## Brazilian Journal of Physics

*On-line version* ISSN 1678-4448

### Braz. J. Phys. vol.34 no.2a São Paulo June 2004

#### http://dx.doi.org/10.1590/S0103-97332004000300013

**Critical behavior of the fully frustrated two dimensional XY model **

**A.B. Lima; B.V. Costa **

Departamento de Física ICEX, Universidade Federal de Minas Gerais, Caixa Postal 702, 30123-970 Belo Horizonte, MG, Brazil

**ABSTRACT**

Using Monte Carlo simulations we have investigated the critical behavior of the classical fully frustrated XY model in two dimensions in a square lattice. There are two phase transitions in the model, one of the Berezinskii-Kosterlitz-Thouless type and a *Z*_{2} transition at higher temperature. We show that the vortex anti-vortex density has a clear signature of the *Z*_{2} phase transition at exactly the percolation threshold.

**1 Introduction**

Classical continuous spin models with short range interactions in two dimensions are a prototype for systems which exhibit topological excitations [1]. It is well known that this kind of model undergoes a phase transition at a finite temperature *T _{BKT}*, from a high-temperature phase where the spin-spin correlations decay exponentially to a low temperature phase where they have a power-law decay. This phase transition is believed to be driven by a vortex-antivortex unbinding mechanism [2]. A vortex (antivortex) is a topological excitation in which spins on a closed path around the excitation have a positive (negative) chirality,

*f*

where q is the angle that the *XY* spin vector component makes with some fixed direction in the plane. The models we are interested in can be described by the following Hamiltonian

where is a spin vector at site *i, J _{i,j}* is an exchange coupling which is ferromagnetic in all lines in the

*x*direction and is alternately ferromagnetic and anti-ferromagnetic in the

*y*direction. The coupling distribution leads the ground state of the model to have a checkerboard pattern of plaquetes with positive (vortex) or negative (anti-vortex) chirality

*f*/p = ±1. Due to this symmetry the model has a

*Z*

_{2}transition at [3]. In a recent work we have shown that

*T*= 0.3655(5)

_{BKT}*J*and = 0.3690(3)

*J*for the

*XY*model [4]. The vortex density at

*T*= 0 is r = 1. Once the temperature grows, pairs vortex anti-vortex begin to annihilate so that we have a diluted Ising model to deal with. Using Monte Carlo simulation we have calculated the vortex anti-vortex density and the percolation probability for the model. The vortex anti-vortex density r is just the number of vortices divided by the lattice volume

*L*

^{2}[1]. The percolation probability

*P*, must be a step function: for r >< r

*,*

_{c}*P*= 0 and

*P*= 1 for r > r

_{c}, where r

*is the critical concentration.[5] We carried out simulations in square lattices of sizes*

_{c}*L*×

*L*with

*L*= 20, 40, 80 and 100. Each point in our simulation is the result of the average over 5 × 10

^{4}independent configurations. Fig. 1 shows the vortex anti-vortex density as a function of temperature.

We observe that close to the transitions there is a steep drop on the vortex density. The insert shows the derivative, *d*r/*dT*, for several lattice sizes. At some value *T _{L}* each curve presents a maximum. An extrapolation for

*L*® ¥ gives

*T*= 0.368(3)

_{L}*J*,which matches inside the error bars. At the Ising transition we expect r() = r

_{c}, where r

_{c}is to be identified with the percolation threshold. Using the percolation probability

*P*we obtain an estimate for the critical point as the intercept of the curves for different lattice sizes.

From Fig. 2 we get *T _{c}* = 0.365(5), in excellent agreement with the results above. In short, we have performed Monte Carlo simulation in the fully frustrated

*XY*model defined by equation 2. The model has two phase transitions, one

*BKT*and other of the Ising type. We have shown that the Ising transition can be obtained from the percolation probability which coincides with the inflection point of the vortex anti-vortex density as a function of temperature. One should notice that the temperatures

*T*= 0.3655(5)

_{BKT}*J*and

*T*= 0.365(5) seems to coincide and we are compelled to say that the

_{c}*BKT*and the Ising transition occur at the same temperature as suggested by earlier works.[6] However, we can not conclude this from our data, since due to the error bars

*T*matches both and

_{c}*T*. A more intensive simulation have to be done to decide about that.

_{BKT}

Financial support from the Brazilian agencies CNPq, CAPES, FAPEMIG and CIAM-02 49.0101/03-8 (CNPq) are gratefully acknowledged.

**References**

[1] J.E.R. Costa, D.P. Landau and B.V. Costa, Phys. Rev.B **57**, (1998)11510. [ Links ]

[2] J. Villan, J. Phys. C**10**, (1977)4793. [ Links ]

[3] S. Teitel and C. Jayaprakash, Phys. Rev. Lett. **51** (1983)1999. [ Links ]

[4] A.B. Lima and B.V. Costa, J. Magn. Magn. Mater. **263**, (2003)324-331. [ Links ]

[5] A. Coniglio, C.R. Nappi, F. Peruggi and L. Russo, Journal of Physics A:Mathematical and General **10**, (1977)205 and references therein. [ Links ]

[6] C. Ebner and D. Stroud, Phys. Rev. B **28** (1983)5053; [ Links ]M.Y. Choi and S. Doniach, Phys. Rev. B **31** (1985)4516. [ Links ]

Received on 8 August, 2004