Higher-order topological phases (HOTPs) feature protected gapless modes on boundaries of higher codimension, such as the corners or hinges of a crystal. They are understood as being protected by lattice symmetries: If the latter are broken, it becomes possible to remove the boundary modes without closing the bulk gap. In this work, we present construction schemes for HOTPs protected solely by the bulk gap, by fundamental symmetries, and by the global topology of the system shape (its genus, or number of holes), independent of any crystalline symmetries. As long as the fundamental local symmetries are preserved, the resulting boundary states cannot be removed by any purely-surface perturbation.

The Chebyshev expansion method is a well-established technique for computing the time evolution of quantum states, particularly in Hermitian systems with a bounded spectrum. Here, we show that the applicability of the Chebyshev expansion method extends well beyond this constraint: It remains valid across the entire complex plane and is thus suitable for arbitrary non-Hermitian matrices. We identify numerical rounding errors as the primary source of errors encountered when applying the method outside the conventional spectral bounds, and they are not caused by fundamental limitations. By carefully selecting the spectral radius and the time step, we show how these errors can be effectively suppressed, enabling accurate time evolution calculations in non-Hermitian systems. We derive an analytic upper bound for the rounding error, which serves as a practical guideline for selecting time steps in numerical simulations. As an application, we illustrate the performance of the method by computing the time evolution of wave packets in the Hatano-Nelson model.

Axial vectors, such as current or magnetization, are commonly used order parameters in time-reversal symmetry breaking systems. These vectors also break isotropy in three dimensional systems, lowering the spatial symmetry. We demonstrate that it is possible to construct a three-dimensional medium with average isotropy and inversion symmetry where time-reversal symmetry is systematically broken. We devise a model of an amorphous material with scalar time-reversal symmetry breaking, implemented by hopping through chiral magnetic clusters along the bonds. The presence of only average spatial symmetries—continuous rotation and inversion—is sufficient to protect a topological phase, yielding a statistical topological insulator. We demonstrate the topological nature of our model by constructing a bulk integer topological invariant for the effective continuum model, which guarantees gapless surface spectrum on any surface with an odd number of Dirac nodes, analogous to crystalline mirror Chern insulators. We also show the expected transport properties of a three-dimensional statistical topological insulator, which remains critical on the surface for odd values of the invariant.

In quantum mechanics, the Schrieffer-Wolff (SW) transformation (also called quasi-degenerate perturbation theory) is known as an approximative method to reduce the dimension of the Hamiltonian. We present a geometric interpretation of the SW transformation: We prove that it induces a local coordinate chart in the space of Hermitian matrices near a k-fold degeneracy submanifold. Inspired by this result, we establish a 'distance theorem': we show that the standard deviation of k neighboring eigenvalues of a Hamiltonian equals the distance of this Hamiltonian from the corresponding k-fold degeneracy submanifold, divided by root k. Furthermore, we investigate one-parameter perturbations of a degenerate Hamiltonian, and prove that the standard deviation and the pairwise differences of the eigenvalues lead to the same order of splitting of the energy eigenvalues, which in turn is the same as the order of distancing from the degeneracy sub-manifold. As applications, we prove the 'protection' of Weyl points using the transversality theorem, and infer geometrical properties of certain degeneracy submanifolds based on results from quantum error correction and topological order.

Parameter-dependent quantum systems often exhibit energy degeneracy points, whose comprehensive description naturally leads to the application of methods from singularity theory. A prime example is an electronic band structure where two energy levels coincide in a point of momentum space. It may happen, and this case is the focus of our work, that three or more levels coincide at a parameter point, called multifold degeneracy. Upon a generic perturbation, such a multifold degeneracy point is dissolved into a set of Weyl points, that is, generic twofold degeneracy points. In this work, we provide an upper bound to the number of Weyl points born from the multifold degeneracy point. To compute this upper bound, we describe the geometric degeneracy variety in the space of complex matrices. We compute its multiplicity at certain singular points corresponding to a multifold degeneracy, and the multiplicity of holomorphic map germs with respect to this variety. Our work covers physics and mathematics aspects in detail and attempts to bridge the two disciplines and communities. For self-containedness, we survey examples of multifold degeneracies in quantum systems and condensed-matter physics, as well as the established tools of local algebraic geometry that we use to identify the upper bound.

A common technique in the study of complex quantum-mechanical systems is to reduce the number of degrees of freedom in the Hamiltonian by using quasi-degenerate perturbation theory. While the Schrieffer--Wolff transformation achieves this and constructs an effective Hamiltonian, its scaling is suboptimal, it is limited to two subspaces, and implementing it efficiently is both challenging and error-prone. We introduce an algorithm for constructing an equivalent effective Hamiltonian as well as a Python package, Pymablock, that implements it. Our algorithm combines an optimal asymptotic scaling and the ability to handle any number of subspaces with a range of other improvements. The package supports numerical and analytical calculations of any order and it is designed to be interoperable with any other packages for specifying the Hamiltonian. We demonstrate how the package handles constructing a k.p model, analyses a superconducting qubit, and computes the low-energy spectrum of a large tight-binding model. We also compare its performance with reference calculations and demonstrate its efficiency.

We theoretically explore the electronic structure of holes in cylindrical Germanium/Silicon core/shell nanowires using a perturbation theory approach. The approach yields a set of interpretable and transferable effective low-energy models for the lowest few sub-bands up to fifth order for experimentally relevant growth directions. In particular, we are able to resolve higher order cross terms e.g., the dependency of the effective mass on the magnetic field. Our study reveals orbital inversions of the lowest sub-bands for low-symmetry growth directions, leading to significant changes of the lower order effective coefficients. We demonstrate a reduction of the direct Rashba spin-orbit interaction due to competing symmetry effects for low-symmetry growth directions. Finally, we find that the effective mass of the confined holes can diverge yielding quasi flat bands interesting for correlated states. We show how one can tune the effective mass of a single spin band allowing one to tune the effective mass selectively to its divergent points.

We introduce a two-dimensional network model that realizes a higher-order topological phase (HOTP). We find that in the HOTP the bulk and boundaries of the system are gapped, and a total of 16 corner states are protected by the combination of a fourfold rotation, a phase-rotation, and a particle-hole symmetry. In addition, the model exhibits a strong topological phase at a point of maximal coupling. This behavior is in opposition to conventional network models, which are gapless at this point. By introducing the appropriate topological invariants, we show how a point group symmetry can protect a topological phase in a network. Our work provides the basis for the realization of HOTP in alternative experimental platforms implementing the network model.

This review is devoted to the different techniques that have been developed to compute the phase-coherent transport properties of quantum nanoelectronic systems connected to electrodes. Beside a review of the different algorithms proposed in the literature, we provide a comprehensive and pedagogical derivation of the two formalisms on which these techniques are based: the scattering approach and the (nonequilibrium) Green's function approach. We show that the scattering problem can be formulated as a system of linear equations and that different existing algorithms for solving this scattering problem amount to different sequences of Gaussian elimination. We explicitly prove the equivalence of the two formalisms. We discuss the stability and numerical complexity of the existing methods. The review ends with a selection of a few applications where numerical calculations were instrumental in shaping our understanding of the physics.

In this work, we describe the phenomenon of Weyl-point teleportation. Weyl points usually move continuously in the configuration parameter space of a quantum system when the control parameters are varied continuously. However, there are special transition points in the control space where the continuous motion of the Weyl points is disrupted. In such transition points, an extended nodal structure (nodal line or nodal surface) emerges, serving as a wormhole for the Weyl points, allowing their teleportation in the configuration space. A characteristic side effect of the teleportation is that the motional susceptibility of the Weyl point diverges in the vicinity of the transition point, and this divergence is characterized by a universal scaling law. We exemplify these effects via a two-spin model and a Weyl Josephson circuit model. We expect that these effects generalize to many other settings including electronic band structures of topological semimetals.

Weyl points are generic and stable features in the energy spectrum of Hamiltonians that depend on a three-dimensional parameter space. Nongeneric isolated twofold degeneracy points, such as multi-Weyl points, split into Weyl points upon a generic perturbation that removes the fine tuning or protecting symmetry. The number of the resulting Weyl points is at least |Q|, where Q is the topological charge associated to the nongeneric degeneracy point. Here, we show that such a nongeneric degeneracy point also has an birth quota, i.e., a maximum number of Weyl points that can be born from it upon any perturbation. This upper bound is a local multiplicity associated to the nongeneric degeneracy point, an invariant of map germs known from singularity theory. This holds not only for the case of a three-dimensional parameter space with a Hermitian Hamiltonian, but also for the case of a two-dimensional parameter space with a chiral-symmetric Hamiltonian. We illustrate the power of this result for band structures of two- and three-dimensional crystals. Our work establishes a strong and powerful connection between singularity theory and topological band structures and, more broadly, parameter-dependent quantum systems.

In our understanding of solids, the formation of highly spatially coherent electronic states, fundamental to command the quantum behavior of materials, relies on the existence of discrete translational symmetry of the crystalline lattice. In contrast, in the absence of long-range order, as in the case of non-crystalline materials, the electronic states are localized and electronic coherence does not develop. This brings forward the fundamental question whether long range order is necessary condition to establish coherence and structured momentum-dependent electronic state, and how to characterize it in the presence of short-range order. Here we study Bi$_2$Se$_3$, a material that exists in its crystalline form with long range order, in amorphous form, with short and medium range order, and in its nanocrystalline form, with reduced short range order. By using angle resolved photoemission spectroscopy to directly access the electronic states in a momentum resolved manner, we reveal that, even in the absence of long-range order, a well-defined real-space length scale is sufficient to produce dispersive band structures. Moreover, we observe for the first time a repeated Fermi surface structure of duplicated annuli, reminiscent of Brillouin zone-like repetitions. These results, together with our simulations using amorphous Hamiltonians, reveal that the typical momentum scale where coherence occurs is the inverse average nearest-neighbor distance, the direct fingerprint of the local order of the underlying atomic structure. These results, not only lead the way to a new understanding of electronic coherence in solids, but also open the way to the realization of novel momentum-dependent quantum phenomena such as momentum pairing and spin-orbit coupling, in a much broader class of materials than the currently studied ones, lacking long range crystalline translational symmetry.

Changes in the number of Weyl nodes in Weyl semimetals occur through merging processes, usually involving a pair of oppositely charged nodes. More complicated processes involving multiple Weyl nodes are also possible, but they typically require fine tuning and are thus less stable. In this Letter, we study how symmetries affect the allowed merging processes and their stability, focusing on the combination of a twofold rotation and time-reversal (C_{2}T) symmetry. We find that, counterintuitively, processes involving a merging of three nodes are more generic than processes involving only two nodes. Our Letter suggests that multi-Weyl merging may be observed in a large variety of quantum materials, and we discuss SrSi_{2} and bilayer graphene as potential candidates.

We derive a \mathbb{Z}_4 ℤ4 topological invariant that extends beyond symmetry eigenvalues and Wilson loops and classifies two-dimensional insulators with a C_4 \mathcal{T} C4𝒯 symmetry. To formulate this invariant, we consider an irreducible Brillouin zone and constrain the spectrum of the open Wilson lines that compose its boundary. We fix the gauge ambiguity of the Wilson lines by using the Pfaffian at high symmetry momenta. As a result, we distinguish the four C_4 \mathcal{T} C4𝒯 -protected atomic insulators, each of which is adiabatically connected to a different atomic limit. We establish the correspondence between the invariant and the obstructed phases by constructing both the atomic limit Hamiltonians and a C_4 \mathcal{T} C4𝒯 -symmetric model that interpolates between them. The phase diagram shows that C_4 \mathcal{T} C4𝒯 insulators allow \pm 1 ±1 and 2 2 changes of the invariant, where the latter is overlooked by symmetry indicators.

Crystalline symmetries have played a central role in the identification and understanding of quantum materials. Here we investigate whether an amorphous analogue of a well known three-dimensional strong topological insulator has topological properties in the solid state. We show that amorphous Bi 2 Se 3 thin films host a number of two-dimensional surface conduction channels. Our angle-resolved photoemission spectroscopy data are 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, confirming the existence of spin-momentum locked surface states. We discuss these experimental results in light of theoretical photoemission spectra obtained with an amorphous topological insulator tight-binding model, contrasting it with alternative explanations. The discovery of spin-momentum locked surface states in amorphous materials opens a new avenue to characterize amorphous matter, and triggers the search for an overlooked subset of quantum materials outside of current classification schemes.

While topological materials are not restricted to crystals, there is no efficient method to diagnose topology in non-crystalline solids such as amorphous materials. Here we introduce the structural spillage, a new indicator that predicts the unknown topological phase of a non-crystalline solid, which is compatible with first-principles calculations. We illustrate its potential with tight-binding and first-principles calculations of amorphous bismuth, predicting a bilayer to be a new topologically nontrivial material. Our work opens up the efficient prediction of non-crystalline solids via first-principles and high-throughput searches.

Phase-change materials are ubiquitous in technology because of their ability to transition between amorphous and crystalline phases fast and reversibly, upon shining light or passing a current. Here we argue that to fully understand their electronic properties it is necessary to define a novel electronic phase: the amorphous obstructed insulator. It differs from an obstructed insulator crystal in that it presents localized edge or surface states irrespective of the sample termination. Consequently, we show that obstructed amorphous insulators in three-dimensions host a surface two-dimensional flat band, detectable using ARPES. Our work establishes basic models for materials where topological and obstructed properties can be switched on and off externally, including two-dimensional surface flatbands.

We show that interactions in quantum spin liquids can result in non-Hermitian phenomenology that differs qualitatively from mean-field expectations. We demonstrate this in two prominent cases through the effects of phonons and disorder on a Kitaev honeycomb model. Using analytic and numerical calculations, we show the generic appearance of exceptional points and rings depending on the symmetry of the system. Their existence is reflected in dynamical observables including the dynamic structure function measured in neutron scattering. The results point to different phenomenological features in realizable spin liquids that must be incorporated into the analysis of experimental data and also indicate that spin liquids could be generically stable to wider classes of perturbations.

We show that topological defects in quadrupole insulators do not host quantized fractional charges, contrary to what their Wannier representation indicates. In particular, we test the charge quantization hypothesis based on the Wannier representation of a parametric defect and a disclination. Against the expectations, we find that the local charge density decays as $\sim 1/r^2$ with distance, leading to a diverging defect charge. We identify sublattice symmetry and not higher order topology as the origin of the previously reported charge quantization.

Weyl points are generic and stable features in the energy spectrum of Hamiltonians that depend on a three-dimensional parameter space. Non-generic isolated two-fold degeneracy points, such as multi-Weyl points, split into Weyl points upon a generic perturbation that removes the fine-tuning or protecting symmetry. The number of the resulting Weyl points is at least $|Q|$, where $Q$ is the topological charge associated to the non-generic degeneracy point. Here, we show that such a non-generic degeneracy point also has a birth quota, i.e., a maximum number of Weyl points that can be born from it upon any perturbation. The birth quota is a local multiplicity associated to the non-generic degeneracy point, an invariant of map germs known from singularity theory. This holds not only for the case of a three-dimensional parameter space with a Hermitian Hamiltonian, but also for the case of a two-dimensional parameter space with a chiral-symmetric Hamiltonian. We illustrate the power of this result for electronic band structures of two- and three-dimensional crystals. Our work establishes a strong connection between singularity theory and topological band structures, and more broadly, parameter-dependent quantum systems.