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.
Publisher Copyright:
© 2020, Springer Nature Switzerland AG.

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 -