Publications

Journal article

Zhou Y, Chaduteau A, Schindler F, 2026,

Topological dislocation response in elementary semiconductors

, Physical Review B, Vol: 114, ISSN: 2469-9950

<jats:p>We study elementary semiconductors and insulators that are symmetric under spatial inversion: silicon, diamond, germanium, and black phosphorene. These materials are ideal candidates for realizing obstructed atomic insulators, which differ from trivial atomic insulators by a quantized spatial shift of their electronic Wannier centers with respect to the atomic lattice. We use symmetry indicator invariants that allow the prediction of nontrivial responses to crystal dislocations with an integer Burgers vector in these materials, and generally, in all inversion-symmetric obstructed atomic insulators. Unlike weak topological insulators where the response only depends on the Burgers vector, the dislocation response of three-dimensional (3D) inversion-symmetric obstructed atomic insulators can also be affected by the line vector of the dislocation. In such insulators, we find that edge dislocations generically exhibit a nontrivial response, while in all 3D obstructed atomic insulators, screw dislocations always display a trivial response. With the aid of numerical simulations of realistic tight-binding models, we confirm the presence of midgap polarization bands localized along dislocations in silicon, diamond, and germanium.</jats:p>

Journal article

Sheppard T, Camacho C, Weidemann S, Szameit A, Feis J, Schindler F, Price HMet al., 2026,

Topological localization in time from parity-time symmetry

, Newton, Pages: 100622-100622, ISSN: 2950-6360

Journal article

Gunawardana TM, Schindler F, Turner AM, Barnett Ret al., 2026,

Microscopic theory of Chern polarization via crystalline defect charge

, Journal of Physics: Condensed Matter, Vol: 38, ISSN: 0953-8984

The modern theory of polarization does not apply in its original form to systems with non-trivial band topology. Chern insulators are one such example. Defining polarization for them is complicated because they are insulating in the bulk but exhibit metallic edge states. Wannier functions formed a key ingredient of the original modern theory of polarization, but it has been considered that these cannot be applied to Chern insulators since they are no longer exponentially localized and the Wannier center, obtained from the Zak phase, is no longer gauge invariant. In this article, we provide an unambiguous definition of absolute polarization for a Chern insulator in terms of the Zak phase. We obtain our expression by studying the non-quantized fractional charge bound to lattice dislocations. Our expression can be computed directly from bulk quantities and makes no assumption on the edge state filling. It is fully consistent with previous results on the quantized charge bound to dislocations in the presence of crystalline symmetry. At the same time, our result is more general since it also applies to Chern insulators which do not have crystalline symmetries other than translations.

Journal article

Terakawa S, Liu B, Schindler F, Neupert T, Ji J, Domaine G, McFarlane EC, Geng D, Yang J, Choueikani F, Ohresser P, Valvidares M, Gargiani P, Polley C, Carbone G, Leandersson M, Parkin SSP, BedoyaPinto A, Schröter NBMet al., 2026,

Proximity induced magnetic anisotropy and trefoil fermiology in monolayer FeCl₂/Bi(111)

, Advanced Materials, Vol: 38, ISSN: 0935-9648

Interfaces between magnetic and non-magnetic materials play a crucial role in various magnetic heterostructures. The emergence of 2D van der Waals (vdW) magnets has introduced new opportunities for exploring proximity effects in vdW heterostructures. While the influence of magnetic layers on nearby non-magnetic materials has been widely studied, it remains unclear whether non-magnetic substrates can similarly modulate the intrinsic magnetic properties of 2D magnets, particularly their magnetic anisotropy. In this work, by analyzing X-ray magnetic circular dichroism spectra of an epitaxially grown FeCl2 monolayer on a Bi(111) surface, a reorientation of magnetic anisotropy is observed – from its natural out-of-plane to a predominantly in-plane alignment. This effect vanishes in bilayer FeCl2/Bi(111), where the magnetic anisotropy reverts to its intrinsic out-of-plane orientation, consistent with the layered antiferromagnetic order of bulk FeCl2. Angle-resolved photoelectron spectroscopy reveals the presence of metallic interface states derived from the Bi surface states, accompanied by charge transfer and emergence of a moiré potential that gives rise to a distinctive trefoil-shaped Fermi surface. These results demonstrate that non-magnetic substrates can exert strong proximity influence on the magnetic and electronic behavior of 2D vdW magnets, offering new strategies for engineering magnetic anisotropy and electronic structure in spintronic heterostructures.

Journal article

Davenport H, Schindler F, Knolle J, 2026,

Berry curvature of low-energy excitons in rhombohedral graphene

, Physical Review B, Vol: 113, ISSN: 2469-9950

We investigate low-energy excitons in rhombohedral pentalayer graphene encapsulated by hexagonal boronnitride (hBN/R5G/hBN), focusing on the regime at the experimental twist angle θ = 0.77◦ and with an appliedelectric field. We introduce a new low-energy two-band model of rhombohedral graphene that captures the band structure more accurately than previous models while keeping the number of parameters low. Using this model, we show that the centres of the exciton Wannier functions are displaced from the moiré unit cell origin by a quantized amount—they are instead localized at C3-symmetric points on the boundary. We also find that the exciton shift is electrically tunable: by varying the electric field strength, the exciton Wannier center can be exchanged between inequivalent corners of the moiré unit cell. Our results suggest the possibility of detecting excitonic corner or edge modes, as well as novel excitonic crystal defect responses in hBN/R5G/hBN. Lastly, we find that the excitons in hBN/R5G/hBN inherit excitonic Berry curvature from the underlying electronic bands, enriching their semiclassical transport properties. Our results position rhombohedral graphene as a compelling tunable platform for probing exciton topology in moiré materials.

Journal article

Davenport H, Knolle J, Schindler F, 2026,

Exciton berryology

, Physical Review B, Vol: 113, ISSN: 2469-9950

In translationally invariant semiconductors that host exciton bound states, one can define an infinite number of possible exciton Berry connections. These correspond to the different ways in which a many-body exciton state, at fixed total momentum, can be decomposed into free electron and hole Bloch states that are entangled by an exciton envelope wave function. Inspired by the modern theory of polarization, we define an exciton projected position operator whose eigenvalues single out two unique choices of exciton Berry phase and associated Berry connection—one for electrons, and one for holes. We clarify the physical meaning of these exciton Berry phases and provide a discrete Wilson loop formulation that allows for their numerical calculation without a smooth gauge. As a corollary, we obtain a gauge-invariant expression for the exciton polarization at a given total momentum, i.e., the mean separation of the electron and hole within the exciton wave function. In the presence of crystalline inversion symmetry, the electron and hole exciton Berry phases are quantized to the same value and we derive how this value can be expressed in terms of inversion eigenvalues of the many-body exciton state. We then consider C2T symmetry, for which no symmetry eigenvalues are available as it is antiunitary, andconfirm that the exciton Berry phase remains quantized and still diagnoses topologically distinct exciton bands.The notion of shift excitons, whose exciton Wannier states are displaced from those of the noninteracting bands by a quantized amount, can therefore be generalized beyond symmetry indicators.

Journal article

Chaduteau A, Lee DKK, Schindler F, 2026,

Lindbladian versus postselected non-hermitian topology

, Physical Review Letters, Vol: 136, ISSN: 0031-9007

The recent topological classification of non-Hermitian “Hamiltonians” is usually interpreted in terms of pure quantum states that decay or grow with time. However, many-body systems with loss and gain are typically better described by mixed-state open quantum dynamics, which only correspond to pure-state non-Hermitian dynamics upon a postselection of measurement outcomes. Since postselection becomes exponentially costly with particle number, we here investigate to what extent the most important example of non-Hermitian topology can survive without it: the non-Hermitian skin effect and its relationship to a bulk winding number in one spatial dimension. After defining the winding number of the Lindbladian superoperator for a quadratic fermion system, we systematically relate it to the winding number of the associated postselected non-Hermitian Hamiltonian. We prove that the two winding numbers are equal (opposite) in the absence of gain (loss), and provide a physical explanation for this relationship. When both loss and gain are present, the Lindbladian winding number typically remains quantized and non-zero, though it can change sign at a phase transition separating the loss and gain-dominated regimes. This transition, which leads to a reversal of the Lindbladian skin effect localization, is rendered invisible by postselection. We also identify a case where removing postselection induces a skin effect from otherwise topologically trivial non-Hermitian dynamics.

Journal article

Schindler F, 2026,

A pedestrian’s guide to the topological phases of free fermions

, Scipost Physics Lecture Notes, Vol: 122

These lecture notes explain the classification of some simple fermionic topological phases of matter in a pedestrian manner, with an aim to be maximally pedagogical = doing things in excruciating detail. We focus on a many-body perspective, even if many of the models we work with are non-interacting. We start out with symmetry protected topological (SPT) phases of free fermions that are protected by U(1) symmetry = topological insulators. We then look at fermion topological phases that don’t even need a symmetry = topological superconductors, and explain how their classification changes in presence of spinless time-reversal symmetry. We close by perturbatively checking which of the 1D topological phases we had found are stable to interactions.

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Journal article

Banerjee N, Bell C, Ciccarelli C, Hesjedal T, Johnson F, Kurebayashi H, Moore TA, Moutafis C, Stern HL, Vera-Marun IJ, Wade J, Barton C, Connolly MR, Curson NJ, Fallon K, Fisher AJ, Gangloff DA, Griggs W, Linfield E, Marrows CH, Rossi A, Schindler F, Smith J, Thomson T, Kazakova Oet al., 2025,

Materials for quantum technologies: a roadmap for spin and topology

, Applied Physics Reviews, Vol: 12, ISSN: 1931-9401

In this perspective article, we explore some of the promising spin and topology material platforms (e.g., spins in semiconductors and superconductors, skyrmionic, topological, and two-dimensional materials) being developed for such quantum components as qubits, superconducting memories, sensing, and metrological standards, and discuss their figures of merit. Spin- and topology-related quantum phenomena have several advantages, including high coherence time, topological protection and stability, low error rate, relative ease of engineering and control, and simple initiation and readout. However, the relevant technologies are at different stages of research and development, and here, we discuss their state-of-the-art, potential applications, challenges, and solutions.

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Journal article

Yang K, Liu Y, Schindler F, Liu C-Xet al., 2025,

Engineering miniband topology via band folding in moiré superlattice materials

, Physical Review B, Vol: 111, ISSN: 2469-9950

The emergence of topologically nontrivial flat bands in moiré materials provides an opportunity to explore the interplay between topological physics and correlation effects, leading to the recent experimental realization of interacting topological phases, e.g., fractional Chern insulators. In this work, we propose a general mechanism of band inversion induced by the moiré Brillouin zone folding of atomic bands for engineering topological minibands in moiré materials. We illustrate this mechanism via two classes of model Hamiltonians, namely the Rashba model and the Bernevig-Hughes-Zhang (BHZ) model, under the moiré superlattice potentials. We find moiré minibands with the nontrivial band topology, including the ℤ2 number, mirror Chern number, and fragile topology, and the topological phase diagram is constructed for these moiré models. A general theory based on band representations in the morié Brillouin zone is also developed for a generalization of this mechanism to all 2D plane groups. Possible experimental realizations of our model Hamiltonian are discussed.

Journal article

Chaduteau A, Raess N, Davenport H, Schindler Fet al., 2025,

Momentum-space modulated symmetries in the Luttinger liquid

, Physical Review B, Vol: 111, ISSN: 2469-9950

The chiral Luttinger liquid develops quantum chaos as soon as a—however slight—nonlinear dispersion is introduced for the microscopic electronic degrees of freedom. For this nonlinear version of the model, we identify an infinite family of translation-invariant interaction potentials with corresponding modulated symmetries. These symmetries are highly unconventional: they are modulated in momentum space (and do not seem to have an easy physical interpretation). We develop a systematic understanding of these symmetries and study the resulting blocks in the Hamiltonian. In particular, this approach allows us to predict the analytic Hamiltonian block sizes and derive asymptotic scaling laws in the limit of large total momentum. These blocks are reminiscent of Hilbert space fragmentation in that, even though they are labeled by a symmetry, this symmetry is highly nonlocal and does not have a simple interpretation. We corroborate this result by studying entanglement entropy and level statistics.

Journal article

Schindler F, Bulchandani VB, Benalcazar WA, 2025,

Nonlinear breathers with crystalline symmetries

, Physical Review B: Condensed Matter and Materials Physics, Vol: 111, ISSN: 1098-0121

Nonlinear lattice models can support “discrete breather” excitations that stay localized in space for all time. By contrast, the localized Wannier states of linear lattice models are dynamically unstable. Nevertheless, symmetric and exponentially localized Wannier states are a central tool in the classification of band structures with crystalline symmetries. Moreover, the quantized transport observed in nonlinear Thouless pumps relies on the fact that—at least in a specific model—discrete breathers recover Wannier states in the limit of vanishing nonlinearity. Motivated by these observations, we investigate the correspondence between nonlinear breathers and exponentially localized Wannier states for a family of discrete nonlinear Schrödinger equations with crystalline symmetries. We develop a formalism to analytically predict the breathers' spectrum, center of mass and symmetry data, and apply this to nonlinear generalizations of the Su-Schrieffer-Heeger chain and the breathing kagome lattice.

Journal article

Davenport H, Knolle J, Schindler F, 2024,

Interaction-induced crystalline topology of excitons

, Physical Review Letters, Vol: 133, ISSN: 0031-9007

We apply the topological theory of symmetry indicators to interaction-induced exciton band structures in centrosymmetric semiconductors. Crucially, we distinguish between the topological invariants inherited from the underlying electron and hole bands and those that are intrinsic to the exciton wave function itself. Focusing on the latter, we show that there exists a class of exciton bands for which the maximally localized exciton Wannier states are shifted with respect to the electronic Wannier states by a quantized amount; we call these excitons shift excitons. Our analysis explains how the exciton spectrum can be topologically nontrivial and sustain exciton edge states in open boundary conditions even when the underlying noninteracting bands have a trivial atomic limit. We demonstrate the presence of shift excitons as the lowest energy neutral excitations of the Su-Schrieffer-Heeger model in its trivial phase when supplemented by local two-body interactions.

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.

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