Topoelectrical circuits are meta-material realizations of topological features of condensed matter systems. In this work, we discuss experimental methods that allow a fast and straightforward detection of the spectral features of these systems from the two-point impedance of the circuit. This allows to deduce the full spectrum of a topoelectrical circuit consisting of N sites from a single two-point measurement of the frequency resolved impedance. In contrast, the standard methods rely on $N^2$ measurements of admittance matrix elements with a subsequent diagonalization on a computer. We experimentally test our approach by constructing a Fibonacci topoelectrical circuit. Although the spectrum of this chain is fractal, i.e., more complex than the spectra of periodic systems, our approach is successful in recovering its eigenvalues. Our work promotes the topoelectrical circuits as an ideal platform to measure spectral properties of various (quasi)crystalline systems.

The discovery of the Hat, an aperiodic monotile, has revealed novel mathematical aspects of aperiodic tilings. However, the physics of particles propagating in such a setting remains unexplored. In this work we study spectral and transport properties of a tight-binding model defined on the Hat. We find that (i) the spectral function displays striking similarities to that of graphene, including six-fold symmetry and Dirac-like features; (ii) unlike graphene, the monotile spectral function is chiral, differing for its two enantiomers; (iii) the spectrum has a macroscopic number of degenerate states at zero energy; (iv) when the magnetic flux per plaquette ($\phi$) is half of the flux quantum, zero-modes are found localized around the reflected `anti-hats'; and (v) its Hofstadter spectrum is periodic in $\phi$, unlike for other quasicrystals. Our work serves as a basis to study wave and electron propagation in possible experimental realizations of the Hat, which we suggest.

The discovery of the Hat, an aperiodic monotile, has revealed novel mathematical aspects of aperiodic tilings. However, the physics of particles propagating in such a setting remains unexplored. In this work we study spectral and transport properties of a tight-binding model defined on the Hat. We find that (i) the spectral function displays striking similarities to that of graphene, including sixfold symmetry and Dirac-like features; (ii) unlike graphene, the monotile spectral function is chiral, differing for its two enantiomers; (iii) the spectrum has a macroscopic number of degenerate states at zero energy; (iv) when the magnetic flux per plaquette (ϕ) is half of the flux quantum, zero modes are found localized around the reflected "anti-hats"; and (v) its Hofstadter spectrum is periodic in ϕ, unlike for other quasicrystals. Our work serves as a basis to study wave and electron propagation in possible experimental realizations of the Hat, which we suggest.

In chiral crystals crystalline symmetries can protect multifold fermions, pseudo-relativistic masless quasiparticles that have no high-energy counterparts. Their realization in transition metal monosilicides has exemplified their intriguing physical properties, such as long Fermi arc surface states and unusual optical responses. Recent experimental studies on amorphous transition metal monosilicides suggest that topological properties may survive beyond crystals, even though theoretical evidence is lacking. Motivated by these findings, we theoretically study a tight-binding model of amorphous transition metal monosilicides. We find that topological properties of multifold fermions survive in the presence of structural disorder that converts the semimetal into a diffusive metal. We characterize this topological diffusive metal phase with the spectral localizer, a real-space topological indicator that we show can signal multifold fermions. Our findings showcase how topological properties can survive in disordered metals, and how they can be uncovered using the spectral localizer.

In recent times the chiral semimetal cobalt monosilicide (CoSi) has emerged as a prototypical, nearly ideal topological conductor hosting giant, topologically protected Fermi arcs. Exotic topological quantum properties have already been identified in CoSi bulk single crystals. However, CoSi is also known for being prone to intrinsic disorder and inhomogeneities, which, despite topological protection, risk jeopardizing its topological transport features. Alternatively, topology may be stabilized by disorder, suggesting the tantalizing possibility of an amorphous variant of a topological metal, yet to be discovered. In this respect, understanding how microstructure and stoichiometry affect magnetotransport properties is of pivotal importance, particularly in case of low-dimensional CoSi thin films and devices. Here we comprehensively investigate the magnetotransport and magnetic properties of ≈25 nm Co1–xSix thin films grown on a MgO substrate with controlled film microstructure (amorphous vs textured) and chemical composition (0.40 < x < 0.60). The resistivity of Co1–xSix thin films is nearly insensitive to the film microstructure and displays a progressive evolution from metallic-like (dρxx/dT > 0) to semiconducting-like (dρxx/dT < 0) regimes of conduction upon increasing the silicon content. A variety of anomalies in the magnetotransport properties, comprising for instance signatures consistent with quantum localization and electron–electron interactions, anomalous Hall and Kondo effects, and the occurrence of magnetic exchange interactions, are attributable to the prominent influence of intrinsic structural and chemical disorder. Our systematic survey brings to attention the complexity and the challenges involved in the prospective exploitation of the topological chiral semimetal CoSi in nanoscale thin films and devices.

Metals and topological insulators have in common that they cannot be described by exponentially localized wave-functions. Here we establish a relationship between these two seemingly unrelated observations. The connection is explicit in the low-lying states of the spectral localizer of trivial metals, an operator that measures the obstruction to finding localized eigenstates. The low-lying spectrum of the spectral localizer of metals is determined by the zero-mode solutions of Dirac fermions with a varying mass parameter. We use this observation, valid in any dimension, to determine the difference between the localizer spectrum of trivial and topological metals, and conjecture the spectrum of the localizer for fractional quantum Hall edges. Because the localizer is a local real-space operator, it may be used as a tool to differentiate between non-crystalline topological and trivial metals, and to characterize strongly correlated systems, for which local topological markers are scarce.

Topological insulators are described by topological invariants that can be computed by integrals over momentum space, but also as traces over local, real-space topological markers. These markers are useful to detect topological insulating phases in disordered crystals, quasicrystals and amorphous systems. Among these markers, only the spectral localizer operator can be used to distinguish topological metals, that show zero-modes of the localizer spectrum. However, it remains unclear whether trivial metals also display zero-modes, and if their localizer spectrum is distinguishable from topological ones. Here, we show that trivial metals generically display zero-modes of the localizer spectrum. The localizer zero-modes are determined by the zero-mode solutions of a Dirac equation with a varying mass parameter. We use this observation, valid in any dimension, to determine the difference between the localizer spectrum of trivial and topological metals, and conjecture the spectrum of the localizer for fractional quantum Hall edges. Because the localizer is a local, real-space operator, it may be used as a tool to differentiate between non-crystalline topological and trivial metals, and characterize strongly correlated systems, for which local topological markers are scarce.

Physically, one tends to think of non-Hermitian systems in terms of gain and loss: the decay or amplification of a mode is given by the imaginary part of its energy. Here, we introduce an alternative avenue to the realm of non-Hermitian physics, which involves neither gain nor loss. Instead, complex eigenvalues emerge from the amplitudes and phase differences of waves backscattered from the boundary of insulators. We show that for any strong topological insulator in a Wigner-Dyson class, the reflected waves are characterized by a reflection matrix exhibiting the non-Hermitian skin effect. This leads to an unconventional Goos-Hänchen effect: due to non-Hermitian topology, waves undergo a lateral shift upon reflection, even at normal incidence. Going beyond systems with gain and loss vastly expands the set of experimental platforms that can access non-Hermitian physics and show signatures associated with non-Hermitian topology.

While periodically-driven phases offer a unique insight into non-equilibrium topology that is richer than its static counterpart, their experimental realization is often hindered by ubiquitous decoherence effects. Recently, we have proposed a decoherence-free approach of realizing these Floquet phases. The central insight is that the reflection matrix, being unitary for a bulk insulator, plays the role of a Floquet time-evolution operator. We have shown that reflection processes off the boundaries of systems supporting higher-order topological phases (HOTPs) simulate non-trivial Floquet phases. So far, this method was shown to work for one-dimensional Floquet topological phases protected by local symmetries. Here, we extend the range of applicability by studying reflection off three-dimensional HOTPs with corner and hinge modes. We show that the reflection processes can simulate both first-order and second-order Floquet phases, protected by a combination of local and spatial symmetries. For every phase, we discuss appropriate topological invariants calculated with the nested scattering matrix method.

The network model is a powerful tool in the study of localization-delocalization transitions and has been used to describe a variety of topological systems without crystalline symmetry. Here, the authors show that network models can also realize topological phases protected by point-group symmetries. The latter lead to the formation of a higher-order topological phase characterized by midgap modes present at the corners of the system.

We introduce a two-dimensional network model that realizes a higher-order topological insulator (HOTI) phase. We find that in the HOTI phase a total of 16 corner states are protected by the combination of a four-fold 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 HOTI systems in alternative experimental platforms implementing the network model.

We show that scattering from the boundary of static, higher-order topological insulators (HOTIs) can be used to simulate the behavior of (time-periodic) Floquet topological insulators. We consider D-dimensional HOTIs with gapless corner states which are weakly probed by external waves in a scattering setup. We find that the unitary reflection matrix describing back-scattering from the boundary of the HOTI is topologically equivalent to a (D-1)-dimensional nontrivial Floquet operator. To characterize the topology of the reflection matrix, we introduce the concept of `nested' scattering matrices. Our results provide a route to engineer topological Floquet systems in the lab without the need for external driving. As benefit, the topological system does not to suffer from decoherence and heating.

Two-dimensional second-order topological superconductors (SOTSCs) have gapped bulk and edge states, with zero-energy Majorana bound states localized at corners. Motivated by recent advances in Majorana nanowire experiments, we propose to realize a tunable SOTSC as a two-dimensional nanowire array. We show that the coupling between the Majorana modes of adjacent wires can be controlled by phase-biasing the device, allowing to access a variety of topological phases. We characterize the system using scattering theory, which provides access to its transport properties and its topological invariants. The setup is robust against disorder, both in the nanowires themselves and in the Josephson junctions formed between adjacent wires. Further, we identify a parameter regime in which an initially trivial system is rendered topological upon adding disorder, providing an example of a second-order topological Anderson phase.

We study numerically the formation of Majorana bound states in a finite-size quasi-one-dimensional square-lattice strip with Rashba spin-orbit coupling, in the presence of a proximity-induced superconducting pairing and a magnetic field perpendicular to the strip. We take into account both the Zeeman and orbital effects of the field. First, using the Majorana polarization, we demonstrate that such a system can host more than one pair of Majorana quasiparticles. We construct the corresponding topological phase diagram and we conclude that the topological regions are extended in the presence of orbital effects, however the gap protecting the topological states is reduced.