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arXiv:2404.16144 · Physical review. B./Physical review. B

Spectral density and sum rules for second-order response functions

Barry Bradlyn, Peter M. Abbamonte

Published Dec 18, 2024• 14 citations
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Abstract

Sum rules for response functions represent some of the few exact results in many-body physics, relating the dynamical response functions to equilibrium ground state properties. To date, however, sum rules have been examined primarily for linear response functions. In this work, the authors introduce a spectral density representation for second-order response functions, using it to systematically derive families of sum rules, including a nonlinear extension of the well-known $f$-sum rule. This work places important constraints on future nonlinear spectroscopic experiments.

DOI
10.1103/physrevb.110.245132

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Sum rules for response functions represent some of the few exact results in many-body physics, relating the dynamical response functions to equilibrium ground state properties. To date, however, sum rules have been examined primarily for linear response functions. In this work, the authors introduce a spectral density representation for second-order response functions, using it to systematically derive families of sum rules, including a nonlinear extension of the well-known fff-sum rule. This work places important constraints on future nonlinear spectroscopic experiments.
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In this work the basic formalism of non-equilibrium Green’s functions is presented and then applied to study a Ward identity in linear response theory, namely the frequency sum-rule. It can be proven that the frequency sum-rule is satisfied when the quantities involved are calculated using perturbation theory within a conserving approximation for the self-energy. To illustrate this equality along with other properties of the response function, a numerical application that solves the Kadanoff-Baym equations for systems of Hubbard chains was used. The results showed that the frequency sum-rule was satisfied to the same extent by all the conserving approximations used as by the exact diagonalization numerical results. The density response function was analyzed diagrammatically for a series of conserving approximations for the self-energy and this demonstrated that even for a first order in perturbation theory approximation for the self-energy, the response function has a corresponding complex, third order in the perturbation diagrammatic structure.
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