Optical Conductivity of Topological Surface States with Emergent Supersymmetry

Abstract

Topological states of electrons present new avenues to explore the rich phenomenology of correlated quantum matter. Topological insulators (TIs) in particular offer an experimental setting to study novel quantum critical points (QCPs) of massless Dirac fermions, which exist on the sample's surface. Here, we obtain exact results for the zero-and finite-temperature optical conductivity at the semimetal-superconductor QCP for these topological surface states. This strongly interacting QCP is described by a scale invariant theory with emergent supersymmetry, which is a unique symmetry mixing bosons and fermions. We show that supersymmetry implies exact relations between the optical conductivity and two otherwise unrelated properties: the shear viscosity and the entanglement entropy. We discuss experimental considerations for the observation of these signatures in TIs.PACS numbers: 05.30. Rt, 74.40.Kb, 71.27.+a, 11.30.Pb Topological insulators [1, 2] allow for the experimental study of new quantum states of matter. The strong spinorbit coupling in these bulk-insulating materials leads to unique gapless Dirac fermion surface states. These can undergo quantum phase transitions forbidden in nontopological systems, and thus constitute a new platform to study the rich physics of quantum criticality [3, 4]. A considerable challenge in the study of interacting QCPs is to determine their dynamical response-that is, their response at finite frequency ω-both at zero and finite temperature T , such as the optical conductivity σ(ω, T ). Here, we focus on the dynamical response of a novel QCP that can appear at the surface of a three-dimensional (3D) topological insulator: it describes the interaction- driven quantum phase transition between a single Dirac cone of electrons and a gapped superconductor [5, 6] (see Fig. 1). As an important step towards observing this transition, recent experiments have reported the discovery of intrinsic superconductivity on the surface of a 3D topological insulator, Sb 2 Te 3 [7]. We emphasize that standard 2D (or layered) systems that do not break timereversal symmetry must have an even number of Dirac cones and thus cannot host this transition. More complex scenarios realizing multiple copies of this QCP can occur via f -wave pairing [8] and pair-density-wave [9] instabilities of spinless Dirac fermions on the 2D honeycomb lattice, or for interacting ultracold atomic gases in optical lattices [10].When the chemical potential is at the Dirac point, a special type of symmetry emerges at the QCP [5, 6, 8, 9, 11]: spacetime supersymmetry (SUSY). SUSY relates bosons and fermions, and has been proposed to exist in extensions of the Standard Model of elementary particle physics, but has not yet been observed. At the QCP of Fig. 1, it emerges naturally by relating the Dirac fermions of the semimetal to the bosonic Cooper pairs of the superconductor. These two become degenerate at the transition and in fact share a deeper relation described by SUSY. We emphasize that this is a consequence of th...

Expanding 3d $$ \mathcal{N} $$ = 2 theories around the round sphere

“…The result nicely matches the analytic result in [54] obtained from direct integration using eq. (A.6).…”

Dynamical Response near Quantum Critical Points

“…Proximity to a QCP alters many observables, even if the (detuned) ground state is otherwise conventional. Of particular importance are dynamical response functions such as the optical conductivity σ(ω) [1][2][3][4][5][6][7][8][9][10][11][12][13][14], where changing the frequency probes physics at different energy scales set by the non-thermal detuning and by the temperature. What often complicates the analysis of the real-time dynamics, especially on short time scales, is the destruction of quasiparticles at the QCP, and the corresponding abundance of incoherent excitations at finite but small detuning.…”

Partition functions on slightly squashed spheres and flux parameters

“…Special mention deserves their rôle in AdS/CFT [25][26][27], where the bulk solutions controlling the corresponding semiclassical partition functions for such boundary metrics correspond to a well-known and important family of gravitational solutions, the so-called Euclidean Taub-NUT/bolt metrics [28][29][30][31][32][33][34][35]. They have been also extensively studied for supersymmetric CFTs 2 -see e.g., [36][37][38][39][40][41][42][43][44].…”