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Progress of Theoretical Physics

Monte Carlo Method for Quantum Spin Systems Based on the Bloch Coherent State Representation

Hiroshi Takano

Published Feb 1, 1985• 11 citations
Publisher

Abstract

On the basis of the path integral representation of the partition function of a quantum spin system, which is derived by Lieb with use of the Bloch coherent state representation, we propose a new Monte Carlo method for quantum spin systems. In this method, the magnitude S of the spin can be varied freely, because S appears in the path integral only as a parameter. The approximation used in this method can be systematically improved by the 1/n-expansion, where n is the additional dimension in the path integral representation. The formulation and some properties of the method are discussed. Simple examples of the application of the method are also presented.

DOI
10.1143/ptp.73.332

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On the basis of the path integral representation of the partition function of a quantum spin system, which is derived by Lieb with use of the Bloch coherent state representation, we propose a new Monte Carlo method for quantum spin systems. In this method, the magnitude S of the spin can be varied freely, because S appears in the path integral only as a parameter. The approximation used in this method can be systematically improved by the 1/n-expansion, where n is the additional dimension in the path integral representation. The formulation and some properties of the method are discussed. Simple examples of the application of the method are also presented.
A simple Monte Carlo high-temperature expansion method to evaluate the partition function of quantum spin systems which was proposed in the previous paper l) is improved so that all of the needed expansion-coefficients are obtained from interpolating points of less than ten with an interpolation formula. By this improvement, this method is applicable to the finite spin-systems of rather large spin size regardless of dimensionality of the system. As an example, the method is applied to the one-dimensional spin-l/2 isotropic Heisenberg ferromagnetic system.
Monte Carlo simulations are useful tools for modeling quantum systems, but in some cases they suffer from a sign problem, leading to an exponential slow down in their convergence to a value. While solving the sign problem is generically NP hard, many techniques exist for mitigating the sign problem in specific cases; in particular, the technique of deforming the Monte Carlo simulation's plane of integration onto Lefschetz thimbles (complex hypersurfaces of stationary phase) has seen significant success in the context of quantum field theories. We extend this methodology to spin systems by utilizing spin coherent state path integrals to reexpress the spin system's partition function in terms of continuous variables. Using some toy systems, we demonstrate its effectiveness at lessening the sign problem in this setting, despite the fact that the initial mapping to spin coherent states introduces its own sign problem. The standard formulation of the spin coherent path integral is known to make use of uncontrolled approximations; despite this, for large spins they are typically considered to yield accurate results, so it is somewhat surprising that our results show significant systematic errors. Therefore, possibly of independent interest, our use of Lefschetz thimbles to overcome the intrinsic sign problem in spin coherent state path integral Monte Carlo enables a novel numerical demonstration of a breakdown in the spin coherent path integral.
In this article we present two phase space path integrals in terms of coherent states. The first one is derived in a standard fashion but using a nonstandard resolution of the identity in terms of coherent states with different width parameters. The second path integral emerges from a novel phase space representation in terms of coherent states distributed on n-dimensional manifolds embedded in the 2n-dimensional phase space of an n-degree-of-freedom system. These states are shown to form locally complete basis sets since we show that fairly smooth and localized functions can be expanded in terms of them in a unique way. In this representation the time evolution operator can be cast in the form of a phase space path integral. Both path integrals can be evaluated by straightforward implementation of Monte Carlo methods. In both cases the probability amplitude between two phase points turns out to be proportional to the average of the phase, eih∫(p dq−H dt), over a Markov process of phase space paths consisting of classical trajectories interrupted at intervals by Gaussian ‘‘quantum jumps.’’ The numerical evaluation of these expressions through importance sampling is demonstrated.
An introduction is given to the methodology of the path-integral formulation of quantum Monte Carlo simulation. This allows us to reformulate a d −dimensional partition function of a quantum mechanical system into the form of a d +1−dimensional path-integral written in classical mechanical variables. In addition, the quantum mechanical partition function is shown to be related by a process of analytical continuation to the time-development operator of the quantum mechanical system. The evaluation of the path-integral representation is discussed for the Maxwell-Boltzmann ideal gas, a chain of atoms, and liquid 4 He . The properties of liquid Helium I and Helium II are discussed in terms of the two-fluid model, the Ginzburg theory, and recent quantum Monte Carlo Simulation studies. The properties of liquid 3 He are discussed in the context of recent analytical p-wave pairing theories which are compared to the theory of s-wave pairing in the theory of superconducting electron.
The cavity method is a well-established technique for solving classical spin models on sparse random graphs (mean-field models with finite connectivity). Laumann et al. [Phys. Rev. B 78, 134424 (2008)] proposed recently an extension of this method to quantum spin-1/2 models in a transverse field, using a discretized Suzuki-Trotter imaginary-time formalism. Here we show how to take analytically the continuous imaginary-time limit. Our main technical contribution is an explicit procedure to generate the spin trajectories in a path-integral representation of the imaginary-time dynamics. As a side result we also show how this procedure can be used in simple heat bath Monte Carlo simulations of generic quantum spin models. The replica symmetric continuous-time quantum cavity method is formulated for a wide class of models and applied as a simple example on the Bethe lattice ferromagnet in a transverse field. The results of the methods are confronted with various approximation schemes in this particular case. On this system we performed quantum Monte Carlo simulations that confirm the exactness of the cavity method in the thermodynamic limit.