Collective modes and electromagnetic response of a chiral superconductor

Abstract

Motivated by the recent controversy surrounding the Kerr effect measurements in strontium ruthenate [1], we examine the electromagnetic response of a clean chiral p-wave superconductor. When the contributions of the collective modes are accounted for, the Hall response in a clean chiral superconductor is smaller by several orders of magnitude than previous theoretical predictions and is too small to explain the experiment. We also uncover some unusual features of the collective modes of a chiral superconductor, namely, that they are not purely longitudinal and couple to external transverse fields. The experiment by Xia et al. found a Kerr angle of approximately 60 nanoradians at a frequency, ω ≃ 0.8 eV, which is small compared to the Kerr angle observed in typical ferromagnets, but can be understood qualitatively if one notes that the superconducting gap (or order parameter) for Sr 2 RuO 4 is substantially reduced from that of a typical ferromagnet [1]. The Kerr angle at high frequencies is related to the ac Hall conductivity. Theoretical work, however, is divided on the issue of whether the experimental observation result is consistent with the linear response theory of a clean chiral p-wave superconductor [8,9,10]. Earlier works on the electromagnetic response of a chiral p-wave superconductor, with an isotropic Fermi surface [10,11], found no quasi-particle contribution in the clean limit, even in the presence of particle-hole asymmetry, but did predict a Kerr angle due to the so-called "flapping" collective mode. The predicted magnitude is smaller by several orders of magnitude than that observed in Sr 2 RuO 4 .These earlier theoretical studies neglected the effect of an anomalous density-current correlation function which vanishes for a non-chiral superconductor. It was recently argued that when this anomalous correlation function is taken into account, the linear response theory does predict an ac Hall conductivity which is large enough to account for the experiments [8,9]. In the effective action language, the anomalous correlation function gives rise to a Chern-Simons-like term which is reminiscent of the quantum Hall effect [12]. A similar term in the Ginzburg Landau free energy leads to a small "spontaneous Hall effect" in a finite system [13]. When the dynamics of the spin degrees of freedom which are unimportant for the purposes of the present paper are also considered, the effective action of a chiral superconductor or superfluid also contains a non-abelian Chern Simons term which is responsible for the spin quantum Hall effect in these systems [14,15,16,17].In this paper, we examine afresh the linear response of a chiral p-wave superconductor taking into account the anomalous density-current correlation functions and also the contributions from collective modes. We find that when both factors are taken into account, the ac Hall response is strictly zero for an idealized beam normally incident on the a-b plane, with no in-plane wave vector. This is in contrast to the recent results which also consid...

Colloquium: Topological band theory

“…A topological superconductor is obtained when the bulk system has a pairing gap, but supports gapless Majorana modes at the boundary. (Roy and Kallin 2008) For instance, the TRS breaking (chiral p+ip) and TRS preserving p±ip pairing states are analogous to the integer quantum Hall and quantum spin-Hall states, respectively. The former case supports chiral propagating Majorana edge modes, while the latter supports the counter-propagating Majorana edge modes (Elliott and Franz 2014) which are topologically protected against time-reversal invariant perturbations.…”

Nontopological nature of the edge current in a chiralp-wave superconductor

“…IV). Note that in a continuum system, reversing the order of limits to evaluate the standard DC Hall conductivity σ xy ≡ lim ω→0 lim q→0 σ xy (q, ω), gives zero 25 . This noncommutativity of limits arises from a subtlety in the effective action (12) which we will discuss later on.…”

“…For lattice models, it depends not just on the chirality or winding of the order parameter, but also the topology of the Fermi surface, but always takes an integer value. One manifestation of a non-zero Chern number is a quantized value of the "static" Hall conductivity [12][13][14][15]25 :…”

Prediction and description of a chiral pseudogap phase

“…These thermally excited quasiparticles will see the magnetic field from the chiral Cooper pairing, and will give rise to a (classical) charge Hall effect (and thermal Hall effect), in complete analogy with Hall effect in an external magnetic field. Note that the chiral pseudo-gap phase displays a bulk charge Hall effect even in the DC limit, with a clean sample, whereas the chiral superconductor cannot display a Hall conductance in this limit [17], because a superconductor cannot sustain a voltage drop. Similar effects should also arise from the effect of the magnetic field on the charged Cooper pairs.…”