Amorphous topological superconductivity in a Shiba glass

Topological states of matter support quantized nondissipative responses and exotic quantum particles that cannot be accessed in common materials. The exceptional properties and application potential of topological materials have triggered a large-scale search for new realizations. Breaking away from the popular trend focusing almost exclusively on crystalline symmetries, we introduce the Shiba glass as a platform for amorphous topological quantum matter. This system consists of an ensemble of randomly distributed magnetic atoms on a superconducting surface. The collection of magnetic moments gives rise to subgap Yu-Shiba-Rusinov states that form a topological superconducting phase at critical density despite a complete absence of spatial order. Experimental signatures of the amorphous topological state can be obtained by STM measurements probing the topological edge mode. Our discovery demonstrates the physical feasibility of amorphous topological quantum matter and presents a concrete route to fabricating new topological systems from nontopological materials with random dopants.Comment: 11 pages, 5 figure

Topological superconductivity in regular and random lattices

In this thesis, we examine a fairly novel area of physics that concerns topological materials, and in particular, topological superconductivity. A goal in the research of topological materials is realizing applications in quantum computing, which could be aided by the emergent quasiparticles that exhibit non-Abelian exchange statistics. These are called Majorana bound states, and they are elusive quasiparticles predicted to be found on the boundary of topological superconductors. We first study a one-dimensional chain of potential impurities placed on the surface of a two-dimensional $p$-wave superconductor. As is usually the case, such chains are composed of perfect lattice structures, which is very challenging to achieve in any laboratory setting. Nevertheless, they serve as a good example of systems where an analytical solution can be well established. We investigate the model without employing any deep-dilute approximation, which gives us an accurate description even far away from the gap center. This is done by formulating the problem as a non-linear eigenvalue equation, which complicates it significantly, but also extends the region of applicability of our theory. We use reciprocal space calculations of two topological invariants to obtain the topological phase diagram of the system. The model is shown to host topological quasiparticle excitations at the ends of the chain, with multiple distinct topological phases. The near-perfect localization of the excitations makes them good candidates for probing Majorana bound states in experimental setups. We then move on to study topological superconductivity in random lattices, as opposed to regular structures which assume arbitrary precision. We frame our work starting with the mathematics of random numbers. Our work is thus in stark contrast with previous studies on topological materials that start off with a perfect lattice structure, and investigate some degree of disorder as perturbations to the regular lattice case. Our work establishes a first-ever realistic candidate for realizing topological superconductivity in an amorphous material. This could enable a novel approach to creating topological materials, and drastically aid in the development of fault-tolerant quantum computing

Criticality in amorphous topological matter : Beyond the universal scaling paradigm

We establish the theory of critical transport in amorphous Chern insulators and show that it lies beyond the current paradigm of topological criticality epitomized by the quantum Hall transitions. We consider models of Chern insulators on percolation-type random lattices where the average density determines the statistical properties of geometry. While these systems display a two-parameter scaling behavior near the critical density, the critical exponents and the critical conductance distributions are strikingly nonuniversal. Our analysis indicates that the amorphous topological criticality results from an interpolation of a geometric-type transition at low density and an Anderson localization-type transition at high density. Our work demonstrates how the recently discovered amorphous topological systems display unique phenomena distinct from their conventionally studied counterparts.Peer reviewe

Quantum Hall effect and Landau levels without spatial long-range correlations

The spectrum of charged particles in translation-invariant systems in a magnetic field is characterized by the Landau levels, which play a fundamental role in the thermodynamic and transport properties of solids. The topological nature and the approximate degeneracy of the Landau levels are known to also survive on crystalline lattices with discrete translation symmetry when the magnetic flux through a primitive cell is small compared to the flux quantum. Here we show that the notion of Landau levels and the quantum Hall effect can be generalized to 2d non-crystalline lattices without spatial long-range order. Remarkably, even when the spatial correlations decay over microscopic distances, 2d systems can exhibit a number of well-resolved Landau-like bands. The existence of these bands imply that non-crystalline systems in magnetic fields can support the hallmark quantum effects which have been typically associated with crystalline solids.Comment: 5 pages + 2.5-page appendix, 4+1 figure

Topological random fractals

The search for novel topological quantum states has recently moved beyond naturally occurring crystalline materials to complex and engineered systems. In this work we generalize the notion of topological electronic states to random lattices in non-integer dimensions. By considering a class D tight-binding model on critical clusters resulting from a two-dimensional site percolation process, we demonstrate that these topological random fractals exhibit the hallmarks of topological insulators. Specifically, our large-scale numerical studies reveal that topological random fractals display a robust mobility gap, support quantized conductance and represent a well-defined thermodynamic phase of matter. The finite-size scaling analysis further suggests that the critical properties are not consistent with the expectations of class D systems in two dimensions, hinting to the nontrivial relationship between fractal and integer-dimensional topological states. Our results establish topological random fractals as the most complex systems known to support nontrivial band topology with their distinct unique properties.publishedVersionPeer reviewe

Quantum walks on random lattices : Diffusion, localization, and the absence of parametric quantum speedup

Discrete-time quantum walks, quantum generalizations of classical random walks, provide a framework for quantum information processing, quantum algorithms, and quantum simulation of condensed-matter systems. The key property of quantum walks, which lies at the heart of their quantum information applications, is the possibility for a parametric quantum speedup in propagation compared to classical random walks. In this work we study propagation of quantum walks on percolation-generated two-dimensional random lattices. In large-scale simulations of topological and trivial split-step walks, we identify distinct prediffusive and diffusive behaviors at different timescales. Importantly, we show that even arbitrarily weak concentrations of randomly removed lattice sites give rise to a complete breakdown of the superdiffusive quantum speedup, reducing the motion to ordinary diffusion. By increasing the randomness, quantum walks eventually stop spreading due to Anderson localization. Near the localization threshold, we find that the quantum walks become subdiffusive. The fragility of quantum speedup implies dramatic limitations for quantum information applications of quantum walks on random geometries and graphs.publishedVersionPeer reviewe

Criticality in amorphous topological matter : Beyond the universal scaling paradigm

We establish the theory of critical transport in amorphous Chern insulators and show that it lies beyond the current paradigm of topological criticality epitomized by the quantum Hall transitions. We consider models of Chern insulators on percolation-type random lattices where the average density determines the statistical properties of geometry. While these systems display a two-parameter scaling behavior near the critical density, the critical exponents and the critical conductance distributions are strikingly nonuniversal. Our analysis indicates that the amorphous topological criticality results from an interpolation of a geometric-type transition at low density and an Anderson localization-type transition at high density. Our work demonstrates how the recently discovered amorphous topological systems display unique phenomena distinct from their conventionally studied counterparts.Peer reviewe

Topological phase transitions in glassy quantum matter

Amorphous systems have rapidly gained attention as promising platforms for topological matter. In this work, we establish a scaling theory of amorphous topological phase transitions driven by the density of lattice points in two dimensions. By carrying out a finite-size scaling analysis of topological invariants averaged over discrete and continuum random geometries, we discover critical properties of Chern and Z(2) glass transitions. Even for short-range hopping models, the Chern glass phase may persist down to the fundamental lower bound given by the classical percolation threshold. While the topological indices accurately satisfy the postulated one-parameter scaling, they do not generally flow to the closest integer value in the thermodynamic limit. Furthermore, the value of the critical exponent describing the diverging localization length varies continuously along the phase boundary and is not fixed by the symmetry class of the Hamiltonian. We conclude that the critical behavior of amorphous topological systems exhibit characteristic features not observed in disordered systems, motivating a wealth of interesting research directions.Peer reviewe

Engineering one-dimensional topological phases on p -wave superconductors

In this paper, we study how, with the aid of impurity engineering, two-dimensional p-wave superconductors can be employed as a platform for one-dimensional topological phases. We discover that, while chiral and helical parent states themselves are topologically nontrivial, a chain of scalar impurities on both systems supports multiple topological phases and Majorana end states. We develop an approach which allows us to extract the topological invariants and subgap spectrum, even away from the center of the gap, for the representative cases of spinless, chiral, and helical superconductors. We find that the magnitude of the topological gaps protecting the nontrivial phases may be a significant fraction of the gap of the underlying superconductor.Peer reviewe

Vaporlike phase of amorphous SiO2 is not a prerequisite for the core/shell ion tracks or ion shaping

When a swift heavy ion (SHI) penetrates amorphous SiO2, a core/shell (C/S) ion track is formed, which consists of a lower-density core and a higher-density shell. According to the conventional inelastic thermal spike (iTS) model represented by a pair of coupled heat equations, the C/S tracks are believed to form via "vaporization" and melting of the SiO2 induced by SHI (V-M model). However, the model does not describe what the vaporization in confined ion-track geometry with a condensed matter density is. Here we reexamine this hypothesis. While the total and core radii of the C/S tracks determined by small angle x-ray scattering are in good agreement with the vaporization and melting radii calculated from the conventional iTS model under high electronic stopping power (S-e) irradiations (>10 keV/nm), the deviations between them are evident at low-S, irradiation (3-5 keV/nm). Even though the iTS calculations exclude the vaporization of SiO2 at the low S-e, both the formation of the C/S tracks and the ion shaping of nanoparticles (NPs) are experimentally confirmed, indicating the inconsistency with the V-M model. Molecular dynamics (MD) simulations based on the two-temperature model, which is an atomic-level modeling extension of the conventional iTS, clarified that the "vaporlike" phase exists at S-e similar to 5 keV/nm or higher as a nonequilibrium phase where atoms have higher kinetic energies than the vaporization energy, but are confined at a nearly condensed matter density. Simultaneously, the simulations indicate that the vaporization is not induced under 50-MeV Si irradiation (S-e similar to 3 keV/nm), but the C/S tracks and the ion shaping of nanoparticles are nevertheless induced. Even though the final density variations in the C/S tracks are very small at the low stopping power values (both in the simulations and experiments), the MD simulations show that the ion shaping can be explained by flow of liquid metal from the NP into the transient low-density phase of the track core during the first similar to 10 ps after the ion impact. The ion shaping correlates with the recovery process of the silica matrix after emitting a pressure wave. Thus, the vaporization is not a prerequisite for the C/S tracks and the ion shaping.Peer reviewe