arX

iv:c

ond-

mat

/010

1187

v1 [

cond

-mat

.sup

r-co

n] 1

2 Ja

n 20

01

Ubiquitous finite-size scaling features inI-V characteristics of various dynamicXY models in twodimensions

Kateryna Medvedyeva, Beom Jun Kim, and Petter MinnhagenDepartment of Theoretical Physics, Umea University, 901 87 Umea, Sweden

Two-dimensional (2D)XY model subject to three different types of dynamics, namely Monte Carlo, resis-tivity shunted junction (RSJ), and relaxational dynamics,is numerically simulated. From the comparisons of thecurrent-voltage (I-V ) characteristics, it is found that up to some constantsI-V curves at a given temperature areidentical to each other in a broad range of external currents. Simulations of the Villain model and the modified2D XY model allowing stronger thermal vortex fluctuations are also performed with RSJ type of dynamics.The finite-size scaling suggested in Medvedyevaet al. [Phys. Rev. B (in press)] is confirmed for all dynamicmodels used, implying that this finite-size scaling behaviors in the vicinity of the Kosterlitz-Thouless transitionare quite robust.

PACS numbers: 74.76.-w, 74.25.Fy, 74.40.+k, 75.40.GbKey words:XY model, current-voltage characteristics, dynamics, finite-size scaling

I. INTRODUCTION

The phase transition between the superconducting andnormal states in many two-dimensional (2D) systems is ofKosterlitz-Thouless (KT) type [1]. The thermally excited vor-tices in a large enough sample, interacting via a logarithmicpotential, are bound in neutral pairs below the KT transitiontemperatureTKT [1,2], and as the temperatureT is increasedacrossTKT from below these pairs start to unbind. The KTtransition has been observed in experiments on superconduct-ing films [3,4], 2D Josephson junction arrays [5], and cupratesuperconductors [6]. In these experiments, the current-voltage(I-V ) characteristics have been commonly measured to de-tect the transition. For small enough currents it has a powerlaw form V ∝ Ia (or equivalently,E ∝ Ja with the electricfield E and the current densityJ), where theI-V exponenta is known to have a universal value 3 precisely at the tran-sition [2]; For T < TKT one hasa > 3, whereasa = 1for T > TKT [7]. The dynamic critical exponentz which re-lates the relaxation timeτ to the vortex correlation lengthξ viaτ ∼ ξz , is connected witha through the relationa = 1+z [8].Consequently,z has the value 2 atTKT .

Indeed, the valuesz = 2 and a = 3 at the KT tran-sition have been confirmed in many numerical simulations,e.g., the lattice Coulomb gas with Monte Carlo dynamics [9],the Langevin-type molecular dynamics of Coulomb gas par-ticles [10], and the 2DXY model both with the resistivelyshunted junction dynamics (RSJD) and the relaxational dy-namics (RD) [11,12]. That is whyz ≈ 6 obtained by Piersonet al. in Ref. [13] at the resistive transition in many 2D sys-tems is very intriguing. We in Ref. [14] have re-analyzed theexperimentalI-V characteristics for an ultra thin YBCO sam-ple in Ref. [4] and compared with the 2D RSJD model. Thisled to the suggestion that a novel finite-size type scaling ef-fect of theI-V characteristics (which can possibly be causednot only by the actual finite size of the sample but also by afinite perpendicular penetration depth or a residual weak mag-netic field [2,14]) is responsible for the large value ofz ≈ 6in Ref. [13], but at the same time it was concluded that thisexponent has nothing to do with the true dynamic critical ex-

ponent which atTKT has value two [14]. Since similar con-clusions were also reached in Ref. [15], we believe that therenow seems to emerge some consensus [14,15], although thephysical mechanism causing this finite-size scaling behavioris still unclear.

In the present paper we search for a possible reason of thisfinite-size scaling behavior. We use the RSJD [11,12,16], theRD (often referred to as time-dependent Ginzburg-Landaudy-namics) [11,12], and the Monte Carlo dynamics (MCD) sim-ulations [9,17,18] to study theI-V characteristics of three dif-ferent models of 2D superconductors: the Villain model [19],theXY model in its original form, and theXY model modi-fied with p-type of potential (see Ref. [20] for details). Thesethree models differ from each other by the density of the ther-mally created vortices. We come to the conclusion that thequalitative features of the scaling suggested in Ref. [14] areubiquitous and independent of both the vortex density and thetype of dynamics. We also demonstrate that theI-V curvesobtained with different types of dynamics, up to constant scalefactors, coincide very well in a broad range of the external cur-rent density. This opens a possibility to use MCD simulation,which from a simulation point of view is more efficient thanRSJD or RD, to examine long-time dynamic properties of themodels.

The layout of the paper is as follows: In Sec. II we reca-pitulate the Hamiltonians for the generalized 2DXY modelwith thep-type of potential (the usualXYmodel is recoveredwhenp = 1) and the Villain model, and describe details ofthe dynamics used (RSJD, RD, and MCD). The results fromthe usualXYmodel withp = 1 subject to different dynamics(RSJD, RD, and MCD) are presented in Sec. III, while the re-sults from RSJD simulations applied to the different types ofmodels, i.e., the usualXY model withp = 1, theXY modelwith p = 2 and the Villain model, are described and analyzedin Sec. IV. We summarize and make final remarks in Sec. V.

II. MODELS AND DETAILS OF DYNAMICS

1

A. Model Hamiltonians

The 2DXY model defined on a square lattice, where eachlattice pointi is associated with the phaseθi of the supercon-ducting order parameter, is often used for studies of the KTtransition. The phase variables in this model interact via theHamiltonian with nearest neighbor coupling, which in the ab-sence of frustration is given by

H =∑

〈ij〉

U(φij = θi − θj), (1)

where〈ij〉 denotes sum over nearest neighbor pairs,φ is theangular difference between nearest neighbors, and the inter-action potentialU(φ) is written as

U(φ) ≡ EJ(1 − cosφ), (2)

with the Josephson coupling strengthEJ .The dominant characteristic physical features close to the

KT transition are associated with vortex pair fluctuations.Oneinteresting aspect is then how the density of the thermally ex-cited vortex fluctuation effects the critical properties. To studythis we generalize the interaction potential by using a param-eterp [20]:

Up(φ) ≡ 2EJ

[

1 − cos2p2

(

φ

2

)]

, (3)

whereUp=1(φ) corresponds to the potential of the usualXYmodel [see Eq. (2)]. The practical point with such gener-alization is that the vortex density increases with increasingp [20]. The variation of the parameterp can also change thenature of the transition: forp exceeding some maximum value(p > pmax ≈ 5) the type of the phase transition changes fromKT to the first order [20,21]. In the present paper we choosep = 2 which is well inside the KT transition region, yet islarge enough to ensure substantially more vortex fluctuationover a temperature region around the phase transition in com-parison with the usualp = 1 XY model given by Eq. (2).While theXY model with thep-type potential withp > 1has more vortices than the usualp = 1 XY model, we alsostudy the Villain model [19] which has less vortex-antivortexpairs [22]. The interaction potentialU(φ) in the Villain modelis given by

e−U(φ)/T ≡

∞∑

n=−∞

exp

[

−EJ

2T(2πn − φ)2

]

. (4)

To simulate the dynamic behaviors of these models we useseveral types of dynamics: RSJD, RD, and MCD. All thesedynamics should result in the same equilibrium static behav-iors if we apply them to the models with the same interactionpotential. However, dynamic properties of the systems can bedifferent. Of course, different types of dynamics have theirown advantages and disadvantages. The RSJD is constructedfrom the elementary Josephson relations for single Josephsonjunction that forms the array units, plus Kirchhoff’s current

conservation condition at each lattice site [11,12]. Therefore,this type of dynamics has a firm physical realization. On theother hand, RSJD is quite slow which leads to the limitationin the time scale one can probe in simulations. Although theRD [11,12,21] is much easier to implement than RSJD, it doesnot converge much faster than RSJD and it does not have asimilar direct physical realization as RSJD. However, a su-perconductor has been argued to have a RD type of dynam-ics rather than a RSJD [8,23]. The MCD simulations [18]are much faster than RSJD or RD, which allows one to in-vestigate dynamic behaviors in much longer time scale (onecan also study dynamic behaviors at much lower temperatureswith MCD). However, since there is no direct physical real-ization of the MCD in practice the applicability of this dy-namics to a specific physical system must then be explicitlydemonstrated. In the following discussions on the details ofthe different dynamics used, we focus on the original 2DXYmodel with the interaction potential in Eq. (2) since the exten-sions to a modified 2DXY model (3) and Villain model (4)are straightforward.

B. Dynamic models

In this section we briefly review the dynamical equationsof motion for RSJD, RD, and MCD, in the presence of thefluctuating twist boundary condition (FTBC) [11,18]. Weperform simulations of unfrustrated squareL × L latticeswith L = 6, 8, and 10 at various temperatures to mea-sure the voltage across the lattice as a function of the exter-nal current. Although the system sizes are relatively small,which is inevitable because of the low temperatures andthe small external currents used here, the FTBC has beenshown to be very efficient in reducing the artifact due tosmall system sizes [11,24], and reliable results can be estab-lished [11,12,18].

In the FTBC, the twist variable∆ = (∆x, ∆y) is in-troduced and the phase differenceφij on the bond(i, j) ischanged intoθi − θj − rij · ∆, with the unit vectorrij

from site i to site j, while the periodicity onθi is imposed:θi = θi+Lx = θi+Ly. The Hamiltonian of 2DL × L XYmodel under FTBC without external current has been intro-duced in Ref. [25], and is written as [compare with Eq. (1)]

H = −EJ

∑

〈ij〉

cos(φij ≡ θi − θj − rij · ∆), (5)

which later in Ref. [18] has been extended to the system in thepresence of an external current and written as

H = −EJ

∑

〈ij〉

cosφij +h

2eL2J∆x, (6)

whereJ the current density in thex direction.

2

1. RSJD and RD

We introduce first the RSJD equations of motion for phasevariables and twist variables, which are generated from thelocal (global) current conservation for the phase (twist) vari-ables (see Ref. [11] for details and discussions). The net cur-rentIij from sitei to sitej is the sum of the supercurrentIs

ij ,the normal resistive currentIn

ij , and the thermal noise cur-rent It

ij : Iij = Isij + In

ij + Itij . The supercurrent is given

by the Josephson current-phase relation,Isij = Ic sin φij ,

whereIc = 2eEJ/h is the critical current of the single junc-tion. The normal resistive current is given byIn

ij = Vij/r,whereVij is the potential difference across the junction, andr is the shunt resistance. Finally the thermal noise currentItij in the shunt at temperatureT satisfies〈It

ij〉 = 0 and〈It

ij(t)Itkl(0)〉 = (2kBT/r)δ(t)(δikδjl − δilδjk), where〈· · ·〉

is thermal average, andδ(t) andδij are Dirac and Kroneckerdelta, respectively. Using the current conservation law ateach site of the lattice together with the Josephson relationφij ≡ dφij/dt = 2eVij/h one can derive the RSJD equationsof motion for phase variables:

θi = −∑

j

Gij

∑

k

′(sin φjk + ηjk), (7)

where the primed summation is over the four nearest neigh-bors ofj, Gij is the lattice Green function for 2D square lat-tice, andηjk is the dimensionless thermal noise current de-fined byηjk ≡ It

jk/Ic. The time, the current, the distance,the energy, and the temperature are normalized in units ofh/2erIc, Ic, the lattice spacinga, the Josephson couplingstrengthEJ , andEJ/kB, respectively. In order to get a closedset of equations we further specify the dynamics of the twistvariable∆ from the condition of the global current conserva-tion that the summation of the all currents through the systemin each direction should vanish [11]:

∆x =1

L2

∑

〈ij〉x

sin φij + η∆x− J, (8)

∆y =1

L2

∑

〈ij〉y

sin φij + η∆y, (9)

where∑

〈ij〉xdenotes the summation over all nearest neigh-

bor links in thex direction, and we apply the external dccurrent with the current densityJ in the x direction. Here,the thermal noise termsη∆x

and η∆yobey the conditions

〈η∆x〉 = 〈η∆y

〉 = 〈η∆xη∆y

〉 = 0, and〈η∆x(t)η∆x

(0)〉 =

〈η∆y(t)η∆y

(0)〉 = (2T/L2)δ(t).In the RD, the equations of motion for the phase variables

are written as [11,12,26]

θi = −Γ∂H

∂θi+ ηi = −

∑

j

′sin φij + ηi, (10)

whereΓ is a dimensionless constant (we setΓ = 1 fromnow one), H is in Eq. (6), t is in units of h/ΓEJ , and

the thermal noiseηi(t) at site i satisfies〈ηi(t)〉 = 0 and〈ηi(t)ηj(0)〉 = 2Tδ(t)δij . The equation of motion for thetwist variables in the absence of an external current is of theform (see Ref. [11,12] for more details)

∆ = −1

L2

∂H

∂∆+ η∆, (11)

which is the same as Eqs. (8) and (9) for RSJD. Accordingly,to some extent the RD may be viewed as a simplified versionof the RSJD where the global current conservation is kept butthe local current conservation is relaxed.

Consequently, the equations for the phase variables are dif-ferent for RSJD and RD [Eqs. (7) and (10), respectively] whilethe same equations (8) and (9) apply to the twist variables forboth dynamics. These coupled equations of motion are dis-cretized in time with the time step∆t = 0.05 and∆t = 0.01for RSJD and RD, respectively, and numerically integratedusing the second order Runge-Kutta-Helfand-Greenside algo-rithm [27]. The voltage dropV across the system in thexdirection is written asV = −L∆x (see Ref. [11]) in unitsof Icr for RSJD and in units ofΓEJ/2e for RD, respectively.We measure the electric fieldE = 〈V/L〉t to obtainI-V char-acteristics, where〈· · ·〉t denotes the time average performedoverO(106) time steps for large currents for both RSJD andRD, andO(5 × 107) andO(109) steps for small currents forRSJD and RD, respectively.

2. MCD

The technique to simulate 2DXYmodel with MCD is basedon the Hamiltonian (6) and the standard Metropolis algo-rithm [28]. The one MC step, which we identify as a timeunit, is composed as follows [18]:

1. Pick one lattice site and try to rotate the phase angle atthe site by an amount randomly chosen in[−δθ, δθ] (wecall δθ the trial angle range). The twist variable∆ iskept constant during the update of the phase variables.

2. Compute the energy difference∆H before and after theabove try; If∆H < 0 or if e−∆H/T is greater than arandom number chosen on the interval[0, 1), accept thetrial move.

3. Repeat steps 1 and 2 for all the lattice sites to update thephase variables.

4. Update the fluctuating twist variables∆x in the similarway thatL∆x is tried to rotate within the angle rangeδ∆ with θi and∆y kept unchanged. (For convenience,we useδ∆ = δθ).

5. Compute the energy difference∆H before and after thetrial step 4 for∆x. Accept the step 4, if∆H < 0 oth-erwise accept it with probabilitye−∆H/T like in step 2.

6. Repeat steps 4 and 5 to update∆y.

3

In the MCD simulation the trial angleδθ = π/6 has beenchosen since it is sufficiently small in order to obtain the cor-rectI-V characteristics while it is big enough to make MCDmuch faster than the other dynamic methods [18]. The time-averaged electric field is obtained after equilibration from theaverages overO(109) (at large currents) toO(1010) (at smallcurrents) MC steps.

III. 2D XYMODEL SUBJECT TO DIFFERENT TYPES OFDYNAMICS

In this section we use three different types of dynamics, theRSJD, the RD, and the MCD to study dynamic behavior ofthe 2DXYmodel withp = 1 under the FTBC. The dynamicbehavior of the system can be obtained from the complex con-ductivity, the flux noise spectrum, as well as theI-V charac-teristics which is commonly measured in experiments. Wewill focus on theI-V characteristics in the present paper. Aspointed out in Sec. II the RD to some extent may be consid-ered as a simplified version of the RSJD. Thus from this pointof view it is perhaps not surprising that these two models (aswe will see) contain similar features of the vortex dynamics.In Ref. [29] from the study of a simple dynamic model of iso-lated magnetic particles in a uniform field it has been shownthat the actual dynamics of the model and MCD are in a goodagreement when the acceptance ratio of the Metropolis step islow enough. This implies that the MCD should give the sameI-V characteristics as the RSJD after an appropriately chosennormalization of time, when the trial angleδθ is sufficientlysmall (it was shown in Ref. [18] thatδθ = π/6 is sufficientlysmall). We will confirm this further in the present simulations.

In Fig. 1 we compareI-V characteristics in the form ofthe electric fieldE versus the current densityJ obtained fromRSJD, RD, and MCD simulations in the temperature range0.70 ≤ T ≤ 1.50 (the temperature interval is 0.05 if0.70 ≤T ≤ 1.00 and 0.10 otherwise) in log scales for the system sizeL = 8. In order to makeI-V data of RSJD and RD simula-tions coincide,E obtained with RD (ERD) is multiplied by atemperature-independent factor represented by the horizontalline in the inset of Fig. 1. Since the measured time-averagedelectric field is inversely proportional to the time scale for agiven dynamics, one can from the ratioERSJD/ERD inferthe correspondence between times of RD and RSJD. ¿Fromthis comparison, we find for0.70 ≤ T ≤ 1.00 that one unitof time in RD approximately corresponds to 0.526 time unitin RSJD, independent of the temperature. Note that theI-V curves corresponding to the temperatures exceeding 1.00are almost a straight lines. Therefore the collapse of data be-tween the different dynamics is trivial forT > 1.00. Alsothe I-V characteristics obtained from MCD can be made tocollapse on top of the corresponding curves for the RSJD andRD, as shown in Fig. 1. However, in this case the time scalefactor, describing how many RSJD time steps one MCD stepcorresponds to, depends on the temperature, as shown in theinset of Fig. 1, where this factor is shown to be a linear func-tion of temperature for system sizeL = 6, 8, and 10 up toT = 1.00. This is in accordance with the model studied in

Ref. [29], where the same linear behavior in terms of the tem-perature has been found. Thus for theI-V curves we havea precise relation between RSJD and MCD: For example, atT = 0.80 we get1 MCS =10.2×RSJD time unit. This opensa practical possibility to study some aspects of dynamic be-havior of theXY model by using MCD simulation, which isusually much more efficient than RSJD and RD.

We have shown in this section that the RSJD, the RD,and the MCD applied to the usualXY model gives basicallythe identicalI-V characteristics up to some constant factors.From this observation, one can conclude that the dynamic crit-ical behaviors of theXY model inferred from theI-V charac-teristics should be identical for all these dynamic models.Wein next section use the RSJD to study the Villain model andXY models withp = 2 and presume from the observation inthe present section that the conclusion drawn in Sec. IV forthe RSJD case should be also valid in the other dynamics (RDand MCD).

IV. VARIOUS XY MODELS SUBJECT TO RSJD

To study the critical behavior of the system in the vicinityof the transition one can use scaling relations. Fisher, Fisher,and Huse (FFH) in Ref. [8] proposed that the nonlinearI-Vcharacteristics in aD-dimensional superconductor scales as

E = JξD−2−zχ±(JξD−1/T ), (12)

whereξ andz are the correlation length and the dynamic crit-ical exponent, respectively, andχ± is the scaling functionabove (+) and below (−) the transition. Piersonet al. inRef. [13] have applied a variant of this FFH scaling approachto the I-V data for thin (D = 2) superconductors and su-perfluids and suggested a phase transition withz ≈ 6. InRef. [14] another scaling relation has been introduced and ithas been shown that a certain finite-size effect which is not in-cluded in the FFH scaling may have caused the largez in spiteof the fact that the finite size effect precludes the possibilityof a real phase transition. This finite-size scaling around andbelow the KT transition is given by the form (see Ref. [14] forthe details)

E

JR= hT

(((

JLgL(T )

)))

, (13)

whereR = limJ→0(E/J) ∝ L−z(T ) is a finite-size inducedresistance without external current,hT (0) = 1, h(x) ∝ xz(T )

for large x, and gL(T ) is a function of at mostT and Lsuch that a finite limit functiong∞(T ) exists in the large-Llimit. For small values of the variablex = JLgL(T ) theT -dependence of the scaling is absorbed in a functiongL(T ) foreach fixed sizeL giving rise to the scaling form

E

JR= h

(((

JLgL(T )

)))

. (14)

In Fig. 2 we demonstrate the existence of the finite-sizescaling given by Eq. (14) forL = 8 within the temperature

4

intervals0.70 ≤ T ≤ 1.10. The data are obtained for MCDand scaled by the appropriate factor so as to correspond toRSJD and RD (compare Fig. 1). Because of the correspon-dence between the three types of dynamics (see Sec. III), thisalso means that the existence of the finite-size scaling givenby Eq. (14) is insensitive to the choice of dynamics.

The FFH scaling given by Eq. (12) is correct only in thethermodynamic limitξ/L → 0. However, from a practicalpoint of view there is a connection between Pierson method,which is based on FFH scaling, and the finite-size scalingintroduced by Eq. (14). If we assume thatξ in Eq. (12) isproportional toR−α, whereα is T -independent constant, theconnection between the two different scaling approach be-comes:gL(T ) = ALR−α with AL being a constant whichmay depend onL. In Fig. 3 there are presented three differentfunctionsgL(T ) corresponding toL = 6, 8, and 10. Thesefunctions are determined from the condition that curves cor-responding to the different temperatures should collapse whenplotted asE/JR vsLJgL(T ). Since it is well established thatthe 2DXY model on the square lattice has the KT transitionat TKT ≈ 0.892 (Ref. [30]), Fig. 3 shows thatgL(T ) over alimited region in the vicinity of the KT transition is very wellrepresented by theR−α with α = 1/6 for all investigatedsystem sizes.

Next we demonstrate how the finite-size scaling given byEq. (14) works for theI-V data obtained by simulations of themodifiedXY model. These simulations are done with RSJD.Fig. 4(a) verifies that this scaling indeed exists for theXYmodel modified with the Villain type of potential introducedby Eq. (4). The inset of Fig. 4(a) shows the scaling functiongL(T ) determined by finding the best data collapse for smallvalues ofLJgL(T ). One can see that in the vicinity ofTKT ,which is approximately equals to1.4 for this model,gL(T )can be fitted byALR−α with α = 1/6. The data collapse inFig. 4(b) shows thatE/JR is only a function of the scalingvariableJLgL(T ) (whenJLgL(T ) is small enough) for theXY model modified withp-type of potential (Eq. (3)), wherep = 2. The inset shows the functiongL(T ) together withALR−α. SinceTKT ≈ 1.15 for this model one can again seethat the scaling functiongL(T ) in the vicinity of KT transitionis proportional toR−α with the same exponentα = 1/6 as forthe originalXY and Villain models.

The crucial difference between the Villain model, the usualXY model and thep = 2 XY model in the present con-text is the vortex density. The KT-transitions for these modelsoccur at the Coulomb gas temperaturesT CG

c = 0.23, 0.2,and 0.1, respectively (T CG = T

2π 〈U′′〉 (see Ref. [2])). Lower

T CGc means higher vortex density. Thus the finite-size scal-

ing property given by Eq. (14) appears to be independent ofvortex density.

V. DISCUSSION AND CONCLUSIONS

We have simulated 2DXY model with three types of dy-namics: RSJD, RD, and MCD. The main conclusion of thepaper is that the qualitative features of the finite-size scaling

given by Eq. (14) are independent of both the vortex densityand the type of dynamics. Therefore the finite-size scaling be-havior given by Eq. (14) of the finite-size induced tails of theI-V characteristics appears to be a robust feature. From thecomparisons of the current-voltage characteristics obtainedfor each type of dynamics we found that, up to some scalefactor, I-V curves at a given temperature are identical overa broad range of external currents. This makes it possible touse MCD simulations, which are more computer efficient thanRSJD and RD simulations, to obtainI-V curves correspond-ing to both RSJD and RD dynamics.

The phase transition for the 2DXY -type models are of KTtype with z = 2. This raises the intriguing question of theorigin of the largez ≈ 6 obtained by Piersonet al. [13]. InRef. [14] it was argued that the Pierson scaling in relation tothe finite-size scaling given by Eq. (14) corresponds to theproportionalitygL(T ) ∼ R−α where1/α is the exponentwhich corresponds to the “z” obtained by the Pierson scal-ing. The reason for the existence of this scaling like behavioris still unclear.

In the present paper we have shown that, within the class of2D XY -type models studied, a value1/α ≈ 6 is obtained in-dependently of the type of dynamics, as well as, of the vortexdensity. The origin of this seemingly robust behavior callsforfurther investigations.

ACKNOWLEDGMENTS

This work was supported by the Swedish Natural ResearchCouncil through Contract No. FU 04040-332.

[1] J.M. Kosterlitz and D.J. Thouless, J. Phys. C6, 1181 (1973);V.L. Berezinskii, Zh. Eksp. Teor. Fiz.61, 1144 (1972) [Sov.Phys. JETP34, 610 (1972)].

[2] For a general review see, e.g., P. Minnhagen, Rev. Mod. Phys.59, 1001 (1987); for connections to high-Tc superconduc-tors see, e.g., P. Minnhagen, inModels and Phenomenologyfor Conventional and High-Temperature Superconductors, Pro-ceedings of the International School of Physics, “Enrico Fermi”Course CXXXVI (IOS Press, Amsterdam, 1998), p. 451.

[3] A.M. Kadin, K.Epstein, and A.M. Goldman, Phys. Rev. B27,6691 (1983).

[4] J.M. Repaci, C. Kwon, Q. Li, X. Jiang, T. Venkatessan, R.E.Glover, C.J. Lobb, and R.S. Newrock, Phys. Rev. B54, R9674(1996).

[5] S.T. Herbert, Y. Jun, R.S. Newrock, C.J. Lobb, K. Ravindran,H.-K. Shin, D.B. Mast, and E. Elhamri, Phys. Rev. B57, 1154(1998).

[6] See, for instance, I.G. Gorlova, and Yu.I. Latyshev, Physica C193, 47 (1992); N.-C. Yeh and C.C. Tsuei, Phys. Rev. B39,9708 (1989); S. Martin, A.T. Fiory, R.M. Fleming, G.P. Es-pinosa and A.S.Cooper, Phys. Rev. Lett.62, 677 (1989); Q.Y.Ying and H.S. Kwok, Phys. Rev. B42, 2242 (1990).

5

[7] D.R. Nelson and J.M. Kosterlitz, Phys. Rev. Lett.39, 1201(1977).

[8] D.S. Fisher, M.P.A. Fisher, and D.A. Huse, Phys. Rev. B43,130 (1991).

[9] J.-R. Lee and S. Teitel, Phys. Rev. B50, 3149 (1994).[10] K. Holmlund and P. Minnhagen, Phys. Rev. B54, 523 (1996).[11] B.J. Kim, P. Minnhagen, and P. Olsson, Phys. Rev. B59, 11506

(1999).[12] L.M. Jensen, B.J. Kim, and P. Minnhagen, Phys. Rev. B61,

15412 (2000).[13] S.W. Pierson, M. Friesen, S.M. Ammirata, J.C. Hunnicutt, and

L.A. Gorham, Phys. Rev. B60, 1309 (1999); S.M. Ammirata,M. Friesen, S.W. Pierson, L.A. Gorham, J.C. Hunnicutt, M.L.Trawick, and C.D. Keener, Physica C313, 225 (1999); S.W.Pierson and M. Friesen, Physica B284, 610 (2000).

[14] K. Medvedyeva, B.J. Kim, and P. Minnhagen, Phys. Rev. B (inpress).

[15] L. Colonna-Romano, S.W. Pierson, and M. Friesen (unpub-lished).

[16] M.V. Simkin and J.M. Kosterlitz, Phys. Rev. B55, 11646(1997).

[17] H. Weber, M. Wallin, and H. Jensen, Phys. Rev. B53, 8566(1996).

[18] B.J. Kim, Phys. Rev. B (in press).[19] J. Villain, J. Phys. C10, 1717 (1977); J.V. Jose, L.P. Kadanoff,

S. Kirkpatrick, and D.R. Nelson, Phys. Rev. B16, 1217 (1977).[20] E. Domani, M. Schick, and R. Swendsen , Phys. Rev. Lett.52,

1535 (1984).[21] A. Jonsson, P. Minnhagen, and M. Nylen, Phys. Rev. Lett.70,

1327 (1993).[22] P. Olsson, private communication.[23] R.A. Wickham and A.T. Dorsey, Phys. Rev. B61, 6945 (2000).[24] M.Y. Choi, G.S. Jeon, and M. Yoon, Phys. Rev. B62, 5357

(2000).[25] P. Olsson, Phys. Rev. B46, 14 598 (1992);52, 4526 (1995);

Ph.D.thesis, Umea University, 1992.[26] J. Houlrik, A. Jonsson, and P. Minnhagen, Phys. Rev. B50,

3953 (1994).[27] See, e.g., G. G. Batrouni, G. R. Katz, A. S. Kronfeld, G. P.

Lepage, B. Svetitsky, and K. G. Wilson, Phys. Rev. D32, 2736(1985) and references therein.

[28] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H.Teller, and E. Teller, J. Chem. Phys.21, 1087(1953).

[29] U. Nowak, R.W. Chantrell, and E.C. Kennedy, Phys. Rev. Lett.84, 163 (2000).

[30] P. Olsson, Phys. Rev. B52, 4511 (1995).

6

10-5

10-3

10-1

10

0.01 0.1 1

E

J

RSJRDMC

5

25

0 1.2

fact

or

T

FIG. 1. Comparisons of theI-V characteristics for the 2DXYmodel with MC, RSJ, and RD dynamics on a square lattice with thefinite sizeL = 8 plotted asE = V/L againstJ = I/L in logscales at temperaturesT = 1.5, 1.3, 1.2, 1.1, 1.0, 0.95, 0.90, 0.85,0.80, 0.75, and 0.70 (from top to bottom). Multiplication bya fac-tor (presented in the inset), which in the case of MCD dependsontemperature, makes the curves coincide in a broad range of externalcurrent density. The straight line from the origin in the inset showsthat the factor in the MCD case is a linear function of temperaturefor system sizes,L = 6, 8, 10 in accordance with expectation [29].However, for higherT there is a deviation. The horizontal full line inthe inset shows that the factor between the RSJD and RDI-V curvesis independent ofT (the factor is multiplied by10 in the inset).

0.1

1

10

100

1000

0.01 0.1 1 10 100

E/J

R

LJgL(T)

FIG. 2. The finite-size scaling of the current-voltage characteris-tics, E/JR vs LJgL(T ), given by Eq. (14). The functiongL(T )is determined from data collapse for small values ofLJgL(T ). Thedata was obtained from MCD and was scaled with aT -dependentfactor so as to correspond to RSJD and RD; system sizeL = 8 and1.10 ≤ T ≤ 0.70 where each curve corresponds to a fixedT .

2

4

6

8

0.7 0.8 0.9 1 1.1

g L(T

)

T

gL(T),L=6

0.73*R-1/6,L=6gL(T),L=8

1.02*R-1/6,L=81.5*gL(T),L=10

1.4*R-1/6,L=10

FIG. 3. The functiongL(T ) determined forI-V characteristicswith sizeL = 6 (open squares),L = 8 (open circles),L = 10(open triangles) obtained as in Fig. 2. One notes that the functiongL for these sizes over a limitedT interval around KT-transition (atT ≈ 0.90) is well approximated bygL ∝ R−α with α ≈ 1/6. Thedata forL = 10 functiongL(T ) as well asALR−α are multipliedby 1.5 for convenience.

1

100

0.01 0.1 1 10

E/J

R

LJgL(T)

(a)

1

2

1.2 1.6

g L(T

)

T

10

100

0.01 0.1 1 10

E/J

R

LJgL(T)

(b)

1

2

1 1.3

g L(T

)

T

FIG. 4. Existence of the scaling in the form Eq. (14) for modified2D XY models corresponding to different vortex density. Systemswith sizeL = 8 have been simulated with RSJD. The finite-size scal-ing of theI-V characteristics obtained for 2DXY model with theVillain potential in the temperature range1.70 ≤ T ≤ 1.15 is shownin (a) whereas (b) shows the same thing for thep = 2-potential for1.30 ≤ T ≤ 0.95. The inset on both (a) and (b) shows the functiongL(T ) determined from the condition of the best data collapse. It isalso shown there that the functiongL for both cases over a limitedTinterval is well approximated bygL ∝ R−α with α ≈ 1/6. In (a)gL ∝ 0.76R−1/6 and in (b)gL ∝ 1.04R−1/6.

7