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arXiv:cond-mat/9612023 · Journal of the Physical Society of Japan

Cluster Heat Bath Method on a Quasi-One-Dimensional Ising Model

Osamu Koseki, Fumitaka Matsubara

Published Feb 15, 1997• 23 citations
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Abstract

We have proposed a cluster heat bath(CHB) method of a quasi-one dimensional Ising model and demonstrated that it reproduces the exact results of the two dimensional Ising model with axially anisotropic exchange interactions. The point of the method is to select one of the equilibrium spin configurations of each of the chains which are subjected by effective fields of surrounding spins. Thus the chains are always in their equilibrium state during the simulation and the $d$ dimensional model effectively becomes as a $d-1$ dimensional model of fictitious spins with their freedom of $2^N$ and the simulation time is drastically reduced, where $N$ is the number of the spins on the chain.

DOI
10.1143/jpsj.66.322

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We have proposed a cluster heat bath(CHB) method of a quasi-one dimensional Ising model and demonstrated that it reproduces the exact results of the two dimensional Ising model with axially anisotropic exchange interactions. The point of the method is to select one of the equilibrium spin configurations of each of the chains which are subjected by effective fields of surrounding spins. Thus the chains are always in their equilibrium state during the simulation and the ddd dimensional model effectively becomes as a d−1d-1d−1 dimensional model of fictitious spins with their freedom of 2N2^N2N and the simulation time is drastically reduced, where NNN is the number of the spins on the chain.
We develop an efficient Monte Carlo algorithm, which accelerates slow Monte Carlo dynamics in quasi-one-dimensional Ising spin systems. The loop algorithm of the quantum Monte Carlo method is applied to the classical spin models with highly anisotropic exchange interactions. Both correlation time and real CPU time are reduced drastically. The algorithm is demonstrated in the layered triangular-lattice antiferromagnetic Ising model. We have obtained the relation between the transition temperature and the exchange interaction parameters, which modifies the result of the chain-mean-field theory.
We study thermodynamic properties of an axial next-nearest-neighbor Ising model in two dimensions using a cluster heat bath Monte Carlo method. The method drastically reduces the relaxation time, and enables us to study equilibrium properties of the model. We show that the model really exhibits two phase transitions at Tc1{T}_{c1}Tc1​ and Tc2(<Tc1),{T}_{c2}(<{T}_{c1}),Tc2​(<Tc1​), as predicted by a free fermion approximation. The one at Tc1{T}_{c1}Tc1​ is shown to be of the Kosterlitz-Thouless type, and the other to be of the Pokrovsky-Talapov type. The higher transition temperature Tc1{T}_{c1}Tc1​ is considerably lower than those estimated by previous authors. Low-temperature properties as well as the lower transition temperature Tc2{T}_{c2}Tc2​ are found to be well described by the fermion approximation.
Critical temperature of quasi-one-dimensional general-spin Ising ferromagnets is investigated by means of the cluster Monte Carlo method performed on infinite-length strips, $L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ or $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$. We find that in the weak interchain coupling regime the critical temperature as a function of the interchain coupling is well-described by a chain mean-field formula with a reduced effective coordination number, as the quantum Heisenberg antiferromagnets recently reported by Yasuda et al. [Phys. Rev. Lett. 94, 217201 (2005)]. It is also confirmed that the effective coordination number is independent of the spin size. We show that in the weak interchain coupling limit the effective coordination number is, irrespective of the spin size, rigorously given by the quantum critical point of a spin-1∕21∕21∕2 transverse-field Ising model.
We introduce a variational implementation of cluster perturbation theory (CPT) to address the dynamics of spin systems driven out of equilibrium. We benchmark the method with the quantum Ising model subject to a sudden quench of the transverse magnetic field across the transition or within a phase. We treat both the one-dimensional case, for which an exact solution is available, as well the two-dimensional case, for which we have to resort to numerical results. Comparison with exact results shows that the approach provides a quite accurate description of the real-time dynamics up to a characteristic timescale $\ensuremath{\tau}$ that increases with the size of the cluster used for CPT. In addition, and not surprisingly, $\ensuremath{\tau}$ is small for quenches across the equilibrium phase transition point, but can be quite larger for quenches within the ordered or disordered phases.
Using Wang-Landau simulation, we investigate the nature of the quasi-one-dimensional(Q1D) Ising model in terms of energy density of states(DOS); In order to treat the large difference of the energy scales between intra- and inter-chain couplings, we particularly introduce DOS in the two dimensional energy space corresponding to the intra- and inter-chain directions, and then discuss size dependence of the peak structure of specific heat. We also analyze the inter-chain coupling dependence of the spin flop transition of the Q1D classical XXZ model with the Ising-like anisotropy using the Wang-Landau simulation.
The heat conductivity of nonperiodic quantum Ising chains whose ends are connected with heat baths at different temperatures are studied numerically by solving the Lindblad master equation. The chains are subjected to a uniform transverse field h, while the exchange coupling J{m} between the nearest-neighbor spins takes the two values J{A} and J{B} arranged in Fibonacci, generalized Fibonacci, Thue-Morse, and period-doubling sequences. We calculate the energy-density profile and energy current of the resulting nonequilibrium steady states to study the heat-conducting behavior of finite but large systems. Although these nonperiodic quantum Ising chains are integrable, it is clearly found that energy gradients exist in all chains and the energy currents appear to scale as the system size ~N{α}. By increasing the ratio of couplings, the exponent α can be modulated from α > -1 to α < -1 corresponding to the nontrivial transition from the abnormal heat transport to the heat insulator. The influences of the temperature gradient and the magnetic field to heat conduction have also been discussed.