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arXiv:1212.3796 · Physical Review B

Z2 anomaly and boundaries of topological insulators

Zohar Ringel, Ady Stern

Published Sep 17, 2013• 19 citations
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Abstract

We study the edge and surface theories of topological insulators from the perspective of anomalies and identify a ${\mathbb{Z}}_{2}$ anomaly associated with charge conservation. The anomaly is manifested through a two-point correlation function involving creation and annihilation operators on two decoupled boundaries. Although charge conservation on each boundary requires this quantity to vanish, we find that it diverges. A corollary result is that under an insertion of a flux quantum, the ground state evolves to an exactly orthogonal state independent of the rate at which the flux is inserted. The anomaly persists in the presence of disorder and imposes sharp restrictions on possible low-energy theories. Being formulated in a many-body, field-theoretical language, the anomaly allows one to test the robustness of topological insulators to interactions in a concise way.

DOI
10.1103/physrevb.88.115307

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We study the edge and surface theories of topological insulators from the perspective of anomalies and identify a Z2{\mathbb{Z}}_{2}Z2​ anomaly associated with charge conservation. The anomaly is manifested through a two-point correlation function involving creation and annihilation operators on two decoupled boundaries. Although charge conservation on each boundary requires this quantity to vanish, we find that it diverges. A corollary result is that under an insertion of a flux quantum, the ground state evolves to an exactly orthogonal state independent of the rate at which the flux is inserted. The anomaly persists in the presence of disorder and imposes sharp restrictions on possible low-energy theories. Being formulated in a many-body, field-theoretical language, the anomaly allows one to test the robustness of topological insulators to interactions in a concise way.
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/2e/2e/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.
Taking the clue from the modern theory of polarization [Rev. Mod. Phys. 66, 899 (1994)], we identify an operator to distinguish between Z2{\mathbb{Z}}_{2}Z2​-even (trivial) and Z2{\mathbb{Z}}_{2}Z2​-odd (topological) insulators in two spatial dimensions. Its definition extends the position operator [Phys. Rev. Lett. 82, 370 (1999)], which was introduced in one-dimensional systems. We first show a few examples of noninteracting models where single-particle wave functions are defined and allow for a direct comparison with standard techniques on large system sizes. Then, we illustrate its applicability for an interacting model on a small cluster where exact diagonalizations are available. Its formulation in the Fock space allows a direct computation of expectation values over the ground-state wave function (or any approximation of it), thus, allowing us to investigate generic interacting systems, such as strongly correlated topological insulators.
We present the correspondence between symmetry-protected topological (SPT) phases and their anomalous boundary states, based on examples in various spacetime dimensions. Through the study of the effect of interactions on these SPT phases, we discovered a new formalism of quantum anomalies, associated with discrete spacetime (such as time-reversal and spatial reflection) symmetries in particular, to classify distinct interacting topological phases. An example is the Z2 classification of the (2+1)d topological insulator protected by charge U(1) and time-reversal (or CP) symmetries, which can be deduced by the form of the global U(1) gauge anomalies on its edge theories defined on closed unorientable manifolds. In this case, the nontrivial phase (in free systems) is robust against electron interactions. Another example is the (3+1)d topological superconductor protected by only time-reversal or reflection symmetry. For this system, we identified the bulk phase by studying the global gravitational anomalies of the surface theories formulated on unorientable spacetime manifolds, and also discussed its connection to the collapse of the non-interacting classification by an integer Z to Z16, in the presence of interactions. We also revisit the problem of gauging a discrete internal symmetry in theories of chiral (Weyl) fermions in 3+1 dimensions – which have not been fully understood so far – from the perspective of fermionic SPT phases in 4+1 dimensions. Comparing with the previous results, we give a complete answer for the anomalies constraints on the discrete symmetry, as our approach is based on purely geometrical considerations, namely, our assumption is more fundamental and general. Furthermore, our result also provides an understanding of gapped states of fermions with anomalous discrete symmetries, and we present a model, based on weak coupling, for constructing these anomalous gapped states.
When time-reversal symmetry is present there is a sharp distinction between topological and trivial band insulators which ensures that, as parameters are varied, these phases are separated by a phase transition at which the bulk gap closes. Surprisingly we find that even in the absence of time-reversal symmetry, gapless regions originating from the phase boundaries persist. Moreover, the critical line generically opens up to enclose Chern insulating phases that are thin but of finite extent in the phase diagram. We explain the topological origin of this effect in terms of quantized charge pumping, showing in particular that it is robust to the effect of disorder and interactions.
We show that the two-dimensional Z2{\mathbb{Z}}_{2}Z2​ invariant for time-reversal invariant insulators can be formulated in terms of the boundary-condition dependence of the ground state wave function for both noninteracting and strongly correlated insulators. By introducing a family of quasi-single-particle states associated to the many-body ground state of an insulator, we show that the Z2{\mathbb{Z}}_{2}Z2​ invariant can be expressed as the integral of a certain Berry connection over half the space of boundary conditions, providing an alternative expression to the formulations that appear in S.-S. Lee and S. Ryu [Phys. Rev. Lett. 100, 186807 (2008)]. We show the equivalence of the different many-body formulations of the invariant and show how they reduce to known band-theoretic results for Slater determinant ground states. Finally, we apply our results to analytically calculate the invariant for the Kane-Mele model with nonlocal (orbital) Hatsugai-Kohmoto (HK) interactions. This rigorously establishes the topological nontriviality of the Kane-Mele model with HK interactions and represents one of the few exact calculations of the Z2{\mathbb{Z}}_{2}Z2​ invariant for a strongly interacting system.
It is often thought that emergent phenomena in topological phases of matter are destroyed when tuning to a critical point. In particular, topologically protected edge states supposedly delocalize when the bulk correlation length diverges. We show that this is not true in general. Edge states of topological insulators or superconductors remain exponentially localized---despite a vanishing band gap---if the transition increases the topological index. This applies to all classes where the topological classification is larger than Z2Z_2Z2​, notably including Chern insulators. Moreover, these edge states are stable to disorder, unlike in topological semi-metals. This new phenomenon is explained by generalizing band (or mass) inversion---a unifying perspective on topological insulators---to kinetic inversion. In the spirit of the bulk-boundary correspondence, we also identify topological invariants at criticality, which take half-integer values and separate topologically-distinct universality classes by a multi-critical point. This work enlarges the scope of topological protection and stability by showing that bulk energy gaps can be unnecessary. Experimental probes and stability to interactions are discussed.