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Topological phases of matter have sparked an immense amount of activity in recent decades. Topological materials are classified by topological invariants that act as a non-local order parameter for any symmetry and condition. As a result,…

The topological invariant of a topological insulator (or superconductor) is given by the number of symmetry-protected edge states present at the Fermi level. Despite this fact, established expressions for the topological invariant require…

Amorphous solids remain outside of the classification and systematic discovery of new topological materials, partially due to the lack of realistic models that are analytically tractable. Here we introduce the topological Weaire-Thorpe…

Topological data analysis has emerged as a powerful tool for extracting the metric, geometric and topological features underlying the data as a multi-resolution summary statistic, and has found applications in several areas where data…

Topological invariants, rigorously defined only in the thermodynamic limit, have been generalized to topological indicators applicable to finite-size disordered systems. However, in many experimentally relevant situations, such as…

We define a class of insulators with gapless surface states protected from localization due to the statistical properties of a disordered ensemble, namely due to the ensemble's invariance under a certain symmetry. We show that these…

Topological invariants have proved useful for analyzing emergent function as they characterize a property of the entire system, and are insensitive to local details, disorder, and noise. They support boundary states, which reduce the system…

Topology in quantum matter is typically associated with gapped phases. For example, in symmetry protected topological (SPT) phases, the bulk energy gap localizes edge modes near the boundary. In this work we identify a new mechanism that…

While topological phases have been extensively studied in amorphous systems in recent years, it remains unclear whether the random nature of amorphous materials can give rise to higher-order topological phases that have no crystalline…

We generalize various notions of stability of invariant sets of dynamical systems to invariant measures, by defining a topology on the set of measures. The defined topology is similar, but not topologically equivalent to weak* topology, and…

This article is meant as a gentle introduction to the "topological terms" that often play a decisive role in effective theories describing topological quantum effects in condensed matter systems. We first take up several prominent examples,…

Data quality is crucial for the successful training, generalization and performance of machine learning models. We propose to measure the quality of a subset concerning the dataset it represents, using topological data analysis techniques.…

We investigate topological properties of density matrices motivated by the question to what extent phenomena like topological insulators and superconductors can be generalized to mixed states in the framework of open quantum systems. The…

One of the hallmarks of topological insulators is the correspondence between the value of its bulk topological invariant and the number of topologically protected edge modes observed in a finite-sized sample. This bulk-boundary…

Topological phases of matter are defined by their nontrivial patterns of ground-state quantum entanglement, which is irremovable so long as the excitation gap and the protecting symmetries, if any, are maintained. Recent studies on…

Symmetry-protected topological (SPT) phases of matter have been interpreted in terms of anomalies, and it has been expected that a similar picture should hold for SPT phases with fermions. Here, we describe in detail what this picture means…

We argue that symmetry-broken phases proximate in phase space to symmetry-protected topological phases can exhibit dynamical signatures of topological physics. This dynamical, symmetry-protected "topological" regime is characterized by…

Topological phases of matter are ubiquitous in crystals, but less is known about their existence in amorphous systems, that lack long-range order. In this perspective, we review the recent progress made on theoretically defining amorphous…

We consider symmetry-protected topological (SPT) phases in 2D protected by linear subsystem symmetries, i.e. those that act along rigid lines. There is a distinction between a "strong" subsystem SPT phase, and a "weak" one, which is…

The classification of electron systems according to their topology has been at the forefront of condensed matter research in recent years. It has been found that systems of the same symmetry, previously thought of as equivalent, may in fact…