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arXiv:2411.14682 · Physical review. B./Physical review. B

Surface topological quantum criticality: Conformal manifolds and discrete strong-coupling fixed points

Saran Vijayan, Fei Zhou

Published May 2, 2025• 0 citations
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Abstract

In this article, we study quantum critical phenomena in surfaces of symmetry-protected topological matter, i.e., surface topological quantum criticality. A generic phase boundary of gapless surfaces in a symmetry-protected state shall be a codimension one manifold in an interaction parameter space of dimension ${D}_{p}$ (where $p$ refers to the parameter space) where the value of ${D}_{p}$ further depends on bulk topologies. In the context of fermionic topological insulators that we focus on, ${D}_{p}$ depends on the number of half-Dirac cones $\mathcal{N}$. We construct such manifolds explicitly for a few interaction parameter spaces with various ${D}_{p}$ values. Most importantly, we further illustrate that in cases with ${D}_{p}=3$ and 6, there are submanifolds of fixed points that dictate the universalities of surface topological quantum criticality. These infrared stable manifolds are associated with emergent symmetries in the renormalization-group-equation flow naturally appearing in the loop expansion. Unlike in the usual order-disorder quantum critical phenomena, typically governed by an isolated Wilson-Fisher fixed point, we find in the one-loop approximation surface topological quantum criticalities are naturally captured by conformal manifolds where the number of marginal operators uniquely determines their codimensions. Isolated strong-coupling fixed points also appear, usually as the endpoints in the phase boundary of surface topological quantum phases. However, their extreme infrared instabilities along multiple directions suggest that they shall be related to multicritical surface topological quantum critical phenomena rather than generic surface topological quantum criticality. We also discuss and classify higher-loop symmetry-breaking effects, which can either distort the conformal manifolds or further break the conformal manifolds down to a few distinct fixed points.

DOI
10.1103/physrevb.111.174504

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In this article, we study quantum critical phenomena in surfaces of symmetry-protected topological matter, i.e., surface topological quantum criticality. A generic phase boundary of gapless surfaces in a symmetry-protected state shall be a codimension one manifold in an interaction parameter space of dimension Dp{D}_{p}Dp​ (where ppp refers to the parameter space) where the value of Dp{D}_{p}Dp​ further depends on bulk topologies. In the context of fermionic topological insulators that we focus on, Dp{D}_{p}Dp​ depends on the number of half-Dirac cones N\mathcal{N}N. We construct such manifolds explicitly for a few interaction parameter spaces with various Dp{D}_{p}Dp​ values. Most importantly, we further illustrate that in cases with Dp=3{D}_{p}=3Dp​=3 and 6, there are submanifolds of fixed points that dictate the universalities of surface topological quantum criticality. These infrared stable manifolds are associated with emergent symmetries in the renormalization-group-equation flow naturally appearing in the loop expansion. Unlike in the usual order-disorder quantum critical phenomena, typically governed by an isolated Wilson-Fisher fixed point, we find in the one-loop approximation surface topological quantum criticalities are naturally captured by conformal manifolds where the number of marginal operators uniquely determines their codimensions. Isolated strong-coupling fixed points also appear, usually as the endpoints in the phase boundary of surface topological quantum phases. However, their extreme infrared instabilities along multiple directions suggest that they shall be related to multicritical surface topological quantum critical phenomena rather than generic surface topological quantum criticality. We also discuss and classify higher-loop symmetry-breaking effects, which can either distort the conformal manifolds or further break the conformal manifolds down to a few distinct fixed points.
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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.
In this article, we study conditions of continuous emergent symmetries in gapless states, either as topological quantum critical points (TQCPs) or a stable phase with protecting symmetries and connections to smooth deformations of the gapped states around. We illustrate that for a wide class of gapless states that can be associated with fully-isolated scale invariant fixed points, there shall always be emergent continuous symmetries that are directly related to smooth deformations of gapped states with symmetries lower than the protecting ones GpG_pGp​. For a 3D TQCP in DIII classes with Gp=Z2TG_p=Z^T_2Gp​=Z2T​, UEM=U(1)U_{EM}=U(1)UEM​=U(1) and Nf=12N_f=\frac{1}{2}Nf​=21​ fermions but without charge U(1)U(1)U(1) symmetry, we explicitly construct a corresponding boundary representation based on a 4D4D4D topological state with lattice symmetry H=Z2T⋉U(1)H=Z^T_2 \ltimes U(1)H=Z2T​⋉U(1) and Nf=1N_f={1}Nf​=1 fermions. Although emergent continuous symmetries appear to be robust at weakly interacting TQCPs, we further show the breakdown of such one-to-one correspondence between deformations of gapped states and emergent continuous symmetries when gapless states become strongly interacting. In a strongly interacting limit, gapless states can be represented by a smooth manifold of conformal-field-theory fixed points rather than a fully isolated one. A smooth manifold of strong coupling fixed points hinders emergence of a continuous emergent symmetry in the strongly interacting gapless limit, as deformations no longer leave a gapless state or a TQCP invariant, unlike in the more conventional weakly interacting case. This typically reduces continuous emergent symmetries to a discrete symmetry originating from duality transformations under the protection symmetry GpG_pGp​.
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.
In this article, we explore conditions of continuous emergent symmetries in gapless states, either as topological quantum critical points (TQCPs) or a stable phase with protecting symmetries and connections to smooth deformations of the gapped states around. For a wide class of gapless states that can be associated with fully isolated scale-invariant fixed points, we illustrate that there shall be emergent continuous symmetries that are directly related to smooth deformations of gapped states with symmetries lower than the protecting ones Gp{G}_{p}Gp​. A short-distance invariance of gapped states under deformations can descend to be an emergent continuous symmetry when approaching the gapless limit. Around one TQCP with Gp=Z2T{G}_{p}={Z}_{2}^{T}Gp​=Z2T​ symmetry, we construct these deformations explicitly and show emergence of symmetries via fully gapped quaternion superconducting states that break the protecting symmetry Gp=Z2T{G}_{p}={Z}_{2}^{T}Gp​=Z2T​. For a 3D TQCP in DIII classes with Gp=Z2T,UEM=U(1){G}_{p}={Z}_{2}^{T}, {U}_{EM}=U(1)Gp​=Z2T​,UEM​=U(1), and Nf=12{N}_{f}=\frac{1}{2}Nf​=21​ fermions but without charge U(1)U(1)U(1) symmetry, we further explicitly construct a corresponding boundary representation based on a 4D4\text{D}4D topological state with lattice symmetry $H={Z}_{2}^{T}\ensuremath{\ltimes}U(1)$ and Nf=1{N}_{f}=1Nf​=1 fermions. The lattice model is shown to be dual to a conventional 4D4\text{D}4D topological insulator. Although emergent continuous symmetries appear to be robust at weakly interacting TQCPs, we further show the breakdown of such one-to-one correspondence between deformations of gapped states and emergent continuous symmetries when gapless states become strongly interacting. In a strongly interacting limit, gapless states can be represented by a smooth manifold of conformal-field-theory fixed points rather than a fully isolated one. A smooth manifold of strong coupling fixed points hinders emergence of a continuous emergent symmetry in the strongly interacting gapless limit, as deformations no longer leave a gapless state or a TQCP invariant, unlike in the more conventional weakly interacting case. This typically reduces continuous emergent symmetries to a discrete symmetry originating from duality transformations under the protection symmetry Gp{G}_{p}Gp​.