On the General Properties of Non-linear Optical Conductivities
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On the General Properties of Non-linear Optical Conductivities Haruki Watanabe1
· Yankang Liu1 · Masaki Oshikawa2,3
Received: 18 April 2020 / Accepted: 7 October 2020 © The Author(s) 2020
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. Keywords Optical conductivity · Response theory · Nonlinear response
1 Introduction The electric conductivity describes the response of the current density ji (t) toward a timej dependent electric field E j (t). In the Fourier space, the linear optical conductivity σi (ω) (i, j are the spatial indices) is the proportionality constant connecting ji (ω) to E i (ω): ji (ω) =
j
σi (ω)E j (ω) + O(E 2 ).
(1)
j j
There has been a long history of studies on the general properties of σi (ω). (See Ref. [1] and the references therein.) For example, the optical conductivity obeys the frequency-sum
Communicated by Keiji Saito.
B
Haruki Watanabe [email protected]
1
Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan
2
Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan
3
Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa 277-8583, Japan
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rule ( f -sum rule) [1]; that is, the integral ∞ −∞
j
dωσi (ω)
(2)
is solely determined by an expectation value in the absence of the electric field. Furthermore, the optical conductivity is known to have the following generic structure: j
σi (ω) =
i j j D + σi (regular) (ω), ω + iη i
j
(3) j
where Di is called the Drude weight that characterizes the singular part of σi (ω) around j ω = 0 and σi (regular) (ω) is the regular part that includes all other terms. The Drude weight is a useful measure distinguishing ideal conductors from insulators and non-ideal conductors [2]. More than a half century ago, Kohn [3] showed that the Drude weight at zero temperature is as a function of the vector potential given by the curvature of the ground state energy E0 ( A) A, j
Di =
1 ∂ 2 E0 ( A) . V ∂ Ai ∂ A j A=0
(4)
This is nowadays known as the Kohn formula [1]. An extension to a finite temperature was achieved in Ref. [4]. Recently, two of us proposed [5] a generalization of the f -sum rule and the Kohn formula to the N -th order optical conductivity σii1 ...i N (ω1 , . . . , ω N ) [defined by Eqs. (10) and (22)] through a heuristic argument utilizing extreme quantum pr
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