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arXiv:cond-mat/0603488 · Physical Review E

Global phase diagram and six-state clock universality behavior in the triangular antiferromagnetic Ising model with anisotropic next-nearest-neighbor coupling: Level-spectroscopy approach

Hiromi Otsuka, Yutaka Okabe, K. Nomura

Published Jul 11, 2006• 8 citations
arXivHTMLPDF

Abstract

We investigate the triangular-lattice antiferromagnetic Ising model with a spatially anisotropic next-nearest-neighbor ferromagnetic coupling, which was first discussed by Kitatani and Oguchi. By employing the effective geometric factor, we analyze the scaling dimensions of the operators around the Berezinskii-Kosterlitz-Thouless (BKT) transition lines, and determine the global phase diagram. Our numerical data exhibit that two types of BKT-transition lines separate the intermediate critical region from the ordered and disordered phases, and they do not merge into a single curve in the antiferromagnetic region. We also estimate the central charge and perform some consistency checks among scaling dimensions in order to provide the evidence of the six-state clock universality. Further, we provide an analysis of the shapes of boundaries based on the crossover argument.

DOI
10.1103/physreve.74.011104

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We investigate the triangular-lattice antiferromagnetic Ising model with a spatially anisotropic next-nearest-neighbor ferromagnetic coupling, which was first discussed by Kitatani and Oguchi. By employing the effective geometric factor, we analyze the scaling dimensions of the operators around the Berezinskii-Kosterlitz-Thouless (BKT) transition lines, and determine the global phase diagram. Our numerical data exhibit that two types of BKT-transition lines separate the intermediate critical region from the ordered and disordered phases, and they do not merge into a single curve in the antiferromagnetic region. We also estimate the central charge and perform some consistency checks among scaling dimensions in order to provide the evidence of the six-state clock universality. Further, we provide an analysis of the shapes of boundaries based on the crossover argument.
We investigate the six-state clock universality of the Ising model on the kagome lattice, considering antiferromagnetic nearest-neighbor (NN) and ferromagnetic next-nearest-neighbor (NNN) interactions. Our comprehensive study employs three approaches: the level-spectroscopy method, Monte Carlo simulations, and a machine-learning phase classification technique. In this system, we observe two Berezinskii-Kosterlitz-Thouless (BKT) transitions. We present a phase diagram consisting of three phases: the low-temperature ordered phase with sublattice magnetizations, the intermediate BKT phase, and the high-temperature disordered phase, as a function of the ratio of the NNN interaction to the NN interaction. We verify the six-state clock universality through the machine-learning study, which uses data from the six-state clock model on the kagome lattice for training.
Abstract Comprehensive studies are made on the six-state clock universality of two models using several approaches. We apply the machine-learning technique of phase classification to the antiferromagnetic (AF) three-state Potts model on the square lattice with ferromagnetic next-nearest-neighbor (NNN) coupling and the triangular AF Ising model with anisotropic NNN coupling to study two Berezinskii–Kosterlitz–Thouless transitions. We also use the Monte Carlo simulation paying attention to the ratio of correlation functions of different distances for these two models. The obtained results are compared with those of the previous studies using the level-spectroscopy method. We directly show the six-state clock universality for totally different systems with the machine-learning study.
The authors study the Ising model on the triangular lattice with nearest-neighbor couplings K(nn) , next-nearest-neighbor couplings K(nnn) >0 , and a magnetic field H . This work is done by means of finite-size scaling of numerical results of transfer matrix calculations, and Monte Carlo simulations. We determine the phase diagram and confirm the character of the critical manifolds. The emphasis of this work is on the antiferromagnetic case K(nn) <0 , but we also explore the ferromagnetic regime K(nn) >/=0 for H=0 . For K(nn) <0 and H=0 we locate a critical phase presumably covering the whole range -infinity< K(nn) <0 . For K(nn) <0 , H not equal 0 we locate a plane of phase transitions containing a line of tricritical three-state Potts transitions. In the limit H-->infinity this line leads to a tricritical model of hard hexagons with an attractive next-nearest-neighbor potential.
The field-induced Berezinskii-Kosterlitz-Thouless (BKT) transition in the ground state of the triangular antiferromagnetic Ising model is studied by the level-spectroscopy method. We analyze dimensions of operators around the BKT line, and estimate the BKT point H(c) approximately equal to 0.5229 +/-0.001, which is followed by the level-consistency check to demonstrate the accuracy of our estimate. Further we investigate the anisotropic case to clarify the stability of the field-induced string-density plateau against an incommensurate liquid state by the density-matrix renormalization-group method.
We study the square-lattice three-state Potts model with the ferromagnetic next-nearest-neighbor coupling at finite temperature. Using the level-spectroscopy method, we numerically analyze the excitation spectrum of the transfer matrices and precisely determine the global phase diagram. Then we find that, contrary to a previous result based on the finite-size scaling, the massless region continues up to the decoupling point with Z3 x Z3 criticality in the antiferromagnetic region. We also check the universal relations among excitation levels to provide the reliability of our result.