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Quantum spin systems

Joseph Boudreau, Eric S. Swanson

Published Dec 21, 2017• 0 citations
Publisher

Abstract

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.

DOI
10.1093/oso/9780198708636.003.0022

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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.
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.
In this paper we describe a numerical method for measuring ground-state expectation values and energies of quantum spin systems. The technique exploits the similarity between the Euclidean-time Schr\"odinger equation and a random walk in state space to generate configurations with a weighted distribution proportional to \ensuremath{\Vert}${\ensuremath{\psi}}_{0}$${\ensuremath{\Vert}}^{2}$. A model application to a single spin is described first, followed by application of the method to the isotropic spin-(1/2 Heisenberg antiferromagnet. For the antiferromagnet we measure the singlet ground-state energy and the singlet-triplet energy gap on chains of up to 48 spins. These results are shown to be in good agreement with energies obtained using Lanczos techniques on chains of up to 20 spins for the energy gap and 24 spins for the ground-state energy.
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.
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.
A quantum Monte Carlo simulation scheme for spin systems is presented. The method is a generalization of Handscomb's method but applicable to any length of the spin, i.e., when the spin traces cannot be evaluated analytically. The Monte Carlo sampling is extended to the space of spin vectors in addition to the usual operator-index sequences. An important technical point is that the index sequences are augmented with the aid of unit operators to a constant, self-consistently determined length. The scheme is applied to the one-dimensional antiferromagnetic spin-S Heisenberg model. Results at low temperatures are reported for S=1 and S=3/2 and system sizes up to N=64. The computed magnetic structure factor in the S=1 chain is in agreement with earlier ground-state calculations. For S=3/2 we find the exponent \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=0.49\ifmmode\pm\else\textpm\fi{}0.04 for the divergence of the antiferromagnetic structure factor. Further, the susceptibility as a function of the wave number is computed. For S=1 the staggered susceptibility \ensuremath{\chi}(\ensuremath{\pi}) at T=0 is found to take the value 20.0\ifmmode\pm\else\textpm\fi{}1.5 in units such that \ensuremath{\chi}(q)\ensuremath{\rightarrow}${\mathit{T}}^{\mathrm{\ensuremath{-}}1}$ at high temperatures (with the temperature scale defined by kB{\mathit{k}}_{\mathit{B}}kB​=1). For S=3/2 we obtain the exponent \ensuremath{\gamma}=1.45\ifmmode\pm\else\textpm\fi{}0.05 for the divergence of the staggered susceptibility.