Spectral Density and Sum Rules for Second-Order Response Functions
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| Format: | Preprint |
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2024
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| _version_ | 1866915203219718144 |
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| author | Bradlyn, Barry Abbamonte, Peter |
| author_facet | Bradlyn, Barry Abbamonte, Peter |
| contents | Sum rules for linear response functions give powerful and experimentally-relevant relations between frequency moments of response functions and ground state properties. In particular, renewed interest has been drawn to optical conductivity and density-density sum rules and their connection to quantum geometry in topological materials. At the same time, recent work has also illustrated the connection between quantum geometry and second-order nonlinear response functions in quantum materials, motivating the search for exact sum rules for second-order response that can provide experimental probes and theoretical constraints for geometry and topology in these systems. Here we begin to address these questions by developing a general formalism for deriving sum rules for second-order response functions. Using generalized Kramers-Kronig relations, we show that the second-order Kubo formula can be expressed in terms of a spectral density that is a sum of Dirac delta functions in frequency. We show that moments of the spectral density can be expressed in terms of averages of equal-time commutators, yielding a family of generalized sum rules; furthermore, these sum rules constrain the large-frequency asymptotic behavior of the second harmonic generation rate. We apply our formalism to study generalized $f$-sum rules for the second-order density-density response function and the longitudinal nonlinear conductivity. We show that for noninteracting electrons in solids, the generalized $f$-sum rule can be written entirely in terms of matrix elements of the Bloch Hamiltonian. Finally, we derive a family of sum rules for rectification response, determining the large-frequency asymptotic behavior of the time-independent response to a harmonic perturbation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_16144 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral Density and Sum Rules for Second-Order Response Functions Bradlyn, Barry Abbamonte, Peter Mesoscale and Nanoscale Physics Statistical Mechanics Strongly Correlated Electrons Sum rules for linear response functions give powerful and experimentally-relevant relations between frequency moments of response functions and ground state properties. In particular, renewed interest has been drawn to optical conductivity and density-density sum rules and their connection to quantum geometry in topological materials. At the same time, recent work has also illustrated the connection between quantum geometry and second-order nonlinear response functions in quantum materials, motivating the search for exact sum rules for second-order response that can provide experimental probes and theoretical constraints for geometry and topology in these systems. Here we begin to address these questions by developing a general formalism for deriving sum rules for second-order response functions. Using generalized Kramers-Kronig relations, we show that the second-order Kubo formula can be expressed in terms of a spectral density that is a sum of Dirac delta functions in frequency. We show that moments of the spectral density can be expressed in terms of averages of equal-time commutators, yielding a family of generalized sum rules; furthermore, these sum rules constrain the large-frequency asymptotic behavior of the second harmonic generation rate. We apply our formalism to study generalized $f$-sum rules for the second-order density-density response function and the longitudinal nonlinear conductivity. We show that for noninteracting electrons in solids, the generalized $f$-sum rule can be written entirely in terms of matrix elements of the Bloch Hamiltonian. Finally, we derive a family of sum rules for rectification response, determining the large-frequency asymptotic behavior of the time-independent response to a harmonic perturbation. |
| title | Spectral Density and Sum Rules for Second-Order Response Functions |
| topic | Mesoscale and Nanoscale Physics Statistical Mechanics Strongly Correlated Electrons |
| url | https://arxiv.org/abs/2404.16144 |