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Physical review. B, Condensed matter

Monte Carlo method for obtaining the ground-state properties of quantum spin systems

Juan Pablo Neirotti, Mário J. de Oliveira

Published Jan 1, 1996• 16 citations
Publisher

Abstract

A Monte Carlo method to find the ground-state properties of quantum spin systems is presented. Transforming a quantum spin Hamiltonian in a matrix with non-negative elements, we set up a Markov process whose stationary probability is dominated by the leading eigenvector of this matrix. From the simulation of the Markov process, by means of a Metropolis algorithm, we obtain the properties and the energy of the ground state. The method is applied to the spin-1 isotropic, Heisenberg antiferromagnet chain. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.53.668

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A Monte Carlo method to find the ground-state properties of quantum spin systems is presented. Transforming a quantum spin Hamiltonian in a matrix with non-negative elements, we set up a Markov process whose stationary probability is dominated by the leading eigenvector of this matrix. From the simulation of the Markov process, by means of a Metropolis algorithm, we obtain the properties and the energy of the ground state. The method is applied to the spin-1 isotropic, Heisenberg antiferromagnet chain. \textcopyright{} 1996 The American Physical Society.
Abstract The quantum mechanical underpinnings of magnetism are explored via the Heisenberg model of antiferromagnetism. The Lanczos algorithm is developed and applied to obtain ground state properties of the anisotropic antiferromagnetic Heisenberg spin chain. In particular, the phase diagram for the system magnetization is determined. A quantum Monte Carlo method that is appropriate for discrete systems is also presented. The method leverages the similarity between the Schrödinger equation and the diffusion equation to compute energy levels. The formalism necessary to compute ground state matrix elements is also developed. Finally, the method is tested with an application to the spin chain.
The Markov-chain Monte Carlo method is at the heart of efficient approximation schemes for a wide range of problems in combinatorial enumeration and statistical physics. It is therefore very natural and important to determine whether quantum computers can speed up classical mixing processes based on Markov chains. To this end, we present a quantum algorithm, making it possible to prepare a quantum sample---i.e., a coherent version of the stationary distribution of a reversible Markov chain. Our algorithm has a significantly better running time than that of a previous algorithm based on adiabatic-state generation. We also show that our methods provide a greater speedup over a recently proposed method for obtaining the ground states of (classical) Hamiltonians.
These lecture notes introduce quantum spin systems and several computational methods for studying their ground-state and finite-temperature properties. Symmetry-breaking and critical phenomena are first discussed in the simpler setting of Monte Carlo studies of classical spin systems, to illustrate finite-size scaling at continuous and first-order phase transitions. Exact diagonalization and quantum Monte Carlo (stochastic series expansion) algorithms and their computer implementations are then discussed in detail. Applications of the methods are illustrated by results for some of the most essential models in quantum magnetism, such as the S=1/2 Heisenberg antiferromagnet in one and two dimensions, as well as extended models useful for studying quantum phase transitions between antiferromagnetic and magnetically disordered states.
We present a new algorithm for world line Monte Carlo simulations of quantum spin systems. The algorithm consists of a stochastic pairing and a stochastic flipping procedure, and the formula for the pairing probability is explicitly given for Heisenberg models. As an illustration of the algorithm, we calculated autocorrelation times for simulations of the S=1S=1S=1 antiferromagnetic Heisenberg chain. The rapid increase in autocorrelation times which is seen in the conventional world line method as the imaginary-time spacing decreases is eliminated by the present algorithm.
We present a universal parameter-free quantum Monte Carlo (QMC) algorithm designed to simulate arbitrary spin- 1/2 Hamiltonians. To ensure the convergence of the Markov chain to equilibrium for every conceivable case, we devise a clear and simple automated protocol that produces QMC updates that are provably ergodic and satisfy detailed balance. We demonstrate the applicability and versatility of our method by considering several illustrative examples, including the simulation of the XY model on a triangular lattice, the toric code, and random k -local Hamiltonians. Published by the American Physical Society 2024