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arXiv:cond-mat/9906155 · Physical review. B, Condensed matter

Maximum entropy method of obtaining thermodynamic properties from quantum Monte Carlo simulations

Carey Huscroft, Richard Gass, Mark Jarrell

Published Apr 1, 2000• 22 citations
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Abstract

We describe a method of obtaining thermodynamic properties of quantum systems using Bayesian inference maximum entropy techniques. The method is applicable to energy values sampled at a discrete set of temperatures from quantum Monte Carlo simulations. The internal energy and the specific heat of the system are easily obtained as are errorbars on these quantities. The entropy and the free energy are also obtainable. No assumptions as to the specific functional form of the energy are made. The use of a priori information, such as a sum rule on the entropy, is built into the method. As a nontrivial example of the method, we obtain the specific heat of the three-dimensional periodic Anderson model.

DOI
10.1103/physrevb.61.9300

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We describe a method of obtaining thermodynamic properties of quantum systems using Bayesian inference maximum entropy techniques. The method is applicable to energy values sampled at a discrete set of temperatures from quantum Monte Carlo simulations. The internal energy and the specific heat of the system are easily obtained as are errorbars on these quantities. The entropy and the free energy are also obtainable. No assumptions as to the specific functional form of the energy are made. The use of a priori information, such as a sum rule on the entropy, is built into the method. As a nontrivial example of the method, we obtain the specific heat of the three-dimensional periodic Anderson model.
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Bayesian probability theory along with the maximum entropy concept is widely used for inferential problems, particularly to infer dynamic properties of strongly correlated fermion systems from quantum Monte Carlo (QMC) imaginary time data. In current QMC applications, however, the error covariance of the QMC data is not treated consistently. Here we present a consistent Bayesian analysis of all the information provided by the QMC simulation. This approach allows us to infer reliable results with the least amount of computer time.
An outstanding problem in the simulation of condensed matter phenomena is how to obtain dynamical information. We consider the numerical analytic continuation of imaginary time Quantum Monte Carlo data to obtain real frequency spectral functions. We suggest an image reconstruction approach which has been widely applied to data analysis in experimental research, the Maximum Entropy Method (MaxEnt). We report encouraging preliminary results for the Fano-Anderson model of an impurity state in a continuum. The incorporation of additional prior information, such as sum rules and asymptotic behavior, can be expected to significantly improve results. We also compare MaxEnt to alternative methods. 17 refs., 4 figs.
We present a history-dependent Monte Carlo scheme for the efficient calculation of the free energy of quantum systems inspired by Wang-Landau and metadynamics. In the two-dimensional quantum Ising model, chosen here for illustration, the accuracy of free energy, critical temperature, and specific heat is demonstrated as a function of simulation time and successfully compared with the best available approaches. The approach is based on a path integral formulation of the quantum problem and can be applied without modifications to quantum Hamiltonians of any level of complexity. The combination of high accuracy and performance with a much broader applicability is a major advance with respect to other available methods.
Imaginary-time correlation functions calculated by quantum Monte Carlo (QMC) are analyzed using the maximum entropy method (MaxEnt) to determine the ground-state energy and spectral overlap function. In contrast to earlier applications of MaxEnt, the data is obtained from importanced-sampled zero-temperature quantum Monte Carlo simulations. The analysis includes two steps. First, that spectral overlap function and ground state energy which maximizes the entropy and agrees with the QMC correlation functions is obtained. Then the errors in the energy are evaluated by averaging over all the possible images (average MaxEnt method), the multidimensional integrals being computed using the Metropolis algorithm. The central feature of this approach is that all the information present in the correlation functions is used in the only way consistent with fundamental probabilistic hypotheses. This allows us to fully exploit the information contained in the correlation functions at small imaginary times, thus avoiding large statistical fluctuations associated with large imaginary times. In addition, the computed errors include both the statistical errors and systematic extrapolation errors. The method is illustrated with a harmonic oscillator and the four-electron LiH molecule.