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The Journal of Physical Chemistry A

Computing Energy Levels by Inversion of Imaginary-Time Cross-Correlation Functions

Arne Lüchow, Daniel Neuhauser, Jaejin Ka, Roi Baer, Jianhan Chen, Vladimir A. Mandelshtam

Published Jun 4, 2003• 1 citation
Publisher

Abstract

As proposed by Ceperley and Bernu, the excited-state energies can be obtained from imaginary-time cross-correlation functions (rather than autocorrelation functions) generated by quantum Monte Carlo (QMC) simulations. We show that, when processed by the filter diagonalization method (FDM), the same cross-correlation functions yield excited-state energies of higher accuracy and greater stability. The reason is that unlike the other methods FDM uses all the time domain information available. The superior performance of the cross-correlation FDM is demonstrated for a two-dimensional harmonic oscillator with closely lying eigenvalues. Because QMC does not take advantage of the separability in the Hamiltonian, this model system provides a challenging and generic test case.

DOI
10.1021/jp034381p

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As proposed by Ceperley and Bernu, the excited-state energies can be obtained from imaginary-time cross-correlation functions (rather than autocorrelation functions) generated by quantum Monte Carlo (QMC) simulations. We show that, when processed by the filter diagonalization method (FDM), the same cross-correlation functions yield excited-state energies of higher accuracy and greater stability. The reason is that unlike the other methods FDM uses all the time domain information available. The superior performance of the cross-correlation FDM is demonstrated for a two-dimensional harmonic oscillator with closely lying eigenvalues. Because QMC does not take advantage of the separability in the Hamiltonian, this model system provides a challenging and generic test case.
A new Monte Carlo method for computing excited state properties of quantum systems is introduced. It is a generalization of the transient estimate method used for fermion Green’s function Monte Carlo and of subspace projection methods used for computing eigenstates of matrices. The time dependent autocorrelation function of a vector of trial functions is calculated for a random walk generated by the imaginary-time Schrödinger equation and estimates of energy levels are determined by the eigenvalues of the matrix of correlation functions. This method is especially useful for treating states with the same symmetry as it automatically keeps higher states orthogonal to lower states. The estimated energy converges to the exact eigenvalue with a rate which decreases with increasing excitation energy, thus limiting the method to relatively low-lying states. The method is zero variance in the sense that as better trial functions are introduced, the statistical error decreases to zero. The method has a nontrivial bias which is analyzed. As an illustration, the eigenvalues of a double well are computed.
Conventional diagonalization methods to calculate nuclear energy levels in the framework of the configuration-interaction (CI) shell model approach are prohibited in very large model spaces. The shell model Monte Carlo (SMMC) is a powerful technique for calculating thermal and ground-state observables of nuclei in very large model spaces, but it is challenging to extract nuclear spectra in this approach. We present a novel method to extract low-lying energy levels for given values of a set of good quantum numbers such as spin and parity. The method is based on imaginary-time one-body density correlation matrices that satisfy asymptotically a generalized eigenvalue problem. We validate the method in a light nucleus that allows comparison with exact diagonalization results of the CI shell model Hamiltonian. The method is applicable to other finite-size quantum many-body systems that can be described within a CI shell model approach.
Nuclear energy levels are usually calculated using conventional diagonalization methods in the framework of the configuration-interaction (CI) shell model but these methods are prohibited in very large model spaces. The shell model Monte Carlo (SMMC) method is a powerful technique for calculating thermal and ground-state observables of nuclei in very large model spaces, but it is challenging to extract nuclear spectra in this approach. We present a novel method to extract within SMMC low-lying energy levels for given values of a set of good quantum numbers such as spin and parity. The method is based on imaginary-time correlation matrices (ITCMs) of one-body densities that satisfy asymptotically a generalized eigenvalue problem. We validate the method in a light nucleus that allows comparison with exact diagonalization results of the CI shell-model Hamiltonian. The method is broadly applicable to quantum many-body systems in other disciplines.
In a previous paper we developed a method, Filter-Diagonalization, for extracting eigenvalues and eigenstates of a given operator at any desired energy range. In essence, the method eliminates correlation between distant eigenstates through a short-time filter while correlations between closely lying states are eliminated by diagonalization. Here we extend Filter-Diagonalization. When used to extract eigenvalues for a given operator H, we show that all eigenvalue information is directly extracted from a short segment of the correlation function C(t)=(ψ(0)‖e−iHt‖ψ(0)), or alternately from a small number of residues (ψ(0)‖Rn(H)‖ψ(0)), where ψ(0) is a random initial function and Rn(H) is any desired polynomial expansion in H. The implications of this feature are twofold. First, in contrast to the previous version the wave packet needs only to be propagated once (to prepare C(t)), and eigenstates at all desired energy windows can then be extracted with negligible extra computation time (and negligible storage requirements). In a simulation presented here, accurate eigenvalues are extracted using propagation times which are only a 0.0041 fraction of the ‘‘natural’’ time, i.e., the time by which the relative phase of the two closest eigenstates reaches 2π. The second and more important feature is that the method is automatically suitable for extracting eigenvalues (or normal modes) using a short-time segment of any signal C(t) which is a sum of (unknown) Fourier components (C(t)=∑ndne−iεnt) regardless of its origin. In addition to its use for determining eigenvalues of known operators, the method may also be utilized to extract normal modes from classical-dynamics simulations, eigenstates from real-time Quantum Monte-Carlo studies, frequencies from experimental optical or electrical signals, or be utilized in any other circumstance where a correlation function or general signal is only known for short times (or expensive to generate at long times).
Combined with numerical analytic continuation techniques, quantum Monte Carlo (QMC) methods enable the extraction of real-frequency dynamical properties from imaginary-time correlation functions. However, the efficient computation of imaginary-time correlation functions by QMC simulations can (depending on the particular model used) be challenging, particularly for operators that are off-diagonal in the computational basis. In this work, we present an efficient and general algorithm within the stochastic series expansion (SSE) framework for evaluating imaginary-time correlation functions of both diagonal and off-diagonal operators. The algorithm builds on a discrete imaginary-time slicing of the SSE operator string, which provides correlation functions on a grid of well-defined imaginary-time points with no discretization error. For off-diagonal operators, we derive estimators that integrate directly into the existing SSE directed-loop or cluster updating schemes, introducing only minimal computational overhead. We benchmark the method on the one-dimensional transverse-field Ising model (sampling with cluster updates) and XXZ spin chain (using directed-loop sampling), demonstrating excellent agreement (with only statistical errors) with exact diagonalization of small systems. We also study larger systems to demonstrate efficiency.