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arXiv:2301.08444 · Physical review. B./Physical review. B

Extraction of the frequency moments of spectral densities from imaginary-time correlation function data

Tobias Dornheim, Damar Wicaksono, Juan-Esteban Suarez Cardona, Panagiotis B. Tolias, Maximilian Böhme, Zhandos A. Moldabekov, Michael Hecht, Jan Vorberger

Published Apr 27, 2023• 30 citations
arXivHTMLPDF

Abstract

We introduce an exact framework to compute the positive frequency moments ${M}^{(\ensuremath{\alpha})}(\mathbf{q})=\ensuremath{\langle}{\ensuremath{\omega}}^{\ensuremath{\alpha}}\ensuremath{\rangle}$ of different dynamic properties from imaginary-time quantum Monte Carlo data. As a practical example, we obtain the first five moments of the dynamic structure factor $S(\mathbf{q},\ensuremath{\omega})$ of the uniform electron gas at the electronic Fermi temperature based on ab initio path integral Monte Carlo simulations. We find excellent agreement with known sum rules for $\ensuremath{\alpha}=1,3$, and present results for $\ensuremath{\alpha}=2,4,5$. Our idea can be straightforwardly generalized to other dynamic properties such as the single-particle spectral function $A(\mathbf{q},\ensuremath{\omega})$, and will be useful for a number of applications, including the study of ultracold atoms, exotic warm dense matter, and condensed matter systems.

DOI
10.1103/physrevb.107.155148

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We introduce an exact framework to compute the positive frequency moments M(\ensuremathα)(q)=\ensuremath⟨\ensuremathω\ensuremathα\ensuremath⟩{M}^{(\ensuremath{\alpha})}(\mathbf{q})=\ensuremath{\langle}{\ensuremath{\omega}}^{\ensuremath{\alpha}}\ensuremath{\rangle}M(\ensuremathα)(q)=\ensuremath⟨\ensuremathω\ensuremathα\ensuremath⟩ of different dynamic properties from imaginary-time quantum Monte Carlo data. As a practical example, we obtain the first five moments of the dynamic structure factor S(q,\ensuremathω)S(\mathbf{q},\ensuremath{\omega})S(q,\ensuremathω) of the uniform electron gas at the electronic Fermi temperature based on ab initio path integral Monte Carlo simulations. We find excellent agreement with known sum rules for \ensuremathα=1,3\ensuremath{\alpha}=1,3\ensuremathα=1,3, and present results for \ensuremathα=2,4,5\ensuremath{\alpha}=2,4,5\ensuremathα=2,4,5. Our idea can be straightforwardly generalized to other dynamic properties such as the single-particle spectral function A(q,\ensuremathω)A(\mathbf{q},\ensuremath{\omega})A(q,\ensuremathω), and will be useful for a number of applications, including the study of ultracold atoms, exotic warm dense matter, and condensed matter systems.
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.
Rigorous diagnostics of experiments with warm dense matter are notoriously difficult. A key method is X-ray Thomson scattering (XRTS), but the interpretation of XRTS measurements is usually based on theoretical models that entail various approximations. Recently, Dornheim et al. [ Nat. Commun. 13 , 7911 (2022)] introduced a new framework for temperature diagnostics of XRTS experiments that is based on imaginary-time correlation functions. On the one hand, switching from the frequency to the imaginary-time domain gives one direct access to a number of physical properties, which facilitates the extraction of the temperature of arbitrarily complex materials without relying on any models or approximations. On the other hand, the bulk of theoretical work in dynamic quantum many-body theory is devoted to the frequency domain, and, to the best of our knowledge, the manifestation of physics properties within the imaginary-time density–density correlation function (ITCF) remains poorly understood. In the present work, we aim to fill this gap by introducing a simple, semi-analytical model for the imaginary-time dependence of two-body correlations within the framework of imaginary-time path integrals. As a practical example, we compare our new model to extensive ab initio path integral Monte Carlo results for the ITCF of a uniform electron gas, and find excellent agreement over a broad range of wavenumbers, densities and temperatures. This article is part of the theme issue ‘Dynamic and transient processes in warm dense matter’.
The ab initio path integral Monte Carlo (PIMC) approach is one of the most successful methods in quantum many-body theory. A particular strength of this method is its straightforward access to imaginary-time correlation functions (ITCFs). For example, the well-known density–density ITCF F(q, τ) allows one to estimate the linear response of a given system for all wave vectors q from a single simulation of the unperturbed system. Moreover, it constitutes the basis for the reconstruction of the dynamic structure factor S(q, ω)—a key quantity in state-of-the-art scattering experiments. In this work, we present analogous relations between the nonlinear density response in the quadratic and cubic order of the perturbation strength and generalized ITCFs measuring correlations between up to four imaginary-time arguments. As a practical demonstration of our new approach, we carry out simulations of the warm dense electron gas and find excellent agreement with previous PIMC results that had been obtained with substantially larger computational effort. In addition, we give a relation between a cubic ITCF and the triple dynamic structure factor S(q1, ω1; q2, ω2), which evokes the enticing possibility to study dynamic three-body effects on an ab initio level.
We report the details of an application of the method of maximum entropy to the extraction of spectral and transport properties from the imaginary-time correlation functions generated from quantum Monte Carlo simulations of the nondegenerate, symmetric, single-impurity Anderson model. We find that these physical properties are approximately universal functions of temperature and frequency when these parameters are scaled by the Kondo temperature. We also found that important details for successful extractions included the generation of statistically independent, Gaussian-distributed data, and a good choice of a default model to represent the state of our prior knowledge about the result in the absence of data. We suggest that our techniques are not restricted to the Hamiltonian and quantum Monte Carlo algorithm used here, but that maximum entropy and these techniques lay the general groundwork for the extraction of dynamical information from imaginary-time data generated by other quantum Monte Carlo simulations.
In the first part of the Thesis we mostly concentrate on spectral properties of strongly correlated systems and on their equilibrium properties. This is accomplished by the general concept of imaginary-time dynamics which we apply to a number of different problems in which different strengths of this approach emerge. In Chapter 1 we introduce the formalism that allows for a connection between the quantum and the classical worlds. The connection is established by means of the imaginary-time quantum evolution which, under certain circumstances, is shown to be equivalent to a classical stochastic process. It is further shown that exact static and spectral properties of correlated systems can be obtained when this mapping is feasible. The relationship between the imaginary-time dynamics in different frameworks such as the path-integral and the perturbative one is also underlined. In Chapter 2 we present a specific implementation of the general ideas previously presented. In particular we introduced an extension to lattice systems of the Reptation Monte Carlo algorithm [30] which benefits of a sampling scheme based on directed updates. Specific improvements over the existing methodologies consist in the unbiased evaluation of the imaginary-time path integrals for bosons and a systematic scheme to improve over the Fixed-node approximation for fermions. Applications to the Hubbard and the Heisenberg models are presented. In Chapter 3 we demonstrate the application of the imaginary-time dynamics to the exact study of spectral properties. Subject of our attention is a highly anharmonic and correlated quantum crystal such as Helium 4 at zero temperature.[33] Concerning this system, we have obtained the first ab-initio complete phonon dispersion in good agreement with neutron spectroscopy experiments. Moreover, we have also studied the density excitations of solid helium in a region of wave-vectors in between the collective (phonon) and the single-particle regimes, where the presence of residual coherence in the dynamics shows analogies between the highly anharmonic crystal and the superfluid phase. In Chapter 4 we introduce a novel method, based on the imaginary-time dynamics, to obtain unbiased estimates of fermionic properties.[34] By means of this method and of a very accurate variational state, we provide strong evidence for the stability of a saturated ferromagnetic phase in the high-density regime of the two-dimensional infinite-U Hubbard model. By decreasing the electron density, we observe a discontinuous transition to a paramagnetic phase, accompanied by a divergence of the susceptibility on the paramagnetic side. This behavior, resulting from a high degeneracy among different spin sectors, is consistent with an infinite-order phase transition scenario. 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In Chapter 6 exact diagonalization schemes and renormalization-based methods for one-dimensional systems are introduced. We identify key phenomenological traits resulting from the many-body correlation in closed systems driven sufficiently away from equilibrium.[31] We provide evidences that the dynamics of interacting lattice bosons away from equilibrium can be trapped into extremely long-lived inhomogeneous metastable states. The slowing down of incoherent density excitations above a threshold energy, much reminiscent of a dynamical arrest on the verge of a glass transition, is identified as the key feature of this phenomenon. In Chapter 7 we present an extension to dynamical properties of the Variational Quantum Monte Carlo method.[32] This is accomplished by introducing a general class of time-dependent variational states which is based on the mapping of the many-body dynamics onto an instantaneous ground-state problem. The application of the method to the experimentally relevant quantum quenches of interacting bosons reveals the accuracy and the reliability of the introduced numerical scheme. We indeed obtain for the first time a consistent variational description of the approach to the equilibrium of local observables and underline the origin of the metastability and glassy behavior previously identified. In the very last part we draw our conclusions and show some possible paths for stimulating future research.
We propose a least-squares-fit method to extract real-frequency correlation functions from imaginary-time Monte Carlo data of a many-particle system. We apply it to the density correlation function in a one-dimensional model of spinless fermions. The spectral functions thus obtained reflect accurately the particle-hole and the soliton-antisoliton excitations of the system in the respective limits of noninteracting and strongly coupled fermions. They are in qualitative agreement with the available analytical results.