Performance analysis of a physically constructed orthogonal representation of imaginary-time Green's function

Chikano, Naoya; Otsuki, Junya; Shinaoka, Hiroshi

doi:10.1103/physrevb.98.035104

Cited by 38 publications

(48 citation statements)

References 27 publications

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“…Alternatives such as uniform power meshes have had some success 45,46 . However, the most compact representations are achieved using a set of (orthogonal) continuous basis functions directly in imaginary time, such as orthogonal polynomials 47,48 or numerical basis functions [49][50][51][52][53] . The convergence of such a representation is faster than exponential, 47,48 and asymptotically superior to any polynomially converging representation.…”

Section: Introductionmentioning

confidence: 99%

Legendre-spectral Dyson equation solver with super-exponential convergence

Dong

Zgid

Gull

et al. 2020

The Journal of Chemical Physics

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Quantum many-body systems in thermal equilibrium can be described by the imaginary time Green's function formalism. However, the treatment of large molecular or solid ab inito problems with a fully realistic Hamiltonian in large basis sets is hampered by the storage of the Green's function and the precision of the solution of the Dyson equation. We present a Legendre-spectral algorithm for solving the Dyson equation that addresses both of these issues. By formulating the algorithm in Legendre coefficient space, our method inherits the known faster-than-exponential convergence of the Green's function's Legendre series expansion. In this basis, the fast recursive method for Legendre polynomial convolution, enables us to develop a Dyson equation solver with quadratic scaling. We present benchmarks of the algorithm by computing the dissociation energy of the helium dimer He 2 within dressed second-order perturbation theory. For this system, the application of the Legendre spectral algorithm allows us to achieve an energy accuracy of 10 −9 E h with only a few hundred expansion coefficients. arXiv:2001.11603v2 [cond-mat.str-el] 5 Apr 2020

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Section: Introductionmentioning

confidence: 99%

Legendre-spectral Dyson equation solver with super-exponential convergence

Dong

Zgid

Gull

et al. 2020

The Journal of Chemical Physics

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“…4 in Ref. 84). These behaviors are in contrast to the power-law increase ∝ β 1/2 observed for the Legendre basis 84,112) and the Chebyshev polynomial basis.…”

Section: Convergence Properties Of Irmentioning

confidence: 91%

“…We now formulate the IR basis functions in the continuous limit. 84,85) The spectral (Lehmann) representation of the single-particle Green's function is…”

Section: Mathematical Properties Of Ir Basis Functionsmentioning

confidence: 99%

“…(78)]. 84) Although the analytic form of the solution is unknown, numerical studies have revealed some interesting properties. When the singular values are ordered in descending order, for even (odd) values of l, U α l (τ) and V α l (ω) are even (odd) functions with respect to the center of the domain, i.e., τ = β/2 or ω = 0.…”

Section: Mathematical Properties Of Ir Basis Functionsmentioning

confidence: 99%

“…84). These behaviors are in contrast to the power-law increase ∝ β 1/2 observed for the Legendre basis 84,112) and the Chebyshev polynomial basis. 113) These results indicate that only a constant number of IR basis functions will suffice in many-body calculations based on the imaginary-time Green's function at low T .…”

Section: Convergence Properties Of Irmentioning

confidence: 99%

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Sparse Modeling in Quantum Many-Body Problems

et al. 2020

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This review paper describes the basic concept and technical details of sparse modeling and its applications to quantum many-body problems. Sparse modeling refers to methodologies for finding a small number of relevant parameters that well explain a given dataset. This concept reminds us physics, where the goal is to find a small number of physical laws that are hidden behind complicated phenomena. Sparse modeling extends the target of physics from natural phenomena to data, and may be interpreted as "physics for data". The first half of this review introduces sparse modeling for physicists. It is assumed that readers have physics background but no expertise in data science. The second half reviews applications. Matsubara Green's function, which plays a central role in descriptions of correlated systems, has been found to be sparse, meaning that it contains little information. This leads to (i) a new method for solving the illconditioned inverse problem for analytical continuation, and (ii) a highly compact representation of Matsubara Green's function, which enables efficient calculations for quantum many-body systems. 1 arXiv:1911.04116v1 [cond-mat.str-el]

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Sparse modeling approach to obtaining the shear viscosity from smeared correlation functions

Itou

Nagai

2020

J. High Energ. Phys.

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We propose the sparse modeling method to estimate the spectral function from the smeared correlation functions. We give a description of how to obtain the shear viscosity from the correlation function of the renormalized energy-momentum tensor (EMT) measured by the gradient flow method (C (t, τ )) for the quenched QCD at finite temperature. The measurement of the renormalized EMT in the gradient flow method reduces a statistical uncertainty thanks to its property of the smearing. However, the smearing breaks the sum rule of the spectral function and the over-smeared data in the correlation function may have to be eliminated from the analyzing process of physical observables. In this work, we demonstrate the sparse modeling analysis in the intermediate-representation basis (IR basis), which connects between the Matsubara frequency data and real frequency data. It works well even using very limited data of C (t, τ ) only in the fiducial window of the gradient flow. We utilize the ADMM algorithm which is useful to solve the LASSO problem under some constraints. We show that the obtained spectral function reproduces the input smeared correlation function at finite flow-time. Several systematic and statistical errors and the flow-time dependence are also discussed.

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Performance analysis of a physically constructed orthogonal representation of imaginary-time Green's function

Cited by 38 publications

References 27 publications

Legendre-spectral Dyson equation solver with super-exponential convergence

Legendre-spectral Dyson equation solver with super-exponential convergence

Sparse Modeling in Quantum Many-Body Problems

Sparse modeling approach to obtaining the shear viscosity from smeared correlation functions

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