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Journal of Physics A Mathematical and Theoretical

Correlation entropy and the Kosterlitz–Thouless transition

Xiaoling Cui, Shi-Jian Gu, Junpeng Cao, Yupeng Wang, Hai-Qing Lin

Published Oct 23, 2007• 8 citations
Publisher

Abstract

A relation between the correlation entropy and the correlation functions for the general spin-1/2 systems is obtained. It is shown that the correlation entropy catches some characters of correlation behavior and can be used to quantify the quantum and finite-temperature phase transitions, including the infinite order or topological ones. As an example, the Kosterlitz–Thouless transition in the quantum two-dimensional XY model is investigated. The critical temperature and the critical exponents are determined from the finite-size scaling analysis of the correlation entropy.

DOI
10.1088/1751-8113/40/45/003

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•Jan 1, 2008•2 citations
A relation between the correlation entropy and the correlation functions for the general spin-1/2 systems is obtained. It is shown that the correlation entropy catches some characters of correlation behavior and can be used to quantify the quantum and finite-temperature phase transitions, including the infinite order or topological ones. As an example, the Kosterlitz–Thouless transition in the quantum two-dimensional XY model is investigated. The critical temperature and the critical exponents are determined from the finite-size scaling analysis of the correlation entropy.
At low temperatures, the two-dimensional continuous-spin systems exhibit large correlation lengths. Some of them show the Berezinskii-Kosterlitz-Thouless-like transitions, and some others show pseudocritical behaviors for which correlation lengths are extremely large but finite. To distinguish pseudo and genuine critical behaviors, it is important to understand the nature of spin-spin correlations and topological defects at low temperatures in continuous-spin systems. In this paper, I develop a finite-size scaling analysis which is suitable for distinguishing the critical behavior and its applications to the two-dimensional XY, Heisenberg, and RP(2) models.
The two-dimensional spin-1/2 XY model is investigated via an extensive quantum Monte Carlo simulation on square lattices as large as 128\ifmmode\times\else\texttimes\fi{}128. The transverse susceptibility and correlation length show precise thermodynamic scaling behaviors of a phase transition of the Kosterlitz-Thouless type at kTc{\mathit{kT}}_{\mathit{c}}kTc​/J=0.353(3). The correlation functions near the transition point exhibit a universal scaling behavior. Various critical exponents are determined and our results demonstrate that the universality class of the two-dimensional quantum system with continuous symmetry is the same as its classical analog. The specific heat exhibits a finite peak around kT/J=0.45, decreases very rapidly near Tc{\mathit{T}}_{\mathit{c}}Tc​, and falls off as T2{\mathit{T}}^{2}T2 near T=0, validating the spin-wave treatment. The ground-state energy is -0.5543(1)J per spin.
We propose a series of scaling theories for Kosterlitz-Thouless (KT) phase transitions on the basis of the hallmark exponential growth of their correlation length. Finite-size scaling, finite-entanglement scaling, short-time critical dynamics, and finite-time scaling, as well as some of their interplay, are considered. Relaxation times of both a normal power-law and an anomalous power-law with a logarithmic factor are studied. Finite-size and finite-entanglement scaling forms somehow similar to but different from a frequently employed ansatz are presented. The Kibble-Zurek scaling of topological defect density for a linear driving across the KT transition point is investigated in detail. An implicit equation for a rate exponent in the theory is derived, and the exponent varies with the distance from the critical point and the driving rate consistent with relevant experiments. To verify the theories, we utilize the KT phase transition of a one-dimensional Bose-Hubbard model. The infinite time-evolving-block-decimation algorithm is employed to solve numerically the model for finite bond dimensions. Both a correlation length and an entanglement entropy in imaginary time and only the entanglement entropy in real-time driving are computed. Both the short-time critical dynamics in imaginary time and the finite-time scaling in real-time driving, both including the finite bond dimension, for the measured quantities are found to describe the numerical results quite well via surface collapses. The critical point is also estimated and confirmed to be 0.302(1) at the infinite bond dimension on the basis of the scaling theories.
Besides the Berezinskii-Kosterlitz-Thouless phase transition, the two-dimensional generalized XY model, identified by a generalization parameter qqq (as proposed by Romano and Zagrebnov), can also support a first-order phase transition, starting from a critical value qc{q}_{c}qc​. However, the value of qc{q}_{c}qc​ at which this transition takes place is unknown. In this paper, we take two approaches to accurately determine the critical parameter qc{q}_{c}qc​. Furthermore, we show that the model is characterized by three distinct regions concerning both first-order and Berezinskii-Kosterlitz-Thouless phase transitions. Finally, the underlying mechanism governing such transitions is presented, along with an estimation of the critical temperatures.
The thermodynamics of the two-dimensional (2D) XY model is formulated by a transfer-matrix method and analyzed by a density-matrix renormalization group. The finite-size scaling and the beta function of the model are studied by the Roomany-Wyld renormalization-group theory. It is found that the 2D XY model has an essential finite-size effect and the Berezinskii-Kosterlitz-Thouless transition with the critical temperature TBKT=0.892{T}_{\mathrm{BKT}}=0.892TBKT​=0.892 appears in a finite system of 2000--3000 spins as a massless to massive transition with the effective critical temperature ${T}_{c}=1.07\ifmmode\pm\else\textpm\fi{}0.01.$
Monte Carlo simulations of the two-dimensional XY model are performed in a square geometry with free and mixed fixed-free boundary conditions. Using a Schwarz-Christoffel conformal mapping, we deduce the exponent η of the order parameter correlation function and its surface equivalent η ‖ at the Kosterlitz-Thouless transition temperature. The well known value η(TKT) = 1/4 is easily recovered even with systems of relatively small sizes, since the shape effects are encoded in the conformal mapping. The exponent associated to the surface correlations is similarly obtained η1(TKT) ≃ 0.54. 1