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Electron-magnon coupling at the interface between a normal metal and a magnetically ordered insulator modifies the electrical conductivity of the normal metal, an effect known as spin-Hall magnetoresistance. It can also facilitate magnon-mediated current drag, the nonlocal electric current response of two normal metal layers separated by a magnetic insulator. In this article, we present a theory of these two spintronic effects and their nonlinear counterparts for time-dependent applied electric fields $E(\omega)$, with driving frequencies $\omega$ up to the THz regime. We compare various mechanisms leading to a quadratic-in-$E$ response $-$ Joule heating, spin-torque diode effect, and magnonic unidirectional spin-Hall magnetoresistance $-$ and show how these can be disentangled by their characteristic dependences on $\omega$ and the magnetization direction., Comment: 22 + 6 pages, 9 figures

The topology of an insulator can be defined even when all eigenstates of the system are localized - an extreme case of Anderson insulators that we call ultra-localized. We derive the classification of such ultra-localized insulators in all symmetry classes and dimensions. We clarify their bulk-boundary correspondence and show that ultra-localized systems are in many instances phases of matter not described by the known classification of topological insulators and superconductors.

We study the spatiotemporal dynamics of ultrafast electron spin transport across nanometer-thick copper layers using broadband terahertz spectroscopy. Our analysis of temporal delays, broadening and attenuation of the spin-current pulse revealed ballistic-like propagation of the pulse peak, approaching the Fermi velocity, and diffusive features including a significant velocity dispersion. A comparison to the frequency-dependent Ficks law identified the diffusion-dominated transport regime for distances larger than 2 nm. The findings lie the groundwork for designing future broadband spintronic devices., Comment: V3: AAM, Licence: CC BY Main text consists of 3 figures and 4 pages of text

Topological insulators and superconductors support extended surface states protected against the otherwise localizing effects of static disorder. Specifically, in the Wigner-Dyson insulators belonging to the symmetry classes A, AI, and AII, a band of extended surface states is continuously connected to a likewise extended set of bulk states forming a ``bridge'' between different surfaces via the mechanism of spectral flow. In this work we show that this mechanism is absent in the majority of non-Wigner-Dyson topological superconductors and chiral topological insulators. In these systems, there is precisely one point with granted extended states, the center of the band, $E=0$. Away from it, states are spatially localized, or can be made so by the addition of spatially local potentials. Considering the three-dimensional insulator in class AIII and winding number $\nu=1$ as a paradigmatic case study, we discuss the physical principles behind this phenomenon, and its methodological and applied consequences. In particular, we show that low-energy Dirac approximations in the description of surface states can be treacherous in that they tend to conceal the localizability phenomenon. We also identify markers defined in terms of Berry curvature as measures for the degree of state localization in lattice models, and back our analytical predictions by extensive numerical simulations. A main conclusion of this work is that the surface phenomenology of non-Wigner-Dyson topological insulators is a lot richer than that of their Wigner-Dyson siblings, extreme limits being spectrum-wide quantum critical delocalization of all states vs. full localization except at the $E=0$ critical point. As part of our study we identify possible experimental signatures distinguishing between these different alternatives in transport or tunnel spectroscopy., Comment: published version, change of the title

The topological classification of gapped band structures depends on the particular definition of topological equivalence. For translation-invariant systems, stable equivalence is defined by a lack of restrictions on the numbers of occupied and unoccupied bands, while imposing restrictions on one or both leads to ``fragile'' and ``delicate'' topology, respectively. In this article, we describe a homotopic classification of band structures -- which captures the topology beyond the stable equivalence -- in the presence of additional lattice symmetries. As examples, we present complete homotopic classifications for spinless band structures with twofold rotation, fourfold rotation and fourfold dihedral symmetries, both in presence and absence of time-reversal symmetry. Whereas the rules of delicate and fragile topology do not admit a bulk-boundary correspondence, we identify a version of stable topology, which restricts the representations of bands, but not their numbers, which does allow for anomalous states at symmetry-preserving boundaries, which are associated with nontrivial bulk topology., Comment: 30+15 pages, 19 figures

If a quantum dot is coupled to a topological superconductor via tunneling contacts, each contact hosts a Majorana zero mode in the limit of zero transmission. Close to a resonance and at a finite contact transparency, the resonant level in the quantum dot couples the Majorana modes, but a ground state degeneracy per fermion parity subspace remains if the number of Majorana modes coupled to the dot is five or larger. Upon varying shape-defining gate voltages while remaining close to resonance, a nontrivial evolution within the degenerate ground-state manifold is achieved. We characterize the corresponding non-Abelian holonomy for a quantum dot with chaotic classical dynamics using random matrix theory and discuss measurable signatures of the non-Abelian time-evolution., Comment: 4 + 32 pages, 3 + 18 figures; V2: published version

We predict a novel metallic state of matter that emerges in a Weyl-semimetal superstructure with spatially varying Weyl-node positions. In the new state, the Weyl nodes are stretched into extended, anisotropic Fermi surfaces, which can be understood as being built from Fermi arc-like states. This "Fermi-arc metal" exhibits the chiral anomaly of the parental Weyl semimetal. However, unlike in the parental Weyl semimetal, in the Fermi-arc metal the "ultra-quantum state", in which the anomalous chiral Landau level is the only state at the Fermi energy, is already reached for a finite energy window at zero magnetic field. The dominance of the ultra-quantum state implies a universal low-field ballistic magnetoconductance and the absence of quantum oscillations, making the Fermi surface "invisible" to de Haas-van Alphen and Shubnikov-de Haas effects, although it signifies its presence in other response properties., Comment: 7 pages, 3 figures

Three-dimensional higher-order topological insulators can have topologically protected chiral modes propagating on their hinges. Hinges with two co-propagating chiral modes can serve as a "beam splitter" between hinges with only a single chiral mode. Here we show how such a crystal, with Ohmic contacts attached to its hinges, can be used to realize a Mach-Zehnder interferometer. We present concrete calculations for a lattice model of a first-order topological insulator in a magnetic field, which, for a suitable choice of parameters, is an extrinsic second-order topological insulator with the required configuration of chiral hinge modes.

The interface between a ferro-/ferrimagnetic insulator and a normal metal can support spin currents polarized collinear with and perpendicular to the magnetization direction. The flow of angular momentum perpendicular to the magnetization direction ("transverse" spin current) takes place via spin torque and spin pumping. The flow of angular momentum collinear with the magnetization ("longitudinal" spin current) requires the excitation of magnons. In this article we extend the existing theory of longitudinal spin transport [Bender and Tserkovnyak, Phys. Rev. B 91, 140402(R) (2015)] in the zero-frequency weak-coupling limit in two directions: We calculate the longitudinal spin conductance non-perturbatively (but in the low-frequency limit) and at finite frequency (but in the limit of low interface transparency). For the paradigmatic spintronic material system YIG|Pt, we find that non-perturbative effects lead to a longitudinal spin conductance that is ca. 40% smaller than the perturbative limit, whereas finite-frequency corrections are relevant at low temperatures < 100 K only, when only few magnon modes are thermally occupied., Comment: 15 pages, 7 figures

Weyl semimetals harbor topological Fermi-arc surface states which determine the nontrivial charge current response to external fields. We here study the quasiparticle decay rate of Fermi arc states arising from their coupling to acoustic phonons, as well as the phonon-limited conductivity tensor for a clean Weyl semimetal slab. Using the phonon modes for an isotropic elastic continuum with a deformation potential coupling to electrons, we determine the temperature dependence of the quasiparticle decay rate, both near and far away from the arc termination points. By solving the coupled Boltzmann equations for the bulk and arc state distribution functions in the slab geometry, we show how the linear response conductivity depends on key parameters such as the temperature, the chemical potential, the geometric shape of the Fermi arcs, or the slab width. The chiral nature of Fermi arc states causes an enhancement of the longitudinal conductivity along the chiral direction at low temperatures, together with a 1/T^2 scaling regime at intermediate temperatures without counterpart for the conductivity along the perpendicular direction., Comment: 27 pages, 6 figures. Accepted version

Spin transport is crucial for future spintronic devices operating at bandwidths up to the terahertz (THz) range. In F|N thin-film stacks made of a ferro/ferrimagnetic layer F and a normal-metal layer N, spin transport is mediated by (1) spin-polarized conduction electrons and/or (2) torque between electron spins. To identify a cross-over from (1) to (2), we study laser-driven spin currents in F|Pt stacks where F consists of model materials with different degrees of electrical conductivity. For the magnetic insulators YIG, GIG and maghemite, identical dynamics is observed. It arises from the THz interfacial spin Seebeck effect (SSE), is fully determined by the relaxation of the electrons in the metal layer and provides an estimate of the spin-mixing conductance of the GIG/Pt interface. Remarkably, in the half-metallic ferrimagnet Fe3O4 (magnetite), our measurements reveal two spin-current components with opposite direction. The slower, positive component exhibits SSE dynamics and is assigned to torque-type magnon excitation of the A- and B-spin sublattices of Fe3O4. The faster, negative component arises from the pyro-spintronic effect and can consistently be assigned to ultrafast demagnetization of e-sublattice minority-spin hopping electrons. This observation supports the magneto-electronic model of Fe3O4. In general, our results provide a new route to the contact-free separation of torque- and conduction-electron-mediated spin currents.

Using a systematic relation between topological gapless phases in three dimensions and topological gapped phases in two dimensions, we identify four types of higher-order topological semimetals or nodal superconductors (HOTS), hosting (i) flat zero-energy "Fermi arcs" at crystal hinges, (ii) flat zero-energy hinge arcs coexisting with surface Dirac cones, (iii) chiral or helical hinge modes, or (iv) flat zero-energy hinge arcs connecting nodes only at finite momentum. Bulk-boundary correspondence relates the hinge states to the bulk topology protecting the nodal point or loop. We classify all HOTS for all tenfold-way classes with an order-two crystalline (anti-)symmetry, such as mirror, twofold rotation, or inversion., Comment: 5 + 13 pages, 1 + 2 figures, v2: published version, added appendix on topological invariants

The exchange stiffness is a central material parameter of all ferromagnetic materials. Its value controls the Curie temperature as well as the dynamic properties of spin waves to a large extend. Using ultrashort spin current pulses we excite perpendicular standing spin waves (PSSW) in ultrathin epitaxial iron layers at frequencies of up to 2.4 THz. Our analysis shows that for the PSSWs the observed exchange stiffness of iron is about 20% smaller compared to the established iron bulk value. In addition, we find an interface-related reduction of the effective exchange stiffness for layers with the thickness below 10 nm. To understand and discuss the possible mechanisms of the exchange stiffness reduction we develop an analytical 1D-model. In doing so we find that the interface induced reduction of the exchange stiffness is mode-dependent.

A heterostructure consisting of a magnetic Weyl semimetal and a conventional superconductor exhibits an equilibrium current parallel to the superconductor interface and perpendicular to the magnetization. Analyzing a minimal model, which as a function of parameters may be in a trivial magnetic insulator phase, a Weyl semimetal phase, or a three-dimensional weak Chern insulator phase, we find that the equilibrium current is sensitive to the presence of surface states, such as the topological Fermi-arc states of the Weyl semimetal or the chiral surface states of the weak Chern insulator. While there is a nonzero equilibrium current in all three phases, the appearance of the surface states in the topological regime leads to a reversal of the direction of the current, compared to the current direction for the trivial magnetic insulator phase. We discuss the interpretation of the surface-state contribution to the equilibrium current as a real-space realization of the superconductivity-enabled equilibrium chiral magnetic effect of a single chirality, predicted to occur in bulk Weyl superconductors., Comment: 14 pages, 4 figures

In bilayers consisting of a normal metal (N) with spin-orbit coupling and a ferromagnet (F), the combination of the spin-Hall effect, the spin-transfer torque, and the inverse spin-Hall effect gives a small correction to the in-plane conductivity of N, which is referred to as spin-Hall magnetoresistance (SMR). We here present a theory of the SMR and the associated off-diagonal conductivity corrections for frequencies up to the terahertz regime. We show that the SMR signal has pronounced singularities at the spin-wave frequencies of F, which identifies it as a potential tool for all-electric spectroscopy of magnon modes. A systematic change of the magnitude of the SMR at lower frequencies is associated with the onset of a longitudinal magnonic contribution to spin transport across the F-N interface., Comment: 21 pages, 9 figures

The interaction of a magnetic insulator with the helical electronic edge of a two-dimensional topological insulator has been shown to lead to many interesting phenomena. One of these is that for a suitable orientation of the magnetic anisotropy axis, the exchange coupling to an easy-plane magnet has no effect on DC electrical transport through a helical edge, despite the fact that it opens a gap in the spectrum of the helical edge [Meng {\em et al.}, Phys.\ Rev.\ B {\bf 90}, 205403 (2014)]. Here, we theoretically consider such a magnet embedded in an interferometer, consisting of a pair of helical edge states connected by two tunneling contacts, at which electrons can tunnel between the two edges. Using a scattering matrix approach, we show that the presence of the magnet in one of the interferometer arms gives rise to AC currents in response to an applied DC voltage. On the other hand, the DC Aharonov-Bohm effect is absent at zero temperature and small DC voltages, and only appears if the applied voltage or the temperature exceeds the magnet-induced excitation gap., Comment: 14 pages, 8 figures

Motivated by recent experiments searching for Majorana zero modes in tripartite semiconductor nanowires with epitaxial superconductor and ferromagnetic-insulator layers, we explore the emergence of topological superconductivity in such devices for paradigmatic arrangements of the three constituents. Accounting for the competition between magnetism and superconductivity, we treat superconductivity self consistently and describe the electronic properties, including the superconducting and ferromagnetic proximity effects, within a direct wave-function approach. We conclude that the most viable mechanism for topological superconductivity relies on a superconductor-semiconductor-ferromagnet arrangement of the constituents, in which spin splitting and superconductivity are independently induced in the semiconductor by proximity and superconductivity is only weakly affected by the ferromagnetic insulator., Comment: 10 pages, 6 figures

Using a simplified microscopic model of coupled spin and lattice excitations in a ferromagnetic insulator we evaluate the magnetic-field dependence of the spin Seebeck effect at low temperatures. The model includes Heisenberg exchange coupling, a harmonic lattice potential, and a pseudo-dipolar exchange interaction. Our approach goes beyond previous work [Phys. Rev. B 98, 134421 (2018)] in that it does not rely on the a priori assumption of a fast equilibration of the magnon and phonon distributions. Our theory shows that singular features in the magnetic-field dependence of the spin Seebeck effect at low temperatures observed by Kikkawa et al. [Phys. Rev. Lett. 117, 207203 (2016)] are independent of the relative strength of magnon-impurity and phonon-impurity scattering., Comment: 34 pages, 14 figures

The interplay of topology and symmetry in a material's band structure may result in various patterns of topological states of different dimensionality on the boundary of a crystal. The protection of these "higher-order" boundary states comes from topology, with constraints imposed by symmetry. We review the bulk-boundary correspondence of topological crystalline band structures, which relates the topology of the bulk band structure to the pattern of the boundary states. Furthermore, recent advances in the K-theoretic classification of topological crystalline band structures are discussed., Comment: Expanded version of the published review article in physica status solidi (b). v2: discussion of experimental realizations of higher-order topological band structures

The realization of integrated quantum circuits requires precise on-chip control of charge carriers. Aiming at the coherent coupling of distant nanostructures at zero magnetic field, here we study the ballistic electron transport through two quantum point contacts (QPCs) in series in a three terminal configuration. We enhance the coupling between the QPCs by electrostatic focusing using a field effect lens. To study the emission and collection properties of QPCs in detail we combine the electrostatic focusing with magnetic deflection. Comparing our measurements with quantum mechanical and classical calculations we demonstrate how the coherent and ballistic dynamics depend on the details of the QPC confinement potentials., Comment: Uploaded final version published on PRL

Quantum point contacts (QPC) are fundamental building blocks of nanoelectronic circuits. For their emission dynamics as well as for interaction effects such as the 0.7-anomaly the details of the electrostatic potential are important, but the precise potential shapes are usually unknown. Here, we measure the one-dimensional subband spacings of various QPCs as a function of their conductance and compare our findings with models of lateral parabolic versus hard wall confinement. We find that a gate-defined QPC near pinch-off is compatible with the parabolic saddle point scenario. However, as the number of populated subbands is increased Coulomb screening flattens the potential bottom and a description in terms of a finite hard wall potential becomes more realistic., Comment: 7+2 pager, 7 figures, v2: published version (minor changes)

We consider the disorder-induced correction to the minimal conductance of an anisotropic two-dimensional Dirac node or a three-dimensional Weyl node. An analytical expression is derived for the correction $\delta G$ to the conductance of a finite-size sample by an arbitrary potential, without taking the disorder average, in second-order perturbation theory. Considering a generic model of a short-range disorder potential, this result is used to compute the probability distribution $P(\delta G)$, which is compared to the numerically exact distribution obtained using the scattering matrix approach. We show that $P(\delta G)$ is Gaussian when the sample has a large width-to-length ratio, and study how the expectation value, the standard deviation, and the probability of finding $\delta G < 0$ depend on the anisotropy of the dispersion., Comment: 10+ pages, 6 figures

Theoretically, a pair of Majorana bound states in a topological superconductor forms a single fermionic level even at large separations, implying that the parity information is stored nonlocally. The nonlocality leads to a long-distance coherence for electrons tunneling through a Coulomb blockaded Majorana wire [Fu, Phys. Rev. Lett. 104, 056402 (2010)], an effect that can be observed, e.g., in an interferometer. Here, we examine theoretically the coherent electron transfer, taking into account that tunneling implies the long-distance transfer of charge, which is carried by one-dimensional plasmons. We show that the charge dynamics does not affect the coherence of the electron tunneling process in a topological superconductor consisting of a semiconductor wire proximitized by a single bulk superconductor. The coherence may be strongly suppressed, however, if the topological superconductivity derives from a semiconductor wire proximitized by a granular superconductor., Comment: 4+ pages, 3 figures

We consider a disordered Hubbard model, and show that, at sufficiently weak disorder, a single spin-down mobile impurity can thermalize an extensive initially localized system of spin-up particles. Thermalization is enabled by resonant processes which involve correlated hops of the impurity and localized particles. This effect indicates that certain localized insulators behave as "supercooled" systems, with mobile impurities acting as ergodic seeds. We provide analytical estimates, supported by numerical exact diagonalization (ED), showing how the critical disorder strength for such mechanism depends on the particle density of the localized system. In the $U\rightarrow\infty$ limit, doublons are stable excitations, and they can thermalize mesoscopic systems by a similar mechanism. The emergence of an additional conservation law leads to an eventual localization of doublons. Our predictions apply to fermionic and bosonic systems and are readily accessible in ongoing experiments simulating synthetic quantum lattices with tunable disorder., Comment: 5 pages + supplementary material (8 pages)

We develop a systematic approach for constructing symmetry-based indicators of a topological classification for superconducting systems. The topological invariants constructed in this work form a complete set of symmetry-based indicators that can be computed from knowledge of the Bogoliubov-de Gennes Hamiltonian on high-symmetry points in Brillouin zone. After excluding topological invariants corresponding to the phases without boundary signatures, we arrive at natural generalization of symmetry-based indicators [H. C. Po, A. Vishwanath, and H. Watanabe, Nature Comm. 8, 50 (2017)] to Hamiltonians of Bogoliubov-de Gennes type., Comment: 51 pages (main text: 29 pages, appendices: 16 pages, supplementary material: 6 pages), 22 figures, v2: Extended discussion to literature, minor corrections

Surface-state contributions to the dc conductivity of most homogeneous metals exposed to uniform electric fields are usually as small as the system size is large compared to the lattice constant. In this work, we show that surface states of topological metals can contribute with the same order of magnitude as the bulk even in large systems. This effect is intimately related to the intrinsic anomalous Hall effect, in which an applied voltage induces chiral surface-state currents proportional to the system size. Unlike the anomalous Hall effect, the large contribution of surface states to the dc conductivity is also present in time-reversal invariant Weyl semimetals, where the surface states come in counter-propagating time-reversed pairs. While the Hall voltage vanishes in the presence of time-reversal symmetry, the twinned chiral surface currents develop similarly as in the time-reversal broken case. For this effect to occur, the relaxation length associated with scattering between time-reversed partner states needs to be larger than the separation of contributing surfaces, which results in a characteristic size dependence of the resistivity and a highly inhomogeneous current-density profile across the sample., Comment: 6+1 pages, 4 figures

Coupling a normal metal wire to a superconductor induces an excitation gap in the normal metal. In the absence of disorder, the induced excitation gap is strongly suppressed by finite-size effects if the thickness of the superconductor is much smaller than the thickness of the normal metal and the superconducting coherence length. We show that the presence of disorder, either in the bulk or at the exposed surface of the superconductor, significantly enhances the magnitude of the induced gap, such that it approaches the superconducting gap in the limit of strong disorder. We also discuss the shift of energy bands inside the normal-metal wire as a result of the coupling to the superconducting shell., Comment: 13 pages, 8 figures

In the presence of crossed electric and magnetic fields, a graphene ribbon has chiral states running along sample edges and along boundaries between $p$-doped and $n$-doped regions. We here consider the scattering of edge states into interface states, which takes place whereever the $pn$ interface crosses the sample boundary, as well as the reverse process. For a graphene ribbon with armchair boundaries, the evolution of edge states into interface states and {\em vice versa} is governed by the conservation of valley isospin. Although valley isospin is not conserved in simplified models of a ribbon with zigzag boundaries, we find that arguments based on isospin conservation can be applied to a more realistic modeling of the graphene ribbon, which takes account of the lifting of electron-hole degeneracy. The valley isospin of interface states is an important factor determining the conductance of a graphene $pn$ junction in a quantizing magnetic field., Comment: 13 pages, 13 figures

Characterizing distinct electron wave packets is a basic task for solid-state electron quantum optics with applications in quantum metrology and sensing. A important circuit element for this task is a non-stationary potential barrier than enables backscattering of chiral particles depending on their energy and time of arrival. Here we solve the quantum mechanical problem of single-particle scattering by a ballistic constriction in an fully depleted quantum Hall system under spatially uniform but time-dependent electrostatic potential modulation. The result describes electrons distributed in time-energy space according to a modified Wigner quasiprobability distribution and scattered with an energy-dependent transmission probability that characterizes constriction in the absence of modulation. Modification of the incoming Wigner distribution due to external time-dependent potential simplifies in case of linear time-dependence and admits semiclassical interpretation. Our results support a recently proposed and implemented method for measuring time and energy distribution of solitary electrons as a quantum tomography technique, and offer new paths for experimental exploration of on-demand sources of coherent electrons., Comment: 13 pages, 10 figures, numerical demonstration added; related to quantum tomography experiment arXiv:1901.10985

Spin-polarized normal-metal wires coupled to a superconductor can host Majorana states at their ends. These decay into the bulk and are protected by a minigap in the spectrum. Previous studies have found that strong coupling between the superconductor and the normal-metal wire renormalizes the properties of this low-energy phase. Here, we develop a semiclassical scattering approach to explain these renormalization effects. We show that a renormalization of the propagation velocity in the normal wire originates from double Andreev reflection processes at the superconductor interface and that it continues to exist in the absence of a proximity-induced minigap in the normal-metal wire. We also show that the renormalization effects exist for arbitrary transparency of the normal-metal--superconductor interface, provided the superconductor coherence length is sufficiently long in comparison to the thickness of the normal metal., Comment: 8+7 pages, 9 figures

We develop a Boltzmann transport theory of coupled magnon-phonon transport in ferromagnetic insulators. The explicit treatment of the magnon-phonon coupling within the Boltzmann approach allows us to calculate the low-temperature magnetic-field dependence of the spin-Seebeck voltage. Within the Boltzmann theory we find that this magnetic field dependence shows similar features as found by Flebus et al. [Phys. Rev. B 95, 144420 (2017)] for a strongly coupled magnon phonon system that forms magnon-polarons, and consistent with experimental findings in yttrium iron garnet by Kikkawa et al. [Phys. Rev. Lett. 117, 207203 (2016)]. In addition to the anomalous magnetic-field dependence of the spin Seebeck effect, we also predict a dependence on the system size., Comment: 12 pages

We study the bulk-boundary correspondence for topological crystalline phases, where the crystalline symmetry is an order-two (anti)symmetry, unitary or antiunitary. We obtain a formulation of the bulk-boundary correspondence in terms of a subgroup sequence of the bulk classifying groups, which uniquely determines the topological classification of the boundary states. This formulation naturally includes higher-order topological phases as well as topologically nontrivial bulk systems without topologically protected boundary states. The complete bulk and boundary classification of higher-order topological phases with an additional order-two symmetry or antisymmetry is contained in this work., Comment: Revised & considerably extended version

Second-order topological insulators and superconductors have a gapped excitation spectrum in bulk and along boundaries, but protected zero modes at corners of a two-dimensional crystal or protected gapless modes at hinges of a three-dimensional crystal. A second-order topological phase can be induced by the presence of a bulk crystalline symmetry. Building on Shiozaki and Sato's complete classification of bulk crystalline phases with an order-two crystalline symmetry [Phys.\ Rev.\ B {\bf 90}, 165114 (2014)], such as mirror reflection, twofold rotation, or inversion symmetry, we classify all corresponding second-order topological insulators and superconductors. The classification also includes antiunitary symmetries and antisymmetries., Comment: 36 pages, 18 figures

$\mathbb{Z}_4$ parafermions can be realized in a strongly interacting quantum spin Hall Josephson junction or in a spin Hall Josephson junction strongly coupled to an impurity spin. In this paper we study a system that has both features, but with weak (repulsive) interactions and a weakly coupled spin. We show that for a strongly anisotropic exchange interaction, at low temperatures the system enters a strong coupling limit in which it hosts two $\mathbb{Z}_4$ parafermions, characterizing a fourfold degeneracy of the ground state. We construct the parafermion operators explicitly, and show that they facilitate fractional $e/2$ charge tunneling across the junction. The dependence of the effective low-energy spectrum on the superconducting phase difference reveals an $8\pi$ periodicity of the supercurrent., Comment: 20 pages, 9 figures

Second-order topological insulators are crystalline insulators with a gapped bulk and gapped crystalline boundaries, but topologically protected gapless states at the intersection of two boundaries. Without further spatial symmetries, five of the ten Altland-Zirnbauer symmetry classes allow for the existence of such second-order topological insulators in two and three dimensions. We show that reflection symmetry can be employed to systematically generate examples of second-order topological insulators and superconductors, although the topologically protected states at corners (in two dimensions) or at crystal edges (in three dimensions) continue to exist if reflection symmetry is broken. A three-dimensional second-order topological insulator with broken time-reversal symmetry shows a Hall conductance quantized in units of $e^2/h$., Comment: 13 pages with supplemental material

Using a dimensional reduction scheme based on scattering theory, we show that the classification tables for topological insulators and superconductors with reflection symmetry can be organized in two period-two and four period-eight cycles, similar to the Bott periodicity found for topological insulators and superconductors without spatial symmetries. With the help of the dimensional reduction scheme the classification in arbitrary dimensions $d \ge 1$ can be obtained from the classification in one dimension, for which we present a derivation based on relative homotopy groups and exact sequences to classify one-dimensional insulators and superconductors with reflection symmetry. The resulting classification is fully consistent with a comprehensive classification obtained recently by Shiozaki and Sato [Phys.\ Rev.\ B {\bf 90}, 165114 (2014)]. The use of a scattering-matrix inspired method allows us to address the second descendant $\bZ_2$ phase, for which the topological nontrivial phase was previously reported to be vulnerable to perturbations that break translation symmetry., Comment: 18 pages, 7 figures

We study the sensitivity of Yu-Shiba-Rusinov states, bound states that form around magnetic scatterers in superconductors, to the presence of nonmagnetic disorder in both two and three dimensional systems. We formulate a scattering approach to this problem and reduce the effects of disorder to two contributions: disorder-induced normal reflection and a random phase of the amplitude for Andreev reflection. We find that both of these are small even for moderate amounts of disorder. In the dirty limit in which the disorder-induced mean free path is smaller than the superconducting coherence length, the variance of the energy of the Yu-Shiba-Rusinov state remains small in the ratio of the Fermi wavelength and the mean free path. This effect is more pronounced in three dimensions, where only impurities within a few Fermi wavelengths of the magnetic scatterer contribute. In two dimensions the energy variance is larger by a logarithmic factor because impurities contribute up to a distance of the order of the superconducting coherence length., Comment: 9 pages, 6 figures

Disorder effects are especially pronounced around nodal points in linearly dispersing bandstructures as present in graphene or Weyl semimetals. Despite the enormous experimental and numerical progress, even a simple quantity like the average density of states cannot be assessed quantitatively by analytical means. We demonstrate how this important problem can be solved employing the functional renormalization group method and, for the two dimensional case, demonstrate excellent agreement with reference data from numerical simulations based on tight-binding models. In three dimensions our analytic results also improve drastically on existing approaches., Comment: Erratum prepended (1 + 4 + 3 pages)

We consider the full nonequilibrium response of a mesoscopic capacitor in the large transparency limit, exactly solving a model with electron-electron interactions appropriate for a cavity in the quantum Hall regime. For a cavity coupled to the electron reservoir via an ideal point contact, we show that the response to any time-dependent gate voltage $V_{\rm g}(t)$ is strictly linear in $V_{\rm g}$. We analyze the charge and current response to a sudden gate voltage shift, and find that this response is not captured by a simple circuit analogy. In particular, in the limit of strong interactions a sudden change in the gate voltage leads to the emission of a sequence of multiple charge pulses, the width and separation of which are controlled by the charge-relaxation time $\tau_{\rm c} = h C_{\rm g}/e^2$ and the time of flight $\tau_{\rm f}$. We also consider the effect of a finite reflection amplitude in the point contact, which leads to nonlinear-in-gate-voltage corrections to the charge and current response., Comment: 14 pages, 11 figures

We calculate the Josephson current density j(phi) for a Weyl superconductor--normal-metal--superconductor junction for which the outer terminals are superconducting Weyl metals and the normal layer is a Weyl (semi)metal. We describe the Weyl (semi)metal using a simple model with two Weyl points. The model has broken time-reveresal symmetry, but inversion symmetry is present. We calculate the Josephson current for both zero and finite temperature for the two pairing mechanisms inside the superconductors that have been proposed in the literature, zero-momentum BCS-like pairing and finite-momentum FFLO-like pairing, and assuming the short-junction limit. For both pairing types we find that the current is proportional to the normal-state junction conductivity, with a proportionality coefficient that shows quantitative differences between the two pairing mechanisms. The current for the BCS-like pairing is found to be independent of the chemical potential, whereas the current for the FFLO-like pairing is not., Comment: 7 pages, 4 figures

Electrical currents at the surface or edge of a topological insulator are intrinsically spin-polarized. We show that such surface/edge currents can be used to switch the orientation of a molecular magnet weakly coupled to the surface or edge of a topological insulator. For the edge of a two-dimensional topological insulator as well as for the surface of a three-dimensional topological insulator the application of a well-chosen surface/edge current can lead to a complete polarization of the molecule if the molecule's magnetic anisotropy axis is appropriately aligned with the current direction. For a generic orientation of the molecule a nonzero but incomplete polarization is obtained. We calculate the probability distribution of the magnetic states and the switching rates as a function of the applied current., Comment: 8+5 pages, 8 figures

Although Lorentz invariance forbids the presence of a term that tilts the energy-momentum relation in the Weyl Hamiltonian, a tilted dispersion is not forbidden and, in fact, generic for condensed matter realizations of Weyl semimetals. We here investigate the combined effect of such a tilted Weyl dispersion and the presence of potential disorder. In particular, we address the influence of a tilt on the disorder-induced phase transition between a quasi-ballistic phase at weak disorder, in which the disorder is an irrelevant perturbation, and a diffusive phase at strong disorder. Our main result is that the presence of a tilt leads to a reduction of the critical disorder strength for this transition or, equivalently, that increasing the tilt at fixed disorder strength drives the system through the phase transition to the diffusive strong-disorder phase. Notably this obscures the tilt induced Lifshitz transition to an over tilted type-II Weyl phase at any finite disorder strength. Our results are supported by analytical calculations using the self-consistent Born approximation and numerical calculations of the density of states and of transport properties., Comment: 8 pages, 8 figures

We study the Josephson effect in a quantum spin Hall system coupled to a localized magnetic impurity. As a consequence of the fermion parity anomaly, the spin of the combined system of impurity and spin-Hall edge alternates between half-integer and integer values when the superconducting phase difference across the junction advances by $2\pi$. This leads to characteristic differences in the splittings of the spin multiplets by exchange coupling and single-ion anisotropy at phase differences, for which time-reserval symmetry is preserved. We discuss the resulting $8\pi$-periodic (or $\mathbb{Z}_4$) fractional Josephson effect in the context of recent experiments., Comment: 5 pages + supplemental materials, 4 figures

In a graphene {\em pn} junction at high magnetic field, unidirectional "snake states" are formed at the {\em pn} interface. In a clean {\em pn} junction, each snake state exists in one of the valleys of the graphene band structure, and the conductance of the junction as a whole is determined by microscopic details of the coupling between the snake states at the {\em pn} interface and quantum Hall edge states at the sample boundaries [Tworzydlo {\em et al.}, Phys. Rev. B {\bf 76}, 035411 (2007)]. Disorder mixes and couples the snake states. We here report a calculation of the full conductance distribution in the crossover between the clean limit and the strong disorder limit, in which the conductance distribution is given by random matrix theory [Abanin and Levitov, Science {\bf 317}, 641 (2007)]. Our calculation involves an exact solution of the relevant scaling equation for the scattering matrix, and the results are formulated in terms of parameters describing the microscopic disorder potential in bulk graphene., Comment: 10 pages, 3 figures

We study Weyl semimetals in the presence of generic disorder, consisting of a random vector potential as well as a random scalar potential. We derive renormalization group flow equations to second order in the disorder strength. These flow equations predict a disorder-induced phase transition between a pseudo-ballistic weak-disorder phase and a diffusive strong-disorder phase for sufficiently strong random scalar potential or for a pure three-component random vector potential. We verify these predictions using a numerical study of the density of states near the Weyl point and of quantum transport properties at the Weyl point. In contrast, for a pure single-component random vector potential the diffusive strong-disorder phase is absent., Comment: published version with minor changes

We numerically investigate the distribution of Drude weights $D$ of many-body states in disordered one-dimensional interacting electron systems across the transition to a many-body localized phase. Drude weights are proportional to the spectral curvatures induced by magnetic fluxes in mesoscopic rings. They offer a method to relate the transition to the many-body localized phase to transport properties. In the delocalized regime, we find that the Drude weight distribution at a fixed disorder configuration agrees well with the random-matrix-theory prediction $P(D) \propto (\gamma^2+D^2)^{-3/2}$, although the distribution width $\gamma$ strongly fluctuates between disorder realizations. A crossover is observed towards a distribution with different large-$D$ asymptotics deep in the many-body localized phase, which however differs from the commonly expected Cauchy distribution. We show that the average distribution width $\langle \gamma\rangle $, rescaled by $L\Delta$, $\Delta$ being the average level spacing in the middle of the spectrum and $L$ the systems size, is an efficient probe of the many-body localization transition, as it increases/vanishes exponentially in the delocalized/localized phase., Comment: 5 pages, 3 figures + 1 page Supplemental Material, 2 figures

Double Weyl nodes are topologically protected band crossing points which carry chiral charge $\pm2$. They are stabilized by $C_{4}$ point group symmetry and are predicted to occur in $\mathrm{SrSi_{2}}$ or $\mathrm{HgCr_{2}Se_{4}}$. We study their stability and physical properties in the presence of a disorder potential. We investigate the density of states and the quantum transport properties at the nodal point. We find that, in contrast to their counterparts with unit chiral charge, double Weyl nodes are unstable to any finite amount of disorder and give rise to a diffusive phase, in agreement with predictions of Goswami and Nevidomskyy [Phys. Rev. B 92, 214504 (2015)] and Bera, Sau, and Roy [Phys. Rev. B 93, 201302(R) (2016)]. However, for finite system sizes a crossover between pseudodiffusive and diffusive quantum transport can be observed., Comment: Supersedes arXiv:1512.07164

Tunneling between a point contact and a one-dimensional wire is usually described with the help of a tunneling Hamiltonian that contains a delta function in position space. Whereas the leading order contribution to the tunneling current is independent of the way this delta function is regularized, higher-order corrections with respect to the tunneling amplitude are known to depend on the regularization. Instead of regularizing the delta function in the tunneling Hamiltonian, one may also obtain a finite tunneling current by invoking the ultraviolet cut-offs in a field-theoretic description of the electrons in the one-dimensional conductor, a procedure that is often used in the literature. For the latter case, we show that standard ultraviolet cut-offs lead to different results for the tunneling current in fermionic and bosonized formulations of the theory, when going beyond leading order in the tunneling amplitude. We show how to recover the standard fermionic result using the formalism of functional bosonization and revisit the tunneling current to leading order in the interacting case., Comment: 17 pages, 5 figures

Transistors, regardless of their size, rely on electrical gates to control the conductance between source and drain contacts. In atomic-scale transistors, this conductance is exquisitely sensitive to single electrons hopping via individual orbitals. Single-electron transport in molecular transistors has been previously studied using top-down approaches to gating, such as lithography and break junctions. But atomically precise control of the gate - which is crucial to transistor action at the smallest size scales - is not possible with these approaches. Here, we used individual charged atoms, manipulated by a scanning tunnelling microscope, to create the electrical gates for a single-molecule transistor. This degree of control allowed us to tune the molecule into the regime of sequential single-electron tunnelling, albeit with a conductance gap more than one order of magnitude larger than observed previously. This unexpected behaviour arises from the existence of two different orientational conformations of the molecule, depending on its charge state. Our results show that strong coupling between these charge and conformational degrees of freedom leads to new behaviour beyond the established picture of single-electron transport in atomic-scale transistors., Comment: 15 pages, 4 figures