Ž .Physica C 337 2000 297–301www.elsevier.nlrlocaterphysc

Phase transition in the frustrated xy spin model

Hui Zheng), Bing-Lin GuCentre for AdÕanced Study and Department of Physics, Tsinghua UniÕersity, 100084 Beijing, People’s Republic of China

Abstract

Ž .Since the high-T superconductor HTSC was discovered in 1986, research in this field has always been a focus ofc

condensed matter physics. Some physical characteristics of the HTSC, a type II anisotropic superconductor, are greatlydetermined by vortex line statistical mechanics. The three major factors for deciding the characteristics of vortex linesystems are the important thermal fluctuation, the anisotropy, and the disorder. A currently predominant framework in probeof this problem is the XY spin model obtained from the ‘‘phase only’’ simplification of the complex superconductor order

Ž .parameter. In our research, we used the frustrated XY FXY spin model for studying the HTSC, where the frustration isŽ .produced by an external transverse magnetic field. As the HTSC has a quasi two-dimensional 2D anisotropic structure, we

studied the 2D XY spin model. We adopted the filling factor fs1r3 which we regard as representative. Some extensiveMonte Carlo simulations with finite-size scaling analysis strongly suggest that the critical exponents of the phase transitionare those given by the pure 2D Ising model. The fluctuation of domain walls of vortex determines the nature of the phasetransition. q 2000 Elsevier Science B.V. All rights reserved.

Keywords: Magnetic system; Frustration; Superconductor

1. Introduction

Ž .The two-dimensional 2D XY model has beenstudied quite extensively. Application of this modelto a number of physical systems including supercon-

w x w xductor 1,2 and superfluid films 3 in general givesresults in agreement with the experiments.

Ž .The 2D frustrated XY FXY model on a squarelattice is defined by Hamiltonian:

H sy Jcos u yu yA , 1Ž .Ž .ÝX Y i j i j² :ij

where the indices i and j numerate the nearest-Ž .neighbor lattice sites, u ypFu -p is the phasei i

) Corresponding author. Tel.: q86-10-627-899-75; fax: q85-10-627-818-86.

Ž .E-mail address: hzheng@phys.tsinghua.edu.cn H. Zheng .

™ ™jŽ .on site i, and A s 2prf H APd l is the integrali j 0 i

of the potential vector from site i to site j, with f0

being the flux quantum. The discrete curl ÝA isi jŽ .equal to 2p f mod 2p for each plaquette.

The 2D FXY model can also serve as a model forthe Josephson junction arrays, which have been stud-

w xied using the mean field analysis 4,5 , the varia-w xtional approach 6 and the numerical simulation

w x5,7,8 .In this paper, we focused on the case fs1r3. It

is a representative situation having ‘‘stair’’ structurein the low temperature phase. We first briefly applythe mean field approximation to this problem. Thesymmetries of the ‘‘stair’’ structure can be clearlyshown by this approach.

One of important methods for FXY model re-search is the Monte Carlo simulation. The essential

0921-4534r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0921-4534 00 00121-0

( )H. Zheng, B.-L. GurPhysica C 337 2000 297–301298

problem of simulations for frustrated models is slowcritical dynamics, so it is very difficult to have aclear conclusion in the critical region. Some exten-sive simulations had been carried out for the FXY

w xfs1r3 model 10–13 . By measuring the XY helic-ity module and the Ising and Potts order parameters,some authors found several phase transitions. Weenvisage only one transition. We tried to explain thispoint by analyzing their data. The nature of the phase

w xtransition is also not clear. Some detailed studies 13show one Ising type transition according to the mea-

Ž .surement of the Ising order Z parameter fluctua-2

tion. Our idea is that if we could also obtain the Isingtype critical exponents by measuring the Potts pa-

Ž .rameter fluctuation Z , we can nearly conclude the3

one Ising type transition scenario.

2. Mean field approximation

The mean field approximation is often used as thefirst approach for studying the symmetries of sys-tems and their dependency on temperature. It is notevident that the mean field calculation always givesout the correct phase diagram because it neglects thefluctuation, especially in low space dimensions; butthe experience shows that this approach often welldescribes the symmetries of phases.

In this work, we first apply the mean field ap-proach to FXY fs1r3 model through the Hub-bard–Stratanovich transformation.

We rewrite Hamiltonian in a matrix form:

˜T ˆ ˜Hsy S J S 2Ž .Ý i i j j² :i , j

˜ TŽ .with S s cos u , sin u , andi i i

Jcos A ql yJsin Ai j i jJ s . 3Ž .i j Jsin A Jcos A qlž /i j i j

Using the Hubbard–Stratanovich transformation,

˜T ˆ ˜exp b S J SÝ i i j jž /² :i , j

˜1 Dhis H '2pˆ(detb J

=y1T T˜ ˆ ˜ ˜ ˜exp y h b J h qb h S 4Ž .Ž .Ý Ýi j i ii j

² : ii , j

˜ ˆ ˜and h sbÝ J C , the partition function is:i j i j j

˜DCi ˜ZZs exp ybF C 5Ž .� 4ŁH ž /i'ž /2pi

with

˜ T ˆ ˜Fs C J CÝ i i j j² :i , j

ˆ ˜< <yT log I b J C q f T , 6Ž . Ž .Ý Ý0 i j jž /i j

Ž .where f T is a function of temperature.We obtained an equivalent system in this way,

then we searched the saddle point of F. The above isthe ‘‘saddle point’’ formalism of the mean fieldmethod.

We used a Monte Carlo algorithm to minimize Fon a square lattice with the periodic boundary condi-tion. Of course the efficiency of the minimizationdepends on the system size NsL2. We have chosenLs3 for the following reason: the fundamental stateof the system FXY fs1r3 has a periodic structure

� Žwith periodicity 3=3 in representation of mod u2p i.4yu yA , and the important spin wave fluctuationj i j

w xmodes also have periodicity 3=3 9 .� Ž .4We found that the spin orientations H T arei

independent of temperature. We have also studiedthe solution for Ls6, and found the same result.

In representation of the gauge invariance phase� Ž .4mod u y u y A , we can clearly see the2p i j i j

Ž .‘‘stair’’ structure Fig. 1 . The stairs persist in adiagonal direction of the square lattice, and they areperiodically placed in this direction with periodicity3.

Ž . � 04Fig. 1. FXY f s1r3 model. a Configuration of the phases u .iŽ . � 0b Configuration of the gauge invariant difference phases u yi

0 4u y A .j i j

( )H. Zheng, B.-L. GurPhysica C 337 2000 297–301 299

Ž Ž . Ž ..Fig. 2. Solution of the mean field equation Eqs. 7 and 8 form and m .0 1

There are three different types of sites for thespins: centers and two sides of the stairs. Consider-ing the symmetry of the system, we can obtain˜ 0< <C sm for the spins on the centers of the stairs,i 0

˜ 0< <and C sm for the others.i 1

It is enough to solve m and m . The equations0 1

for m and m are:0 1

m sR b J 2lm q2m , 7Ž . Ž .Ž .0 0 1

m sR b J m q2 lq1 m , 8Ž . Ž .Ž .Ž .1 0 1

Ž . Ž . Ž . Ž .where R x s I x rI x , and I x is the modi-1 0 n

fied Bessel function of order n.We put ls0. The numerical solution showed a

phase transition of second order, i.e., m , m ™00 1y Ž .when T™T Fig. 2 .c

Ž . Ž .With the asymptotic behavior of R x : R x ;'3 q1

Ž .Ž .xr2 x™0 , we have T s J.c 2The mean field calculation gives several impor-

tant indications:

Ž .i the low temperature phase has a stair structure,Ž .ii there is only one phase transition,Ž .iii the transition is continue.

3. Monte Carlo simulation

The FXY system fs1r3 shows us a ‘‘stair’’Ž .structure Fig. 1 . One of its remarkable characters is

anisotropic, i.e., the equivalence of two diagonaldirections of the square lattice is broken. Ising pa-rameter can be introduced because of the anisotropy,and 3-Potts parameter can be introduced because ofperiodicity 3 in the orthogonal direction of stairs.

The definitions of Ising and Potts parameters arew xbased on ‘‘vortex-charge’’ configurations 12 . In

this paper, we used some new definitions for the twoparameters. They seem to be simpler, and the equiva-lence has been confirmed.

We considered all neighboring vortex pairs indiagonal directions of the square lattice, and drewsegments between these pairs, then we counted num-bers of different types of segments.

There are six segment types, n , n , n , n , n ,1 2 3 4 5Ž .n , according to the orientation two possibilities6

Ž .and the position mod 3 of a segment.We define the Ising parameter as:

3² < <:p s n qn qn y n qn qn ,Ž . Ž .Ising 1 2 3 4 5 6N

9Ž .

²:where means thermal average. We made a countevery 100 Monte Carlo steps in the simulation. ThePotts parameter is defined as:

3p s max max 3n¦ �ŽPotts ig �1 ,2,34 i2 N

y n qn qn , max 3nŽ . Ž. � 41 2 3 ig 4,5,6 i

y n qn qn , 10Ž . Ž .;4.4 5 6

i.e., we define a ‘‘Potts’’ parameter for each classŽ .diagonal direction , and take the larger one as thetrue Potts parameter.

As a comparison, we also define an additionalquantity:

3Xp s min max 3n¦ �ŽPotts ig �1 ,2,34 i2 N

y n qn qn , max 3nŽ . Ž. � 41 2 3 ig 4,5,6 i

y n qn qn . 11Ž . Ž .;4.4 5 6

We found that this quantity has a finite peak atT , but it tends to 0 when N™`. It means that therec

is no Potts order in the non-preferential diagonalw xdirection 13 .

( )H. Zheng, B.-L. GurPhysica C 337 2000 297–301300

For studying the superconductor phase coherence,we considered the helicity module g . The componentg is defined as the second derivation of freex x

energy in relation to a distortion in direction x. Forthe FXY spin model, g is expressed as:x x

1 2g s J x yx cos u yu yAŽ . Ž .Ýx x i j i j i j¦ ;2L ² :i , j

by J x yxŽ .Ý i j½2¦L ² :i , j

=

2

sin u yu yA , 12Ž .Ž .i j i j 5;² :where i, j represent two nearest neighboring sites.

Some details of the Monte Carlo procedure usedin the work are as follows.

Because the calculation of functions sin and cosŽ x y.costs time considerably, we used the form s ,s toi i

represent the orientation of spin i. We randomlychose a spin, and a number c in a uniform intervalw xye ,e . The relation between c and proposed Du isi

2'sin Du sc, so cosDu s 1yc . In this way, wei iŽ . Ž .avoided the calculation of the type arc sin– arc cos.

The proposition of the angle change Du would bei

accepted or refused according to the standardw xMetropolis algorithm 15 .

To compare with existing simulation results, wehave calculated the susceptibility and the Binder

w xcumulant 15 of the Potts parameter, but found theIsing class universality of its critical exponents. Thefluctuation of domain walls of vortex determines the

w xnature of the phase transition 13 .Some details of our work are presented as follow-

ing.We gave first here a remark on the conclusion

about one phase transition. Some extensive simula-w xtions 13 also gave the one phase transition scenario.

However, some others showed several transitions.We think that an important possible reason comes

Ž .from the Weber–Minnhagen W–M scaling relationw x X Y14 used for determining T in the later works.c

The validity of the W–M scaling relation for theFXY system remains to be checked. BecauseŽ Ž X Y -.. X Yg T rT G2rp is always satisfied in 2D XYc c

systems, we traced the line ys2rp T to cross

Ž . w xg T , L obtained from Ref. 12 , and extrapolatedx x

the intersection points by a simple law relation. Wefinally found T X Y s0.217J. It is different with thec

value obtained by using the W–M scaling relationw x12 .

The error estimation for critical exponents is oftendelicate. In the following, we have not given statisti-cal errors at first, but gave the final estimation in theend combining all data.

ŽThe Binder cumulant is defined as U s1r2 3yL² 4: ² 4:2p r p , where p is the Potts parameter. Forsystems of enough large sizes, this quantity is inde-

w xpendent of the system size L at TsT 15 . We havecŽ .calculated U T , L for Ls12, 24, 36, and found

T s0.22.cŽ . ŽŽWith the scaling relation U T , L s u T y

. 1rn .T L , we found that T s0.222, 1rns0.85.c cŽ .We have also traced p T , L . With the scalingPotts

yb rn ŽŽ . 1rn .relation p sL p TyT L , we foundPotts c

that T s0.22 J, 1rns1.1, brns0.20, so bfc

0.18.The susceptibility x associating to the Potts pa-

rameter is calculated. With the help of the scalingŽ . g rn ŽŽ . 1rn .relation x T , L sL x TyT L , we foundc

that T s0.21, 1rns0.8, grns1.37, so gf1.7cŽ .Fig. 3 .

Some authors found that g satisfies the scalingx xw xrelation of the order parameters 13 , i.e.,

g T , L sLyb rn y TyT L1rn . 13Ž . Ž . Ž .Ž .x x c

It seems that this point is confirmed well in ourŽ .calculation. From Eq. 13 , we found that T s0.217,cŽ .1rns1.0, brns0.13 Fig. 4 .

Fig. 3. The susceptibility of the Potts parameter for systems ofsizes Ls12, 24, 36.

( )H. Zheng, B.-L. GurPhysica C 337 2000 297–301 301

Fig. 4. The helicity module for systems of sizes Ls12, 24, 36.

Using the hyperscaling relation 2yhsgrn , weobtain h s1r4. The index ‘‘chiral’’ means thatchiral

h is defined for chiral variables. h is definedchiral X Yw xfor XY spin variables. The data of Lee and Lee 12

Ž .shows that h TrJs0.217 s0.25, so we getX Y

h sh at the transition temperature.chiral X Y

4. Conclusion

The FXY fs1r3 model has been studied by themean field approach and the Monte Carlo simulation.A ‘‘stair’’ structure is found in the low temperaturephase. Some new definitions for Ising and Pottsorder parameters have been made. We have ex-plained the one phase transition conclusion, andcalculated the critical exponents corresponding to the

Ž .Potts order fluctuation, and found that T s0.22 1 ,cŽ . Ž .ns1.0 2 , bs0.13 5 and gf1.7, which are in

agreement with Ising transition critical exponents. AtŽ . Ž .last, n T sh T s1r4 is also obtained.chiral c X Y c

Acknowledgements

The authors wish to acknowledge Z.L. Cao, W.H.Duan, J.S. Liu and X.B. Wang for their help inpreparing the manuscript. H.Z. would like also tothank M. Gabay for helpful discussions.

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