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Register a new userSpM: Sparse modeling tool for analytic continuation of imaginary-time Greens function
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We present SpM, a sparse modeling tool for the analytic continuation of imaginary-time Greens function, licensed under GNU General Public License version 3. In quantum Monte Carlo simulation, dynamic physical quantities such as single-particle and magnetic excitation spectra can be obtained by applying analytic continuation to imaginary-time data. However, analytic continuation is an ill-conditioned inverse problem and thus sensitive to noise and statistical errors. SpM provides stable analytic continuation against noise by means of a modern regularization technique, which automatically selects bases that contain relevant information unaffected by noise. This paper details the use of this program and shows some applications.
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We present a new charge self-consistent scheme combining Density Functional and Dynamical Mean Field Theory, which uses Greens function of multiple scattering-type. In this implementation the many-body effects are incorporated into the Kohn-Sham iterative scheme without the need for the numerically ill-posed analytic continuation of the Greens function and of the self-energy. This is achieved by producing the Kohn-Sham Hamiltonian in the sub-space of correlated partial waves and allows to formulate the Greens function directly on the Matsubara axis. The spectral moments of the Matsubara Greens function enable us to put together the real space charge density, therefore the charge self-consistency can be achieved. Our results for the spectral functions (density of states) and equation of state curves for transition metal elements, Fe, Ni and FeAl compound agree very well with those of Hamiltonian based LDA+DMFT implementations. The current implementation improves on numerical accuracy, requires a minimal effort besides the multiple scattering formulation and can be generalized in several ways that are interesting for applications to real materials.
Bayesian parametric analytic continuation (BPAC) is proposed for the analytic continuation of noisy imaginary-time Greens function data as, e.g., obtained by continuous-time quantum Monte Carlo simulations (CTQMC). Within BPAC, the spectral function is inferred from a suitable set of parametrized basis functions. Bayesian model comparison then allows to assess the reliability of different parametrizations. The required evidence integrals of such a model comparison are determined by nested sampling. Compared to the maximum entropy method (MEM), routinely used for the analytic continuation of CTQMC data, the presented approach allows to infer whether the data support specific structures of the spectral function. We demonstrate the capability of BPAC in terms of CTQMC data for an Anderson impurity model (AIM) that shows a generalized Kondo scenario and compare the BPAC reconstruction to the MEM as well as to the spectral function obtained from the real-time fork tensor product state impurity solver where no analytic continuation is required. Furthermore, we present a combination of MEM and BPAC and its application to an AIM arising from the ab initio treatment of SrVO$_3$.
The imaginary-time Greens 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 Greens function can be constructed by decomposing its spectral representation. We investigate the performance of this so-called textit{intermedaite representation} (IR) from several points of view. First, for two simple models, we study the number of coefficients necessary to achieve a given tolerance in expanding the Greens function. We show that the number of coefficients grows only as $O(log beta)$ for fermions, and converges to a constant for bosons as temperature $T=1/beta$ decreases. Second, we show that this remarkable feature is ascribed to the properties of the physically constructed basis functions. The fermionic basis functions have features in the spectrum whose width is scaled as $O(T)$, which are consistent with the low-$T$ 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 $T$. These results demonstrate the potential wide application of the IR to calculations of correlated systems.
This lecture note reviews recently proposed sparse-modeling approaches for efficient ab initio many-body calculations based on the data compression of Greens functions. The sparse-modeling techniques are based on a compact orthogonal basis representation, intermediate representation (IR) basis functions, for imaginary-time and Matsubara Greens functions. A sparse sampling method based on the IR basis enables solving diagrammatic equations efficiently. We describe the basic properties of the IR basis, the sparse sampling method and its applications to ab initio calculations based on the GW approximation and the Migdal-Eliashberg theory. We also describe a numerical library for the IR basis and the sparse sampling method, irbasis, and provide its sample codes. This lecture note follows the Japanese review article [H. Shinaoka et al., Solid State Physics 56(6), 301 (2021)].
Many-body calculations at the two-particle level require a compact representation of two-particle Greens functions. In this paper, we introduce a sparse sampling scheme in the Matsubara frequency domain as well as a tensor network representation for two-particle Greens functions. The sparse sampling is based on the intermediate representation basis and allows an accurate extraction of the generalized susceptibility from a reduced set of Matsubara frequencies. The tensor network representation provides a system independent way to compress the information carried by two-particle Greens functions. We demonstrate efficiency of the present scheme for calculations of static and dynamic susceptibilities in single- and two-band Hubbard models in the framework of dynamical mean-field theory.
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