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arXiv:1410.0822 · Physical Review B

Reentrance of Berezinskii-Kosterlitz-Thouless-like transitions in a three-state Potts antiferromagnetic thin film

Chengxiang Ding, Wenan Guo, Youjin Deng

Published Oct 27, 2014• 10 citations
arXivHTMLPDF

Abstract

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 ${T}_{c2}\ensuremath{\le}T\ensuremath{\le}{T}_{c1}$.

DOI
10.1103/physrevb.90.134420

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We study the q-state Potts model on the simple cubic lattice with ferromagnetic interactions along one lattice direction, and antiferromagnetic interactions along the two other directions. As the temperature T decreases, the system undergoes a second-order phase transition that fits in the universality class of the 3D O(n) model with n=q-1. This conclusion is based on the estimated critical exponents, and histograms of the order parameter. At even smaller T we find, for q=4 and 5, a first-order transition to a phase with a different type of long-range order. This long-range order dissolves at T=0, and the system effectively reduces to a disordered two-dimensional Potts antiferromagnet. These results are obtained by means of Monte Carlo simulations and finite-size scaling.
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We study the triangular lattice Ising model with a finite number of vertically stacked layers. By combining analytical and numerical results, we demonstrate a low temperature reentrance of two Berezinskii-Kosterlitz-Thouless transitions that results in an extended disordered regime down to T=0. We introduce an effective theory that quantitatively describes the low temperature physics emerging from the extensively degenerate ground state manifold and explain the global qualitative features of the phase diagram by mapping the classical spin model into the single-layer quantum Ising model.
Abstract In this paper, we thoroughly examined the Berezinskii-Kosterlitz-Thouless (BKT) phase transition in the two-dimensional XY model on the honeycomb lattice. To address its thermodynamical behavior, we combined standard numerical Monte Carlo simulations with the simulated annealing (SA) protocol and entropic simulations based on the Wang-Landau (WL) algorithm. The transition temperature was determined using the second (ϒ) and fourth-order (ϒ4) helicity modulus as the order parameter. Our best finite-size scaling analysis suggests T BKT = 0.575(8) from SA and T BKT = 0.576(3) from WL. These values deviate significantly from the expected theoretical value of 1 / 2 . We believe that this discrepancy suggests that the theoretical assumptions regarding the analytical calculation may need to be revisited. Additionally, we calculated the vortex density and the formation energy of the vortex-antivortex pairs, where the obtained vortex formation energy is 2μ = 5.80(12). Upon comparison with the square lattice, our results support the notion of instability of the honeycomb lattice to support the spin long-range order and give additional backing to the critical behavior we found.
The frustrated antiferromagnetic XY model on the triangular lattice is a simple model describing the critical phenomena of magnetic systems. By using the Monte Carlo simulation method associated with the Metropolis algorithm, we investigate phase transitions of the system at critical temperatures and near critical temperatures. To achieve a scaling form of important parameters, the finite size effect must be taken into account. Our results show clearly that the broken symmetries of the system occur at two separate critical temperatures, the Kosterlitz-Thouless temperature and the Ising-like temperature (TI). However these temperatures lie close to each other about 0.6% compared with the lower temperature. At TI, the reflection symmetry breaking transition occurs and it is known as the Ising-like transition. While at the lower temperature, the rotational symmetry breaking transition occurs and it is known as the Kosterlitz-Thouless transition. Moreover, we found the asymptotic behavior of the correlation length of the Ising-like transition with the critical exponent, ν = 0.74. Taken into account the finite size effect, this value approaches 0.83 which differs from the value, ν = 1, for the Ising universality class. It is likely that the Ising-like transition may not belong to the Ising universality class. Also the other parameters such as the energy of the system, the order parameters, the specific heat, the magnetic susceptibility and the correlation functions have been calculated and their properties near the critical temperatures and at the critical temperatures have been studied.
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.