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Journal of Applied Physics

Ferromagnetic phase boundary in the bond frustrated Heisenberg model

A. D. Beath, Dominic H Ryan

Published Apr 26, 2005• 6 citations
Publisher

Abstract

We determine the ferromagnetic phase boundary for the short range ±J bond frustrated Heisenberg model in three dimensions using a very efficient Monte Carlo algorithm which eliminates the critical slowing down usually experienced at a second order phase transition. The phase boundary is identified by measuring the correlation length directly, a method which we show to be superior to more conventional methods such as the crossing of the Binder cumulant. The critical concentration of antiferromagnetic bonds beyond which ferromagnetism is lost is xc=0.208(2).

DOI
10.1063/1.1851916

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We determine the ferromagnetic phase boundary for the short range ±J bond frustrated Heisenberg model in three dimensions using a very efficient Monte Carlo algorithm which eliminates the critical slowing down usually experienced at a second order phase transition. The phase boundary is identified by measuring the correlation length directly, a method which we show to be superior to more conventional methods such as the crossing of the Binder cumulant. The critical concentration of antiferromagnetic bonds beyond which ferromagnetism is lost is xc=0.208(2).
We investigate the boundary phases of a (2+1)-dimensional quantum critical Heisenberg model with a dangling spin chain. By introducing a multispin QQQ-term along the boundary, we drive a continuous boundary transition from an antiferromagnetic (AF) order to a valence-bond solid (VBS) order. Using large-scale quantum Monte Carlo simulations, we locate the critical point at Qc=0.310(11)Q_{c}=0.310(11)Qc​=0.310(11), and obtain the critical exponents at QcQ_{c}Qc​, including ys=0.81(4)y_{s}=0.81(4)ys​=0.81(4) and the scaling dimensions of AF and VBS order parameters Δs=0.660(15)Δ_{s}=0.660(15)Δs​=0.660(15) and Δv=0.204(14)Δ_{v}=0.204(14)Δv​=0.204(14). The weak long-range AF order for $Q
Phase transitions and critical properties in the antiferromagnetic Heisenberg model on a layered cubic lattice with allowance for intralayer next nearest neighbor interactions have been studied using the replica Monte Carlo algorithm. The character of phase transitions has been analyzed using the histogram method and the Binder cumulant method. It has been found that a transition from the collinear to paramagnetic phase in the model under study occurs as a second order phase transition. The statistical critical exponents of the specific heat α, susceptibility γ, order parameter β, and correlation radius ν, as well as the Fisher index η, have been calculated using the finite-size scaling theory. It has been shown that the three-dimensional Heisenberg model on the layered cubic lattice with allowance for the next nearest neighbor interaction belongs to the same universality class of the critical behavior as the antiferromagnetic Heisenberg model on a layered triangular lattice.
The phase transitions in 2D ferro- and antiferromagnetic Potts models with number of spin states q = 3 on a triangular lattice are investigated by the cluster and classical Monte Carlo methods. Systems with linear sizes L = 20–120 are considered. Fourth-order Binder cumulants and histogram data analysis are used to show that second- and first-order phase transitions are observed in the ferromagnetic and antiferromagnetic Potts models, respectively. The static critical indices are calculated for specific heat α, susceptibility γ, magnetization β, and correlation length ν on the basis of finite-size scaling theory for a ferromagnetic Potts model.
Phase transitions in the antiferromagnetic Heisenberg model on a cubic lattice with intralayer next-nearest neighbor interactions are studied using the replica Monte Carlo algorithm. The magnitude of next-nearest neighbor interactions varies in the range of 0.0 ≤ r ≤ 1.0. The characteristics of the phase transitions are analyzed by the histogram and Binder cumulant techniques. The phase diagram relating the transition temperature and the magnitude of next-nearest neighbor interactions is constructed. It is shown that a second order phase transition occurs in the r range under study. In this model, it is found that the intralayer next-nearest neighbor interactions do not change the order of the phase transition.
The Binder cumulant at the phase transition of Ising models on square lattices with ferromagnetic couplings between nearest neighbors and with competing antiferromagnetic couplings between next-nearest neighbors, along only one diagonal, is determined using Monte Carlo techniques. In the phase diagram a disorder line occurs separating regions with monotonically decaying and with oscillatory spin-spin correlations. Findings on the variation of the critical cumulant with the ratio of the two interaction strengths are compared to related recent results based on renormalization-group calculations.