Crystalline symmetries have played a central role in the identification of topological materials. The use of symmetry indicators and band representations have enabled a classification scheme for crystalline topological materials, leading to large sca
le topological materials discovery. In this work we address whether amorphous topological materials, which lie beyond this classification due to the lack of long-range structural order, exist in the solid state. We study amorphous Bi$_2$Se$_3$ thin films, which show a metallic behavior and an increased bulk resistance. The observed low field magnetoresistance due to weak antilocalization demonstrates a significant number of two dimensional surface conduction channels. Our angle-resolved photoemission spectroscopy data is consistent with a dispersive two-dimensional surface state that crosses the bulk gap. Spin resolved photoemission spectroscopy shows this state has an anti-symmetric spin texture resembling that of the surface state of crystalline Bi$_2$Se$_3$. These experimental results are consistent with theoretical photoemission spectra obtained with an amorphous tight-binding model that utilizes a realistic amorphous structure. This discovery of amorphous materials with topological properties uncovers an overlooked subset of topological matter outside the current classification scheme, enabling a new route to discover materials that can enhance the development of scalable topological devices.
The kernel polynomial method allows to sample overall spectral properties of a quantum system, while sparse diagonalization provides accurate information about a few important states. We present a method combining these two approaches without loss of
performance or accuracy. We apply this hybrid kernel polynomial method to improve the computation of thermodynamic quantities and the construction of perturbative effective models, in a regime where neither of the methods is sufficient on its own. To achieve this we develop a perturbative kernel polynomial method to compute arbitrary order series expansions of expectation values. We demonstrate the efficiency of our approach on three examples: the calculation of supercurrent and inductance in a Josephson junction, the interaction of spin qubits defined in a two dimensional electron gas, and the calculation of the effective band structure in a realistic model of a semiconductor nanowire.
We present an algorithm to determine topological invariants of inhomogeneous systems, such as alloys, disordered crystals, or amorphous systems. Based on the kernel polynomial method, our algorithm allows us to study samples with more than $10^7$ deg
rees of freedom. Our method enables the study of large complex compounds, where disorder is inherent to the system. We use it to analyse Pb$_{1-x}$Sn$_{x}$Te and tighten the critical concentration for the phase transition.
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 qu
asicrystalline 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.
Symmetry is a guiding principle in physics that allows to generalize conclusions between many physical systems. In the ongoing search for new topological phases of matter, symmetry plays a crucial role because it protects topological phases. We addre
ss two converse questions relevant to the symmetry classification of systems: Is it possible to generate all possible single-body Hamiltonians compatible with a given symmetry group? Is it possible to find all the symmetries of a given family of Hamiltonians? We present numerically stable, deterministic polynomial time algorithms to solve both of these problems. Our treatment extends to all continuous or discrete symmetries of non-interacting lattice or continuum Hamiltonians. We implement the algorithms in the Qsymm Python package, and demonstrate their usefulness with examples from active research areas in condensed matter physics, including Majorana wires and Kekule graphene.
Lattice translation symmetry gives rise to a large class of weak topological insulators (TIs), characterized by translation-protected gapless surface states and dislocation bound states. In this work we show that space group symmetries lead to constr
aints on the weak topological indices that define these phases. In particular we show that screw rotation symmetry enforces the Hall conductivity along the screw axis to be quantized in multiples of the screw rank, which generally applies to interacting systems. We further show that certain 3D weak indices associated with quantum spin Hall effects (class AII) are forbidden by the Bravais-lattice and by glide or even-fold screw symmetries. These results put a strong constraints on candidates of weak TIs in the experimental and numerical search for topological materials, based on the crystal structure alone.
