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arXiv:1907.01212 · Physical review. B./Physical review. B

Generalized f−sum rules and Kohn formulas on nonlinear conductivities

Haruki Watanabe, Masaki Oshikawa

Published Oct 20, 2020• 32 citations
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Abstract

The $f$-sum rule and the Kohn formula are well-established general constraints on the electric conductivity in quantum many-body systems. We present their generalization to nonlinear conductivities at all orders of the response in a unified manner by considering two limiting quantum time-evolution processes: a quench process and an adiabatic process. Our generalized formulas are valid in any stationary state, including the ground state and finite-temperature Gibbs states, regardless of the details of the system such as the specific form of the kinetic term, the strength of the many-body interactions, or the presence of disorders.

DOI
10.1103/physrevb.102.165137

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The fff-sum rule and the Kohn formula are well-established general constraints on the electric conductivity in quantum many-body systems. We present their generalization to nonlinear conductivities at all orders of the response in a unified manner by considering two limiting quantum time-evolution processes: a quench process and an adiabatic process. Our generalized formulas are valid in any stationary state, including the ground state and finite-temperature Gibbs states, regardless of the details of the system such as the specific form of the kinetic term, the strength of the many-body interactions, or the presence of disorders.
Abstract The optical conductivity is the basic defining property of materials characterizing the current response toward time-dependent electric fields. In this work, following the approach of Kubo’s response theory, we study the general properties of the nonlinear optical conductivities of quantum many-body systems both in equilibrium and non-equilibrium. We obtain an expression of the second- and the third-order optical conductivity in terms of correlation functions and present a perturbative proof of the generalized Kohn formula proposed recently. We also discuss a generalization of the f-sum rule to a non-equilibrium setting by focusing on the instantaneous response.
The f-sum rules set general constraints on the response of a quantum many-body system to an external probe and hold significant relevance in the realm of various spectroscopy measurements. In practical conditions, a system unavoidably couples with the environment and acquires effective dissipation. In this Letter, we derive and prove a set of f-sum rules for dissipative systems. Within the framework of linear response theory, we obtain the system response in both linear order of probe field and dissipation parameter. We formulate and prove one first-order and two second-order dissipative f-sum rules. These rules are validated numerically for some example models, and the realization schemes are proposed. In addition, the potential applications are discussed for two interacting many-body systems.
The many-body linear response expressions obtained in previous papers [J. Math. Phys. 19, 1345 (1978); 20, 2573 (1979)] are applied to systems of weakly interacting particles. General expressions for the susceptibility and conductivity in such systems are obtained. The diagonal parts depend on the scattering processes, for which we consider interactions with bosons with mass and electron-phonon interaction. For elastic collisions simple closed forms result. For general two-body collisions, the closed expressions are cumbersome, except when the current is due to collisional current through localized states, such as Landau orbits; in that event a generalized Adams-Holstein result is obtained. The nondiagonal electrical conductivity is shown to be of paramount importance for the quantum mechanical Hall effect. We also derive quantum mechanical Boltzmann equations, both for the diagonal occupancy operator 〈nζ〉t and for the nondiagonal operator 〈c+ζ′ cζ\〉t. The total Boltzmann equation is shown to be fully equivalent with the linear response results. Finally, in the last part we derive the Boltzmann equation for the Wigner function of inhomogeneous systems. In the classical limit this yields the usual Boltzmann transport equation. This equation has therefore been obtained by first principles from the von Neumann equation.
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.
For many-electron systems, we consider a nonequilibrium state (NES) that is driven by a pump field(s), which is either an optical field or a longitudinal electric field. For the differential optical conductivity describing the differential response of the NES to a probe optical field, we derive exact sum rules and asymptotic behaviors, which open wide possibilities for experiments. In deriving these results, we have also derived universal properties of general differential response functions of time-dependent NESs of general systems.
Geometry plays a fundamental role in a wide range of physical responses, from anomalous transport coefficients to their related sum rules. Notable examples include the quantization of the Hall conductivity and the Souza-Wilkens-Martin (SWM) sum rule -- both valid at zero temperature, independent of interactions and disorder. The finite-temperature generalization of the SWM sum rule has been explored in the literature, revealing deep connections to the geometry of density matrices. Building on recent advances in time-dependent geometric frameworks, we propose a time-dependent quantum geometric tensor for thermal density matrices. This formalism provides a unified interpretation of known sum rules within the framework of the fluctuation-dissipation theorem, further elucidating their fundamental geometric origin. In addition, it provides experimentally accessible methods to probe quantum geometry beyond the zero-temperature regime.