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Journal of Physics Conference Series

Wang-Landau simulation of the quasi-one-dimensional spin systems

Kouichi Okunishi, Takayuki Tanabe

Published Jan 1, 2010• 1 citation
Publisher

Abstract

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.

DOI
10.1088/1742-6596/200/2/022043

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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.
We revisit the nature of the quasi-one-dimensional Ising model on the basis of Wang-Landau simulation. We introduce the density of states in the two-dimensional energy space corresponding to the intra- and inter-chain directions. We then analyze the interchain coupling dependence of the specific heat of the anistropic two-dimensional Ising model in the context of the density of states, and further discuss the size dependences of the peak temperature. We also discuss the feature of the phase transition in the three-dimensional case.
We study spin-S Ising models with p-spin interactions on the one-dimensional chain and the two-dimensional square lattice. Here, S denotes the magnitude of the spin and p represents the number of spins involved in each interaction. The analysis is performed for S=1/2,1,3/2,2, and p=3,4,5. For the one-dimensional model, we formulate transfer matrices and numerically diagonalize them to analyze the temperature dependence of the free energy and spin-spin correlations. In the case of S=1/2, the free energy does not depend on p, and the spin-spin correlations are uniformly enhanced across all temperature scales as p increases. In contrast, for S≥1, the free energy varies with p, and the spin-spin correlations are significantly enhanced at lower temperatures as p increases. For the two-dimensional model, by using multicanonical simulations, we analyze physical quantities such as an order parameter, internal energy, and specific heat. In addition, we define and examine an order parameter to distinguish ordered and disordered phases. It is found that a first-order phase transition occurs at finite temperatures for all S and p≥3, and increasing p strengthens its nature. We present S and p dependence of the transition temperature and latent heat, and discuss effects of higher-order interactions on the nature of phase transitions.
We performed extensive simulations, using the Replica Exchange-Wang-Landau method, of the clock model for orders 3 and 4 on a square lattice, where critical behaviors are expected to belong to the Ising universality class. Though order 2 represents the Ising model, thus, being exactly solvable in two-dimensions, we still provide such results for comparison to the other two orders. Results for various energy related quantities such as the mean energy per spin, specific heat, as well as logarithm scaling of the peak of the specific heat are presented and shown to follow Ising behavior. Additionally, we also present results related to magnetic quantities, such as the magnetization, magnetic susceptibility, and corresponding scaling behavior of the peak of the magnetic susceptibility. Again, our results show scaling in conformity to Ising critical behavior.
We study the one-dimensional quantum Ising model in a transverse magnetic field where the exchange couplings are ordered according to the Thue-Morse (TM) sequence. At zero temperature, this model is equivalent to a two-dimensional classical Ising model in a magnetic field with TM aperiodicity along one direction. We compute the order parameter (magnetization) of the chain and the scaling behavior of the energy spectrum when the system undergoes a phase transition. Analogous to the quasiperiodic (QP) quantum Ising chain, the onset of long-range order is signaled by a nonanaliticity in the exponent \ensuremath{\delta} which describes the scaling of the total bandwidth with the size of the chain. The critical spin-coupling can be computed analytically and it is found to be lower than the QP case. Furthermore, the energy bands are found to be narrower than the corresponding QP chain. The former and latter results are consistent with the fact that the present structure has a degree of ordering intermediate between QP and random.
We explore the phase diagram of Ising spins on one-dimensional chains which criss-cross in two perpendicular directions and are connected by interchain couplings. This system is of interest as a simpler, classical analog of a quantum Hamiltonian which has been proposed as a model of magnetic behavior in Nb12_{12}12​O29_{29}29​ and also, conceptually, as a geometry which is intermediate between one and two dimensions. Using mean field theory as well as Metropolis Monte Carlo and Wang-Landau simulations, we locate quantitatively the boundaries of four ordered phases. Each becomes an effective Ising model with unique effective couplings at large interchain coupling. Away from this limit we demonstrate non-trivial critical behavior, including tricritical points which separate first and second order phase transitions. Finally, we find that the crossover of the magnetization critical exponent from the Ising to tricritical Ising value shows an unusual non-monotonic behavior.
The authors develop a quantum Monte Carlo procedure for the xxz spin chain. The quantum model in one dimension is mapped by a path integral method based on the Trotter formula into the infinite anisotropy limit of a classical two-dimensional spin system. The expectation values of the energy density, susceptibility and static s z -s z structure function are computed in the classical model for finite but increasing anisotropy. The computation method is a Monte Carlo simulation of the classical system. The results are systematically extrapolated to the infinite anisotropy limit.