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arXiv:2307.10310

Simulation of Kosterlitz-Thouless (KT) Transition with Classical Monte-Carlo Simulation

Nepal Banerjee

Published Jul 19, 2023• 0 citations
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Abstract

Spontaneous symmetry breaking of 2D isotropic Heisenberg magnet is restricted by Mermin-Wagner theorem at any finite temperature in presence of short-range exchange interaction.Kosterlitz and Thouless using XY spin model showed that how an order state could developed in 2D spin system in presence of short range isotropic interaction.Very recent discovery of several van der waals magnet revised and redefined our understanding on 2D Heisenberg magnet and its ground state properties.After a rigorous and careful study of several 2D magnetic material we have realized from both experimentally and numerically that the finite size of a 2D system has great impact on the ground state symmetry breaking.Because of that finite size effect more often an anisotropic residual magnetic moment is generated and trigger the spontaneous symmetry breaking at finite temperature(T) and even only presence of short-range interaction we observed the phase transition of that Heisenberg spin system.In this present work we have shown the basic role of finite size,anisotropy during the symmetry breaking of 2D Heisenberg XY magnet.Here we have simulated Kosterlitz-Thouless transition using classical Monte-carlo simulation and study the effect of anisotropy during the phase transition.We presented the behaviour of different thermodynamic properties of 2D XY spin model system during the Kosterlitz-Thouless(KT) transition.The generic characteristic of KT transition which make it distinct from other critical phenomena is that the peak of heat capacity is not diverging with increase of system size rather peak is decreasing with the increasing of system size near at transition temperature.Here we are observing that behaviour in our present simulation and that specific behaviour help us for classifying the present transition as Kosterlitz-Thouless(KT) transition.

DOI
10.48550/arxiv.2307.10310

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Spontaneous symmetry breaking of 2D isotropic Heisenberg magnet is restricted by Mermin-Wagner theorem at any finite temperature in presence of short-range exchange interaction.Kosterlitz and Thouless using XY spin model showed that how an order state could developed in 2D spin system in presence of short range isotropic interaction.Very recent discovery of several van der waals magnet revised and redefined our understanding on 2D Heisenberg magnet and its ground state properties.After a rigorous and careful study of several 2D magnetic material we have realized from both experimentally and numerically that the finite size of a 2D system has great impact on the ground state symmetry breaking.Because of that finite size effect more often an anisotropic residual magnetic moment is generated and trigger the spontaneous symmetry breaking at finite temperature(T) and even only presence of short-range interaction we observed the phase transition of that Heisenberg spin system.In this present work we have shown the basic role of finite size,anisotropy during the symmetry breaking of 2D Heisenberg XY magnet.Here we have simulated Kosterlitz-Thouless transition using classical Monte-carlo simulation and study the effect of anisotropy during the phase transition.We presented the behaviour of different thermodynamic properties of 2D XY spin model system during the Kosterlitz-Thouless(KT) transition.The generic characteristic of KT transition which make it distinct from other critical phenomena is that the peak of heat capacity is not diverging with increase of system size rather peak is decreasing with the increasing of system size near at transition temperature.Here we are observing that behaviour in our present simulation and that specific behaviour help us for classifying the present transition as Kosterlitz-Thouless(KT) transition.
Early theoretical studies of the two-dimensional XY model with four-fold anisotropy (XY h 4 ) predicted that any finite anisotropy should completely suppress Kosterlitz–Thouless (KT) transition of the isotropic system and replace it with a second-order phase transition. This prediction is inconsistent with Monte Carlo (MC) simulations, which suggested KT-like behavior at weak anisotropy. To resolve this discrepancy, we performed large-scale MC simulations to systematically examine the XY h 4 model across a wide range of anisotropic strengths. Our results indicate that while weak anisotropy can mimic KT behavior in small systems, a detailed finite-size scaling analysis of larger systems reveals that the temperature derivative of Binder ratio and the correlation length ratio displays scaling behaviors characteristic of a second-order transition even at the weak h 4 values. The critical exponent ν is non-universal and varies continuously with h 4 from weak to strong anisotropy. These findings confirm that XY h 4 model exhibits a second-order phase transition for weak finite anisotropy, effectively reconciling theoretical predictions with numerical observations.
The Kitaev model on the honeycomb lattice has been receiving substantial attention due to the discovery of quantum spin liquid state associated with this model. Consequently, its classical partners such as the Kitaev-Heisenberg (KH) model and associated phase transitions become concerned. Specifically, an intermediate Kosterlitz-Thouless (KT) phase engaged in the transition from the high-temperature (T) disordered state to the low-T sixfold degenerate state is predicted in the isotropic KH model [Phys. Rev. Lett. 109, 187201 (2012)10.1103/PhysRevLett.109.187201], but so far no sufficient experimental proof has been reported. In this work, we consider an essential extension of this KH model on the honeycomb lattice by including the Kitaev exchange anisotropy that is non-negligible in realistic materials. The associated phase transitions are thus investigated using the Monte Carlo simulations. It is found that such an anisotropy will result in a degradation of the sixfold degeneracy of the ground state in the isotropic KH model down to the fourfold or twofold degenerate ground state, and the finite-T phase transitions will also be modified remarkably. Interestingly, the intermediate KT phase can be suppressed by this Kitaev exchange anisotropy. This work thus provides a more realistic description of the physics ingredient with the KH model and presents a possible explanation on absence of the intermediate phase in real materials where the Kitaev exchange anisotropy can be more or less available.
A two-dimensional triangular Ising lattice with the nearest-neighbor antiferromagnetic interaction J 1 and the next-nearest-neighbor ferromagnetic interaction - J 2 is studied by the Monte Carlo simulation. It is shown that the system with a finite J 2 undergoes the successive phase transition, paramagnetic→Kosterlitz-Thouless (KT)-like→ferrimagnetic phases, as temperature decreases. The sub-lattice switching phenomenon observed in our previous work is shown to be an intrinsic property inherent to the present system which has an XY character. The marginal exponent of the order-parameter correlation function at the KT-like transition point is seen to increase from 1/4 as the ratio J 2 / J 1 decreases, namely, it reveals a deviation from the universal behavior of a genuine XY model.
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.
Convincing numerical evidence is obtained via extensive quantum Monte Carlo simulations on square lattices as large as 96\ifmmode\times\else\texttimes\fi{}96 that the spin-1/2 XY model undergoes a Kosterlitz-Thouless phase transition at kTc{\mathit{kT}}_{\mathit{c}}kTc​/J=0.350(4). Correlation length and in-plane susceptibility diverge at Tc{\mathit{T}}_{\mathit{c}}Tc​ precisely according to the form predicted by Kosterlitz and Thouless for the classical XY model. The specific heat increases very rapidly near Tc{\mathit{T}}_{\mathit{c}}Tc​ and exhibits a peak around kT/J=0.45.