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The Philosophical Magazine A Journal of Theoretical Experimental and Applied Physics

Quantum Monte Carlo simulations revisited: The case of anisotropic Heisenberg chains

Petros Androvitsaneas, Nikolaos G. Fytas, Emmanuel Paspalakis, A.F. Terzis

Published Aug 31, 2012• 2 citations
Publisher

Abstract

We revisit one of the first quantum Monte Carlo methods using a combination of the Suzuki–Trotter transformation and Taylor series expansion for the XYZ Heisenberg chain with a magnetic field along the z-axis. We show a substantial gain in accuracy and computational time. This combination is especially useful for small magnetic fields and lower temperatures.

DOI
10.1080/14786435.2012.716200

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We revisit one of the first quantum Monte Carlo methods using a combination of the Suzuki–Trotter transformation and Taylor series expansion for the XYZ Heisenberg chain with a magnetic field along the z-axis. We show a substantial gain in accuracy and computational time. This combination is especially useful for small magnetic fields and lower temperatures.
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Recently, the stochastic series expansion (SSE) has been proposed as a powerful Monte Carlo method, which allows simulations at low T for quantum-spin systems. We show that the SSE allows us to compute the magnetic conductance for various one-dimensional spin systems without further approximations. We consider various modifications of the anisotropic Heisenberg chain. We recover the Kane-Fisher scaling for one impurity in a Luttinger liquid and study the influence of noninteracting leads for the conductance of an interacting system.
Generative neural samplers offer a complementary approach to Monte Carlo methods for problems in statistical physics and quantum field theory. This paper tests the ability of generative neural samplers to estimate observables for real-world low-dimensional spin systems. It maps out how autoregressive models can sample configurations of a quantum Heisenberg chain via a classical approximation based on the Suzuki-Trotter transformation. We present results for energy, specific heat, and susceptibility for the isotropic XXX and the anisotropic XY chain are in good agreement with Monte Carlo results within the same approximation scheme.
We introduce the concept of directed loops in stochastic series expansion and path-integral quantum Monte Carlo methods. Using the detailed balance rules for directed loops, we show that it is possible to smoothly connect generally applicable simulation schemes (in which it is necessary to include backtracking processes in the loop construction) to more restricted loop algorithms that can be constructed only for a limited range of Hamiltonians (where backtracking can be avoided). The "algorithmic discontinuities" between general and special points (or regions) in parameter space can hence be eliminated. As a specific example, we consider the anisotropic S=1/2 Heisenberg antiferromagnet in an external magnetic field. We show that directed-loop simulations are very efficient for the full range of magnetic fields (zero to the saturation point) and anisotropies. In particular, for weak fields and anisotropies, the autocorrelations are significantly reduced relative to those of previous approaches. The back-tracking probability vanishes continuously as the isotropic Heisenberg point is approached. For the XY model, we show that back tracking can be avoided for all fields extending up to the saturation field. The method is hence particularly efficient in this case. We use directed-loop simulations to study the magnetization process in the two-dimensional Heisenberg model at very low temperatures. For LxL lattices with L up to 64, we utilize the step structure in the magnetization curve to extract gaps between different spin sectors. Finite-size scaling of the gaps gives an accurate estimate of the transverse susceptibility in the thermodynamic limit: chi( perpendicular )=0.0659+/-0.0002.
A cluster update (the ``operator loop'') is developed within the framework of a numerically exact quantum Monte Carlo method based on the power series expansion of $\mathrm{exp}(\ensuremath{-}\ensuremath{\beta}H)$ (stochastic series expansion). The method is generally applicable to a wide class of lattice Hamiltonians for which the expansion is positive definite. For some important models the operator-loop algorithm is more efficient than loop updates previously developed for ``worldline'' simulations. The method is here tested on a two-dimensional anisotropic Heisenberg antiferromagnet in a magnetic field.