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Blockspin Cluster Algorithms for Quantum Spin Systems

U.‐J. Wiese, H. -p. Ying

Published Jan 1, 2008• 30 citations

Abstract

Cluster algorithms are developed for simulating quantum spin systems like the one- and two-dimensional Heisenberg ferro- and anti-ferromagnets. The corresponding two- and three-dimensional classical spin models with four-spin couplings are maped to blockspin models with two-blockspin interactions. Clusters of blockspins are updated collectively. The efficiency of the method is investigated in detail for one-dimensional spin chains. Then in most cases the new algorithms solve the problems of slowing down from which standard algorithms are suffering. Two-dimensional quantum spin systems are relevant for the description of the undoped anti-ferromagnetic precursor insulators of high-Tc superconductors. Presumably understanding the physics of the superconductors requires an understanding of their precursor insulators. This already is a nontrivial problem, which most likely does not allow for a complete analytic solution. Therefore it is natural to use a numerical approach to compute the properties of these materials. At present different

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Cluster algorithms are developed for simulating quantum spin systems like the one- and two-dimensional Heisenberg ferro- and anti-ferromagnets. The corresponding two- and three-dimensional classical spin models with four-spin couplings are maped to blockspin models with two-blockspin interactions. Clusters of blockspins are updated collectively. The efficiency of the method is investigated in detail for one-dimensional spin chains. Then in most cases the new algorithms solve the problems of slowing down from which standard algorithms are suffering. Two-dimensional quantum spin systems are relevant for the description of the undoped anti-ferromagnetic precursor insulators of high-Tc superconductors. Presumably understanding the physics of the superconductors requires an understanding of their precursor insulators. This already is a nontrivial problem, which most likely does not allow for a complete analytic solution. Therefore it is natural to use a numerical approach to compute the properties of these materials. At present different

In this paper, we proposed a blockspin scheme and developed the usual cluster-updating algorithms for the one-and two-dimensional quantum spin systems, saying s=1/2 isotropic Hei-senberg models. The results for he ferro-and antiferro-magnet chains showed that the criti-cal slowing down (CSD) was indeed eliminated, and so it offered a possibility to improve the estimations in the lower-temperature region. We expect that the algorithms could also be ap-plied to the anisotropic or/and the higher-spin cases, as will as some other interesting models, such as eight-vortex lattice model and low-dimensional Fermi-systems.

We present a numerical study using a cluster algorithm for the 1-d

S=1/2

quantum Heisenberg models. The dynamical critical exponent for anti-ferromagnetic chains is

z=0.0(1)

such that critical slowing down is eliminated.

Numerical simulations of numerous quantum systems suffer from the notorious sign problem. Meron-cluster algorithms lead to an efficient solution of sign problems for both fermionic and bosonic models. Here we apply the meron concept to quantum spin systems in an arbitrary external magnetic field, in which case standard cluster algorithms fail. As an example, we simulate antiferromagnetic quantum spin ladders in a uniform external magnetic field that competes with the spin-spin interaction. The numerical results are in agreement with analytic predictions for the magnetization as a function of the external field.

Abstract We extend a semiclassical numerical method, bosonic auxiliary-field Monte Carlo, to quantum spin systems. This method breaks the lattice into clusters, solves each cluster precisely and couples them with classical auxiliary fields through classical Monte Carlo simulation. We test the method with antiferromagnetic spin models in one-dimensional chains, square lattices and triangular lattices, and obtain reasonable results at finite temperatures. This algorithm builds a bridge between classical Monte Carlo method and quantum methods. The algorithm can be improved with either progress in classical Monte Carlo sampling or the development of quantum solvers, and can also be further applied to systems with different lattices or interactions.

We construct an efficient cluster algorithm for ferromagnetic SU(N)-symmetric quantum spin systems.Such systems provide a new regularization for CP(N -1) models in the framework of D-theory, which is an alternative non-perturbative approach to quantum field theory formulated in terms of discrete quantum variables instead of classical fields.Despite several attempts, no efficient cluster algorithm has been constructed for CP(N -1) models in the standard formulation of lattice field theory.In fact, there is even a no-go theorem that prevents the construction of an efficient Wolff-type embedding algorithm.We present various simulations for different correlation lengths, couplings and lattice sizes.We have simulated correlation lengths up to 250 lattice spacings on lattices as large as 640×640 and we detect no evidence for critical slowing down.