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arXiv:2109.01593 · Physical review. B./Physical review. B

Causal optimization method for imaginary-time Green's functions in interacting electron systems

Mancheon Han, Hyoung Joon Choi

Published Sep 7, 2021• 4 citations
arXivHTMLPDF

Abstract

We develop a causal optimization method that ensures causality in numerical calculations of Green's functions in interacting electron systems. Our method removes noncausality of numerical data by finding causal functions closest to the data. By testing our method with an exactly calculable model and applying it to practical dynamical mean-field calculations, we find that intermediate-frequency behaviors of Green's functions are determined solely by causality, and noncausal statistical errors are removed very efficiently. Furthermore, we demonstrate that numerical calculations of the physical branch of the Luttinger-Ward functional can be stabilized by ensuring causality of the noninteracting Green's function. Our method and findings provide a basis for improving stability and efficiency of numerical simulations of quantum many-body systems.

DOI
10.1103/physrevb.104.115112

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We develop a causal optimization method that ensures causality in numerical calculations of Green's functions in interacting electron systems. Our method removes noncausality of numerical data by finding causal functions closest to the data. By testing our method with an exactly calculable model and applying it to practical dynamical mean-field calculations, we find that intermediate-frequency behaviors of Green's functions are determined solely by causality, and noncausal statistical errors are removed very efficiently. Furthermore, we demonstrate that numerical calculations of the physical branch of the Luttinger-Ward functional can be stabilized by ensuring causality of the noninteracting Green's function. Our method and findings provide a basis for improving stability and efficiency of numerical simulations of quantum many-body systems.
We develop a reliable parameter-free analytic continuation method for quantum many-body calculations. Our method is based on a kernel grid, a causal spline, a regularization using the second-derivative roughness penalty, and the L-curve criterion. We also develop the L-curve averaged deviation to estimate the precision of our analytic continuation. To deal with statistically obtained data more efficiently, we further develop a bootstrap-averaged analytic continuation method. In the test using the exact imaginary-frequency Green's function with added statistical error, our method produces the spectral function that converges systematically to the exact one as the statistical error decreases. As an application, we simulate the two-orbital Hubbard model for various electron numbers with the dynamical-mean field theory in the imaginary time and obtain the real-frequency self-energy with our analytic continuation method, clearly identifying a non-Fermi-liquid behavior as the electron number approaches the half filling from the quarter filling. Our analytic continuation can be used widely, and it will facilitate drawing clear conclusions from imaginary-time quantum many-body calculations.
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.
This work contains a computationally efficient Green's function scheme for treating correlated electron-boson systems out of equilibrium. The numerical effort scales linearly with the simulation time, thus competing with that of the fastest quantum methods available. On a more fundamental level the proposed scheme fulfills energy balance as well as other fundamental conservation laws, and it can be systematically improved using standard many-body perturbation theory. We demonstrate its accuracy and stability through benchmarks against exact results in the Holstein model. However, our formulation equally applies to molecules and solids for first-principles transient calculations with quantum bosons, such as photons and phonons.
We present a method to obtain dynamical correlation functions of quantum many-body systems in the real-frequency domain from Monte Carlo data of the corresponding imaginary-time Green's functions. The method is based on a least-squares-fit procedure to solve the integral equation relating imaginary-time and real-frequency correlation functions. To demonstrate its feasibility, we have applied our method to imaginary-time Monte Carlo data for the density correlation function of a simple model of interacting spinless fermions in one dimension. We compare our results to the analytical results, available in the limits of zero and strong interaction. We find that despite the presence of noise in the input data, the real-frequency spectra obtained with our method are qualitatively correct. They reflect accurately the particle-hole and the soliton-antisoliton excitations, present in the noninteracting and in the strongly coupled fermion system, respectively. We also compare our method to other approaches that have been proposed for obtaining dynamical correlation functions via Monte Carlo simulation.
Causality is a fundamental requirement for physical consistency in many-body approximations. In the context of Green's functions, causality requires a positive spectral function and analyticity in the upper half of the complex-frequency plane. These properties are not guaranteed by approximate methods, and formal proofs of causality for the underlying approximation schemes are often difficult to establish. The recently rediscovered Nevanlinna--Pick criterion provides a practical way to test causality directly from Matsubara-frequency data, without performing analytic continuation to the real-frequency axis. In this work, we use the Nevanlinna--Pick criterion to investigate the causality of the ladder dual-fermion approach, a diagrammatic extension of dynamical mean-field theory designed to incorporate nonlocal correlations. The analysis is performed for the spinless Falicov--Kimball model, which provides a useful testing ground because several quantities can be computed exactly or semi-analytically, avoiding Monte Carlo noise. This is particularly important since the Nevanlinna--Pick criterion is highly sensitive to noise in the input data. In addition, we derive several analytical properties of the Pick matrix, providing insight into the structure of the criterion when applied to Green's functions. For small systems, apparent causality violations are observed, but these are found to be associated with finite-size effects. For larger system sizes, no robust signs of causality violations are found, neither within the full ladder dual-fermion approach nor within the leading dual-fermion correction.
The imaginary-time Green's function is a building block of various numerical methods for correlated electron systems. Recently, it was shown that a model-independent compact orthogonal representation of the Green's function can be constructed by decomposing its spectral representation. We investigate the performance of this so-called intermediate representation (IR) from several points of view. First, we develop an efficient algorithm for computing the IR basis functions of arbitrary high degree. Second, for two simple models, we study the number of coefficients required to represent the Green's function within a given tolerance. We show that the number of coefficients grows only as $O(ln\ensuremath{\beta})$ for fermions and converges to a constant for bosons as temperature $T=1/\ensuremath{\beta}$ decreases. Third, we show that this remarkable feature is ascribed to the properties of the physically constructed basis functions. The fermionic basis functions on the real-frequency axis have features whose width is scaled as O(T)O(T)O(T), which is consistent with the low-TTT properties of quasiparticles in a Fermi liquid state. On the other hand, the properties of the bosonic basis functions are consistent with those of spin/orbital susceptibilities at low TTT. These results demonstrate the potential wide applications of the IR to calculations of correlated systems.