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arXiv:cond-mat/0209415 · Journal of Physics A Mathematical and General

Kosterlitz Thouless transition in three-state mixed Potts ferro antiferromagnets

Miguel Quartin, S. L. A. de Queiroz

Published Jan 15, 2003• 7 citations

Abstract

We study three-state Potts spins on a square lattice, in which all bonds are ferromagnetic along one of the lattice directions, and antiferromagnetic along the other. Numerical transfer-matrix methods are used on infinite strips of width L sites, 4 ⩽ L ⩽ 14. Based on the analysis of the ratio of scaled mass gaps (inverse correlation lengths) and scaled domain-wall free energies, we provide strong evidence that a critical (Kosterlitz–Thouless) phase is present, whose upper limit is, in our best estimate, T c = 0.29 ± 0.01. From an analysis of the (extremely anisotropic) nature of excitations below T c , we argue that the critical phase extends all the way down to T = 0. While domain walls parallel to the ferromagnetic direction are soft for the whole extent of the critical phase, those along the antiferromagnetic direction seem to undergo a softening transition at a finite temperature. Assuming a bulk correlation length varying, for T > T c , as ξ( T ) = a ξ exp[ b ξ ( T − T c ) −σ ], σ ≃ 1/2, we attempt finite-size scaling plots of our finite-width correlation lengths. Our best results are for T c = 0.50 ± 0.01. We propose a scenario in which such inconsistency is attributed to the extreme narrowness of the critical region.

DOI: 10.1088/0305-4470/36/4/307

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We study three-state Potts spins on a square lattice, in which all bonds are ferromagnetic along one of the lattice directions, and antiferromagnetic along the other. Numerical transfer-matrix methods are used on infinite strips of width L sites, 4 ⩽ L ⩽ 14. Based on the analysis of the ratio of scaled mass gaps (inverse correlation lengths) and scaled domain-wall free energies, we provide strong evidence that a critical (Kosterlitz–Thouless) phase is present, whose upper limit is, in our best estimate, T c = 0.29 ± 0.01. From an analysis of the (extremely anisotropic) nature of excitations below T c , we argue that the critical phase extends all the way down to T = 0. While domain walls parallel to the ferromagnetic direction are soft for the whole extent of the critical phase, those along the antiferromagnetic direction seem to undergo a softening transition at a finite temperature. Assuming a bulk correlation length varying, for T > T c , as ξ( T ) = a ξ exp[ b ξ ( T − T c ) −σ ], σ ≃ 1/2, we attempt finite-size scaling plots of our finite-width correlation lengths. Our best results are for T c = 0.50 ± 0.01. We propose a scenario in which such inconsistency is attributed to the extreme narrowness of the critical region.

We study phase transitions of the Potts model on the centered-triangular lattice with two types of couplings, namely, K between neighboring triangular sites, and J between the centered and the triangular sites. Results are obtained by means of a finite-size analysis based on numerical transfer-matrix calculations and Monte Carlo simulations. Our investigation covers the whole (K,J) phase diagram, but we find that most of the interesting physics applies to the antiferromagnetic case K<0, where the model is geometrically frustrated. In particular, we find that there are, for all finite J, two transitions when K is varied. Their critical properties are explored. In the limits J→±∞ we find algebraic phases with infinite-order transitions to the ferromagnetic phase.

We investigate the finite-temperature corrections to scaling in the three-state square-lattice Potts antiferromagnet, close to the critical point at T=0. Numerical diagonalization of the transfer matrix on semi-infinite strips of width L sites, 4 < or = L < or = 14, yields finite-size estimates of the corresponding scaled gaps, which are extrapolated to L --> infinity. Owing to the characteristics of the quantities under study, we argue that the natural variable to consider is x identical with L e(-2 beta). For the extrapolated scaled gaps we show that square-root corrections, in the variable x, are present, and provide estimates for the numerical values of the amplitudes of the first- and second-order correction terms, for both the first and second scaled gaps. We also calculate the third scaled gap of the transfer matrix spectrum at T=0, and find an extrapolated value of the decay-of-correlations exponent, eta(3)=2.00(1). This is at odds with earlier predictions, to the effect that the third relevant operator in the problem would give eta(P(stagg))=3, corresponding to the staggered polarization.

We investigate an antiferromagnetic triangular Ising model with anisotropic ferromagnetic interactions between further neighbors, originally proposed by Kitatani and Oguchi [J. Phys. Soc. Jpn. 57, 1344 (1988)]. The phase diagram as a function of temperature and the ratio between first- and second-neighbor interaction strengths is thoroughly examined. We search for a Kosterlitz-Thouless transition to a state with algebraic decay of correlations, calculating the correlation lengths on strips of width up to 15 sites by transfer-matrix methods. Phenomenological renormalization, conformal invariance arguments, the Roomany-Wyld approximation, and a direct analysis of the scaled mass gaps are used. Our results provide limited evidence that a Kosterlitz-Thouless phase is present. Alternative scenarios are discussed.

Finite-size corrections to scaling of critical correlation lengths and free energies of Ising and three-state Potts ferromagnets are analysed by numerical methods, on strips of width N sites of square, triangular and honeycomb lattices. Strong evidence is given that the amplitudes of the `analytical' correction terms, N -2 , are identically zero for triangular and honeycomb Ising systems. For Potts spins, our results are broadly consistent with this lattice-dependent pattern of cancellations, though for correlation lengths non-vanishing (albeit rather small) amplitudes cannot be entirely ruled out.

Using Monte Carlo simulations and finite-size scaling, we study a three-state Potts antiferromagnet on a layered square lattice with two and four layers:

{L}_{z}=2

and 4, respectively. As temperature decreases, the system develops quasi-long-range order via a Berezinskii-Kosterlitz-Thouless transition at finite temperature

{T}_{c1}

. For

{L}_{z}=4

, as temperature is further lowered, a long-range order breaking the

{Z}_{6}

symmetry develops at a second transition at

{T}_{c2}<{T}_{c1}

. The transition at

{T}_{c2}

is also Berezinskii-Kosterlitz-Thouless-like, but has a magnetic critical exponent $\ensuremath{\eta}=1/9$ instead of the conventional value $\ensuremath{\eta}=1/4$. The emergent

U(1)

symmetry is clearly demonstrated in the quasi-long-range ordered region ${T}_{c2}\ensuremath{\le}T\ensuremath{\le}{T}_{c1}$.

We study the phase transition of thin films in the three-dimensional XY universality class. To this end, we perform a Monte Carlo study of the improved two-component ϕ 4 model, the improved dynamically diluted XY model and the standard XY model on the simple cubic lattice. We study films of a thickness up to L 0 = 32 lattice spacings. In the short direction of the lattice, free boundary conditions are employed. Using a finite size scaling (FSS) method, proposed recently, we determine the transition temperature with high accuracy. The effectively two-dimensional finite size scaling behaviour of the Binder cumulant U 4 , the second-moment correlation length over the lattice size ξ 2nd / L , the ratio of the partition functions with anti-periodic and periodic boundary conditions Z a / Z p and the helicity modulus ϒ clearly confirm the Kosterlitz–Thouless nature of the transition. We analyse the scaling of the transition temperature with the thickness L 0 of the film. We compute the universal ratio of the thickness of the film L 0 and the transverse correlation length ξ T in the three-dimensional thermodynamic limit at the Kosterlitz–Thouless transition temperature of a film of thickness L 0 : [ L 0,KT /ξ T ] * = 1.595(7). This results can be compared with experimental results on thin films of 4 He near the λ-transition.