Session R55: Higher-Order Topological Phase

Characterization for higher-order symmetry-protected topological phases by quantized Berry phases

We propose the Z_Q Berry phase as a topological invariant for higher-order symmetry-protected topological (HOSPT) phases for two- and three-dimensional systems. It is topologically stable for electron-electron interactions assuming the gap remains open. The integer Q is determined by the rotational symmetry the system has. As a concrete example, we show that the Berry phase is quantized in Z₄ and characterizes the HOSPT phase of the extended Benalcazar-Bernevig-Hughes (BBH) model, which contains the next-nearest neighbor hopping and the intersite Coulomb interactions. In addition, we introduce the Z₄ Berry phase for the spin-model-analog of the BBH model. Furthermore, we demonstrate the Berry phase is quantized in Z₄ for the three-dimensional version of the BBH model. We also confirm the bulk-corner correspondence between the Z_Q Berry phase and the corner states in the HOSPT phases.   

Evolution of planar topological phases in topocircuits

Liberation of topological phases from the realm of electronic materials has been recently initiated by their realizations in highly tunable matematerials, such as phononic and photonic systems, as well as in electric circuits. These classical systems facilitate engineering of various seemingly complex phases of matter, and unveil their topological or geometric properties. We here present a systematic evolution of two-dimensional topological insulators, starting from the analog of quantum anomalous and spin Hall insulators, supporting one-dimensional edge states in electric circuits, also known as topocircuits. Subsequently, we construct rotational symmetry breaking higher-order topological insulators (HOTIs), supporting pointlike corner modes protected by an antiunitary symmetry. The systemic reduction in localization of the topological boundary modes from the edge to the corners is detected from site and orbital selective measurement of the impedance. We also present a simple circuit model for three-dimensional HOTI.   

Magnetotransport in a second-order topological insulator

The most salient feature distinguishing topological insulators from ordinary band insulators is their bulk-boundary correspondence: the topologically-nontrival nature of a bulk sample is signalled by the presence of protected gap-crossing electronic states, localized to the boundary of the sample. Higher-order topological insulators exhibit a somewhat more subtle bulk-boundary correspondence. They still possess protected gap-crossing states, but these are localized to a particular submanifold of the boundary. In the case of a three-dimensional, second-order topological insulator (SOTI), the protected states are localized to the one-dimensional "hinges" of a rectangular nanowire

We theoretically examine the effects of an applied magnetic field in a model of an SOTI, considering the case where the field couples to the orbital electronic motion. Based on numerical calculations, we predict a clear magnetotransport signature reflecting the interplay between Landau level physics and the hinge modes of the SOTI.   

Many-Body Invariants for Topological Insulators: Multipole, Chern, and Hinge States

We propose many-body invariants for the broad classes of topological insulators including Chern insulators, chiral hinge insulators, and multipole insulators. Unlike band indices which only work for non-interacting band insulators, our invariants can detect non-trivial topology of quantum many-body wave functions hence applicable to fully interacting quantum systems. To this end, we design several unitaries whose expectation values on many-body ground states serve as the invariants. We show that the unitaries detect the coefficients of the topological field theory, which are the defining characteristics of topological insulators. This allows us to develop a new way of evaluating Chern numbers, and also the many-body invariant for chiral hinge insulator. Furthermore, we will also show that boundary observables such as the edge-localized polarizations and the corner charge can be measured purely by the many-body unitaries when endowed with appropriate background geometry.   

Gapless hinge states from adiabatic pumping of axion coupling

We demonstrate that gapless chiral hinge states naturally emerge in insulating crystals undergoing a slow cyclic evolution that changes the Chern-Simons axion coupling θ by 2π. This happens when the surface (not just the bulk) returns to its initial state at the end of the cycle, in which case it must pass through a metallic state to dispose of the excess quantum of surface anomalous Hall conductivity pumped from the bulk. If two adjacent surfaces become metallic at different points along the cycle, there is an interval where they are in topologically distinct insulating states, with gapless chiral modes propagating along the connecting hinge. We illustrate these ideas for a 3D tight-binding model consisting of coupled Haldane-model layers. The surface topology is determined in a slab geometry using two different markers, surface anomalous Hall conductivity and surface polarization, and we find that both correctly predict the appearance of gapless hinge states in a rod geometry.   

Transport signature of helical hinge states of quasi-one-dimensional topological insulators

Higher-order topological insulators exhibit protected corner or hinge states generalizing the bulk-boundary correspondence. The quasi-one-dimensional materials Bi₄Br₄ and Bi₄I₄ have been predicted to be prototypical examples of such topological matter with helical hinge states along the atomic chain direction. We examine the electronic behavior of their helical hinge states and predict unique signatures in transport experiment. We further show how these signatures depend on the applied electric and magnetic fields.   

Flux Response of Higher-Order Topological Insulators

One of the highlights of the past few years of topological condensed matter physics has been the discovery of new forms of 3D topological crystalline insulators that are characterized by gapped bulks and surfaces and gapless hinges. Incipient experimental signatures of the gapless hinge states of these “higher-order” topological insulators (HOTIs) have been observed in bismuth, MoTe₂, and WTe₂. However, these signatures have also attracted other explanations. Therefore, it is of intense interest to establish additional indicators of higher-order topology beyond anomalous hinge states. In this work, we use threaded π-flux to probe HOTIs with and without time-reversal symmetry. We establish a general framework that captures the 0D and 1D bound states, charge, and spin that accumulate on the flux cores and tubes. Our framework captures all previous results, such as the fractional charges bound to flux cores in Chern insulators. However, we also discover new examples. Specifically, we demonstrate that π-flux tubes in inversion- and time-reversal-symmetric HOTIs bind Kramers pairs of end states, which represent observable signatures of the anomalous “half” quantum spin Hall effect present on the surfaces of weak topological insulators and HOTIs.   

Disordered intrinsic higher-order topological insulators

Higher order topological insulators are a novel phase of matter in which topologically protected modes appear at corners or hinges rather than surfaces. Unlike conventional topological insulators, higher-order topological insulators can be protected by either a bulk gap or a Wannier gap. Here, we study disordered models of higher-order topological insulators whose topological modes are protected by internal symmetries and a Wannier gap. Like conventional topological insulators, we find that the topological modes are stable against weak disorder. However, we also find that increasing disorder leads to a transition to a trivial state without the mobility gap closing, in striking contrast to the case of conventional disordered topological insulators, in which topological transitions only occur at mobility gap closings. Instead, the topological transition occurs during a real-space Wannier gap closing.   

Higher Order Topological Insulators in Amorphous Solids

We identify the possibility of realizing higher order topological (HOT) phases in noncrystalline or amorphous materials. Specifically, starting from two and three dimensional crystalline HOT insulators, accommodating topological corner states, we gradually enhance structural randomness in the system. Within a parameter regime, as long as amorphousness is confined by outer crystalline boundary, the system continues to host corner states, realizing an amorphous HOT insulator. However, as structural disorder percolates to the edges, corner states start to dissolve into amorphous bulk, and ultimately the system becomes a trivial insulator (devoid of corner modes), when amorphousness plagues the entire system. These outcomes are further substantiated from the scaling of the quadrupolar (octupolar) moment in two (three) dimensions with the scrambling radius. Therefore, HOT phases can be realized in amorphous solids, when wrapped by a thin crystalline layer.   

Pfaffian formalism for higher-order topological insulators

We generalize the Pfaffian formalism, which has been playing an important role in the study of time-reversal invariant topological insulators (TIs), to 3D chiral higher-order topological insulators (HOTIs) protected by the product of four-fold rotational symmetry C₄ and the time-reversal symmetry T. This Pfaffian description reveals a deep and fundamental link between TIs and HOTIs, and allows important conclusions about TIs to be generalized to HOTIs. As examples, we demonstrate in the Letter how to generalize Fu-Kane's parity criterion for TIs to HOTIs, and also present a general method to efficiently compute the Z₂ index of 3D chiral HOTIs without a global gauge.   

Identifying Higher Order Topology and Fractional Corner Charge Using Entanglement Spectra

We study the entanglement spectrum (ES) of two-dimensional Cn-symmetric second-order topological insulators (TIs). We show that some characteristic higher order topological, e.g., the filling anomaly and its associated fractional corner charge, can be determined from the ES of atomic and fragile TIs. By constructing the relationship between the configuration of Wannier orbitals and the number of protected in-gap states in the ES for different symmetric cuts in real space, we express the fractional corner charge in terms of the number of protected in-gap states of the ES. We show that our formula is robust in the presence of electron-electron interactions as long as the interactions preserve rotation symmetry and charge-conservation symmetry. Moreover, we discuss the possible signatures higher order topology in the many-body ES. Our methods allow the identification of some classes of higher order topology without requiring the of nested Wilson loops or nested entanglement spectra.