The boundary of a topological insulator (TI) hosts an anomaly restricting its possible phases: e.g., three-dimensional (3D) strong and weak TIs maintain surface conductivity at any disorder if symmetry is preserved on average at least when electron interactions on the surface are weak. However, the interplay of strong interactions and disorder with the boundary anomaly has not yet been theoretically addressed. Here we study this combination for the edge of a two-dimensional TI and the surface of a 3D weak TI, showing how it can lead to an ``Anomalous Many Body Localized'' (AMBL) phase that preserves the anomaly. We discuss how the anomalous Kramers parity switching with $\ensuremath{\pi}$ flux arises in the bosonized theory of the localized helical state. The anomaly can be probed in localized boundaries by electrostatically sensing nonlinear hopping transport with e/2 shot noise. Our AMBL construction in 3D weak TIs fails for 3D strong TIs, which suggests that their anomaly restrictions are distinguished by strong interactions.
Symmetry-protected topological (SPT) phases host gapless surface states that are robust against weak, symmetry-preserving interactions. However, strong interactions can drive spontaneous breaking of the protecting symmetries, leading to gapped ordered surfaces. The phase boundary separating the gapless and ordered phases can host exotic conformal field theory states/universality classes typically not observed in conventional quantum phase transitions. We refer to these exotic quantum critical states as surface topological quantum critical points (sTQCPs). In this dissertation, we study attractive interaction-driven quantum criticality on the surface of three-dimensional topological insulators, a class of SPT phases hosting Dirac fermion surface states. When multiple Dirac cones are present, the interaction parameter space becomes intrinsically multi-dimensional, and the phase boundary separating the gapless and ordered phases forms a continuous manifold. Using renormalization group (RG) analysis, we show that in the limit of suppressed quantum fluctuations, the universality of this phase boundary is governed by conformal manifolds: continuous families of interacting conformal field theories characterized by exactly marginal operators. These manifolds are featureless in the sense that they host a single universality class. Their emergence is traced to continuous symmetries of the RG equations that are otherwise absent in the interaction Hamiltonian. We further demonstrate that higher-order quantum fluctuations break these emergent symmetries, fragmenting the conformal manifolds into isolated fixed points of varying infrared stabilities. Remarkably, we find that along the RG flow within the manifold, an EPR-like entanglement entropy in fermion flavor space always increases. Infrared stable conformal field theories correspond to maximally entangled interaction operators, while weakly entangled fixed points are unstable. These results establish the central role of entangled conformal operators and their entropy in shaping the universality classes of sTQCPs. Finally, we propose an experimentally accessible route to realizing surface quantum criticality by coupling a topological insulator surface to a metallic thin film. Strong electron–phonon interactions in the metal mediate effective attractive interactions between surface Dirac fermions, enhanced via a quantum-well resonance mechanism, providing a concrete pathway for accessing sTQCP physics in real materials.
Symmetry-protected topological (SPT) phases host gapless surface states that are robust against weak, symmetry-preserving interactions. However, strong interactions can drive spontaneous breaking of the protecting symmetries, leading to gapped ordered surfaces. The phase boundary separating the gapless and ordered phases can host exotic conformal field theory states/universality classes typically not observed in conventional quantum phase transitions. We refer to these exotic quantum critical states as surface topological quantum critical points (sTQCPs). In this dissertation, we study attractive interaction-driven quantum criticality on the surface of three-dimensional topological insulators, a class of SPT phases hosting Dirac fermion surface states. When multiple Dirac cones are present, the interaction parameter space becomes intrinsically multi-dimensional, and the phase boundary separating the gapless and ordered phases forms a continuous manifold. Using renormalization group (RG) analysis, we show that in the limit of suppressed quantum fluctuations, the universality of this phase boundary is governed by conformal manifolds: continuous families of interacting conformal field theories characterized by exactly marginal operators. These manifolds are featureless in the sense that they host a single universality class. Their emergence is traced to continuous symmetries of the RG equations that are otherwise absent in the interaction Hamiltonian. We further demonstrate that higher-order quantum fluctuations break these emergent symmetries, fragmenting the conformal manifolds into isolated fixed points of varying infrared stabilities. Remarkably, we find that along the RG flow within the manifold, an EPR-like entanglement entropy in fermion flavor space always increases. Infrared stable conformal field theories correspond to maximally entangled interaction operators, while weakly entangled fixed points are unstable. These results establish the central role of entangled conformal operators and their entropy in shaping the universality classes of sTQCPs. Finally, we propose an experimentally accessible route to realizing surface quantum criticality by coupling a topological insulator surface to a metallic thin film. Strong electron–phonon interactions in the metal mediate effective attractive interactions between surface Dirac fermions, enhanced via a quantum-well resonance mechanism, providing a concrete pathway for accessing sTQCP physics in real materials.