Quantum Criticality in Topological Insulators and Superconductors: Emergence of Strongly Coupled Majoranas and Supersymmetry

TL;DR

The paper analyzes symmetry-breaking quantum phase transitions in topological insulators and superconductors where the bulk remains gapped, focusing on how gapless boundary modes interact with critical fluctuations. It shows that in most higher-dimensional cases, surface Majorana/Dirac modes decouple from bulk criticality, remaining gapless, while in 1D TSC edges they couple strongly but nonperturbatively, yielding non-Fermi-liquid behavior. Remarkably, surface criticality in several setups exhibits emergent supersymmetry, linking topological boundary modes to SUSY fixed points and enabling exact predictions for critical exponents. The fate of Majorana modes at point and line defects at bulk criticality reveals strongly-coupled, yet gapless, defect states with anomalous dimensions or marginal behavior. The work outlines experimental pathways in magnetic TI/TSC materials and proposes concrete signatures in STM/ARPES, NMR, and neutron scattering to probe these phenomena.

Abstract

We study symmetry breaking quantum phase transitions in topological insulators and superconductors where the single electron gap remains open in the bulk. Specifically, we consider spontaneous breaking of the symmetry that protects the gapless boundary modes, so that in the ordered phase these modes are gapped. Here we determine the fate of the topological boundary modes right at the transition where they are coupled to the strongly fluctuating order parameter field. Using a combination of exact solutions and renormalization group techniques, we find that the surface fermionic modes either decouple from the bulk fluctuations, or flow to a strongly coupled fixed point which remains gapless. In addition, we study transitions where the critical fluctuations are confined only to the surface and find that in several cases the critical point is naturally supersymmetric. This allows a determination of critical exponents and points to an underlying connection between band topology and supersymmetry. Finally, we study the fate of gapless Majorana modes localized on point and line defects in topological superconductors at bulk criticality, which is analogous to a quantum impurity problem. Again, an interplay of topology and strong correlations causes these modes to remain gapless but in a strongly coupled state. Experimental candidates for realizing these phenomena are discussed.

Figures (7)

• Figure 1: The two class of problems studied in this paper that concern the fate of fermionic surface modes in TI/TSC at a critical point. $T$ denotes temperature while $r$ is the tuning parameter for the quantum phase transition. The counter-propagating fermionic surface modes (blue lines) interact with the critical fluctuations (headed vectors). The critical point may correspond to a bulk transition as depicted in (a) or it could be a surface transition where the bulk is unaffected as shown in (b).
• Figure 2: (above) Illustration of Majorana boundary modes $\eta_1, \eta_2$ in a one dimensional time-reversal invariant TSC coupled to critical Ising fluctuations with interaction strength $g$. (below) Renormalization group flow for the same problem. $h$ is the transverse field for the Ising magnet. The $g=0$ fixed point is unstable to $g^{*} = 1$ fixed point where Majorana modes are strongly coupled to the critical fluctuations and have correlation function $\langle \eta_\alpha(0) \eta_\alpha(\tau)\rangle \sim 1/\tau$ for $\alpha = 1,2$.
• Figure 3: The coupling to bulk critical fluctuations is irrelevant for two and three dimensional DIII TSCs and three dimensional AII TIs. This means that at low energies, the boundary fermionic modes completely decouple from the bulk critical fluctuations.
• Figure 4: A schematic drawing of a point defect carrying two Majorana modes (blue circles) coupled to critical magnetic fluctuations at the transition between a non-magnetic two-dimensional DIII TSC and a magnetically ordered ordinary insulator.
• Figure 5: A schematic drawing of a line defect carrying helical Majorana modes (blue lines) coupled to critical magnetic fluctuations at the transition between a non-magnetic three-dimensional DIII TSC and a magnetically ordered ordinary insulator.