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arXiv:2004.04561 · Journal of Statistical Physics

On the General Properties of Non-linear Optical Conductivities

Haruki Watanabe, Yankang Liu, Masaki Oshikawa

Published Oct 18, 2020• 22 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1007/s10955-020-02654-5

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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.
Abstract Motivated by the recent developments in terahertz spectroscopy using pump-probe setups to study correlated electronic materials, we review the field theoretical formalism to compute finite frequency nonlinear electro-optical responses in centrosymmetric systems starting from basic time dependent perturbation theory. We express the nonlinear current kernel as a sum of several causal response functions. These causal functions cannot be evaluated using perturbative field theory methods, since they are not contour ordered. Consequently, we associate each response function with a corresponding imaginary time ordered current correlation function, since the latter can be factorized using Wick’s theorem. The mapping between the response functions and the correlation functions, suitably analytically continued to real frequencies, is proven exactly. We derive constraints satisfied by the nonlinear current kernel and we prove a generalized f-sum rule for the nonlinear conductivity, all of which are consequences of particle number conservation. The constraints guarantee that the nonlinear static responses are free from spurious divergences. We apply the theory to compute the gauge invariant nonlinear conductivity of a system of noninteracting electrons in the presence of weak disorder. As special cases of this generalized nonlinear response, we discuss its third harmonic and its instantaneous terahertz Kerr signals. The formalism can be used to compute the nonlinear conductivity in symmetry broken phases of electronic systems such as superconductors, density waves and nematic states.
Motivated by the recent developments in terahertz spectroscopy using pump-probe setups, we develop the theory of finite frequency nonlinear electro-optical responses in centrosymmetric metals starting from basic time dependent perturbation theory. We express the nonlinear current kernel as a sum of several causal response functions. These functions cannot be evaluated using perturbative field theory methods. Consequently, we associate each response function with an imaginary time ordered current correlation function, which can be factorized using Wick's theorem. The mapping between the response functions and the correlation functions, suitably analytically continued to real frequencies, is proven exactly. We derive constraints satisfied by the nonlinear current kernel and we prove a generalized fff-sum rule for the nonlinear conductivity, all of which are consequences of particle number conservation. We apply the theory to compute the gauge invariant nonlinear conductivity of a system of noninteracting electrons in the presence of weak disorder. As special cases of this generalized nonlinear response, we discuss its third harmonic and its instantaneous terahertz Kerr signals. Our formalism can be used to compute the nonlinear conductivity in symmetry broken phases such as density waves and nematic states.
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.
Motivated by the recent developments in terahertz spectroscopy using pump-probe setups to study correlated electronic materials, we review the field theoretical formalism to compute finite frequency nonlinear electro-optical responses in centrosymmetric systems starting from basic time dependent perturbation theory. We express the nonlinear current kernel as a sum of several causal response functions. These causal functions cannot be evaluated using perturbative field theory methods, since they are not contour ordered. Consequently, we associate each response function with a corresponding imaginary time ordered current correlation function, since the latter can be factorized using Wick's theorem. The mapping between the response functions and the correlation functions, suitably analytically continued to real frequencies, is proven exactly. We derive constraints satisfied by the nonlinear current kernel and we prove a generalized fff-sum rule for the nonlinear conductivity, all of which are consequences of particle number conservation. The constraints guarantee that the nonlinear static responses are free from spurious divergences. We apply the theory to compute the gauge invariant nonlinear conductivity of a system of noninteracting electrons in the presence of weak disorder. As special cases of this generalized nonlinear response, we discuss its third harmonic and its instantaneous terahertz Kerr signals. The formalism can be used to compute the nonlinear conductivity in symmetry broken phases of electronic systems such as superconductors, density waves and nematic states.
We describe microscopic theory for the quantum transport through finite interacting systems connected to noninteracting leads. It can be applied to small systems such as quantum dots, quantum wires, atomic chain, molecule, and so forth. The Keldysh formalism is introduced to study the nonlinear current-voltage characteristics, and general properties of the nonequilibrium Green's functions are provided. We apply the formulated to an out-of-equilibrium Anderson model that has been used widely for the Kondo effect in quantum dots. In the linear-response regime the Kubo formalism still has an advantage, because it relates the transport coefficients directly to the correlation functions defined with respect to the thermal equilibrium, and it has no ambiguities about a profile of the inner electric field. We introduce a many-body transmission coefficient, by which the dc conductance can be expressed in a Landauer-type form, quite generally. We also discuss transport properties of the Tomonaga-Luttinger liquid, which has also been an active field of research in this decade.
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.