Topological insulators (TIs) are an exciting discovery because of their robustness against disorder and interactions. Recently, second-order TIs have been attracting increasing attention, because they host topologically protected 1D hinge states in 3D or 0D corner states in 2D. A significantly critical issue is whether the second-order TIs also survive interactions, but it is still unexplored. We study the effects of weak Coulomb interactions on a 3D second-order TI, with the help of renormalization-group calculations. We find that the 3D second-order TIs are always unstable, suffering from two types of topological phase transitions. One is from second-order TI to TI, the other is to normal insulator. The first type is accompanied by emergent time-reversal and inversion symmetries and has a dynamical critical exponent κ=1. The second type does not have the emergent symmetries but has nonuniversal dynamical critical exponents κ<1. Our results may inspire more inspections on the stability of higher-order topological states of matter and related novel quantum criticalities.
Based on renormalization-group (RG) calculations, a recent Letter by Zhao et al [Phys. Rev. Lett. 127, 176601 (2021)] claimed that three-dimensional second-order topological insulators (SOTIs) are always unstable to the Coulomb interaction and they will thus undergo topological phase transitions to either topological insulators (TIs) or normal insulators. While the calculations in this paper are correct, the conclusion about the topological phase transition from SOTI to TI is not true. The reason behind this false conclusion lies in that the authors use a wrong criterion to identify the phase transition. In this Comment we would like to remind that, to locate the possible transitions from SOTI to TI, an appropriate quantity is the sign-changing mass gap
msurf for the surface states. By examining its behavior within RG approach, the stability of SOTI against weak Coulomb interactions is demonstrated.
In this article, we study the influence of long-range Coulomb interaction on three-dimensional second-order topological insulator (TI) by renormalization group theory. We find that both the analysis method and conclusions in the recent Letter Phys. Rev. Lett. {\bf 127}, 176601 (2021) are unreliable. There are two problems in this Letter. Firstly, the characteristic described by the RG flows
m→∞ and
D→0 can not be used as the criterion for transition from second-order TI to TI, since this characteristic could be essentially not induced by Coulomb interaction but only results from the trivial power counting contribution of fermion action. Indeed, this characteristic is satisfied even for free second-order TI. Second, the flow of
B is not paid attention, which is very important and should be seriously studied. In this article, we analyze carefully the corrections for the flows of the model parameters induced by Coulomb interaction. We find that the sign of
m changes but the sign of
B holds if the initial Coulomb strength is large enough, while the sign of
m holds but the sign of
B changes if the initial Coulomb strength takes small values. These results indicate that second-order TI is unstable to trivial band insulator not only under strong Coulomb interaction but also under weak Coulomb interaction. We also study the effects of disorder scattering in second-order TI by renormalization group theory. According to the criterion in Phys. Rev. Lett. {\bf 127}, 176601 (2021), weak disorder drives second-order TI to TI. However, we find that second-order TI is robust against weak disorder, since weak disorder does not give qualitative modification for second-order TI. This result is consistent with recent studies based on other methods. Interplay of Coulomb interaction and disorder in second-order TI is also investigated.
For
D-dimensional weakly interacting topological insulators in certain symmetry classes, the topological invariant can be calculated from a
D- or
(D+1)-dimensional integration over a certain curvature function that is expressed in terms of single-particle Green's functions. Based on the divergence of curvature function at the topological phase transition, we demonstrate how a renormalization group approach circumvents these integrations and reduces the necessary calculation to that for the Green's function alone, rendering a numerically efficient tool to identify topological phase transitions in a large parameter space. The method further unveils a number of statistical aspects related to the quantum criticality in weakly interacting topological insulators, including correlation function, critical exponents, and scaling laws, that can be used to characterize the topological phase transitions driven by either interacting or noninteracting parameters. We use 1D class BDI and 2D class A Dirac models with electron-electron and electron-phonon interactions to demonstrate these principles and find that interactions may change the critical exponents of the topological insulators.
The surfaces of three-dimensional topological insulators (3D TIs) are generally described as Dirac metals, with a single Dirac cone. It was previously believed that a gapped surface implied breaking of either time-reversal
T or
U(1) charge conservation symmetry. Here, we discuss a possibility in the presence of interactions, a surface phase that preserves all symmetries but is nevertheless gapped and insulating. Then, the surface must develop topological order of a kind that can not be realized in a two-dimensional (2D) system with the same symmetries. We discuss candidate surface states, non-Abelian quantum Hall states which, when realized in 2D, have ${\ensuremath{\sigma}}_{xy}=\frac{1}{2}$ and hence break
T symmetry. However, by constructing an exactly soluble 3D lattice model, we show they can be realized as
T-symmetric surface states. The corresponding 3D phases are confined, and have $\ensuremath{\theta}=\ensuremath{\pi}$ magnetoelectric response. Two candidate states have the same 12-particle topological order, the (Read-Moore) Pfaffian state with the neutral sector reversed, which we term T-Pfaffian topological order, but differ in their
T transformation. Although we are unable to connect either of these states directly to the superconducting TI surface, we argue that one of them describes the 3D TI surface, while the other differs from it by a bosonic topological phase. We also discuss the 24-particle Pfaffian-antisemion topological order (which can be connected to the superconducting TI surface) and demonstrate that it can be realized as a
T-symmetric surface state.
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.
The Bloch wave functions have been playing a crucial role in the diagnosis of topological phases in noninteracting systems. However, the Bloch waves are no longer applicable in the presence of finite Coulomb interaction and alternative approaches are needed to identify the topological indices. In this paper, we focus on three-dimensional higher-order topological insulators protected by
C4T symmetry and show that the topological index can be computed through eigenstates of inverse Green's function at zero frequency. If there is an additional
S4 rotoinversion symmetry, the topological index
P3 can be determined by eigenvalues of
S4 at high-symmetry momenta, similar to the Fu-Kane parity criterion. We verify this method using many-body exact diagonalization in higher-order topological insulators with interaction. We also discuss the realization of this higher-order topological phase in tetragonal lattice structure with
C4T-preserving magnetic order. Finally, we discuss the boundary conditions necessary for the hinge states to emerge and show that these hinge states exist even when the boundary is smooth and without a sharp hinge.