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arXiv:cond-mat/9905247 · Physical review. B, Condensed matter

Auxiliary-field Monte Carlo for quantum spin and boson systems

M. Ulmke, Richard T. Scalettar

Published Apr 1, 2000• 4 citations
arXivHTMLPDF

Abstract

We describe an algorithm for the numerical simulation of quantum spin and boson systems. The method is based on the Trotter decomposition in imaginary time and a decoupling by auxiliary Ising spins. It can be applied, in principle, to arbitrary (random) spin systems, however, in general it suffers from the ``minus-sign problem.'' This problem is absent in the case of the Ising model in a transverse field in arbitrary dimensions and geometries. We show test results for the spin-1/2 $\mathrm{XY}$ model, the one-dimensional transverse Ising model with disorder, and the phase transition induced by a transverse field in the two-dimensional Ising model.

DOI
10.1103/physrevb.61.9607

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We describe an algorithm for the numerical simulation of quantum spin and boson systems. The method is based on the Trotter decomposition in imaginary time and a decoupling by auxiliary Ising spins. It can be applied, in principle, to arbitrary (random) spin systems, however, in general it suffers from the ``minus-sign problem.'' This problem is absent in the case of the Ising model in a transverse field in arbitrary dimensions and geometries. We show test results for the spin-1/2 XY\mathrm{XY}XY model, the one-dimensional transverse Ising model with disorder, and the phase transition induced by a transverse field in the two-dimensional Ising model.
Abstract Over the past decades, quantum Monte Carlo methods have proven to be very valuable for the study of quantum‐mechanical many‐body systems. The basic ingredient of theses methods is a decomposition of the imaginary‐time evolution operator into path integrals. These paths can then be sampled using Monte Carlo techniques. Though very successful for bosons, these methods often have troubles in dealing with fermions: the fermionic antisymmetry leads to a ‘sign problem’. One approach, the auxiliary‐field method, has yielded slightly better results in dealing with the sign problem. Some recent developments on this method are highlighted. As a case study, it is applied to a pairing model for Fe nuclei, where indications are found for a phase transition related to nucleon–nucleon pairing.
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.
Abstract We present a quantum Monte Carlo algorithm for the simulation of general quantum and classical many-body models within a single unifying framework. The algorithm builds on a power series expansion of the quantum partition function in its off-diagonal terms and is both parameter-free and Trotter error-free. In our approach, the quantum dimension consists of products of elements of a permutation group. As such, it allows for the study of a very wide variety of models on an equal footing. To demonstrate the utility of our technique, we use it to clarify the emergence of the sign problem in the simulations of non-stoquastic physical models. We showcase the flexibility of our algorithm and the advantages it offers over existing state-of-the-art by simulating transverse-field Ising model Hamiltonians and comparing the performance of our technique against that of the stochastic series expansion algorithm. We also study a transverse-field Ising model augmented with randomly chosen two-body transverse-field interactions.
Frustrated spin systems generically suffer from the negative sign problem inherent in Monte Carlo methods. Since the severity of this problem is formulation dependent, optimization strategies can be put forward. We introduce a phase pinning approach in the realm of the auxiliary field quantum Monte Carlo algorithm. If we can find an antiunitary operator that commutes with the one-body Hamiltonian coupled to the auxiliary field, then the phase of the action is pinned to 0 and $\ensuremath{\pi}$. For generalized Kitaev models, we can successfully apply this strategy and observe a remarkable improvement of the average sign. We use this method to study the thermodynamical and dynamical properties of the Kitaev-Heisenberg model down to temperatures corresponding to half of the exchange coupling constant. Our dynamical data reveal finite temperature properties of ordered and spin-liquid phases inherent in this model.
The auxiliary-field quantum Monte Carlo method is reviewed. The Hubbard-Stratonovich transformation converts an interacting Hamiltonian into a non-interacting Hamiltonian in a time-dependent stochastic field, allowing calculation of the resulting functional integral by Monte Carlo methods. The method is presented in a sufficiently general form to be applicable to any Hamiltonian with one- and two-body terms, with special reference to the Heisenberg model and one- and many-band Hubbard models. Many physical correlation functions can be related to correlation functions of the auxiliary field; general results are given here. Issues relating to the choice of auxiliary fields are addressed; operator product identities change the relative dimensionalities of the attractive and repulsive parts of the interaction. Frequently the integrand is not positive-definite, rendering numerical evaluation unstable. If the auxiliary field violates time-reversal invariance, the integrand is complex and this sign problem becomes a phase problem. The origin of this sign or phase is examined from a number of geometrical and other viewpoints and illustrated by simple examples: the phase problem by the spin (1/2) Heisenberg model, and the sign problem by the attractive SU(N) Hubbard model on a triangular molecule with negative hopping integrals. In the latter case, widely studied in the Jahn-Teller literature, the sign is due neither to fermions nor spin, but to frustration. This system is used to illustrate a number of suggested interpretations of the sign problem.
We introduce the transverse Ising model as a prototype for discussing quantum phase transitions. Mean field theory and its application to superconductivity by BCS are discussed, as well as real space renormalization group techniques in one dimension. Next, we introduce the Suzuki-Trotter formalism to show the correspondence between d-dimensional quantum systems and (d + 1)-dimensional classical systems. We then discuss transverse Ising spin glass models, namely the Sherrington-Kirkpatrick (SK) model, the Edwards-Anderson (EA) model and the ±J model. We briefly discuss the mean field, exact diagonalization and quantum Monte Carlo results for their phase diagrams. We discuss the question of replica symmetry restoration in quantum spin glasses due to the possibility of tunneling through the barriers. Finally, we discuss the quantum annealing technique and indicate its relationship with replica symmetry restoration in quantum spin glasses.