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arXiv:1311.6893 · Physical Review B

Continuous-pole-expansion method to obtain spectra of electronic lattice models

Peter Staar, Bart Ydens, Anton Kozhevnikov, Jean‐Pierre Locquet, T. C. Schulthess

Published Jun 11, 2014• 7 citations
arXivHTMLPDF

Abstract

We present a new algorithm to analytically continue the self-energy of quantum many-body systems from Matsubara frequencies to the real axis. The method allows straightforward, unambiguous computation of electronic spectra for lattice models of strongly correlated systems from self-energy data that has been collected with state-of-the-art continuous time solvers within dynamical mean-field simulations. Using well-known analytical properties of the self-energy, the analytic continuation is cast into a constrained minimization problem that can be formulated as a quadratic programmable optimization with linear constraints. The algorithm is validated against exactly solvable finite-size problems, showing that all features of the spectral function near the Fermi level are very well reproduced and coarse features are reproduced for all energies. The method is applied to two well-known lattice problems, the two-dimensional Hubbard model at half filling, where the momentum dependence of the gap formation is studied, as well as a multiband model of NiO, for which the spectral function can be directly compared to experiment. Agreement with published results is very good.

DOI
10.1103/physrevb.89.245114

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We present a new algorithm to analytically continue the self-energy of quantum many-body systems from Matsubara frequencies to the real axis. The method allows straightforward, unambiguous computation of electronic spectra for lattice models of strongly correlated systems from self-energy data that has been collected with state-of-the-art continuous time solvers within dynamical mean-field simulations. Using well-known analytical properties of the self-energy, the analytic continuation is cast into a constrained minimization problem that can be formulated as a quadratic programmable optimization with linear constraints. The algorithm is validated against exactly solvable finite-size problems, showing that all features of the spectral function near the Fermi level are very well reproduced and coarse features are reproduced for all energies. The method is applied to two well-known lattice problems, the two-dimensional Hubbard model at half filling, where the momentum dependence of the gap formation is studied, as well as a multiband model of NiO, for which the spectral function can be directly compared to experiment. Agreement with published results is very good.
In this paper, we propose an analytic continuation method to extract real-frequency spectral functions from imaginary-frequency Green's functions of quantum many-body systems. This method is based on the pole representation of Matsubara Green's function and a stochastic sampling procedure is utilized to optimize the amplitudes and locations of poles. In order to capture narrow peaks and sharp band edges in the spectral functions, a constrained sampling algorithm and a self-adaptive sampling algorithm are developed. To demonstrate the usefulness and performance of this method, we at first apply it to study the spectral functions of representative fermionic and bosonic correlators. Then we employ this method to tackle the analytic continuation problems of matrix-valued Green's functions. The synthetic Green's functions, as well as realistic correlation functions from finite-temperature quantum many-body calculations, are used as input. The benchmark results demonstrate that this method is capable of reproducing most of the key characteristics in the spectral functions. The sharp, smooth, and multipeak features in both low- and high-frequency regions of spectral functions could be accurately resolved, which overcomes one of the main limitations of the traditional maximum entropy method. More importantly, it exhibits excellent robustness with respect to noisy and incomplete input data. The causality of spectral function is always satisfied even in the presence of sizable noises. As a by-product, this method could derive a fitting formula for the Matsubara data, which provides a compact approximation to the many-body Green's functions. Hence, we expect that this method could become a pivotal workhorse for numerically analytic continuation and be broadly useful in many applications.
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.
The investigation of quantum many-particle systems is a central goal of modern condensed-matter physics. Complex physical phenomena such as high-temperature superconductivity, heavy fermion behavior, and correlation-driven metal-insulator transitions are central aspects of this particular field of science. The Hubbard model is a paradigm of correlated-electron physics. It is one of the simplest quantum-mechanical lattice models capable of capturing relevant physical aspects of strongly correlated electron systems. The Hubbard model also plays an important role in the explanation of many-body phenomena observed in ultra-cold atoms trapped in optical lattices. In this thesis, we study the three-dimensional Hubbard model using quantum Monte Carlo simulations (QMC). Our aim is a precise numerical study of its properties in the thermodynamic limit. However, a direct investigation of lattice models in the thermodynamic limit is generally impossible because QMC simulations cannot treat infinite lattice sizes. Therefore, approximative schemes that allow calculations directly in the thermodynamic limit are advantageous. A widely applied approximation of this kind is the dynamical mean-field theory (DMFT) which maps the infinite-lattice problem onto a single-site impurity model embedded in a mean field. We apply cluster mean-field theories which represent a systematic extension of the DMFT by expanding the single impurity to a finite cluster. The second focus of this thesis is the calculation of spectral properties of the Hubbard model. QMC methods map the quantum-mechanical lattice model on a classical one at the expense of an additional imaginary time dimension. For this reason, QMC algorithms can only provide dynamical data on the imaginary time or frequency axis. The necessary analytic continuation to the physically relevant real axis has proven to be a difficult problem that has to be approached by advanced data analysis tools. The maximum-entropy method (MEM) is the standard technique to handle this problem. We apply the MEM to the problem of extracting momentum-resolved single particle spectra from QMC simulations of the 3D Hubbard model. Furthermore, we discuss and expand algorithms that were recently proposed as alternatives to the standard MEM approach.
The self-energy Pad\'e approach is discussed as an analytic continuation of numerical methods that evaluate correlation functions in imaginary time. Instead of direct analytic continuation of the correlation functions, the Pad\'e method is applied for the self-energy in this approach, which is then used for deriving the Green's functions at real energies. We demonstrate that the self-energy Pad\'e approach is more stable and robust against statistical errors compared to the direct way. The characteristics and success of the self-energy Pad\'e approach are analyzed by actual calculations for the illustrative examples of the noninteracting Anderson lattice and interacting Hubbard model. A zero-gap Kondo lattice model with linearly vanishing conduction electron density of states at the Fermi level is also studied by using the self-energy Pad\'e approach as an analytic continuation. We investigate the properties of the Kondo insulating state including the dependence of the insulating gap on the Kondo coupling and coherence effects. Furthermore, we identify two energy scales from dynamic and thermodynamic quantities that are associated as a direct and an indirect gap in a band hybridization picture.
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.
A new algorithm for analytic continuation of noisy quantum Monte Carlo (QMC) data from the Matsubara domain to real frequencies is proposed. Unlike the widely used maximum-entropy (MaxEnt) procedure, our method is linear with respect to input data and can therefore be applied to off-diagonal components of a thermal Green's function, or to a self-energy function. The latter possibility is used to analyze QMC results for the half-filled single-band Hubbard model on a Bethe lattice at a low temperature. Our method qualitatively resolves peaks near the inner edges of the Hubbard bands in the vicinity of a Mott transition, whereas a MaxEnt procedure does not. An existence of such structures has been clearly established before in a high-precision D-DMRG calculation by Karski et al. We also analyze a stability of the new method subject to changes of adjustable parameters.