Erratum: Kerr effect as evidence of gyrotropic order in the cuprates [Phys. Rev. B87, 115116 (2013)]

Abstract

Recent analysis has confirmed earlier general arguments that the Kerr response vanishes in any time-reversal invariant system which satisfies the Onsager relations. Thus, the widely cited relation between natural optical activity (gyrotropy) and the Kerr response, employed in Hosur et al, Phys. Rev. B 87, 115116 (2013), is incorrect. However, there is increasingly clear experimental evidence that, as argued in our paper, the onset of an observable Kerr signal in the cuprates reflects pointgroup symmetry rather than time-reversal symmetry breaking.Measurements of the rotation of the polarization of normal-incidence light upon reflection (Kerr effect) provide an extremely useful probe of any order which breaks time-reversal symmetry and all mirror symmetries, e.g. ferromagnetic order with a component of the moment perpendicular to the surface. However, arguments have been put forward that a combination of gyrotropy (handedness) and dissipation can lead to a Kerr response, even in the absence of time-reversal symmetry breaking. In Hosur et al. 1 we adopted these arguments and on this basis interpreted the onset of the Kerr effect below a welldefined, doping-dependent onset temperature in cuprate superconductors in terms of spontaneous generation of handedness induced by charge ordering. As outlined below, we are now persuaded by a review of the literature and by our own explicit calculations, that Kerr rotation is forbidden for a system subject to the combined constraints of linear response, thermal equilibrium, and time-reversal symmetry. We therefore wish to retract our suggestion that chiral charge density wave order, by itself, provides a basis for understanding the Kerr rotation observed in the cuprates.Assuming linear response, the Kerr effect can be inferred from the dielectric tensor of the material. It was a key element of the design of the Sagnac Interferometer employed 2-4 in the experiments in question, that it should reject all "non-reciprocal" effects. Reciprocity relations are implicit in linear response theory, which were used by Halperin 5 to demonstrate that the Kerr effect requires time-reversal symmetry breaking beyond simple dissipation. However, controversy existed before the Halperin paper 6-12 and continued to linger after it was published [13][14][15][16][17] .A source of the confusion can be traced to a subtlety in the boundary conditions on the electromagnetic field at the boundary between two media with different gyrotropic constants. Specifically, at the interface between two isotropic media, the requisite boundary conditions are:with α = 1/2, where ∆E , ∆B , and ∆γ are, respectively, the discontinuity in the parallel components of the electric and magnetic fields, and of the gyrotropy. The term proportional to ∆γ can be viewed as the contribution of an induced surface current, which for α = 1/2, but only for this value cancels the contributions of induced bulk currents to the Kerr effect. We will sketch below an analysis in one simple limit in which the value of α = 1/2 can be derived...

Constraints on Jones transmission matrices from time-reversal invariance and discrete spatial symmetries

“…Such experiments have been interpreted as being consistent with the onset of a chiral phase, but this appears to not be a possible explanation (at least at the level of linear response and in thermal equilibrium). See also Hosur et al 32,40 , Cho et al 41 for further discussion.…”

“…A practical matter that arises when considering reflection, is the long standing controversy regarding whether or not a Kerr-like rotation can occur in reflection from a chiral TRS media 26,[37][38][39][40][41] . A straightforward application of Fresnel's equation using the bulk (antisymmetric) dielectric constant predicts that a Kerr rotation should be observed if there is finite dissipation 36 .…”

“…Moreover for many complex materials the precise issue of electromagnetic boundary conditions can be complicated. For instance, there is still ongoing discussion about the proper extension of the Fresnel equations to chiral materials 26,[36][37][38][39][40][41] . In the present approach, one avoids all this and concentrates on the experimentally measured quantities directly.…”

“…In the cuprates, a small but significant signal was seen at a temperature close to (but slightly below) that where the enigmatic pseudogap state of the cuprates onsets [29][30][31] . It was originally attributed to a ferromagnetic-type time-reversal symmetry (TRS) breaking 29 , and later proposed to be consistent with a chiral gyrotropic (TRS preserving) or mag-netoelectric (TRS breaking) effects [32][33][34][35] . As discussed below and elsewhere, it is a matter of current debate whether or not a chiral material as such can exhibit a Kerr rotation 26,[36][37][38][39][40][41] .…”

“…It was originally attributed to a ferromagnetic-type time-reversal symmetry (TRS) breaking 29 , and later proposed to be consistent with a chiral gyrotropic (TRS preserving) or mag-netoelectric (TRS breaking) effects [32][33][34][35] . As discussed below and elsewhere, it is a matter of current debate whether or not a chiral material as such can exhibit a Kerr rotation 26,[36][37][38][39][40][41] . Another case where optics have been used to find new correlated states of matter is in the case of La 2−x Sr x CuO 4 , where a factor of two difference in the a and b direction optical conductivities was found in an underdoped crystal that had only a 1% difference in the in-plane lattice constants 42 .…”