIG. 1. Topological-sector fluctuations of lattice electric fieldsin the two-dimensional Coulomb gas on a torus. Shown is thex-component of the normalized total harmonic mode LEx/2πand winding field LEw,x/2π versus Monte Carlo time for anL × L system of linear size L = 16 at T = 1.34 (top) andT = 2.0 (bottom). The system was simulated using the MRalgorithm with local moves only. At the lower temperature(top) harmonic-mode fluctuations are finite (black) but thereis no winding-field component (blue), while at the higher tem-perature (bottom) the winding-field component becomes fi-nite, indicating topological-sector fluctuations.

mode of the electric field: infinitely many topological sec-tors describe any given charge configuration. In additionto the local updates of the MR algorithm (see AppendixB), we also consider global updates, which correspond toindependently sampling this winding field.

Henceforth, we set the elementary charge q = 2π, thelattice spacing a = 1, the electric permittivity ε0 = 1, andBoltzmann’s constant kB = 1. The choice q = 2π recog-nizes the standard BKT theory, where a charge emergesas a local 2π winding in an associated lattice field, suchas the spin differences in the 2D-XY model [28].

The BKT transition drives the deconfinement of chargepairs in the two-dimensional lattice Coulomb gas, whichgenerates topological-sector fluctuations. The transition

occurs at TBKT = 1.351 (to four significant figures) [46]in the thermodynamic limit [a value specific to a gasof elementary charges with εc(m = 1) = 0], which isscaled to higher temperatures in finite-size systems (seebelow). Fig. 1 shows the evolution of the (normalized)x-component of the harmonic mode of a system of linearsize L = 16, simulated using local moves only (numeri-cal simulation details are described in Appendix D). Notopological-sector fluctuations are visible just below theBKT transition temperature TBKT = 1.351, but they be-come important at temperatures above TBKT.

III. ERGODICITY BREAKING

A convenient measure of the topological-sector fluctu-ations is the winding-field susceptibility χw, defined by

χw(L, T ) := βε0L2(〈E2

w〉 − 〈Ew〉2). (13)

In a fully ergodic system, χw is nonzero, even in the ab-sence of charge fluctuations, as can be seen by limitingthe Gibbs ensemble contributing to ZCoul to field config-urations of zero charge. In this case it is straightforwardto show, using Eqs. (9) and (12), that the constrainedsusceptibility is given by

χglobalw (T ) = βε0L

2 4q2 exp(−βq2/2ε0

)/ε20L

2 + . . .

1 + 4 exp (−βq2/2ε0) + . . .

' 4βq2

ε0exp

(−βq2/2ε0

), (14)

for kBT � q2/2ε0. The system-size dependence falls outof this expression so that a fully ergodic system wouldshow small but finite topological-sector fluctuations inthe low-temperature phase.

Assuming local charge dynamics, a topological-sectorfluctuation requires the separation of a pair of chargesover a distance greater than L/2 in either the x or they direction [see Eq. (8) and the subsequent discussion].As the charge concentration falls to zero at low temper-ature, screening becomes negligible and the energy bar-rier against such configurations diverges logarithmicallywith the linear system size L [11, 12, 31]. As the chargeconcentration increases with temperature, however, en-tropy and charge screening break down the free-energybarrier, making it finite at the BKT transition. Abovethe transition, charge pairs are free to unbind and traceclosed paths around the torus, giving finite-valued wind-ing fields, as observed in Fig. 1. In contrast, in the low-temperature phase, the probability of separation througha distance L/2 becomes strictly zero in the thermody-namic limit.

The BKT transition is therefore an ergodicity break-ing: a change in the phase space explored by a systemwith local dynamics. In detail, it is an ergodicity break-ing between topological sectors, signalled by the strictsuppression of topological-sector fluctuations in the elec-tric field at T < TBKT. If the dynamics were non-local

5

1.2 1.3 1.4 1.5 1.6 1.7

T

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6χ

loca

lw

/χal

lw

FIG. 2. The susceptibility quotient χlocalw /χall

w versus temper-ature for an L × L Coulomb gas of linear size L = 64. Inthe region T < 1.2, the quotient is zero, while for T > 1.6,the quotient approaches unity. This divergence between theresults of the local-update and the all-updates simulations,accompanied by striking fluctuations in the intermediate re-gion, signals an ergodicity breaking as the system is cooledthrough the BKT transition. The line is a guide to the eye.

(including global updates of the winding component ofthe harmonic mode [33–38]), χw would remain finite atall temperatures.

To explore this ergodicity breaking, we have simulatedthe two-dimensional Coulomb gas, first with local fieldupdates only, and second with both local and globalfield updates [33–38]. Corresponding to each case, wedefine the winding-field susceptibilities χlocal

w and χallw ,

respectively. Differences between χlocalw and χall

w reflectthe inability of local moves to explore a fully represen-tative phase space on the time scale of the simulation.To quantify this, we introduce the susceptibility quotientχlocal

w /χallw , which may be used to analyse the ergodicity

of the system.Fig. 2 clearly shows that ergodicity is broken in the

vicinity of the BKT transition. For T > 1.6, χlocalw = χall

w ,indicating that the free-energy barrier for a topological-sector fluctuation via local moves is small. For T < 1.2,the quotient is zero, indicating that the energy barrierprevents topological-sector fluctuations via local chargemoves. In between these low- and high-temperature re-gions there are strong fluctuations in the quotient be-cause charge deconfinement via local updates representsincreasingly rare events, an inevitable precursor to lossof ergodicity. In Section IV, this ergodicity breaking isshown to occur precisely at TBKT in the thermodynamiclimit.

Our analysis thus leads to a precise definition oftopological order for the two-dimensional Coulomb gasthrough the ergodic freezing of the topological sector toits lowest absolute value. Two-dimensional systems withU(1) symmetry are often associated with an absence of

an ordering field at finite temperature [47]. Here we ex-plicitly show that, in the case of the BKT transition, theordering of a conventional order parameter is replacedby topological ordering through an ergodicity breakingbetween the topological sectors. The topological order isdirectly related to the confinement-deconfinement tran-sition of the charges, the local topological defects of theelectric field. This type of ergodicity breaking is dis-tinct from either the symmetry breaking that character-izes a standard phase transition, or that due to the roughfree-energy landscape that develops at a spin-glass tran-sition [48].

IV. FINITE-SIZE SCALING

In order to explore the approach to the thermodynamiclimit, the two-dimensional Coulomb gas was simulated bythe Monte Carlo method as a function of system size, us-ing the MR algorithm. The global update was employedin order to improve the statistics (numerical simulationdetails are described in Appendix D).

Fig. 3 shows the simulated winding-field susceptibil-ity χw as a function of temperature for L × L Coulombgases of linear sizes between L = 8 and L = 64. There isa marked increase in the winding-field susceptibility χw

as the system passes through the BKT transition tem-perature TBKT = 1.351 [46] for all system sizes. Sus-ceptibility curves for successive values of L intersect attemperatures above T = 1.8 and below T = 1.5. Betweenthese two temperatures, the winding-field susceptibilityincreases for a given temperature as the linear system sizeL increases. These results are consistent with the finite-size scaling of the BKT transition temperature [24]: asthe system size decreases the effective transition temper-ature T ∗(L) increases.

Below TBKT, the probability of a charge pair sepa-rating over a distance greater than L/2 increases withdecreasing system size. This, combined with the finite-size transition temperature T ∗(L) also increasing withdecreasing system size, results in the winding-field sus-ceptibility curves for successive values of L intersectingin the vicinity of TBKT. The inset in Fig. 3 shows thatthe low-temperature crossover points of the susceptibil-ity curves are at T = 1.45, T = 1.40 and T = 1.37 (towithin estimated error). To extrapolate the trend of thedata shown in Fig. 3 to the thermodynamic limit, we de-fine the crossover temperature TCross(L) to be the lowertemperature at which χw(L) = χw(L/2).

Fig. 4 shows the crossover temperature TCross as afunction of inverse linear system size 1/L, along withstraight-line fits to the data. In the thermodynamic limit,TCross extrapolates to the value TCross = 1.351 to withinthe estimated error of the extrapolation, that is, it ex-trapolates to the BKT transition temperature [46]:

TCross(L→∞) = TBKT. (15)

6

1.4 1.6 1.8 2.0 2.2 2.4

T

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6χ

w

L=8L=16L=32L=64

1.35 1.40 1.450.00

0.01

0.02

FIG. 3. The winding-field susceptibility χw as a functionof temperature for L × L Coulomb gases of linear size L =8, 16, 32 and 64 (using local and global MR moves). Thecurves intersect at low and high temperature. Inset: An ex-panded plot of the data in the region of the low-temperatureintersections (with error bars representing two standard de-viations). The indicated crossover temperatures are givenby TCross(L = 16) = 1.45, TCross(L = 32) = 1.40 andTCross(L = 64) = 1.37 (to within estimated error), based ona data fit.

The 1/L scaling of TCross is unusual for the BKT tran-sition, for which the finite-size BKT transition temper-ature typically scales as a simple function of the loga-rithm of L [24, 49]. However, Minnhagen and Kim [50]found that a fourth-order cumulant that measures fluc-tuations of the helicity modulus in the 2D-XY modelalso scales as 1/L: as this closely relates to fluctuationsin the harmonic-mode susceptibility [32], it seems likelythat we are observing the same finite-size scaling here.The magnitude of the winding-field susceptibility at thecrossover points χCross

w (L → ∞) similarly extrapolatesto ∼ 5 × 10−4 in the thermodynamic limit, with an es-timated error of the same order. This small number isnot measurably different to the winding-field susceptibil-ity due to global moves only, which, at TBKT, evaluatesto ∼ 5 × 10−5 for all system sizes [see Eq. (14)]. Theinference is that topological-sector fluctuations due to lo-cal moves only turn on precisely at the universal pointTCross(L → ∞) = TBKT in the thermodynamic limit.This confirms that topological-sector fluctuations signalcharge deconfinement and the high-temperature phase ofthe BKT transition.

Given that the topological-sector fluctuations turnon at the temperature at which the system experi-ences the famous universal jump in the helicity modu-lus [13, 28, 32, 50], it is interesting to estimate the con-tribution that topological-sector fluctuations make to theuniversal jump. To do this, we define the harmonic-mode susceptibility χE and the polarization susceptib-lity χp by replacing Ew in Eq. (13) with E and Ep,

0.00 0.01 0.02 0.03 0.04 0.05 0.06

1/L

1.34

1.36

1.38

1.40

1.42

1.44

1.46

TCro

ss

Temperature fitTemperature data

0

5

10

15

20

25

30

χCro

ssw

×103

Susceptibility fitSusceptibility data

FIG. 4. The crossover temperature TCross (black data; left-hand y axis) and crossover susceptibility χCross

w (red data;right-hand y axis) as functions of inverse linear system size1/L, with error bars representing two standard deviations.Lines are weighted (with respect to the error bars) linear-regression fits to each data set, from which the y-intercept(L → ∞) was calculated. TCross(L → ∞) = 1.351(2), equalto the BKT transition temperature TBKT [46]. The crossoversusceptibility χCross

w (L→∞) = 5×10−4 with estimated errorof the same order: there is no measurable difference betweenthis quantity and the winding-field susceptibility due to globalupdates only at T = 1.351.

respectively. The helicity modulus is then given byΥ = ε−1

0 (1− χE/2) [32], so that χE makes a jump of or-der unity at TBKT. We find that the ratio (χE−χp)/χE isless than 5×10−2 for all T ≤ 1.6 for systems of linear sizeL = 8 to 64, showing that the contribution to the uni-versal jump from topological-sector fluctuations is verysmall. This reflects the near-cancellation of 〈E2

w〉 withthe coupling term 2〈Ep · Ew〉 in the evaluation of χE, re-flecting strong correlations between the polarization andwinding fields at the transition.

V. CONCLUSIONS

In conclusion, the BKT transition has long been aparadigm for the importance of topological defects incondensed-matter physics [1]. Vallat and Beck showedthat XY-type systems on the torus generate global topo-logical defects at the BKT transition that reflect thetoroidal topology [32]. Here we have used lattice-fieldsimulations to reveal topological-sector fluctuations inthe electric field of a two-dimensional lattice Coulomb gason a torus. We have shown how these provide a strikingand sensitive measure of the topological and ergodicity-breaking character of the BKT transition, allowing a pre-cise definition of topological order in terms of this brokenergodicity.

The topological-sector fluctuations at the BKT tran-sition are very clearly revealed in the lattice electricfield description of the two-dimensional Coulomb gas,

7

but we expect them to be equally relevant to any sys-tem that has a BKT transition. In suitable systems,the winding-field susceptibility that signals the onset oftopological-sector fluctuations will contribute to experi-mentally measurable responses of the system. For exam-ple, in a cylindrical or toroidal magnetic film with XYsymmetry, winding-field fluctuations in the Coulomb gasrepresentation correspond to measurable spin configura-tions in the magnetic representation. As we will showin future work [51], fluctuations of an appropriate topo-logical sector accompany the destruction of the finite-size magnetization of an XY spin system through vortexdeconfinement. They could therefore be observable inultrathin ferromagnetic metallic films [52] or magneticLangmuir-Blodgett films [53, 54].

Another promising system on which to measure thesetopological-sector fluctuations is the one-dimensionalquantum lattice Bose gas. When the system is placedon a ring, its angular momentum is no longer a goodquantum number. The angular momentum can there-fore fluctuate quantum mechanically, and the systemshould undergo a dramatic increase in these fluctua-tions as it passes through the superfluid – Mott in-sulator quantum phase transition [55, 56]. This dra-matic increase in the fluctuations corresponds to finite-valued global topological defects in the quantum system,and therefore, via the Feynman path-integral mapping,to topological-sector fluctuations in the two-dimensionalclassical lattice Coulomb gas on a torus. Murray et al.measured the angular momentum of ring-shaped Bose-Einstein condensates via the vortex-density profile of thesystem [57]. Our measure of the BKT transition couldtherefore correspond to equivalent, experimentally mea-surable topological-sector fluctuations in the cold-atomsystem [58].

Finally, it is worth noting that it is natural to asso-ciate a conducting phase with the excitation of windingfields, as may be seen by considering a loop of wire ina changing magnetic field. Recalling that the magneticfield does no work on a test charge, the induced elec-tromotive force must arise from a divergence-free electricfield running round the loop. The curl of this field obeysthe Maxwell-Faraday law, ∇ × E = ∂B/∂t. Hence, inthree dimensions, electromagnetic induction provides apractical method of exciting topological winding fieldsanalogous to those discussed here.

ACKNOWLEDGMENTS

It is a pleasure to thank A. C. Maggs, S. T. Banks,V. Kaiser and G. B. Davies for valuable discussions, andA. Gormanly for help with automating repeated simula-tions. We are also grateful to T. Roscilde for pointing outthe possible application to the one-dimensional latticeBose gas. M.F.F. is grateful for financial support fromthe CNRS and University College London. P.C.W.H.acknowledges financial support from the Institut Univer-

sitaire de France.

Appendix A: Dimensional analysis of thetwo-dimensional Coulomb gas

In the following, [ · ] denotes the dimensions of somequantity, L denotes the dimensions of length, d is thespatial dimensionality of the system, and ε0 is the vacuumpermittivity in d−dimensional space. Consider Gauss’law for the MR algorithm,

∇ ·E(x) = ρ(x)/ε0, (A1)

and the dimensions of electric charge density,

[ρ(x)] = [q]L−d, (A2)

which generates

[E(x)] = [q]L(1−d) [ε0]−1. (A3)

From a consideration of the exponent of the Boltzmannfactor (with β = 1/kBT ) we find a dimensionless group,

Π =adβε0

2

∑x∈D|E(x)|2 (A4)

⇒ [ε0] =[q]2L(2−d) [β] (A5)

⇒ [E(x)] =[q]−1L−1 [β]−1. (A6)

Setting the charge to be dimensionless, it follows that

[ε0] = [β] (A7)

and

[E(x)] = [β]−1L−1 (A8)

in d = 2.

Appendix B: The MR electrostatic model and thepartition function

The MR electrostatic model is a lattice-field modelfrom which it is possible to form the lattice partitionfunction of electrostatics. To show this, we describe theMR algorithm in terms of microscopic variables that rep-resent the local field updates. A conjugate lattice D′ isdefined such that each of its sites is at the centre of eachplaquette of D. Each site in D′ is associated with a real-valued variable ϕ whose adjustment corresponds to anupdate of the auxiliary field, while each pair of nearest-neighbour sites is associated with an integer-valued vari-able s whose adjustment corresponds to a charge-hop up-date. Both sets of variables are subject to PBCs.

Component-wise, we now define the field

[∆θ]i

(x +

a

2ei

):=

ϕ(x+aei)−ϕ(x)+qs(x+aei,x)

a,

(B1)

8

and identify

E(x) ≡ 1

ε0

[∆θ]y(x + a2 ex)

−[∆θ]x(x + a2 ey)

. (B2)

The x coordinates in Eq. (B1) are in D′; the x coordi-nates in Eq. (B2) are in D.

A charge hop in the positive x/y direction correspondsto a decrease/increase in the relevant s variable by anamount q, as shown in Fig. 5 (where sij represents the svariable between sites i and j of the conjugate lattice).

+α β

7−−−−→

i

j

α+

β

i

j

FIG. 5. A charge-hop update in the positive x direction: Thesij variable (red arrow) has its value decreased by an amountq. The value of the electric field flux Eαβ (black arrow) flowingfrom site α to site β then decreases by q/ε0, corresponding toa charge-hop update. Red circles represent positive charges;white circles represent empty charge sites.

Fig. 6 depicts the microscopic-variable representationof the auxiliary-field updates, with an alteration of aparticular ϕ variable rotating the field around its sur-rounding plaquette. In the figure, the ϕ variables arerepresented by spin-like arrows in order to emphasize therotation of the electric field.

1 2

3 4

7−−−−−−→

1 2

3 4

FIG. 6. An update of the rotational degrees of freedom of theelectric field: The value of the ϕ variable at the centre of arandomly chosen lattice plaquette decreases by an amount ∆.This rotates the electric flux by an amount ∆/ε0 around theplaquette, leaving Gauss’ law satisfied. Red arrows representϕ variables, black arrows represent the electric field, dashedred lines represent the conjugate lattice D′, the blue arrowrepresents the direction of the field rotation and grey circlesrepresent sites of arbitrary charge.

With the internal energy of the electric fields corre-sponding to a given charge and auxiliary-field configura-tion given by U0 in Eq. (4), it is possible to write thepartition function in the microscopic-variable represen-tation. For ease of manipulation, we allow charge-hop

updates to create charge pairs out of the vacuum andinclude the possibility of all integer-valued multiples ofthe elementary charge. Combining Eqs. (4) and (B2),the partition function in the microscopic-variable repre-sentation is given by

Z =∑{s}

∫Dϕ exp

− β

2ε0

∑〈x,x′〉

|ϕ(x)−ϕ(x′)+qs(x,x′)|2

× exp (−βUCore) , (B3)

where ∫Dϕ :=

∏x∈D′

[∫ q/2

−q/2dϕ(x)

], (B4)

and∑{s} :=

∑{s(x,x′)∈Z}. Here, the grand-canonical

energy of the Coulombic system U = U0 +UCore is used.This representation reproduces Gauss’ law:∑

x∈∂Γ

∆θ(x) · l(x) = QΓ, (B5)

where QΓ is the charge enclosed within some subset of thelattice Γ ⊆ D, ∂Γ ⊂ D′ is the boundary enclosing Γ, and ltraces an anticlockwise path along ∂Γ and has dimensionsof length. This equation results from the ϕ variablescancelling and the s variables being integer valued. Itfollows that∑

x∈∂Γ

∆θ(x) · l(x) =a2∑x∈Γ

ε0∇ ·E(x) (B6)

⇒ ∇ ·E(x) =ρ(x)/ε0, (B7)

recovering Eq. (3), as required.The constraints imposed upon the electric field (Gauss’

law and the form of the harmonic mode, the latter ofwhich is derived in detail in Appendix C) are combinedwith the grand-canonical energy of the system to writethe partition function in terms of the electric field. Wedefine the set X := qZ/a2, such that the partition func-tion is given by

Z = |J|∑

{ρ(x)∈X}

∑w0∈Z2

∫DE exp

[−βε0a

2

2

∑x∈D|E(x)|2

]

× exp (−βUCore)∏x∈D

[δ(∇ ·E(x)− ρ(x)/ε0

)]× δ

(∑x∈D

E(x) +

(N

ε0P− Lq

ε0a2w0

)), (B8)

where the functional integral∫DF :=

∏x∈D

[∫RdFx(x + aex/2)

∫RdFy(x + aey/2)

](B9)

9

for any vector field F, and |J| is the Jacobian determi-nant.

This partition function may be separated into two com-ponents by defining the new rotational field

e(x) := E(x) + ∇φ(x)− E. (B10)

The partition function is then given by

Z = ZCoul ZRot, (B11)

where

ZCoul :=∑

{∇2φ(x)∈Y }

exp

[−βε0a

2

2

∑x∈D|∇φ(x)|2

]

×∑

w0∈Z2

exp

(− β

2ε0|LP− qw0|2

)× exp (−βUCore) , (B12)

and

ZRot := |J|∫De exp

[−βε0a

2

2

∑x∈D|e(x)|2

]

×∏x∈D

[δ(∇ · e(x)

)]δ

(∑x∈D

e(x)

)(B13)

are the Coulombic and auxiliary-field components of thepartition function, respectively. Here Y := qZ/ε0a2 andwe have used the fact that all coupling terms in thegrand-canonical energy sum to zero. The MR algorithmtherefore reproduces Coulombic physics since the sepa-ration of the auxiliary-field partition function from theCoulombic partition function ensures that the charge-charge correlations are independent of the auxiliary field.

The lattice Green’s function G(x,x′) between twocharge-lattice sites x and x′ is defined such that

∇2xG(x,x′) = −δx,x′ , (B14)

where the subscript x denotes with respect to which co-ordinate system the lattice Laplacian is applied.

We define the k-space lattice Green’s function G,

Gx′(k) :=∑x∈D

e−ik·xG(x,x′), (B15)

and the set∑

k∈B :=∏i∈{x,y}

[∑ki∈Bi

], where Bi :=

{0,± 2πNia

,±2 2πNia

, · · · ,±(Ni

2 − 1) 2πNia

, Ni

22πNia} is the set of

k-space values in the i direction, and Ni :=√N . Com-

bining Eqs. (B14) and (B15), it then follows that∑k∈B

eik·(x−x′) = 2∑k∈B

eik·x [2− cos(kxa)− cos(kya)]

× Gx′(k). (B16)

This is solved by

Gx′(k) =e−ik·x

′

2 [2−cos(kxa)−cos(kya)]∀k 6= 0, (B17)

where the k = 0 part of the lattice Green’s function is setto zero since the harmonic component of E is attributedto E. It follows that

G(x,x′) =1

2N

∑k6=0

eik·(x−x′)

2− cos(kxa)− cos(kya). (B18)

The internal energy of the Poisson component of theelectric field is given by

UPoisson :=ε0a

2

2

∑x∈D|∇φ(x)|2 (B19)

=− ε0a2

2

∑x∈D

φ(x)∇2φ(x) (B20)

=a4

2ε0

∑xi,xj∈D

ρ(xi)G(xi,xj)ρ(xj), (B21)

hence, the Coulombic partition function can be writtenas

ZCoul =∑

{ρ(x)∈X}

exp

−βa4

2ε0

∑xi,xj∈D

ρ(xi)G(xi,xj)ρ(xj)

×∑

w0∈Z2

exp

(− β

2ε0|LP− qw0|2

)

× δ(∑

x

ρ(x)

)exp (−βUCore) , (B22)

where the δ function enforces charge neutrality in theGreen’s function representation.

Appendix C: Polarization

We consider the sum of each component of the electricfield over the entire lattice in order to analyse the har-monic mode. The sum of the x/y-component is split intoseparate sums over all x/y-components that enter a par-ticular strip of plaquettes of width a that wrap aroundthe torus in the y/x direction. Each component of theharmonic mode Ex/y is then expressed in terms of the

10

charge enclosed along each of the strips of plaquettes:

L2Ex =a2∑x∈D

Ex

(x +

a

2ex

)(C1)

=a

L−2a∑x=0

(x+a)

L−a∑y=0

[Ex

(x+

a

2, y)−Ex

(x+

3a

2, y

)]

+ La

L−a∑y=0

[Ex

(L− a

2, y)− Ex

(a2, y)]

+ La

L−a∑y=0

Ex

(a2, y)

(C2)

=− a2

ε0

L∑x=a

x

L∑y=a

ρ(x) + La

L∑y=a

Ex

(a2, y), (C3)

which follows from applying Gauss’ law to each strip ofplaquettes that wrap around the torus in the y direction.The same argument holds for the y component, hence,the harmonic mode is given by

E = − 1

ε0P +

q

Lε0w0, (C4)

where P :=∑

x∈D xρ(x)/N is the origin-dependentpolarization vector of the system and w0,x :=

ε0a∑Ly=aEx(a/2, y)/q is the x component of the origin-

dependent winding field, with the y component definedanalogously. Here, P and w0 are measured from a spe-cific origin. Note that the above applies to systems com-posed of either single- or multi-valued charges.

We have thus shown that E, which is origin-independent, is given by the sum of two origin-dependentterms. One of these is attributed to the polarizationof the system, while the other describes the winding ofcharges around the torus given that the polarization ismeasured with respect to the chosen origin.

Restricting our attention to the gas of elementarycharges, we now devise an origin-independent measure ofthe topological sector of the system. First, we note thatadding ω windings to either component of the harmonicmode E corresponds to

Ex/y 7→ Ex/y +q

Lε0ω, (C5)

and that this results in a change in the grand-canonical

energy of the system given by

∆U =Lq

2ω

(q

Lε0ω + 2Ex/y

). (C6)

Hence, given an arbitrary charge distribution, the lowest-energy harmonic mode that describes the charge distri-bution is an element of the set in Eq. (10). We thereforedefine a convention in which the harmonic mode is givenby Eq. (8), where the polarization component of the har-monic mode is an element of the set in Eq. (10) and thewinding component of the harmonic mode is given by Eq.(9).

Appendix D: Simulation details

The system was simulated using the MR algorithm onan L×L lattice of lattice spacing a = 1. One charge-hopsweep corresponded to picking a charge site at random,picking the x or y direction at random, then proposing acharge hop in the positive or negative direction (at ran-dom), repeating this 2N times (replacing each site / fieldbond after each proposal). One auxiliary-field sweep cor-responded to picking a charge site at random and propos-ing a field rotation around the site, repeating this Ntimes. One global sweep corresponded to proposing awinding update in the positive or negative (at random)x and y directions. For all simulations, we performedfive auxiliary-field sweeps per charge-hop sweep, and, forthose simulations that also employed the global update,we performed one global update per charge-hop sweep.One charge-hop sweep corresponds to one Monte Carlotime step.

The data sets in Sections III and IV were averaged overmultiple runs of 106 charge-hop sweeps per lattice site.The data set in Fig. 2 was averaged over 608 and 446runs between T = 1.15 and 1.45 with the global updateoff and on, respectively, over 384 runs between T = 1.5and 1.6, and over 256 runs between T = 1.65 and 1.75.

The L = 8 data set in Fig. 3 was averaged over 128(T = 0.1−1.1), 256 (T = 1.15−1.39;T = 1.41−1.44;T =1.46 − 1.49), 768 (T = 1.4;T = 1.45;T = 1.5 − 1.75),and 256 (T = 1.8 − 2.5) runs; the L = 16 data set wasaveraged over 128 (T = 0.1−1.1) and 256 (T = 1.15−2.5)runs; the L = 32 data set was averaged over 128 (T =0.1− 1.1), 256 (T = 1.15− 2.0), and 128 (T = 2.0− 2.5)runs; the L = 64 data set was averaged over 128 (T =0.1 − 1.1), 448 (T = 1.15 − 1.45), 384 (T = 1.5 − 1.6),256 (T = 1.65− 2.0), and 128 (T = 2.05− 2.5) runs.

We also simulated the L = 10, L = 20, and L = 40systems over small temperature ranges to calculate ad-ditional crossover points for Fig. 4: all data sets wereaveraged over 512 runs.

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