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Journal of Statistical Mechanics Theory and Experiment

Ising antiferromagnets in a nonzero uniform magnetic field

Chi‐Ok Hwang, Seung‐Yeon Kim, Daeseung Kang, J. M. Kim

Published May 8, 2007• 9 citations
Publisher

Abstract

We evaluate the density of states g ( M , E ) as a function of energy E and magnetization M of Ising models on square and triangular lattices, using the exact enumeration method for small systems and the Wang–Landau method for larger systems. From the density of states the average magnetization per spin, m ( T , h ), of the antiferromagnets has been obtained for any values of temperature T and uniform magnetic field h . Also, based on g ( M , E ), the behaviour of m ( T , h ) is understood microcanonically. The microcanonical approach reveals the differences between the unfrustrated model (on the square lattice) and the frustrated one (on the triangular lattice).

DOI
10.1088/1742-5468/2007/05/l05001

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We evaluate the density of states g ( M , E ) as a function of energy E and magnetization M of Ising models on square and triangular lattices, using the exact enumeration method for small systems and the Wang–Landau method for larger systems. From the density of states the average magnetization per spin, m ( T , h ), of the antiferromagnets has been obtained for any values of temperature T and uniform magnetic field h . Also, based on g ( M , E ), the behaviour of m ( T , h ) is understood microcanonically. The microcanonical approach reveals the differences between the unfrustrated model (on the square lattice) and the frustrated one (on the triangular lattice).
The exact integer values for the density of states of the Ising model on an equilateral triangular lattice with free boundary conditions are evaluated up to L = 24 spins on a side for the first time by using the microcanonical transfer matrix. The total number of states is 2 N s = 2300 ≈ 2.037 × 1090 for L = 24, where N s = L(L+1)/2 is the number of spins. Classifying all 2300 spin states according to their energy values is an enormous work. From the density of states, the exact partition function zeros in the complex temperature plane of the triangular-lattice Ising model are evaluated. Using the density of states and the partition function zeros, we investigate the properties of the triangularlattice Ising antiferromagnet. The scaling behavior of the ground-state entropy and the form of the correlation length at T = 0 are studied for the triangular-lattice Ising antiferromagnet with free boundary conditions. Also, the scaling behavior of the Fisher edge singularity is investigated.
Abstract We study the critical properties of the antiferromagnetic spin-1/2 Ising model in an external field on the square lattice. Using tomographic entropic sampling, a flat-histogram simulation method, we estimate the number of configurations, , and related microcanonical averages in the energy-magnetization space, for system sizes L = 10–30. The critical line and exponents are calculated using finite-size scaling analysis in the temperature-external field plane. With these estimates in hand, we perform detailed studies of critical behavior using Metropolis sampling of larger systems ( ). These results are compared to several approximate theoretical methods. Our estimates of critical exponents and Binder’s reduced fourth cumulant along the critical line are in very good agreement with their respective literature values for the two-dimensional Ising universality class. We verify as well that the specific heat scales along the critical line, as expected for an Ising-like critical point.
The quantum antiferromagnetic spin-1/2 Ising model on a triangular lattice and analogous fully frustrated Ising model on a square lattice with quantum fluctuation induced by the application of the transverse magnetic field are studied at finite temperatures by constructing an exact mapping onto a purely classical model with a more complex interaction. It is shown that in weak fields the temperatures of the phase transitions separating the critical phase from the ordered and disordered phases in both models are proportional to the magnitude of the field.
We report on a systematic study of two-dimensional, periodic, frustrated Ising models with quantum dynamics introduced via a transverse magnetic field. The systems studied are the triangular and kagom\'e lattice antiferromagnets, fully frustrated models on the square and hexagonal (honeycomb) lattices, a planar analog of the pyrochlore antiferromagnet, a pentagonal lattice antiferromagnet, as well as two quasi-one-dimensional lattices that have considerable pedagogical value. All of these exhibit a macroscopic degeneracy at T=0T=0T=0 in the absence of the transverse field, which enters as a singular perturbation. We analyze these systems with a combination of a variational method at weak fields, a perturbative Landau-Ginzburg-Wilson approach from large fields, as well as quantum Monte Carlo simulations utilizing a cluster algorithm. Our results include instances of quantum order arising from classical criticality (triangular lattice) or classical disorder (pentagonal and probably hexagonal) as well as notable instances of quantum disorder arising from classical disorder (kagom\'e). We also discuss the effect of finite temperature, as well as the interplay between longitudinal and transverse fields---in the kagom\'e problem the latter gives rise to a nontrivial phase diagram with bond-ordered and bond-critical phases in addition to the disordered phase. We also note connections to quantum-dimer models and thereby to the physics of Heisenberg antiferromagnets in short-ranged resonating valence-bond phases that have been invoked in discussions of high-temperature superconductivity.