TY - JOUR

T1 - Sparsity Pattern of the Self-energy for Classical and Quantum Impurity Problems

AU - Lin, Lin

AU - Lindsey, Michael

N1 - Funding Information:
This work was partially supported by the Department of Energy under Grant Nos. DE-SC0017867, DE-AC02-05CH11231, by the Air Force Office of Scientific Research under Award No. FA9550-18-1-0095 (L. L.), by the National Science Foundation Graduate Research Fellowship Program under Grant DGE-1106400 (M. L.), and by the National Science Foundation under Award No. 1903031. We thank Garnet Chan, Jianfeng Lu, Nicolai Reshetikhin, Reinhold Schneider, and Lexing Ying for helpful discussions.

PY - 2020/7/1

Y1 - 2020/7/1

N2 - We prove that for various impurity models, in both classical and quantum settings, the self-energy matrix is a sparse matrix with a sparsity pattern determined by the impurity sites. In the quantum setting, such a sparsity pattern has been known since Feynman. Indeed, it underlies several numerical methods for solving impurity problems, as well as many approaches to more general quantum many-body problems, such as the dynamical mean field theory. The sparsity pattern is easily motivated by a formal perturbative expansion using Feynman diagrams. However, to the extent of our knowledge, a rigorous proof has not appeared in the literature. In the classical setting, analogous considerations lead to a perhaps less known result, i.e., that the precision matrix of a Gibbs measure of a certain kind differs only by a sparse matrix from the precision matrix of a corresponding Gaussian measure. Our argument for this result mainly involves elementary algebraic manipulations and is in particular non-perturbative. Nonetheless, the proof can be robustly adapted to various settings of interest in physics, including quantum systems (both fermionic and bosonic) at zero- and finite-temperature, non-equilibrium systems, and superconducting systems.

AB - We prove that for various impurity models, in both classical and quantum settings, the self-energy matrix is a sparse matrix with a sparsity pattern determined by the impurity sites. In the quantum setting, such a sparsity pattern has been known since Feynman. Indeed, it underlies several numerical methods for solving impurity problems, as well as many approaches to more general quantum many-body problems, such as the dynamical mean field theory. The sparsity pattern is easily motivated by a formal perturbative expansion using Feynman diagrams. However, to the extent of our knowledge, a rigorous proof has not appeared in the literature. In the classical setting, analogous considerations lead to a perhaps less known result, i.e., that the precision matrix of a Gibbs measure of a certain kind differs only by a sparse matrix from the precision matrix of a corresponding Gaussian measure. Our argument for this result mainly involves elementary algebraic manipulations and is in particular non-perturbative. Nonetheless, the proof can be robustly adapted to various settings of interest in physics, including quantum systems (both fermionic and bosonic) at zero- and finite-temperature, non-equilibrium systems, and superconducting systems.

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U2 - 10.1007/s00023-020-00917-1

DO - 10.1007/s00023-020-00917-1

M3 - Article

AN - SCOPUS:85086357075

VL - 21

SP - 2219

EP - 2257

JO - Annales Henri Poincare

JF - Annales Henri Poincare

SN - 1424-0637

IS - 7

ER -