In the presence of crystalline symmetries, certain topological insulators present a filling anomaly: a mismatch between the number of electrons in an energy band and the number of electrons required for charge neutrality. In this paper, we show that a filling anomaly can arise when corners are introduced in $C_n$-symmetric crystalline insulators with vanishing polarization, having as consequence the existence of corner-localized charges quantized in multiples of $\frac{e}{n}$. We characterize the existence of this charge systematically and build topological indices that relate the symmetry representations of the occupied energy bands of a crystal to the quanta of fractional charge robustly localized at its corners. When an additional chiral symmetry is present, $\frac{e}{2}$ corner charges are accompanied by zero-energy corner-localized states. We show the application of our indices in a number of atomic and fragile topological insulators and discuss the role of fractional charges bound to disclinations as bulk probes for these crystalline phases.
Two-dimensional higher-order topological insulators can display a number of exotic phenomena such as half-integer charges localized at corners or disclination defects. In this paper, we analyze these phenomena, focusing on the paradigmatic example of the quadrupole insulator with $C_4$ rotation symmetry, and present a topological field theory description of the mixed geometry-charge responses. Our theory provides a unified description of the corner and disclination charges in terms of a physical geometry (which encodes disclinations), and an effective geometry (which encodes corners). We exten…
Motivated by recent progress in crystalline symmetry protected topological (SPT) phases of interacting bosons, we study topological crystalline insulators/superconductors (TCIs) of strongly interacting fermions. We construct a class of intrinsically interacting fermionic TCIs, and show that they are beyond both free-fermion TCIs and bosonic crystalline SPT phases. We also show how these phases can be characterized by symmetry protected gapless fermion modes on the corners/hinges of an open system.
A wide variety of higher-order symmetry protected topological phase(HOSPT) with gapless corners or hinges had been proposed as a descendant of topological crystalline insulator protected by spatial symmetry. In this work, we address a new class of higher-order topological state which does not require crystalline symmetries but instead relies on subsystem symmetry for protection. We propose several strong interacting models with gapless hinge or corner based on a `decorated hinge-wall condensate' picture. The hinge-wall, which appears as the defect configuration of $Z_2$ paramagnet is decorated…
When the crystalline symmetries that protect a higher-order topological phase are not preserved at the boundaries of the sample, gapless hinge modes or in-gap corner states cannot be stabilized. Therefore, careful engineering of the sample termination is required. Similarly, magnetic textures, whose quantum fluctuations determine the supported magnonic excitations, tend to relax to new configurations that may also break crystalline symmetries when boundaries are introduced. Here we uncover that antiskyrmion crystals provide an experimentally accessible platform to realize a magnonic topologica…
We show that in crystalline insulators point group symmetry alone gives rise to a topological classification based on the quantization of electric polarization. Using C3 rotational symmetry as an example, we first prove that the polarization is quantized and can only take three inequivalent values. Therefore, a Z3 topological classification exists. A concrete tight-binding model is derived to demonstrate the Z3 topological phase transition. Using first-principles calculations, we identify graphene on BN substrate as a possible candidate to realize the Z3 topological states. To complete our ana…
We prove a topological criterion for the existence of zero-energy Majorana bound-state on a disclination, a rotation symmetry breaking point defect, in 4-fold symmetric topological crystalline superconductors (TCS). We first establish a complete topological classification of TCS using the Chern invariant and three integral rotation invariants. By analytically and numerically studying disclinations, we algebraically deduce a Z_2-index that identifies the parity of the number of Majorana zero-modes at a disclination. Surprisingly, we also find weakly-protected Majorana fermions bound at the corn…
We study two-dimensional spinful insulating phases of matter that are protected by time-reversal and crystalline symmetries. To characterize these phases we employ the concept of corner charge fractionalization: Corners can carry charges that are fractions of even multiples of the electric charge. The charges are quantized and topologically stable as long as all symmetries are preserved. We classify the different corner charge configurations for all point groups, and match them with the corresponding bulk topology. For this we employ symmetry indicators and (nested) Wilson loop invariants. We …
Recent years saw the complete classification of topological band structures, revealing an abundance of topological crystalline insulators. Here we theoretically demonstrate the existence of topological materials beyond this framework, protected by quasicrystalline symmetries. We construct a higher-order topological phase protected by a point group symmetry that is impossible in any crystalline system. Our tight-binding model describes a superconductor on a quasicrystalline Ammann-Beenker tiling which hosts localized Majorana zero modes at the corners of an octagonal sample. The Majorana modes …
Coupled-wire constructions have been widely applied to quantum Hall systems and symmetry-protected topological (SPT) phases. In this Letter, we use the coupled one-dimensional nonchiral Luttinger liquids with domain-wall structured mass terms as quantum wires to construct the crystalline higher-order topological superconductors (HOTSC) in two-dimensional interacting fermionic systems by two representative examples: $D_4$-symmetric class-D HOTSC and $C_4$-symmetric class-BD\1 HOTSC, with Majorana corner modes on the edge. Furthermore, based on the coupled-wire constructions, the quantum phase t…
We propose many-body order parameters for bulk multipoles in crystalline systems, which originate from the non-trivial topology of the nested Wilson loop spectrum. The many-body order parameters are designed to measure multipolar charge distribution in a crystalline unit cell, and they match the localized corner charge originating from the multipoles. We provide analytic arguments and numerical proof for the order parameters. Furthermore, we show that the many-body order parameters faithfully measure the physical multipole moments even when the symmetries quantizing multipoles are lost and thu…
We show that lattices with higher-order topology can support corner-localized bound states in the continuum (BICs). We propose a method for the direct identification of BICs in condensed matter settings and use it to demonstrate the existence of BICs in a concrete lattice model. Although the onset for these states is given by corner-induced filling anomalies in certain topological crystalline phases, additional symmetries are required to protect the BICs from hybridizing with their degenerate bulk states. We demonstrate the protection mechanism for BICs in this model and show how breaking this…
One of the cornerstones for topological quantum computations is the Majorana zero mode, which has been intensively searched in fractional quantum Hall systems and topological superconductors. Several recent works suggest that such an exotic mode can also exist in a one-dimensional (1D) interacting double-wire setup even without long-range superconductivity. A notable instability in these proposals comes from interchannel single-particle tunneling that spoils the topological ground state degeneracy. Here we show that a 1D Dirac semimetal (DSM) nanowire is an ideal number-conserving platform to …
Real-space construction and classification for time-reversal symmetric crystalline superconductors in 2D interacting fermionic systems
Southern University of Science and Technology（南方科技大学）；Graduate School of China Academy of Engineering Physics（中国工程物理研究院研究生院）；The Chinese University of Hong Kong（香港中文大学）
Recent developments in higher-order topological phases have elucidated on the relationship between nontrivial multipole moments in the bulk and the emergence of fractionally quantized charges at the boundary. Here, we put on the table a proposal of the three-dimensional octupole insulator with fractionally quantized corner charges $\pm e/8$: sodium chloride, commonly known as table salt. The fractional quantization of the corner charge is unaffected by the quantum fluctuation of the electric charge per ion, as we demonstrate explicitly with ab initio calculations. We further show that the elec…
Recent studies revealed that the electric multipole moments of insulators result in fractional electric charges localized to the hinges and corners of the sample. We here explore the magnetic analog of this relation. We show that a collinear antiferromagnet with spin $S$ defined on a $d$-dimensional cubic lattice features fractionally quantized magnetization $M_{\text{c}}^z=S/2^d$ at the corners. We find that the quantization is robust even in the presence of gapless excitations originating from the spontaneous formation of the Néel order, although the localization length diverges, suggesting …
Crystalline topological phases have recently attracted a lot of experimental and theoretical attention. Key advances include the complete elementary band representation analyses of crystalline matter by symmetry indicators and the discovery of higher-order hinge and corner states. However, current classification schemes of such phases are either implicit or limited in scope. We present a new scheme for the explicit classification of crystalline topological insulators and superconductors. These phases are protected by crystallographic point group symmetries and are characterized by bulk topolog…
Three-dimensional, crystalline Hopf insulators are generic members of unitary Wigner-Dyson class, which can break all global discrete symmetries and point group symmetries. In the absence of first Chern number for any two-dimensional plane of Brillouin zone, the Hopf invariant $N_H \in \mathbb{Z}$. But in the presence of Chern number $N_H \in \mathbb{Z}_{2q}$, where $q$ is the greatest common divisor of Chern numbers for $xy$, $yz$, and $xz$ planes of Brillouin zone. How does $N_H$ affect topological quantization of isotropic, magneto-electric coefficient? We answer this question with calculat…
The modern theory of electric polarization has recently been extended to higher multipole moments, such as quadrupole and octupole moments. The higher electric multipole insulators are essentially topological crystalline phases protected by underlying crystalline symmetries. Henceforth, it is natural to ask what are the consequences of symmetry breaking in these higher multipole insulators. In this work, we investigate topological phases and the consequences of symmetry breaking in generalized electric quadrupole insulators. Explicitly, we generalize the Benalcazar-Bernevig-Hughes model by add…
Recent formal classifications of crystalline topological insulators predict that the combination of time-reversal and rotational symmetry gives rise to topological invariants beyond the ones known for other lattice symmetries. Although the classification proves their existence, it does not indicate a way of calculating the values of those invariants. Here, we show that a specific set of concentric Wilson loops and line invariants yields the values of all topological invariants in two-dimensional systems with pure rotation symmetry in class AII. Examples of this analysis are given for specific …
We propose and prove a family of generalized Lieb-Schultz-Mattis (LSM) theorems for symmetry protected topological (SPT) phases on boson/spin models in any dimensions. The "conventional" LSM theorem, applicable to e.g. any translation invariant system with an odd number of spin-1/2 particles per unit cell, forbids a symmetric short-range-entangled ground state in such a system. Here we focus on systems with no LSM anomaly, where global/crystalline symmetries and fractional spins within the unit cell ensure that any symmetric SRE ground state must be a nontrivial SPT phase with anomalous bounda…
Monolayer WTe$_2$, a centrosymmetric transition metal dichacogenide, has recently been established as a quantum spin Hall insulator and found superconducting upon gating. Here we study the pairing symmetry and topological nature of superconducting WTe$_2$ with a microscopic model at mean-field level. Surprisingly, we find that the spin-triplet phases in our phase diagram all host Majorana modes localized on two opposite corners. Even when the conventional pairing is favored, we find that an intermediate in-plane magnetic field exceeding the Pauli limit stabilizes an unconventional equal-spin p…
Clean isotropic quantum Hall fluids in the continuum possess a host of symmetry-protected quantized invariants, such as the Hall conductivity, shift and Hall viscosity. Here we develop a theory of symmetry-protected quantized invariants for topological phases defined on a lattice, where quantized invariants with no continuum analog can arise. We develop topological field theories using discrete crystalline gauge fields to fully characterize quantized invariants of (2+1)D Abelian topological orders with symmetry group $G = U(1) \times G_{\text{space}}$, where $G_{\text{space}}$ consists of orie…
Topological crystalline superconductors are known to have possible higher-order topology, which results in Majorana modes on $d-2$ or lower-dimensional boundaries. Given the rich possibilities of boundary signatures, it is desirable to have topological invariants that can predict the type of Majorana modes from band structures. Although symmetry indicators, a type of invariant that depends only on the band data at high-symmetry points, have been proposed for certain crystalline superconductors, there exist symmetry classes in which symmetry indicators fail to distinguish superconductors with d…
For three-dimensional (3D) crystalline insulators, preserving space-inversion ($\mathcal{P}$) and time-reversal ($\mathcal{T}$) symmetries, the third homotopy class of two-fold, Kramers-degenerate bands is described by a 3D winding number $n_{3,j} \in \mathbb{Z}$, where $j$ is the band index. It governs space group symmetry-protected, instanton or tunneling configurations of $SU(2)$ Berry connection, and the quantization of magneto-electric coefficient $θ_j = n_{3,j} π$. We show that $|n_{3,j}|$ for realistic, \emph{ab initio} band structures can be identified from a staggered symmetry-indicat…
