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Journal of the Physical Society of Japan

Cluster-Spin Quantum Monte Carlo Study of One-Dimensional Heisenberg Model

Yutaka Okabe, Macoto Kikuchi

Published Jun 15, 1987• 8 citations
Publisher

Abstract

The cluster-spin decomposition method is studied for the quantum Monte Carlo simulation based on the Suzuki-Trotter formula. A new vectorized algorithm for the speed-up of the calculation is implemented. We investigate the one-dimensional ferromagnetic and antiferromagnetic Heisenberg models, paying attention to the cluster size of decomposition.

DOI
10.1143/jpsj.56.1963

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The cluster-spin decomposition method is studied for the quantum Monte Carlo simulation based on the Suzuki-Trotter formula. A new vectorized algorithm for the speed-up of the calculation is implemented. We investigate the one-dimensional ferromagnetic and antiferromagnetic Heisenberg models, paying attention to the cluster size of decomposition.
I present a new and efficient disconnected cluster decomposition to perform Monte Carlo Simulations based on the Trotter-Suzuki method for 1-d quantum systems. As opposed to the normal checkerboard world line algorithm this mapping allows one to simulate models with interactions up to next nearest neighbours or even one-dimensional double layers. I have used this cluster decomposition to simulate the one-dimensional t-J-J' model at α=J'/J=0.5 and have investigated the sign problem in this model. At half band filling the system is dimerised into singlet spin pairs when α=0.5 due to frustration. I believe that the spin gap is no longer present around quarter band filling
The Suzuki-Trotter transformation has been used to transform NNN-spin S=12S=\frac{1}{2}S=21​ Heisenberg and XY\mathrm{XY}XY chains into two-dimensional ($N\ifmmode\times\else\texttimes\fi{}2m$) classical Ising models with complicated interactions. A Monte Carlo method using multispin flipping techniques is then used to study these models for $\frac{\mathrm{kT}}{|J|}\ensuremath{\ge}0.0125$ and $m\ensuremath{\le}10$. Finite-size scaling is used to analyze the low-temperature susceptibility of the ferromagnetic Heisenberg chain, and we find a power-law divergence $\ensuremath{\chi}={T}^{\ensuremath{-}\ensuremath{\gamma}}$ with $\ensuremath{\gamma}=1.32\ifmmode\pm\else\textpm\fi{}0.13$. Difficulties in the method, particularly due to singular bonding, are discussed.
We study the spin-correlation functions of the one-dimensional S=1 antiferromagnetic Heisenberg-Ising model by the large-cluster-decomposition Monte Carlo method. It is found that this model shows a second-order transition at the anisotropy parameter ${\ensuremath{\lambda}}_{c}$=1.167\ifmmode\pm\else\textpm\fi{}0.007. The obtained critical exponents (\ensuremath{\eta}=0.253\ifmmode\pm\else\textpm\fi{}0.002, \ensuremath{\beta}=0.126\ifmmode\pm\else\textpm\fi{}0.007, \ensuremath{\nu}=0.98\ifmmode\pm\else\textpm\fi{}0.02) strongly suggest that this transition belongs to the same universality class as the two-dimensional Ising model.