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  • Journal article
    Hossain MS, Schindler F, Islam R, Muhammad Z, Jiang Y-X, Cheng Z-J, Zhang Q, Hou T, Chen H, Litskevich M, Casas B, Yin J-X, Cochran TA, Yahyavi M, Yang XP, Balicas L, Chang G, Zhao W, Neupert T, Hasan MZet al., 2024,

    A hybrid topological quantum state in an elemental solid

    , Nature, Vol: 628, Pages: 527-533, ISSN: 0028-0836

    Topology1-3 and interactions are foundational concepts in the modern understanding of quantum matter. Their nexus yields three important research directions: (1) the competition between distinct interactions, as in several intertwined phases, (2) the interplay between interactions and topology that drives the phenomena in twisted layered materials and topological magnets, and (3) the coalescence of several topological orders to generate distinct novel phases. The first two examples have grown into major areas of research, although the last example remains mostly unexplored, mainly because of the lack of a material platform for experimental studies. Here, using tunnelling microscopy, photoemission spectroscopy and a theoretical analysis, we unveil a 'hybrid' topological phase of matter in the simple elemental-solid arsenic. Through a unique bulk-surface-edge correspondence, we uncover that arsenic features a conjoined strong and higher-order topology that stabilizes a hybrid topological phase. Although momentum-space spectroscopy measurements show signs of topological surface states, real-space microscopy measurements unravel a unique geometry of topologically induced step-edge conduction channels revealed on various natural nanostructures on the surface. Using theoretical models, we show that the existence of gapless step-edge states in arsenic relies on the simultaneous presence of both a non-trivial strong Z2 invariant and a non-trivial higher-order topological invariant, which provide experimental evidence for hybrid topology. Our study highlights pathways for exploring the interplay of different band topologies and harnessing the associated topological conduction channels in engineered quantum or nano-devices.

  • Journal article
    Yang K, Xu Z, Feng Y, Schindler F, Xu Y, Bi Z, Bernevig BA, Tang P, Liu C-Xet al., 2024,

    Topological minibands and interaction driven quantum anomalous Hall state in topological insulator based moiré heterostructures

    , Nature Communications, Vol: 15, ISSN: 2041-1723

    The presence of topological flat minibands in moiré materials provides an opportunity to explore the interplay between topology and correlation. In this work, we study moiré minibands in topological insulator films with two hybridized surface states under a moiré superlattice potential created by two-dimensional insulating materials. We show the lowest conduction (highest valence) Kramers' pair of minibands can be Z 2 non-trivial when the minima (maxima) of moiré potential approximately form a hexagonal lattice with six-fold rotation symmetry. Coulomb interaction can drive the non-trivial Kramers' minibands into the quantum anomalous Hall state when they are half-filled, which is further stabilized by applying external gate voltages to break inversion. We propose the monolayer Sb2 on top of Sb2Te3 films as a candidate based on first principles calculations. Our work demonstrates the topological insulator based moiré heterostructure as a potential platform for studying interacting topological phases.

  • Journal article
    Denner MM, Neupert T, Schindler F, 2023,

    Infernal and exceptional edge modes: non-hermitian topology beyond the skin effect

    , Journal of Physics: Materials, Vol: 6, Pages: 1-9

    The classification of point gap topology in all local non-Hermitian (NH) symmetry classes has been recently established. However, many entries in the resulting periodic table have only been discussed in a formal setting and still lack a physical interpretation in terms of their bulk-boundary correspondence. Here, we derive the edge signatures of all two-dimensional phases with intrinsic point gap topology. While in one dimension point gap topology invariably leads to the NH skin effect, NH boundary physics is significantly richer in two dimensions. We find two broad classes of non-Hermitian edge states: (1) infernal points, where a skin effect occurs only at a single edge momentum, while all other edge momenta are devoid of edge states. Under semi-infinite boundary conditions, the point gap thereby closes completely, but only at a single edge momentum. (2) NH exceptional point dispersions, where edge states persist at all edge momenta and furnish an anomalous number of symmetry-protected exceptional points. Surprisingly, the latter class of systems allows for a finite, non-extensive number of edge states with a well defined dispersion along all generic edge terminations. Concomitantly, the point gap only closes along the real and imaginary eigenvalue axes, realizing a novel form of NH spectral flow.

  • Journal article
    Schindler F, Gu K, Lian B, Kawabata Ket al., 2023,

    Hermitian bulk  –  non-hermitian boundary correspondence

    , PRX Quantum, Vol: 4, Pages: 1-18, ISSN: 2691-3399

    Non-Hermitian band theory distinguishes between line gaps and point gaps. While point gaps can give rise to intrinsic non-Hermitian band topology without Hermitian counterparts, line-gapped systems can always be adiabatically deformed to a Hermitian limit. Here, we show that line-gap topology and point-gap topology can be intricately connected: topological line-gapped systems in d dimensions induce nontrivial point-gap topology on their (d−1)-dimensional boundaries when suitable internal and spatial symmetries are present. Since line-gapped systems essentially realize Hermitian topological phases, this establishes a correspondence between Hermitian bulk topology and intrinsic non-Hermitian boundary topology. For the correspondence to hold, no non-Hermitian perturbations are required in the bulk itself, so that the bulk can be purely Hermitian. Concomitantly, the presence of non-Hermitian perturbations in the bulk does not affect any results as long as they do not close the bulk line gap. On the other hand, non-Hermitian perturbations are essential on the boundary to open a point gap. The non-Hermitian boundary topology then further leads to higher-order skin modes, as well as chiral and helical hinge modes, that are protected by point gaps and hence unique to non-Hermitian systems. We identify all the internal symmetry classes where bulk line-gap topology induces boundary point-gap topology as long as an additional spatial symmetry is present and establish the correspondence between their topological invariants. There also exist some symmetry classes where the Hermitian edge states remain stable, in the sense that even a point gap cannot open on the boundary.

  • Journal article
    Denner MM, Schindler F, 2023,

    Magnetic flux response of non-Hermitian topological phases

    , SciPost Physics, Vol: 14, Pages: 1-49, ISSN: 2542-4653

    We derive the response of non-Hermitian topological phases with intrinsic point gap topology to localized magnetic flux insertions. In two spatial dimensions, we identify the necessary and sufficient conditions for a flux skin effect that localizes an extensive number of in-gap modes at a flux core. In three dimensions, we furthermore establish the existence of: a flux spectral jump, where flux tube insertion fills up the entire point gap only at a single parallel crystal momentum; a higher-order flux skin effect, which occurs at the ends of flux tubes in presence of pseudo-inversion symmetry; and a flux Majorana mode that represents a spectrally isolated mid-gap state in the complex energy plane. We uniquely associate each non-Hermitian symmetry class with intrinsic point gap topology with one of these cases or a trivial flux response, and discuss possible experimental realizations.

  • Journal article
    Schindler F, Tsirkin SS, Neupert T, Andrei Bernevig B, Wieder BJet al., 2022,

    Topological zero-dimensional defect and flux states in three-dimensional insulators

    , Nature Communications, Vol: 13, Pages: 1-18, ISSN: 2041-1723

    In insulating crystals, it was previously shown that defects with two fewer dimensions than the bulk can bind topological electronic states. We here further extend the classification of topological defect states by demonstrating that the corners of crystalline defects with integer Burgers vectors can bind 0D higher-order end (HEND) states with anomalous charge and spin. We demonstrate that HEND states are intrinsic topological consequences of the bulk electronic structure and introduce new bulk topological invariants that are predictive of HEND dislocation states in solid-state materials. We demonstrate the presence of first-order 0D defect states in PbTe monolayers and HEND states in 3D SnTe crystals. We relate our analysis to magnetic flux insertion in insulating crystals. We find that π-flux tubes in inversion- and time-reversal-symmetric (helical) higher-order topological insulators bind Kramers pairs of spin-charge-separated HEND states, which represent observable signatures of anomalous surface half quantum spin Hall states.

  • Journal article
    Schindler F, Vafek O, Bernevig BA, 2022,

    Trions in twisted bilayer graphene

    , Physical Review B, Vol: 105, ISSN: 2469-9950

    The strong-coupling phase diagram of magic-angle twisted bilayer graphene (TBG) predicts a series of exact one-particle charge ±1 gapped excitations on top of the integer-filled ferromagnetic ground states. Finite-size exact diagonalization studies showed that these are the lowest charge ±1 excitations in the system (for 10 nm screening length), with the exception of charge +1 at filling −1 in the chiral limit. In the current paper we show that this “trion bound state,” a 3-particle, charge 1 excitation of the insulating ferromagnetic ground state of the projected Hamiltonian of TBG, is the lowest charge +1 overall excitation at ν=−1, and also for some large (≈20 nm) screening lengths at ν=−2 in the chiral limit and with very small binding energy. At other fillings, we show that trion bound states do exist, but only for momentum ranges that do not cover the entire moiré Brillouin zone. The trion bound states (at different momenta) exist for finite parameter range w0/w1 but they all disappear in the continuum far below the realistic values of w0/w1=0.8. We find the conditions for the existence of the trion bound state, a good variational wave function for it, and investigate its behavior for different screening lengths, at all integer fillings, on both the electron and hole sides.

  • Journal article
    Herzog-Arbeitman J, Peri V, Schindler F, Huber SD, Bernevig BAet al., 2022,

    Superfluid weight bounds from symmetry and quantum geometry in flat bands

    , Physical Review Letters, Vol: 128, ISSN: 0031-9007

    Flat-band superconductivity has theoretically demonstrated the importance of band topology to correlated phases. In two dimensions, the superfluid weight, which determines the critical temperature through the Berezinksii-Kosterlitz-Thouless criteria, is bounded by the Fubini-Study metric at zero temperature. We show this bound is nonzero within flat bands whose Wannier centers are obstructed from the atoms-even when they have identically zero Berry curvature. Next, we derive general lower bounds for the superfluid weight in terms of momentum space irreps in all 2D space groups, extending the reach of topological quantum chemistry to superconducting states. We find that the bounds can be naturally expressed using the formalism of real space invariants (RSIs) that highlight the separation between electronic and atomic degrees of freedom. Finally, using exact Monte Carlo simulations on a model with perfectly flat bands and strictly local obstructed Wannier functions, we find that an attractive Hubbard interaction results in superconductivity as predicted by the RSI bound beyond mean field. Hence, obstructed bands are distinguished from trivial bands in the presence of interactions by the nonzero lower bound imposed on their superfluid weight.

  • Journal article
    Schindler F, Regnault N, Bernevig BA, 2022,

    Exact quantum scars in the chiral nonlinear Luttinger liquid

    , Physical Review B, Vol: 105, Pages: 1-23, ISSN: 2469-9950

    While the chiral linear Luttinger liquid is integrable via bosonization, its nonlinear counterpart does not admit for an analytic solution. In this work, we find a subextensive number of exact eigenstates for a large family of density-density interaction terms. These states are embedded in a continuum of strongly correlated excited states. The real-space entanglement entropy of some exact states scales logarithmically with system size while that of others has volume-law scaling. We introduce momentum-space entanglement as an unambiguous differentiator between these exact states and the remaining excited states. With regard to momentum space, the exact states behave as bona fide quantum many-body scars: they exhibit identically zero momentum-space entanglement, while typical eigenstates behave thermally. We corroborate this finding by a level statistics analysis. Furthermore, we detail the general formalism for systematically finding all interaction terms and associated exact states, and present a number of infinite exact state sequences extending to arbitrarily high energies. Unlike many previous examples of quantum many-body scars, the exact states uncovered here do not lie at equidistant energies and do not follow from a special operator algebra. Instead, they are uniquely enabled by the interplay of Fermi statistics and chirality.

  • Journal article
    Schindler F, Bernevig BA, 2021,

    Noncompact atomic insulators

    , Physical Review B, Vol: 104, Pages: 1-5, ISSN: 2469-9950

    We study the conditions for Bloch bands to be spanned by symmetric and strictly compact Wannier states that have zero overlap with all lattice sites beyond a certain range. Similar to the characterization of topological insulators in terms of an algebraic (rather than exponential) localization of Wannier states, we find that there may be impediments to the compact localization even of topologically “trivial” obstructed atomic insulators. These insulators admit exponentially localized Wannier states centered at unoccupied orbitals of the crystalline lattice. First, we establish a sufficient condition for an insulator to have a compact representative. Second, for C2 rotational symmetry, we prove that the complement of fragile topological bands cannot be compact, even if it is an atomic insulator. Third, for C4 symmetry, our findings imply that there exist fragile bands with zero correlation length. Fourth, for a C3-symmetric atomic insulator, we explicitly derive that there are no compact Wannier states overlapping with less than 18 lattice sites. We conjecture that this obstruction generalizes to all finite Wannier sizes. Our results can be regarded as the stepping stone to a generalized theory of Wannier states beyond dipole or quadrupole polarization.

  • Journal article
    Schindler F, Prem A, 2021,

    Dislocation non-Hermitian skin effect

    , Physical Review B, Vol: 104, Pages: 1-6, ISSN: 2469-9950

    We demonstrate that crystal defects can act as a probe of intrinsic non-Hermitian topology. In particular, in point-gapped systems with periodic boundary conditions, a pair of dislocations may induce a non-Hermitian skin effect, where an extensive number of Hamiltonian eigenstates localize at only one of the two dislocations. An example of such a phase are two-dimensional systems exhibiting weak non-Hermitian topology, which are adiabatically related to a decoupled stack of Hatano-Nelson chains. Moreover, we show that strong two-dimensional point-gap topology may also result in a dislocation response, even when there is no skin effect present with open boundary conditions. For both cases, we directly relate their bulk topology to a stable dislocation non-Hermitian skin effect. Finally, and in stark contrast to the Hermitian case, we find that gapless non-Hermitian systems hosting bulk exceptional points also give rise to a well-localized dislocation response.

  • Journal article
    Denner MM, Skurativska A, Schindler F, Fischer MH, Thomale R, Bzdušek T, Neupert Tet al., 2021,

    Exceptional topological insulators

    , Nature Communications, Vol: 12, Pages: 1-7, ISSN: 2041-1723

    We introduce the exceptional topological insulator (ETI), a non-Hermitian topological state of matter that features exotic non-Hermitian surface states which can only exist within the three-dimensional topological bulk embedding. We show how this phase can evolve from a Weyl semimetal or Hermitian three-dimensional topological insulator close to criticality when quasiparticles acquire a finite lifetime. The ETI does not require any symmetry to be stabilized. It is characterized by a bulk energy point gap, and exhibits robust surface states that cover the bulk gap as a single sheet of complex eigenvalues or with a single exceptional point. The ETI can be induced universally in gapless solid-state systems, thereby setting a paradigm for non-Hermitian topological matter.

  • Journal article
    Vecsei PM, Denner MM, Neupert T, Schindler Fet al., 2021,

    Symmetry indicators for inversion-symmetric non-Hermitian topological band structures

    , Physical Review B, Vol: 103, Pages: 1-5, ISSN: 2469-9950

    We characterize non-Hermitian band structures by symmetry indicator topological invariants. Enabled by crystalline inversion symmetry, these indicators allow us to short-cut the calculation of conventional non-Hermitian topological invariants. In particular, we express the three-dimensional winding number of point-gapped non-Hermitian systems, which is defined as an integral over the whole Brillouin zone, in terms of symmetry eigenvalues at high-symmetry momenta. Furthermore, for time-reversal symmetric non-Hermitian topological insulators, we find that symmetry indicators characterize the associated Chern-Simons form, whose evaluation usually requires a computationally expensive choice of smooth gauge. In each case, we discuss the non-Hermitian surface states associated with nontrivial symmetry indicators.

  • Journal article
    Schindler F, 2020,

    Dirac equation perspective on higher-order topological insulators

    , Journal of Applied Physics, Vol: 128, Pages: 1-13, ISSN: 0021-8979

    In this Tutorial, we pedagogically review recent developments in the field of non-interacting fermionic phases of matter, focusing on the low-energy description of higher-order topological insulators in terms of the Dirac equation. Our aim is to give a mostly self-contained treatment. After introducing the Dirac approximation of topological crystalline band structures, we use it to derive the anomalous end and corner states of first- and higher-order topological insulators in one and two spatial dimensions. In particular, we recast the classical derivation of domain wall bound states of the Su–Schrieffer–Heeger (SSH) chain in terms of crystalline symmetry. The edge of a two-dimensional higher-order topological insulator can then be viewed as a single crystalline symmetry-protected SSH chain, whose domain wall bound states become the corner states. We never explicitly solve for the full symmetric boundary of the two-dimensional system but instead argue by adiabatic continuity. Our approach captures all salient features of higher-order topology while remaining analytically tractable.

  • Journal article
    Bösch C, Dubček T, Schindler F, Fichtner A, Serra-Garcia Met al., 2020,

    Discovery of topological metamaterials by symmetry relaxation and smooth topological indicators

    , Physical Review B, Vol: 102, Pages: 1-6, ISSN: 2469-9950

    Physical properties of a topological origin are known to be robust against small perturbations. This robustness is both a source of theoretical interest and a driver for technological applications, but presents a challenge when looking for new topological systems: Small perturbations cannot be used to identify the global direction of change in the topological indices. Here, we overcome this limitation by breaking the symmetries protecting the topology. The introduction of symmetry-breaking terms causes the topological indices to become smooth, nonquantized functions of the system parameters, which are amenable to efficient design algorithms based on gradient methods. We demonstrate this capability by designing discrete and continuous phononic systems realizing conventional and higher-order topological insulators.

  • Journal article
    Schindler F, Jermyn AS, 2020,

    Algorithms for tensor network contraction ordering

    , Machine Learning: Science and Technology, Vol: 1, Pages: 1-13, ISSN: 2632-2153

    Contracting tensor networks is often computationally demanding. Well-designed contraction sequences can dramatically reduce the contraction cost. We explore the performance of simulated annealing and genetic algorithms, two common discrete optimization techniques, to this ordering problem. We benchmark their performance as well as that of the commonly-used greedy search on physically relevant tensor networks. Where computationally feasible, we also compare them with the optimal contraction sequence obtained by an exhaustive search. Furthermore, we present a systematic comparison with state-of-the-art tree decomposition and graph partitioning algorithms in the context of random regular graph tensor networks. We find that the algorithms we consider consistently outperform a greedy search given equal computational resources, with an advantage that scales with tensor network size. We compare the obtained contraction sequences and identify signs of highly non-local optimization, with the more sophisticated algorithms sacrificing run-time early in the contraction for better overall performance.Contracting tensor networks is often computationally demanding. Well-designed contraction sequences can dramatically reduce the contraction cost. We explore the performance of simulated annealing and genetic algorithms, two common discrete optimization techniques, to this ordering problem. We benchmark their performance as well as that of the commonly-used greedy search on physically relevant tensor networks. Where computationally feasible, we also compare them with the optimal contraction sequence obtained by an exhaustive search. Furthermore, we present a systematic comparison with state-of-the-art tree decomposition and graph partitioning algorithms in the context of random regular graph tensor networks. We find that the algorithms we consider consistently outperform a greedy search given equal computational resources, with an advantage that scales with tensor network size. We co

  • Journal article
    Schindler F, Bradlyn B, Fischer MH, Neupert Tet al., 2020,

    Pairing obstructions in topological superconductors

    , Physical Review Letters, Vol: 124, ISSN: 0031-9007

    The modern understanding of topological insulators is based on Wannier obstructions in position space. Motivated by this insight, we study topological superconductors from a position-space perspective. For a one-dimensional superconductor, we show that the wave function of an individual Cooper pair decays exponentially with separation in the trivial phase and polynomially in the topological phase. For the position-space Majorana representation, we show that the topological phase is characterized by a nonzero Majorana polarization, which captures an irremovable and quantized separation of Majorana Wannier centers from the atomic positions. We apply our results to diagnose second-order topological superconducting phases in two dimensions. Our work establishes a vantage point for the generalization of topological quantum chemistry to superconductivity.

  • Journal article
    Hofmann T, Helbig T, Schindler F, Salgo N, Brzezińska M, Greiter M, Kiessling T, Wolf D, Vollhardt A, Kabaši A, Lee CH, Bilušić A, Thomale R, Neupert Tet al., 2020,

    Reciprocal skin effect and its realization in a topolectrical circuit

    , Physical Review Research, Vol: 2, Pages: 1-11, ISSN: 2643-1564

    A system is non-Hermitian when it exchanges energy with its environment and nonreciprocal when it behaves differently upon the interchange of input and response. Within the field of metamaterial research on synthetic topological matter, the skin effect describes the conspiracy of non-Hermiticity and nonreciprocity to yield extensive anomalous localization of all eigenmodes in a (quasi) one-dimensional geometry. Here, we introduce the reciprocal skin effect, which occurs in non-Hermitian but reciprocal systems in two or more dimensions: Eigenmodes with opposite longitudinal momentum exhibit opposite transverse anomalous localization. We experimentally demonstrate the reciprocal skin effect in a passive RLC circuit, suggesting convenient alternative implementations in optical, acoustic, mechanical, and related platforms. Skin mode localization brings forth potential applications in directional and polarization detectors for electromagnetic waves.

  • Journal article
    Schindler F, Brzezińska M, Benalcazar WA, Iraola M, Bouhon A, Tsirkin SS, Vergniory MG, Neupert Tet al., 2019,

    Fractional corner charges in spin-orbit coupled crystals

    , Physical Review Research, Vol: 1, Pages: 1-20, ISSN: 2643-1564

    We study two-dimensional spinful insulating phases of matter that are protected by time-reversal and crystalline symmetries. To characterize these phases, we employ the concept of corner charge fractionalization: corners can carry charges that are fractions of even multiples of the electric charge. The charges are quantized and topologically stable as long as all symmetries are preserved. We classify the different corner charge configurations for all point groups, and match them with the corresponding bulk topology. For this we employ symmetry indicators and (nested) Wilson loop invariants. We provide formulas that allow for a convenient calculation of the corner charge from Bloch wave functions and illustrate our results using the example of arsenic and antimony monolayers. Depending on the degree of structural buckling, these materials can exhibit two distinct obstructed atomic limits. We present density functional theory calculations for open flakes to support our findings.

  • Journal article
    Wan B, Schindler F, Wang K, Wu K, Wan X, Neupert T, Lu H-Zet al., 2018,

    Theory for the negative longitudinal magnetoresistance in the quantum limit of Kramers Weyl semimetals.

    , Journal of Physics: Condensed Matter, Vol: 30, Pages: 1-9, ISSN: 0953-8984

    Negative magnetoresistance is rare in non-magnetic materials. Recently, negative magnetoresistance has been observed in the quantum limit of β-Ag2Se, where only one band of Landau levels is occupied in a strong magnetic field parallel to the applied current. β-Ag2Se is a material that hosts a Kramers Weyl cone with band degeneracy near the Fermi energy. Kramers Weyl cones exist at time-reversal invariant momenta in all symmorphic chiral crystals, and at a subset of these momenta, including the Γ point, in non-symmorphic chiral crystals. Here, we present a theory for the negative magnetoresistance in the quantum limit of Kramers Weyl semimetals. We show that, although there is a band touching similar to those in Weyl semimetals, negative magnetoresistance can exist without a chiral anomaly. We find that it requires screened Coulomb scattering potentials between electrons and impurities, which is naturally the case in β-Ag2Se.

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