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 -