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## Optical activity of noncentrosymmetric metals 

1 Commissariat ` a l’Energie Atomique, INAC/SPSMS, 38054 Grenoble, France 

V. P. Mineev 1 and Yu Yoshioka 2 

2 Osaka University, Graduate School of Engineering Science, Toyonaka, Osaka 560-8531, Japan 

(Dated: November 8, 2018) 

We describe the phenomenon of optical activity of noncentrosymmetric metals in their normal and superconducting states. The found conductivity tensor contains the linear in wave vector off diagonal part responsible for the natural optical activity. Its value is expressed through the ratio of light frequency to the band splitting due to the spin-orbit interaction. The Kerr rotation of polarization of light reflected from the metal surface is calculated. 

The metals without inversion symmetry have re- cently become a subject of considerable interest mostly due to the discovery of superconductivity in CePt$_{3}$Si. 1 Now the list of noncentrosymmetric superconductors has grown to include UIr $^{2}$, CeRhSi$_{3}$$^{3}$, CeIrSi$_{3}$$^{4}$, Y$_{2}$ C$_{3}$ $^{5}$, Li$_{2}$(Pd$_{1}$$_{−}$$_{x}$,Pt$_{x}$)$_{3}$B $^{6}$, KOs$_{2}$ O$_{6}$ $^{7}$, and other compounds. The spin-orbit coupling of electrons in noncentrosymmet- ric crystal lifts the spin degeneracy of the electron energy band causing a noticeable band splitting. The Fermi sur- face splitting can be observed by the de Haas-van Alphen effect discussed theoretically in the paper Ref. 8. The band splitting reveals itself in the large residual value of the spin susceptibility of noncentrosymmetric supercon- ductors at zero temperature. 9 It also makes possible the existence of nonuniform superconducting states that can be traced to the Lifshitz invariants in the free energy. 10 Another significant manifestation of the band splitting is the natural optical activity. 

The natural optical activity or natural gyrotropy is well known phenomenon typical for the bodies having no centre of symmetry $^{11}$. The optical properties of a nat- urally active body resemble those of the magneto-active media having no time reversal symmetry. It exhibits dou- ble circular refraction, the Faraday and the Kerr effects. In this case, the tensor of dielectric permeability has lin- ear terms in the expansion in powers of wave vector 

where γ$_{ikl}$ ( ω ) is an antisymmetric third rank tensor called the tensor of gyrotropy. 

The spacial dispersion term in permeability has been derived by Arfi and Gor’kov 12 in frame of model where ”a conductor lacking a center of inversion is simulated by an ordered arrangement of impurities whose scattering po- tential is asymmetric” (see also $^{13}$). We shall be interested in gyrotropy properties of a clean noncentrosymmetric metal. In metals, it is more natural to describe optical properties in terms of spacial dispersion of conductivity tensor having the following form: 

PACS numbers: 78.20.Ek, 74.25.Nf, 74.20.Fg 

## I. INTRODUCTION 

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Here, the gyrotropy tensor λ$_{ijl}$ ( ω ) is an odd function of frequency. 

The gyrotropic tensor has the most simple form in the metals with cubic symmetry. In this case, the Drude part of the conductivity tensor is isotropic σ$_{ij}$ ( ω, 0) = σ ( ω ) δ$_{ij}$ and the gyrotropic conductivity tensor λ$_{ikl}$ ( ω ) = λ ( ω ) e$_{ikl}$ is determined by the single complex function λ ( ω ) = λ $^{′}$( ω ) + iλ $^{′′}$( ω ) such that a normal state density of current is 

In time representation λ is an operator being an odd function of operation of time derivative ∂/∂t . 

In the superconducting state besides λ rot E the gy- rotropic part of current density contains also an addi- tional term proportional to the magnetic induction B 

Here ν = ν ( T ) is a constant coefficient being equal to zero in the normal state. 14,15 

In this paper we present the derivation of current re- sponse to the electro-magnetic field with finite frequency and wave vector valid for the normal and the supercon- ducting state of noncentrosymmetric metals with cubic symmetry. We find that the gyrotropy conductivity λ is directly proportional to the ratio of the light frequency to the energy of the band splitting. Then making use the Maxwell equations and corresponding boundary con- ditions at the surface of noncentrosymmetric metal de- rived in the Section V we calculate the Kerr rotation of polarization of light reflected from the surface of metal. 

The current response of a clean metal to the electro- magnetic field at finite q and ω can be written following the textbook procedure $^{16}$. In application to our situation one has to remember that due to spin-orbital coupling determined by the dot product of the Pauli matrix vec- tor σ and pseudovector γ ( k ), which is odd in respect to 

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## II. CURRENT RESPONSE TO ELECTRO-MAGNETIC FIELD 

