A Fourier-Matsubara series expansion is derived for imaginary-time correlation functions that constitutes the imaginary-time generalization of the infinite Matsubara series for equal-time correlation functions. The expansion is consistent with all known exact properties of imaginary-time correlation functions and opens up new avenues for the utilization of quantum Monte Carlo simulation data. Moreover, the expansion drastically simplifies the computation of imaginary-time density-density correlation functions with the finite temperature version of the self-consistent dielectric formalism. Its existence underscores the utility of imaginary-time as a complementary domain for many-body physics.
A Fourier-Matsubara series expansion is derived for imaginary-time correlation functions that constitutes the imaginary-time generalization of the infinite Matsubara series for equal-time correlation functions. The expansion is consistent with all known exact properties of imaginary-time correlation functions and opens up new avenues for the utilization of quantum Monte Carlo simulation data. Moreover, the expansion drastically simplifies the computation of imaginary-time density-density correlation functions with the finite temperature version of the self-consistent dielectric formalism. Its existence underscores the utility of imaginary-time as a complementary domain for many-body physics.
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.
An accurate theoretical description of the dynamic properties of correlated quantum many-body systems, such as the dynamic structure factor S(q, ω), is important in many fields. Unfortunately, highly accurate quantum Monte Carlo methods are usually restricted to the imaginary time domain, and the analytic continuation of the imaginary-time density–density correlation function F(q, τ) to real frequencies is a notoriously hard problem. Here, it is argued that often no such analytic continuation is required because by definition, F(q, τ) contains the same physical information as does S(q, ω), only represented unfamiliarly. Specifically, it is shown how one can directly extract key information such as the temperature or quasi-particle excitation energies from the τ domain, which is highly relevant for equation-of-state measurements of matter under extreme conditions [T. Dornheim et al., Nat. Commun. 13, 7911 (2022)]. As a practical example, ab initio path-integral Monte Carlo results for the uniform electron gas (UEG) are considered, and it is shown that even nontrivial processes such as the roton feature of the UEG at low density [T. Dornheim et al., Commun. Phys. 5, 304 (2022)] are manifested straightforwardly in F(q, τ). A comprehensive overview is given of various useful properties of F(q, τ) and how it relates to the usual dynamic structure factor. In fact, working directly in the τ domain is advantageous for many reasons and opens up multiple avenues for future applications.
We present a way of combining real-time path integral Monte Carlo simulations with a maximum entropy numerical analytic continuation scheme in a new approach for calculating time correlation functions for finite temperature many body quantum systems. The real-time dynamics is expressed in the form of the symmetrized time correlation function, which is suitable for Monte Carlo methods, and several simulation techniques are presented for evaluating this function accurately up to moderate values of time. The symmetrized time correlation function is then analytically continued in combination with imaginary time data to obtain the real-time correlation function. We test this approach on several exactly solvable problems, including two one-dimensional systems, as well two cases of vibrational relaxation of a system coupled to a dissipative environment. The computed time correlation functions are in good agreement with exact results over several multiples of the thermal time βℏ, and exhibit a significant improvement over analytic continuation of imaginary time correlation functions. Moreover, we show how the method can be systematically improved.
Measuring vibronic spectra probes dynamical processes in molecular systems. When interpreted via suitable theoretical tools, the experimental data provides comprehensive information about the system in question. For complex many-body problems, such an approach usually requires the formulation of proper classical-like approximations, which is particularly challenging if multiple electronic states are involved. In this work, we express the imaginary-time shifted time correlation function and, thus, the vibronic spectrum in terms of the so-called Matsubara dynamics, which combines quantum statistics and classical-like dynamics. By applying the Matsubara approximation in the adiabatic limit, we derive a formal generalization of the existing Matsubara dynamics formalism to multiple potential energy surfaces (PESs), which, however, does not feature all the defining properties of its single-PES counterpart though suffering equally from the sign problem. The mathematical analysis for two shifted harmonic oscillators suggests a new modified method to practically simulate the standard correlation function via Matsubara-like dynamics. Importantly, this modified method samples the thermal Wigner function without suffering from the sign problem and yields an accurate approximation to the vibronic absorption spectrum, not only for the harmonic system but also for the anharmonic one.
Diagrammatic expansions are a central tool for treating correlated electron systems. At thermal equilibrium, they are most naturally defined within the Matsubara formalism. However, extracting any dynamic response function from a Matsubara calculation ultimately requires the ill-defined analytical continuation from the imaginary- to the real-frequency domain. It was recently proposed [A. Taheridehkordi et al., Phys. Rev. B 99, 035120 (2019)] that the internal Matsubara summations of any interaction-expansion diagram can be performed analytically by using symbolic algebra algorithms. The result of the summations is then an analytical function of the complex frequency rather than Matsubara frequency. Here we apply this principle and develop a diagrammatic Monte Carlo technique which yields results directly on the real-frequency axis. We present results for the self-energy $\mathrm{\ensuremath{\Sigma}}(\ensuremath{\omega})$ of the doped $32\ifmmode\times\else\texttimes\fi{}32$ cyclic square-lattice Hubbard model in a nontrivial parameter regime, where signatures of the pseudogap appear close to the antinode. We discuss the behavior of the perturbation series on the real-frequency axis and in particular show that one must be very careful when using the maximum entropy method on truncated perturbation series. Our approach holds great promise for future application in cases when analytical continuation is difficult and moderate-order perturbation theory may be sufficient to converge the result.