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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 are protected by particle-hole symmetry and by the combination of an 8-fold rotation and in-plane reflection symmetry. We find a bulk topological invariant associated with the presence of these zero modes, and show that they are robust against large symmetry preserving deformations, as long as the bulk remains gapped. The nontrivial bulk topology of this phase falls outside all currently known classification schemes.
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Non-Hermitian topological phases bear a number of exotic properties, such as the non-Hermitian skin effect and the breakdown of conventional bulk-boundary correspondence. In this paper, we introduce an unsupervised machine learning approach to classify non-Hermitian topological phases based on diffusion maps, which are widely used in manifold learning. We find that the non-Hermitian skin effect will pose a notable obstacle, rendering the straightforward extension of unsupervised learning approaches to topological phases for Hermitian systems ineffective in clustering non-Hermitian topological phases. Through theoretical analysis and numerical simulations of two prototypical models, we show that this difficulty can be circumvented by choosing the on-site elements of the projective matrix as the input data. Our results provide a valuable guidance for future studies on learning non-Hermitian topological phases in an unsupervised fashion, both in theory and experiment.
Robust fractional charge localized at disclination defects has been recently found as a topological response in $C_{6}$ symmetric 2D topological crystalline insulators (TCIs). In this article, we thoroughly investigate the fractional charge on disclinations in $C_n$ symmetric TCIs, with or without time reversal symmetry, and including spinless and spin-$frac{1}{2}$ cases. We compute the fractional disclination charges from the Wannier representations in real space and use band representation theory to construct topological indices of the fractional disclination charge for all $2D$ TCIs that admit a (generalized) Wannier representation. We find the disclination charge is fractionalized in units of $frac{e}{n}$ for $C_n$ symmetric TCIs; and for spin-$frac{1}{2}$ TCIs, with additional time reversal symmetry, the disclination charge is fractionalized in units of $frac{2e}{n}$. We furthermore prove that with electron-electron interactions that preserve the $C_n$ symmetry and many-body bulk gap, though we can deform a TCI into another which is topologically distinct in the free fermion case, the fractional disclination charge determined by our topological indices will not change in this process. Moreover, we use an algebraic technique to generalize the indices for TCIs with non-zero Chern numbers, where a Wannier representation is not applicable. With the inclusion of the Chern number, our generalized fractional disclination indices apply for all $C_n$ symmetric TCIs. Finally, we briefly discuss the connection between the Chern number dependence of our generalized indices and the Wen-Zee term.
Topological phase transitions in a three-dimensional (3D) topological insulator (TI) with an exchange field of strength $g$ are studied by calculating spin Chern numbers $C^pm(k_z)$ with momentum $k_z$ as a parameter. When $|g|$ exceeds a critical value $g_c$, a transition of the 3D TI into a Weyl semimetal occurs, where two Weyl points appear as critical points separating $k_z$ regions with different first Chern numbers. For $|g|<g_c$, $C^pm(k_z)$ undergo a transition from $pm 1$ to 0 with increasing $|k_z|$ to a critical value $k_z^{tiny C}$. Correspondingly, surface states exist for $|k_z| < k_z^{tiny C}$, and vanish for $|k_z| ge k_z^{tiny C}$. The transition at $|k_z| = k_z^{tiny C}$ is acompanied by closing of spin spectrum gap rather than energy gap.
We consider a three-dimensional topological insulator (TI) wire with a non-uniform chemical potential induced by gating across the cross-section. This inhomogeneity in chemical potential lifts the degeneracy between two one-dimensional surface state subbands. A magnetic field applied along the wire, due to orbital effects, breaks time-reversal symmetry and lifts the Kramers degeneracy at zero-momentum. If placed in proximity to an $s$-wave superconductor, the system can be brought into a topological phase at relatively weak magnetic fields. Majorana bound states (MBSs), localized at the ends of the TI wire, emerge and are present for an exceptionally large region of parameter space in realistic systems. Unlike in previous proposals, these MBSs occur without the requirement of a vortex in the superconducting pairing potential, which represents a significant simplification for experiments. Our results open a pathway to the realisation of MBSs in present day TI wire devices.
Symmetry is fundamental to topological phases. In the presence of a gauge field, spatial symmetries will be projectively represented, which may alter their algebraic structure and generate novel topological phases. We show that the $mathbb{Z}_2$ projectively represented translational symmetry operators adopt a distinct commutation relation, and become momentum dependent analogous to twofold nonsymmorphic symmetries. Combined with other internal or external symmetries, they give rise to many exotic band topology, such as the degeneracy over the whole boundary of the Brillouin zone, the single fourfold Dirac point pinned at the Brillouin zone corner, and the Kramers degeneracy at every momentum point. Intriguingly, the Dirac point criticality can be lifted by breaking one primitive translation, resulting in a topological insulator phase, where the edge bands have a M{o}bius twist. Our work opens a new arena of research for exploring topological phases protected by projectively represented space groups.
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