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In this Letter, we investigate the fate of the non-Hermitian skin effect in one-dimensional systems that conserve the dipole moment and higher moments of an associated global U(1) charge. Motivated by field theoretical arguments and lattice model calculations, we demonstrate that the key feature of the non-Hermitian skin effect for m-pole conserving systems is the generation of an (m+1)th multipole moment. For example, in contrast to the conventional skin effect where charges are anomalously localized at one boundary, the dipole-conserving skin effect results in charges localized at both boundaries, in a configuration that generates an extremal quadrupole moment. In addition, we explore the dynamical consequences of the m-pole skin effect, focusing on charge and entanglement propagation. Both numerically and analytically, we provide evidence that long-time steady states have Fock-space localization and an area-law scaling of entanglement entropy, which serve as quantum indicators of the skin effect.

We propose heterostructures that realize first and second order topological superconductivity with vanishing net magnetization by utilizing altermagnetism. Such platforms may offer a significant improvement over conventional platforms with uniform magnetization since the latter suppresses the superconducting gap. We first introduce a 1D semiconductor-superconductor structure in proximity to an altermagnet that realizes end Majorana zero modes (MZMs) with vanishing net magnetization. Additionally, a coexisting Zeeman term provides a tuning knob to distinguish topological and trivial zero modes. We then propose 2D altermagnetic platforms that can realize chiral Majorana fermions or higher order corner MZMs. Our Letter paves the way toward realizing Majorana boundary states with an alternative source of time-reversal breaking and zero net magnetization.

Weyl fermions are hypothetical chiral particles that can also manifest as excitations near three-dimensional band crossing points in lattice systems. These quasiparticles are subject to the Nielsen-Ninomiya "no-go" theorem when placed on a lattice, requiring the total chirality across the Brillouin zone to vanish. This constraint results from the topology of the (orientable) manifold on which they exist. Here, we ask to what extent the concepts of topology and chirality of Weyl points remain well defined when the underlying manifold is nonorientable. We show that the usual notion of chirality becomes ambiguous in this setting, allowing for systems with a nonzero total chirality. This circumvention of the Nielsen-Ninomiya theorem stems from a generic discontinuity of the vector field whose zeros are Weyl points. Furthermore, we discover that Weyl points on nonorientable manifolds carry an additional Z_{2} topological invariant which satisfies a different no-go theorem. We implement such Weyl points by imposing a nonsymmorphic symmetry in the momentum space of lattice models. Finally, we experimentally realize all aspects of their phenomenology in a photonic platform with synthetic momenta. Our work highlights the subtle but crucial interplay between the topology of quasiparticles and of their underlying manifold.

We study the geometric response of three-dimensional non-Hermitian crystalline systems with nontrivial point-gap topology. For systems with fourfold rotation symmetry, we show that in the presence of disclination lines with a total Frank angle, which is an integer multiple of 2π, there can be nontrivial one-dimensional point-gap topology along the direction of the disclination lines. This results in disclination-induced non-Hermitian skin effects. By doubling a non-Hermitian Hamiltonian to a Hermitian three-dimensional chiral topological insulator, we show that the disclination-induced skin modes are zero modes of the effective surface Dirac fermion(s) in the presence of a pseudomagnetic flux induced by disclinations. Furthermore, we find that our results have a field theoretic description, and the corresponding geometric response actions (e.g., the Euclidean Wen-Zee action) enrich the topological field theory of non-Hermitian systems.

We investigate higher-order Weyl semimetals (HOWSMs) having bulk Weyl nodes attached to both surface and hinge Fermi arcs. We identify a new type of Weyl node, which we dub a 2nd-order Weyl node, that can be identified as a transition in momentum space in which both the Chern number and a higher order topological invariant change. As a proof of concept we use a model of stacked higher order quadrupole insulators (QI) to identify three types of WSM phases: 1st order, 2nd order, and hybrid order. The model can also realize type-II and hybrid-tilt WSMs with various surface and hinge arcs. After a comprehensive analysis of the topological properties of various HOWSMs, we turn to their physical implications that show the very distinct behavior of 2nd-order Weyl nodes when they are gapped out. We obtain three remarkable results: (i) the coupling of a 2nd-order Weyl phase with a conventional 1st-order one can lead to a hybrid-order topological insulator having coexisting surface cones and flat hinge arcs that are independent and not attached to each other. (ii) A nested 2nd-order inversion-symmetric WSM by a charge-density wave (CDW) order generates an insulating phase having coexisting flatband surface and hinge states all over the Brillouin zone. (iii) A CDW order in a time-reversal symmetric higher-order WSM gaps out a 2nd-order node with a 1st-order node and generates an insulating phase having coexisting surface Dirac cone and hinge arcs. Moreover, we show that a measurement of charge density in the presence of magnetic flux can help to identify some classes of 2nd-order WSMs. Finally, we show that periodic driving can be utilized as a way for generating HOWSMs. Our results are relevant to metamaterials as well as various phases of Cd_{3}As_{2}, KMgBi, and rutile-structure PtO_{2} that have been predicted to realize higher order Dirac semimetals.

Topological insulators, having intrinsic or proximity-coupled s-wave superconductivity, host Majorana zero modes (MZMs) at the ends of vortex lines. The MZMs survive up to a critical doping of the TI at which there is a vortex phase transition that eliminates the MZMs. In this work, we show that the phenomenology in higher-order topological insulators (HOTIs) can be qualitatively distinct. In particular, we find two distinct features. (i) We find that vortices placed on the gapped (side) surfaces of the HOTI, exhibit a pair of phase transitions as a function of doping. The first transition is a surface phase transition after which MZMs appear. The second transition is the well-known vortex phase transition. We find that the surface transition appears because of the competition between the superconducting gap and the local T-breaking gap on the surface. (ii) We present numerical evidence that shows strong variation of the critical doping for the vortex phase transition as the center of the vortex is moved toward or away from the hinges of the sample. We believe our work provides new phenomenology that can help identify HOTIs, as well as illustrating a promising platform for the realization of MZMs.

We study nonreciprocity in spatiotemporally modulated 1D resonator chains from the perspective of equivalent 2D resonator arrays with a synthetic dimension and transverse synthetic electric and magnetic fields. The synthetic fields are respectively related to temporal and spatial modulation of the resonator chain, and we show that their combination can induce strong transmission nonreciprocity, i.e., complete isolation with only a weak perturbative modulation. This nonreciprocal effect is analogous to the Hall effect for charged particles. We experimentally implement chains of two and three spatiotemporally modulated resonators and measure over 58 dB of isolation contrast.

Topological crystalline insulators are a new class of materials that have metallic surface states on select surfaces due to point group crystalline symmetries. In this Letter, we consider a model for a three-dimensional topological crystalline insulator with Dirac nodes occurring on a surface that are protected by the mirror symmetry. We demonstrate that the electromagnetic response for such a system is characterized by a 1-form b_{μ}. The value of b_{μ} can be inferred from the locations of the surface Dirac nodes in energy-momentum space. From both the effective action and analytical band structure calculations, we show that the vortex core of b[over →] or a domain wall of a component of b[over →] can trap surface charges.

We explore a scenario where local interactions form one-dimensional gapped interfaces between a pair of distinct chiral two-dimensional topological states-referred to as phases 1 and 2-such that each gapped region terminates at a domain wall separating the chiral gapless edge states of these phases. We show that this type of T junction supports pointlike fractionalized excitations obeying parafermion statistics, thus implying that the one-dimensional gapped interface forms an effective topological parafermionic wire possessing a nontrivial ground state degeneracy. The physical properties of the anyon condensate that gives rise to the gapped interface are investigated. Remarkably, this condensate causes the gapped interface to behave as a type of anyon "Andreev reflector" in the bulk, whereby anyons from one phase, upon hitting the interface, can be transformed into a combination of reflected anyons and outgoing anyons from the other phase. Thus, we conclude that while different topological orders can be connected via gapped interfaces, the interfaces are themselves topological.

Existing proximity effects stem from systems with a local order parameter, such as a local magnetic moment or a local superconducting pairing amplitude. Here, we demonstrate that despite lacking a local order parameter, topological phases also may give rise to a proximity effect of a distinctively inverted nature. We focus on a general construction in which a topological phase is extensively coupled to a second system, and we argue that, in many cases, the inverse topological order will be induced on the second system. To support our arguments, we rigorously establish this "bulk topological proximity effect" for all gapped free-fermion topological phases and representative integrable models of interacting topological phases. We present a terrace construction which illustrates the phenomenological consequences of this proximity effect. Finally, we discuss generalizations beyond our framework, including how intrinsic topological order may also exhibit this effect.

We used low-energy, momentum-resolved inelastic electron scattering to study surface collective modes of the three-dimensional topological insulators Bi_{2}Se_{3} and Bi_{0.5}Sb_{1.5}Te_{3-x}Se_{x}. Our goal was to identify the "spin plasmon" predicted by Raghu and co-workers [Phys. Rev. Lett. 104, 116401 (2010)]. Instead, we found that the primary collective mode is a surface plasmon arising from the bulk, free carriers in these materials. This excitation dominates the spectral weight in the bosonic function of the surface χ^{"}(q,ω) at THz energy scales, and is the most likely origin of a quasiparticle dispersion kink observed in previous photoemission experiments. Our study suggests that the spin plasmon may mix with this other surface mode, calling for a more nuanced understanding of optical experiments in which the spin plasmon is reported to play a role.

The chiral AIII symmetry class in the classification table of topological insulators contains topological phases classified by a winding number ν for each odd space dimension. An open problem for this class is the characterization of the phases and phase boundaries in the presence of strong disorder. In this work, we derive a covariant real-space formula for ν and, using an explicit one-dimensional disordered topological model, we show that ν remains quantized and nonfluctuating when disorder is turned on, even though the bulk energy spectrum is completely localized. Furthermore, ν remains robust even after the insulating gap is filled with localized states, but when the disorder is increased even further, an abrupt change of ν to a trivial value is observed. Using exact analytic calculations, we show that this marks a critical point where the localization length diverges. As such, in the presence of disorder, the AIII class displays markedly different physics from everything known to date, with robust invariants being carried entirely by localized states and bulk extended states emerging from an absolutely localized spectrum. Detailed maps and a clear physical description of the phases and phase boundaries are presented based on numerical and exact analytic calculations.

We apply the geometric theory of swimming at low Reynolds number to the study of nearly circular swimmers in two-dimensional fluids with nonvanishing "odd," or Hall, viscosity. The odd viscosity gives an off-diagonal contribution to the fluid stress tensor, which results in a number of striking effects. In particular, we find that a swimmer whose area is changing will experience a torque proportional to the rate of change of the area, with the constant of proportionality given by the coefficient ηo of odd viscosity. After working out the general theory of swimming in fluids with odd viscosity for a class of simple swimmers, we give a number of example swimming strokes which clearly demonstrate the differences between swimming in a fluid with conventional viscosity and a fluid which also has an odd viscosity. We also include a discussion of the extension of the famous Scallop theorem of low Reynolds number swimming to the case where the fluid has a nonzero odd viscosity. A number of more technical results, including a proof of the torque-area relation for swimmers of more general shape, are explained in a set of Appendixes.

We prove a topological criterion for the existence of a zero-energy Majorana bound state on a disclination, a rotation symmetry breaking point defect, in fourfold symmetric topological crystalline superconductors (TCS) in two dimensions. We first establish a complete topological classification of TCS using the Chern invariant and three integral rotation invariants. By analytically and numerically studying disclinations, we algebraically deduce a Z2 index that identifies the parity of the number of Majorana zero modes at a disclination. Surprisingly, we also find weakly protected Majorana fermions bound at the corners of superconductors with trivial Chern and weak invariants.

We consider how fractional excitations bound to a dislocation evolve as the dislocation is separated into a pair of disclinations. We show that some dislocation-bound excitations (such as Majorana modes and half-quantum vortices) are possible only if the elementary dislocation consists of two inequivalent disclinations, as is the case for stripes or square lattices but not for triangular lattices. The existence of multiple inequivalent disclination classes governs the two-dimensional melting of quantum liquid crystals (i.e., nematics and hexatics), determining whether superfluidity and orientational order can simultaneously vanish at a continuous transition.

The use of quantum entanglement to study condensed matter systems has been flourishing in critical systems and topological phases. Additionally, using real-space entanglement one can characterize localized and delocalized phases of disordered fermion systems. Here we instead propose the momentum-space entanglement spectrum as a means of characterizing disordered models. We show that localization in one dimension can be characterized by the momentum space entanglement between left and right movers and illustrate our methods using explicit models with spatially correlated disorder that exhibit phases which avoid complete Anderson localization. The momentum space entanglement spectrum clearly reveals the location of delocalized states in the energy spectrum, can be used as a signature of the phase transition between a delocalized and localized phase, and only requires a single numerical diagonalization to yield clear results.

It has been shown that doped topological insulators, up to a certain level of doping, still preserve some topological signatures of the insulating phase such as axionic electromagnetic response and the presence of a Majorana mode in the vortices of a superconducting phase. Multiple topological insulators such as HgTe, ScPtBi, and other ternary Heusler compounds have been identified and generically feature the presence of a topologically trivial band between the two topological bands. In this Letter we show that the presence of such a trivial band can stabilize the topological signature over a much wider range of doping. Specifically, we calculate the structure of vortex modes in the superconducting phase of doped topological insulators, a model that captures the features of HgTe and the ternary Heusler compounds. We show that, due to the hybridization with the trivial band, Majorana modes are preserved over a large, extended doping range for p doping. In addition to presenting a viable system where much less fine-tuning is required to observe the Majorana modes, our analysis opens a route to study other topological features of doped compounds that cannot be modeled using the simple Bi(2)Se(3) Dirac model.

We provide a characterization of tunneling between coupled topological insulators in 2D and 3D under the influence of a ferromagnetic layer. We explore conditions for such systems to exhibit integer quantum Hall physics and localized fractional charge, also taking into account interaction effects for the 2D case. We show that the effects of tunneling are topologically equivalent to a certain deformation or folding of the sample geometry. Our key advance is the realization that the quantum Hall or fractional charge physics can appear in the presence of only a single magnet unlike previous proposals which involve magnetic domain walls on the surface or edges of topological insulators, respectively. We give illustrative topological folding arguments to prove our results and show that for the 2D case our results are robust even in the presence of interactions.

We consider the viscoelastic response of the electronic degrees of freedom in 2D and 3D topological insulators (TI's). Our primary focus is on the 2D Chern insulator which exhibits a bulk dissipationless viscosity analogous to the quantum Hall viscosity predicted in integer and fractional quantum Hall states. We show that the dissipationless viscosity is the response of a TI to torsional deformations of the underlying lattice geometry. The viscoelastic response also indicates that crystal dislocations in Chern insulators will carry momentum density. We briefly discuss generalizations to 3D which imply that time-reversal invariant TI's will exhibit a quantum Hall viscosity on their surfaces.

Torsional degrees of freedom play an important role in modern gravity theories as well as in condensed matter systems where they can be modeled by defects in solids. Here we isolate a class of torsion models that support torsion configurations with a localized, conserved charge that adopts integer values. The charge is topological in nature, and the torsional configurations can be thought of as torsional "monopole" solutions. We explore some of the properties of these configurations in gravity models with a nonvanishing curvature and discuss the possible existence of such monopoles in condensed matter systems. To conclude, we show how the monopoles can be thought of as a natural generalization of the Cartan spiral staircase.

We investigate the behavior of a topological Chern insulator in the presence of disorder, with a focus on its entanglement spectrum (EtS) constructed from the ground state. For systems with symmetries, the EtS was shown to contain explicit information about the topological universality class revealed by sorting the EtS against the conserved quantum numbers. In the absence of any symmetry, we demonstrate that statistical methods such as the level statistics of the EtS can be equally insightful, allowing us to distinguish when an insulator is in a topological or trivial phase and to map the boundary between the two phases. The phase diagram of a Chern insulator is explicitly computed as function of Fermi level (EF) and disorder strength using the level statistics of the EtS and energy spectrum, together with a computation of the Chern number (C) via a new, efficient real-space formula.

We generalize the topological entanglement entropy to a family of topological Rényi entropies parametrized by a parameter alpha, in an attempt to find new invariants for distinguishing topologically ordered phases. We show that, surprisingly, all topological Rényi entropies are the same, independent of alpha for all nonchiral topological phases. This independence shows that topologically ordered ground-state wave functions have reduced density matrices with a certain simple structure, and no additional universal information can be extracted from the entanglement spectrum.

We construct time-reversal invariant topological superconductors and superfluids in two and three dimensions. These states have a full pairing gap in the bulk, gapless counterpropagating Majorana states at the boundary, and a pair of Majorana zero modes associated with each vortex. The superfluid 3He B phase provides a physical realization of the topological superfluidity, with experimentally measurable surface states protected by the time-reversal symmetry. We show that the time-reversal symmetry naturally emerges as a supersymmetry, which changes the parity of the fermion number associated with each time-reversal invariant vortex and connects each vortex with its superpartner.

The quantum spin Hall (QSH) state is a topologically nontrivial state of quantum matter which preserves time-reversal symmetry; it has an energy gap in the bulk, but topologically robust gapless states at the edge. Recently, this novel effect has been predicted and observed in HgTe quantum wells and in this Letter we predict a similar effect arising in Type-II semiconductor quantum wells made from InAs/GaSb/AlSb. The quantum well exhibits an "inverted" phase similar to HgTe/CdTe quantum wells, which is a QSH state when the Fermi level lies inside the gap. Due to the asymmetric structure of this quantum well, the effects of inversion symmetry breaking are essential. Remarkably, the topological quantum phase transition between the conventional insulating state and the quantum spin Hall state can be continuously tuned by the gate voltage, enabling quantitative investigation of this novel phase transition.

We predict the existence of a three-dimensional quantum Hall effect plateau in a graphite crystal subject to a magnetic field. The plateau has a Hall conductivity quantized at 4e2/variant Planck's over 2pi 1/c0 with c0 the c-axis lattice constant. We analyze the three-dimensional Hofstadter problem of a realistic tight-binding Hamiltonian for graphite, find the gaps in the spectrum, and estimate the critical value of the magnetic field above which the Hall plateau appears. When the Fermi level is in the bulk Landau gap, Hall transport occurs through the appearance of chiral surface states. We estimate the magnetic field necessary for the appearance of the effect to be 15.4 T for electron carriers and 7.0 T for holes.

The spin Hall effect depends crucially on the intrinsic spin-orbit coupling of the energy band. Because of the smaller spin-orbit coupling in silicon, the spin Hall effect is expected to be much reduced. We show that an electric field in p-doped silicon can induce a dissipationless orbital current in a fashion reminiscent of the spin Hall effect. The vertex correction from impurity scattering vanishes and the effect is robust against disorder. The orbital Hall effect leads to accumulation of local orbital momentum at the edge of the sample, and can be detected by the Kerr effect.

A semiclassical analysis of a two-dimensional electron droplet in a high, nonuniform magnetic field predicts that the droplet will form "fingered" patterns upon increasing the number of electrons. We construct explicit examples of these patterns using methods first developed for the flow of two-dimensional viscous fluids. We complement our analytical results with Monte Carlo simulations of the droplet wave function, and find that at the point where the semiclassical analysis predicts a cusp on the interface, the droplet fissions-a type of "quantum breakup" phenomenon.