Session Q17: Topological Crystalline Insulators: Theory and Experiment

Topological real-space responses and stable integer invariants from dimensional reduction

Dimensional reduction of topological response is a powerful tool for understanding topological field theory of insulating states at various spatial dimensions. Thus far this procedure has been implemented by Qi et al. for time-reversal symmetric, spin-orbit coupled insulators (symplectic, class AII). In this work, we show that the dimensional reduction of four-dimensional quantum Hall effect can be performed for all three Wigner Dyson classes to understand stable topological states at three and two spatial dimensions, which are not predicted by the ten-fold classification scheme. We accomplish this goal by using magnetic monopoles and flux tubes as non-perturbative, real-space probes. We show that many topological phases of matter, which are often called ''fragile topological insulators" actually support stable invariants and quantized response.   

Detecting SPT Signatures and Projective Representations in Free-Fermion Topological Crystalline Insulators

The many-body quantum numbers of 0D collective excitations bound to crystal and electromagnetic defects (such as magnetic fluxes and monopoles) provide powerful indicators of bulk topology, in that they can indicate the presence of quantized responses in the bulk that are governed by long-wavelength topological field theories that are stable to symmetric interactions. In interacting symmetry-protected topological phases (SPTs), defect quantum numbers frequently indicate nontrivial bulk topology if they transform in projective representations of the local symmetry group. We here introduce numerical methods for computing defect quantum numbers in stable and fragile free-fermion topological crystalline insulators (TCIs) via the reduced density matrix, revealing a deep connection between defect quantum numbers and the entanglement spectrum. We surprisingly find that when crystal symmetries are included in the local symmetry group, defects can appear to transform projectively even in Wannierizable (fragile) insulators. Our framework systematically characterizes stable and fragile TCIs beyond filling anomalies and higher-order topology, allowing direct connections between free-fermion TCIs and interacting SPTs.   

Uncovering Meaning for the Z4 Classification of Inversion-Symmetric Nonmagnetic 3D Insulators through Domain-Wall Symmetry Fractionalization

It was unexpectedly recently discovered that inversion-symmetric 3D insulators have a stable Z₄ classification, as opposed to the Z₂ predicted by Fu and Kane. Confusingly, the Z₄ = 1, 3 states are both standard 3D topological insulators governed by θ = π axion electrodynamics, whereas the Z₄ = 2 topological crystalline insulator (TCI) states, despite having abundant and accessible material candidates, have unknown quantized response signatures. We here theoretically and numerically demonstrate that the full Z₄ group can be understood via an anomalous sequence of quantized symmetry fractionalization exhibited by domain-wall fluxes that is stable to symmetric interactions. We further uncover a link to other Z₄ invariants in 1D and 3D, generalizing the well-established relationship in insulators with U(1) charge-conservation symmetry between the Z₂ Berry phase in 1D and θ angle in 3D. To aid our analysis, we introduce a generalization of spin-resolved topology that characterizes U(1)-broken insulators with appropriately chosen interactions. Our results concretely link the long-wavelength properties of interacting symmetry-protected topological phases and real-material free-fermion TCIs.   

Transport Study of Bismuth Halide Series

The novel one-dimensional van der Waals materials Bi4X4 (X=Bi, I) constitute a promising platform to explore the topological phases. Theoretical calculation has revealed large bulk band gaps, gapless edge states, and higher order topological phases, which have also been experimentally proved. Despite the progress achieved by scanning tunneling microscopy and the angle-resolved photoemission spectroscopy, transport study on this platform remains elusive. Here we demonstrate the preliminary transport features of bismuth halide series with different components and discuss the evidence of edge states in this material.   

Evidence of One-dimensional Edge States in Quasi-one-dimensional Topological Insulators Bismuth halogenides

Bi4Br4 is predicted to host gapless monolayer step edge and a large bulk gap (~210meV) which indicates that it is a promising quantum spin Hall insulator up to room temperature. Due to the weak coupling between layers, even in multilayer Bi4Br4, the quantum spin hall edge can still be preserved. To date, the study on transport properties of Bi4Br4 is still rare. In this talk, I will discuss our recent work on transport study of multilayer Bi4Br4 devices and preliminary evidence that indicates the presence of one-dimensional edge states.   

Revealing higher-order topology through loop of spin-helical hinge states in Bi nanocrystal with proximity superconductivity

Higher-order topological insulators (HOTI) are newly proposed topological materials which host gapless boundary states in a codimension of order index. E.g., a 2nd order time-reversal invariant 3D TI will possess 1D boundary states residing in a series of hinges encircling the crystal. These hinge states are spin-helical which resembles the edge state of a quantum spin hall insulator. Experimentally, verifying a 3D HOTI would require systematical investigation on all the boundaries of a 3D crystal, which is challenging. Here we studied the element of Bismuth (Bi), a promising candidate of 2nd order TI, in the form of nanocrystals. The nanocrystals were fabricated on superconducting V3Si (111) substrate. Via scanning tunneling microscopy/spectroscopy (STM/S) measurement, we revealed dispersive 1D states on various hinges of the crystal, which are consistent with first-principle band calculation. After introducing ferromagnetic clusters, we found new scattering channels opened at certain hinges, which suggest they are spin-helical. Remarkably, these spin-helical states on different hinges formed a closed loop surrounding the nanocrystal, further supporting their topological origin. Thus our study provided direct evidence on the existence of HOTI state in nature. Moreover, we detected proximity-induced superconductivity in the hinge states, which enables HOTI as a novel platform for generating Majorana quasi-particles.   

Counterintuitive Emergence of the π Zak Phase on the (111) Surface of Bismuth

Bismuth (Bi) has emerged as a crucial element in the realm of topological materials due to its strong spin-orbit coupling, leading to band inversions and various topologically nontrivial phases. This study unveils an additional nontrivial π Zak phase in Bi, inducing metallic in-gap states on the (111) surface. The Zak phase, one of the fundamental topological invariants, is shown to depend on surface orientation and termination (i.e., the sign of dimerization). In sharp contrast to the Su-Schrieffer-Heeger model, the one-dimensional counterpart of Bi, where nontrivial edge modes emerge at ends cutting strong bonds, the nontrivial surface states of Bi appear on the (111) surface where weak bonds are cut, which is counterintuitive. These findings elucidate the underlying origins of the intriguing (111) surface states of Bi, often interpreted as an analog of antimony, exhibiting a similar but topologically distinct electronic structure.   

Crystalline-electromagnetic responses of Dirac semimetal-charge density wave insulators

We show that insulators arising from coupling a three-dimensional Dirac semimetal to charge density wave (CDW) order can exhibit a three-dimensional generalization of the Wen-Zee response. This three-dimensional Wen-Zee response binds a quantized change per length to disclination defects in an amount proportional to the separation of the Dirac nodes in momentum space. Furthermore, when the CDW respects the inversion symmetry of the underlying Dirac semimetal, the Dirac-CDW insulator exhibits a disclination filling anomaly that manifests as a quantized difference in the charge bound to a disclination for systems with periodic and open boundary conditions. We construct an effective topological response theory that captures these crystalline responses of Dirac-CDW insulators and confirm our predictions numerically with lattice model calculations.   

Topological Markers for C_{n}-symmetric Topological Crystalline Insulators with and without Translation Symmetry

Bulk and surface topological properties of non-interacting topological phases can be diagnosed via symmetry-eigenvalue analysis of Bloch states at high symmetry points in the Brillouin zone, in the presence of crystalline symmetry. This symmetry-based diagnosis has allowed for a simplification of the calculation of topological invariants. However, when open boundaries are present, only the point group part of the symmetry group remains, and it is unclear how to utilize crystalline symmetries to diagnose bulk topology. In this work, we introduce basis-independent topological crystalline markers to characterize bulk topology in C_{n}-symmetric (n=2,3,4,6) topological crystalline insulators with and without translation symmetry. These markers are expressed solely in terms of the trace of the product of the crystalline symmetry operator and the ground state projector. First, we provide a general method for calculating topological markers in periodic systems with an arbitrary number of unit cells in each direction. This is because it is not possible to obtain the symmetry eigenvalues of Bloch states at all high-symmetry points in the Brillouin zone even if periodic boundaries are present; the quantization of the crystalline momentum along each direction spanning the Brillouin zone depends on the unit cell number along each direction spanned by the primitive lattice vector on the real-space lattice, thereby determining the presence of high-symmetry points. Second, we construct explicit mappings from the markers to the Chern number, bulk polarization, and sector charge for two-dimensional $C_{n}$-symmetric insulators in symmetry classes A, AI, AII, and D. Finally, we demonstrate how to numerically calculate the marker in finite-size systems with open boundaries, and how to diagnose the bulk topology from the marker.   

Aharonov-Bohm interference of topological hinge states

The transport response of topological surface states in strong topological insulators has undergone extensive investigation, yet the behavior of topological hinge modes remains enigmatic. Here, we uncover phase-coherent transport facilitated by the topological hinge states in a higher-order topological insulator. These conducting hinge states reside within the insulating bulk and surface, both gapped across the entire Brillouin zone. Our magnetoresistance measurements reveal pronounced h/e periodic (where h denotes Planck’s constant and e represents the electron charge) Aharonov–Bohm oscillation. Notably, the observed periodicity, which directly reflects the enclosed area of phase-coherent electron propagation, matches the area enclosed by the sample hinges. Furthermore, the h/e oscillations evolve as a function of magnetic field orientation, following the interference paths along the hinge modes allowed by topology and symmetry. These findings offer compelling evidence for the quantum interference phenomena arising from electron motion around the hinges.   

Polymorphic and dual topological insulating phases of van der Waals layered materials

We present a study that explores the exotic characteristics of van der Waals (vdW) layered systems, which exhibit both topological insulators and topological crystalline insulators. This intriguing behavior gives rise to the emergence of a dual topological insulator (DTI) within this vdW layered system, characterized by a nontrivial invariant and a nontrivial mirror Chern number . By considering the stacking order of InTe, chosen as an example of vdW layered materials, we discovered the formation of an additional mirror symmetry that enhances the robustness of the surface state against perturbations. These polymorphic variations lead to a triply-protected surface state, combining the newly formed mirror symmetry with the existing mirror symmetry and the time-reversal symmetry. Furthermore, we found that the Dirac states in InTe not only withstand time-reversal symmetry-breaking magnetic fields but also exhibit a displacement away from the time-reversal invariant momentum points, with the direction of movement determined by the spin texture. The dual topological nature induced by vdW stacking offers exciting possibilities for controlling the stability and mobility of Dirac surface states, thus opening avenues for the development of highly versatile and reliable DTI-based devices.   

Quantum Phase Transitions in Quasicrystalline Higher-Order Topological Insulators

Topological phase transitions in disordered systems exhibit rich critical behaviors, including the emergence of multifractal wavefunctions. However, topological phase transitions in quasicrystalline systems have received relatively limited attention, despite the rapidly growing research on quasicrystalline topological phases. This study investigates the phase transition between a quasicrystalline higher-order topological insulator (HOTI) and a trivial band insulator. We perform model studies on two distinct systems: an Ammann-Beenker (AB) tiling quasicrystal and a square lattice with a quasiperiodic potential. Our findings reveal common low-energy physics shared by these systems: (i) We observe the presence of a gapless, critical region that separates the HOTI and trivial phases. This critical region is characterized by a high density of states and multifractal wavefunctions near the Fermi level. (ii) The topological phase transition, i.e., jump of fractional corner charge, occurs at a specific point in the critical region where the eigenstates at the Fermi level are maximally extended. Our findings suggest the existence of a novel gapless phase with simultaneous fractional corner charge and multifractal states at the Fermi level, shedding light on intriguing and previously unexplored facets of topological transitions within quasicrystalline systems.   

Magnetoconductance oscillations in topological crystalline insulator Pb1-xSnxTe nanowires

SnTe is a topological crystalline insulator (TCI) with gapless topological states at the surface. However, electrically probing these states is a challenge due to the large charge carrier density in the bulk, which dominates the signal. Therefore, we use the ternary alloy Pb_1-xSn_xTe, for which x can be varied to reduce the carrier density but at the same time undergoes a topological to trivial phase transition for x ≈ 0.35. Here, we report magnetoconductance measurements on Pb_1-xSn_xTe nanowire devices with x = 0.4. We observe periodic oscillations of the magnetoconductance as a function of in-plane magnetic field. The Fourier spectrum after subtracting the background shows three distinct sets of peaks, corresponding to flux values of h/e, h/2e and h/3e. This indicates the Aharonov-Bohm (AB) effect with higher harmonics and the Altschuler-Aronov-Spivak (AAS) effect, which have previously been reported for topological insulator (TI) nanowires. The magnitude of the oscillations is on the order of conductance quantum 2e²/h at  mK, and decays exponentially with increasing temperature. These results indicate that Pb_1-xSn_xTe nanowires are a promising platform for electrically probing the topological states in a TCI in quantum transport experiments.   

Spin-charge separation and spin-Chern number of disordered, higher-order topological insulators

According to the ten-fold classification scheme of topological insulators, only the time-reversal-symmetric, spin-orbit-coupled systems (class AII) can support stable, quantum spin Hall states at two spatial dimensions. In this work, we employ magnetic flux tube as a real-space probe to show that all three Wigner Dyson symmetry classes (A, AI, and AII) can display stable, quantum spin (or pseudo-spin) Hall insulators in the presence of disorder. This goal is accomplished by demonstrating spin-charge separation in the presence of flux tube. Our analysis is a manifestly gauge invariant method for addressing the magnitude of spin Chern number of first order and higher order topological insulators.   

Curvature effects in surface states of topological materials

In recent years, researchers have revealed the pivotal role that curvature plays in 3D topological materials. For instance, in the context of the spherical topological insulators, curvature-induced magnetic monopole effects provide the basis for the emerging Landau levels. Another noteworthy example can be found in the realm of the spherical topological superconductors, where the same curvature effects result in the generation of uniform magnetic fields, leading to the formation of vortices. These instances collectively underscore the profound impact of surface curvature in topological materials, manifesting as the production of magnetic fields. The surface states of topological insulators can be effectively described using the two-dimensional massless Dirac equation. We consider the parallel transport of the Dirac spinor field on the curved surface and show that the spin-connection must be introduced. The spin connection plays the role as a gauge field, which gives rise to an effective magnetic field. We show that the effective magnetic field is proportional to the Gauss curvature of the surface, which is a topological invariance upon integration over closed surfaces. Furthermore, we analyze the effect of curvature on transport properties of electrons on the surface of the topological materials. In particular, it is shown that surface roughness on the surface of topological insulators generally generate the spin Hall effect.