TY - JOUR
T1 - Spectral density and sum rules for second-order response functions
AU - Bradlyn, Barry
AU - Abbamonte, Peter
N1 - The authors thank C. L. Kane, J. E. Moore, and P. Phillips for fruitful discussions. This work is supported by the U.S. DOE, Office of Basic Energy Sciences, Energy Frontier Research Center for Quantum Sensing and Quantum Materials through Grant No. DE-SC0021238. B.B. received additional support for theoretical development from the Alfred P. Sloan foundation, and the National Science Foundation under Grant No. DMR-1945058. P.A. gratefully acknowledges additional support from the EPiQS program of the Gordon and Betty Moore Foundation, Grant No. GBMF9452.
PY - 2024/12/15
Y1 - 2024/12/15
N2 - 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 studies have 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.
AB - 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 studies have 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.
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U2 - 10.1103/PhysRevB.110.245132
DO - 10.1103/PhysRevB.110.245132
M3 - Article
AN - SCOPUS:85213695104
SN - 2469-9950
VL - 110
JO - Physical Review B
JF - Physical Review B
IS - 24
M1 - 245132
ER -