Imaginary-time Correlation Function Research Articles

SmoQyDEAC.jl: A differential evolution package for the analytic continuation of imaginary time correlation functions

We introduce the SmoQyDEAC.jl package, a Julia implementation of the Differential Evolution Analytic Continuation (DEAC) algorithm [N. S. Nichols et al., Phys. Rev. E 106, 025312 (2022)] for analytically continuing noisy imaginary time correlation functions to the real frequency axis. Our implementation supports fermionic and bosonic correlation functions on either the imaginary time or Matsubara frequency axes, and treatment of the covariance error in the input data. This paper presents an overview of the DEAC algorithm and the features implemented in the SmoQyDEAC.jl package. It also provides detailed benchmarks of the package’s output against the popular maximum entropy and stochastic analytic continuation methods. The code for this package can be downloaded from our GitHub repository at https://github.com/SmoQySuite/SmoQyDEAC.jl or installed using the Julia package manager. The online documentation, including examples, can be accessed at https://smoqysuite.github.io/SmoQyDEAC.jl/stable/.

Quantum delocalization, structural order, and density response of the strongly coupled electron liquid

We investigate the impact of electronic correlations and quantum delocalization onto the static structure factor and static density response function of the strongly coupled electron liquid. In contrast to a classical system, the density response of the electron liquid vanishes on small length scales due to quantum delocalization effects, which we rigorously quantify in terms of imaginary-time correlation functions and dynamic Matsubara response functions. This allows us to analyze the interplay of structural order and dynamic quantum effects as it manifests itself in the dynamic Matsubara local field correction. Finally, we identify an effective electronic attraction in the spin-offdiagonal static density response when the wavelength of the perturbation is commensurate with the average interparticle distance.

X-ray Thomson scattering absolute intensity from the f-sum rule in the imaginary-time domain

We present a formally exact and simulation-free approach for the normalization of X-ray Thomson scattering (XRTS) spectra based on the f-sum rule of the imaginary-time correlation function (ITCF). Our method works for any degree of collectivity, over a broad range of temperatures, and is applicable even in nonequilibrium situations. In addition to giving us model-free access to electronic correlations, this new approach opens up the intriguing possibility to extract a plethora of physical properties from the ITCF based on XRTS experiments.

Fourier-Matsubara series expansion for imaginary-time correlation functions.

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.

Effective field theories and finite-temperature properties of zero-dimensional superradiant quantum phase transitions

The existence of zero-dimensional superradiant quantum phase transitions seems inconsistent with conventional statistical physics. This work clarifies this apparent inconsistency. We demonstrate the corresponding effective field theories and finite-temperature properties of light-matter interacting systems, and show how this zero-dimensional quantum phase transition occurs. We first focus on the Rabi model, which is a minimum model that hosts a superradiant quantum phase transition. With the path-integral method, we derive the imaginary-time action of the photon degrees of freedom. We also define a dynamical critical exponent as the rescaling between the temperature and the photon frequency, and perform dimensional analysis to the effective action. The dynamical critical exponent shows that the effective theory of the Rabi model is a free scalar field, where a true second-order quantum phase transition emerges. These results are also verified by numerical simulations of imaginary-time correlation functions of the order parameter. Furthermore, we also generalize this method to the Dicke model. Our results make the zero-dimensional superradiant quantum phase transition compatible with conventional statistical physics, and pave the way to understand it in the perspective of effective field theories. Published by the American Physical Society 2024

Fermionic physics from abinitio path integral Monte Carlo simulations of fictitious identical particles.

The abinitio path integral Monte Carlo (PIMC) method is one of the most successful methods in statistical physics, quantum chemistry and related fields, but its application to quantum degenerate Fermi systems is severely hampered by an exponential computational bottleneck: the notorious fermion signproblem. Very recently, Xiong and Xiong [J. Chem. Phys. 157, 094112 (2022)] have suggested to partially circumvent the signproblem by carrying out simulations of fictitious systems guided by an interpolating continuous variable ξ ∈ [-1, 1], with the physical Fermi- and Bose-statistics corresponding to ξ = -1 and ξ = 1. It has been proposed that information about the fermionic limit might be obtained by calculations within the bosonic sector ξ > 0 combined with an extrapolation throughout the fermionic sector ξ < 0, essentially bypassing the signproblem. Here, we show how the inclusion of the artificial parameter ξ can be interpreted as an effective penalty on the formation of permutation cycles in the PIMC simulation. We demonstrate that the proposed extrapolation method breaks down for moderate to high quantum degeneracy. Instead, the method constitutes a valuable tool for the description of large Fermi-systems of weak quantum degeneracy. This is demonstrated for electrons in a 2D harmonic trap and for the uniform electron gas (UEG), where we find excellent agreement (∼0.5%) with exact configuration PIMC results in the high-density regime while attaining a speed-up exceeding 11 orders of magnitude. Finally, we extend the idea beyond the energy and analyze the radial density distribution (2D trap), as well as the static structure factor and imaginary-time density-density correlation function (UEG).

Analysing the dynamic structure of warm dense matter in the imaginary-time domain: theoretical models and simulations.

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'.

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

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

Imaginary-time correlation function thermometry: A new, high-accuracy and model-free temperature analysis technique for x-ray Thomson scattering data

The accurate interpretation of experiments with matter at extreme densities and pressures is a notoriously difficult challenge. In a recent work [Dornheim et al., Nat. Commun. 13, 7911 (2022)], we have introduced a formally exact methodology that allows extracting the temperature of arbitrary complex materials without any model assumptions or simulations. Here, we provide a more detailed introduction to this approach and analyze the impact of experimental noise on the extracted temperatures. In particular, we extensively apply our method both to synthetic scattering data and to previous experimental measurements over a broad range of temperatures and wave numbers. We expect that our approach will be of high interest to a gamut of applications, including inertial confinement fusion, laboratory astrophysics, and the compilation of highly accurate equation-of-state databases.

Electronic density response of warm dense matter

Matter at extreme temperatures and pressures—commonly known as warm dense matter (WDM)—is ubiquitous throughout our Universe and occurs in astrophysical objects such as giant planet interiors and brown dwarfs. Moreover, WDM is very important for technological applications such as inertial confinement fusion and is realized in the laboratory using different techniques. A particularly important property for the understanding of WDM is given by its electronic density response to an external perturbation. Such response properties are probed in x-ray Thomson scattering (XRTS) experiments and are central for the theoretical description of WDM. In this work, we give an overview of a number of recent developments in this field. To this end, we summarize the relevant theoretical background, covering the regime of linear response theory and nonlinear effects, the fully dynamic response and its static, time-independent limit, and the connection between density response properties and imaginary-time correlation functions (ITCF). In addition, we introduce the most important numerical simulation techniques, including path-integral Monte Carlo simulations and different thermal density functional theory (DFT) approaches. From a practical perspective, we present a variety of simulation results for different density response properties, covering the archetypal model of the uniform electron gas and realistic WDM systems such as hydrogen. Moreover, we show how the concept of ITCFs can be used to infer the temperature from XRTS measurements of arbitrary complex systems without the need for any models or approximations. Finally, we outline a strategy for future developments based on the close interplay between simulations and experiments.

Chiral symmetry restoration and properties of Goldstone bosons at finite temperature* *Supported in part by the National Natural Science Foundation of China (12075117, 11775112, 11535005, 11690030, 11905104, 11805024, 11947406) and Jiangsu Provincial Natural Science Foundation of China (BK20180323)

We study chiral symmetry restoration by analyzing thermal properties of QCD's(pseudo-) Goldstone bosons, especially the pion. The meson properties are obtained from the spectral densities of mesonic imaginary-time correlation functions. To obtain the correlation functions, we solve the Dyson-Schwinger equations and the inhomogeneous Bethe-Salpeter equations in the leading symmetry-preserving rainbow-ladder approximation. In chiral limit, the pion and its partner sigma degenerate at the critical temperature . At , it was found that the pion rapidly dissociates, which signals deconfinement phase transition. Beyond the chiral limit, the pion dissociation temperature can be used to define the pseudo-critical temperature of the chiral phase crossover, which is consistent with that obtained by the maximum point of chiral susceptibility. A parallel analysis for kaon and pseudoscalar suggests that heavy mesons may survive above .

Parameter-free differential evolution algorithm for the analytic continuation of imaginary time correlation functions.

We report on differential evolution for analytic continuation: a parameter-free evolutionary algorithm to generate the dynamic structure factor from imaginary time correlation functions. Our approach to this long-standing problem in quantum many-body physics achieves enhanced spectral fidelity while using fewer compute (CPU) hours. The need for fine-tuning of algorithmic control parameters is eliminated by embedding them within the genome to be optimized for this evolutionary computation-based algorithm. Benchmarks are presented for models where the dynamic structure factor is known exactly and experimentally relevant results are included for quantum Monte Carlo simulations of bulk ^{4}He below the superfluid transition temperature.

Application of the imaginary time hierarchical equations of motion method to calculate real time correlation functions.

We investigate the application of the imaginary time hierarchical equations of motion method to calculate real time quantum correlation functions. By starting from the path integral expression for the correlated system-bath equilibrium state, we first derive a new set of equations that decouple the imaginary time propagation and the calculation of auxiliary density operators. The new equations, thus, greatly simplify the calculation of the equilibrium correlated initial state that is subsequently used in the real time propagation to obtain the quantum correlation functions. It is also shown that a periodic decomposition of the bath imaginary time correlation function is no longer necessary in the new equations such that different decomposition schemes can be explored. The applicability of the new method is demonstrated in several numerical examples, including the spin-Boson model, the Holstein model, and the double-well model for proton transfer reaction.

Hybrid quantum-classical algorithm for computing imaginary-time correlation functions

Quantitative descriptions of strongly correlated materials pose a considerable challenge in condensed matter physics and chemistry. A promising approach to address this problem is quantum embedding methods. In particular, the dynamical mean-field theory (DMFT) maps the original system to an effective quantum impurity model comprising correlated orbitals embedded in an electron bath. The biggest bottleneck in DMFT calculations is numerically solving the quantum impurity model, i.e., computing the Green's function. Past studies have proposed theoretical methods to compute the Green's function of a quantum impurity model in polynomial time using a quantum computer. So far, however, efficient methods for computing the imaginary-time Green's functions have not been established despite the advantages of the imaginary-time formulation. We propose a quantum-classical hybrid algorithm for computing imaginary-time Green's functions on quantum devices with limited hardware resources by applying the variational quantum simulation. Using a quantum circuit simulator, we verified this algorithm by computing Green's functions for a dimer model as well as a four-site impurity model obtained by DMFT calculations of the single-band Hubbard model, although our method can be applied to general imaginary-time correlation functions.

Minimally entangled typical thermal state algorithms for finite temperature Matsubara Green functions

We extend finite-temperature tensor network methods to compute Matsubara imaginary-time correlation functions, building on the minimally entangled typical thermal states (METTS) and purification algorithms. While imaginary-time correlation functions are straightforward to formulate with these methods, care is needed to avoid convergence issues that would result from naive estimators. As a benchmark, we study the single-band Anderson impurity model, even though the algorithm is quite general and applies to lattice models. The special structure of the impurity model benchmark system and our choice of basis enable techniques such as reuse of high-probability METTS for increasing algorithm efficiency. The results are competitive with state-of-the-art continuous time Monte Carlo. We discuss the behavior of computation time and error as a function of the number of purified sites in the Hamiltonian.

Nonlinear density response from imaginary-time correlation functions: Ab initio path integral Monte Carlo simulations of the warm dense electron gas.

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.

Dynamical spin excitations of the topological Haldane gapped phase in the S=1 Heisenberg antiferromagnetic chain with single-ion anisotropy

We study the dynamical spin excitations of the one-dimensional $S=1$ Heisenberg antiferromagnetic chain with single-ion anisotropy by using quantum Monte Carlo simulations and stochastic analytic continuation of imaginary-time correlation function. Using the transverse dynamic spin structure factor, we observe the quantum phase transition with a critical point between the topological Haldane gapped phase and the trivial phase. At the quantum critical point, we find a broad continuum characterized by the Tomonaga-Luttinger liquid similar to a $S=1/2$ Heisenberg antiferromagnetic chain. We further identify that the elementary excitations are fractionalized spinons.

Measuring the imaginary-time dynamics of quantum materials

ABSTRACT Theoretical analysis typically involves imaginary-time correlation functions. Inferring real-time dynamical response functions from this information is notoriously difficult. However, as we articulate here, it is straightforward to compute imaginary-time correlators from the measured frequency dependence of (real-time) response functions. In addition to facilitating comparison between theory and experiment, the proposed approach can be useful in extracting certain aspects of the (long-time relaxational) dynamics from a complex data set. We illustrate this with an analysis of the nematic response inferred from Raman scattering spectroscopy on the iron-based superconductor , which includes a new method for identifying a putative quantum-critical contribution to that response.

Ab initio path integral Monte Carlo approach to the static and dynamic density response of the uniform electron gas

In a recent Letter [T. Dornheim et al., Phys. Rev. Lett. 121, 255001 (2018)] we have presented ab initio results for the dynamic structure factor $S(\mathbf{q},\ensuremath{\omega})$ of the uniform electron gas for conditions ranging from the warm dense matter regime to the strongly correlated electron liquid. This was achieved on the basis of exact path integral Monte Carlo data by stochastically sampling the dynamic local field correction $G(\mathbf{q},\ensuremath{\omega})$. In this paper we introduce in detail this reconstruction method and provide several practical demonstrations. Moreover, we thoroughly investigate the associated imaginary-time density-density correlation function $F(\mathbf{q},\ensuremath{\tau})$. The latter also gives us access to the static density-response function $\ensuremath{\chi}(\mathbf{q})$ and static local field correction $G(\mathbf{q})$, which are compared to standard dielectric theories like the widespread random phase approximation. In addition, we study the high-frequency limit of $G(\mathbf{q},\ensuremath{\omega})$ and provide extensive results for the dynamic structure factor for different densities and temperatures. Finally, we discuss the implications of our findings for warm dense matter research and the interpretation of experiments.

Out-of-time-ordered correlators of the Hubbard model: Sachdev-Ye-Kitaev strange metal in the spin-freezing crossover region

The Sachdev-Ye-Kitaev (SYK) model describes a strange metal that shows peculiar non-Fermi liquid properties without quasiparticles. It exhibits a maximally chaotic behavior characterized by out-of-time-ordered correlators (OTOCs), and is expected to be a holographic dual to black holes. Recently, a striking similarity between the SYK model and the Hund-coupling induced spin-freezing crossover in multi-orbital Hubbard models has been pointed out. To further explore this connection, we study OTOCs for fermionic Hubbard models, which are prototypical models for strongly correlated electrons in solids. We introduce an imaginary-time four-point correlation function with an appropriate time ordering, which by means of the spectral representation and the out-of-time-order fluctuation-dissipation theorem can be analytically continued to real-time OTOCs. Based on this approach, we numerically evaluate real-time OTOCs for Hubbard models in the thermodynamic limit, using the dynamical mean-field theory in combination with a numerically exact continuous-time Monte Carlo impurity solver. The results for the single-orbital model show that a certain spin-related OTOC captures local moment formation in the vicinity of the metal-insulator transition, while the self-energy does not show SYK-like non-Fermi liquid behavior. On the other hand, for the two- and three-orbital models with nonzero Hund coupling we find that the OTOC exhibits a rapid damping at short times and an approximate power-law decay at longer times in the spin-freezing crossover regime characterized by fluctuating local moments and a non-Fermi liquid self-energy $\Sigma(\omega) \sim \sqrt{\omega}$. These results are in a good agreement with the behavior of the SYK model, providing firm evidence for the close relation between the spin-freezing crossover physics of multi-orbital Hubbard models and the SYK strange metal.