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Journal articleHwang Y, Gupta V, Schindler F, et al., 2026,
Stable real-space invariants and topology beyond symmetry indicators
, Nature Communications, Vol: 17, ISSN: 2041-1723We introduce stable real-space invariants (SRSIs), topological invariants defined from adiabatic deformations between Wannier states, generalizing previously discovered local and composite real-space invariants. SRSIs are Z-and Zn-valued (n = 2, 4) linear combinations of Wannier state multiplicities characterizing the stable equivalence of atomic insulators. We enumerate all SRSIs in nonmagnetic space groups with and without spin-orbit coupling. ZSRSIs are in one-to-one correspondence with momentum-space symmetry data and thus determine symmetry indicators of topology (SIs). ZnSRSIs capture real-space information beyond momentum-space symmetry data and SIs. Applying SRSIs to split elementary band representations (EBRs) whose symmetry data decomposes into positive sums of other EBR symmetry data, we diagnose the topology of all 211 cases across 51 space groups except for 8 exceptions in 5 space groups. Our results solidify Topological Quantum Chemistry beyond SIs and momentum-space symmetry data. Finally, we use SRSIs to diagnose an obstructed atomic insulator in a realistic material.
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Journal articleSheppard T, Camacho C, Weidemann S, et al., 2026,
Topological localization in time from parity-time symmetry
, Newton, ISSN: 2950-6360Time has entered the domain of topological phases in the field of non-Hermitian physics. Previous studies have relied on periodic modulation in time to make an intuitive connection to established spatial topological invariants, albeit with energy and momentum exchanged. This connection has revealed the potential for topological interface states along the time axis, analogous to those in spatial models. In this work, we uncover a theoretical framework describing such topological interface states along the time axis, with no underlying connection to spatial models nor need for periodic driving. This new framework uncovers that this phenomenon—the robust localization of waves at an interface—appears in every system that has parity-time symmetry and two coupled modes or bands, regardless of its spatial dimensionality. The topological nature of this localization is understood by the identification of certain topological phases that are specific to parity-time-symmetric models of two coupled modes. Our theoretical framework can be applied to all existing experimental observations, notably including photonic time crystals, and serves as a foundation for future experiments in areas in which the topological localization of waves in time has yet to be studied.
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Journal articleZhou Y, Chaduteau A, Schindler F, 2026,
Topological dislocation response in elementary semiconductors
, Physical Review B, Vol: 114, ISSN: 2469-9950We 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.
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Journal articleGunawardana TM, Schindler F, Turner AM, et al., 2026,
Microscopic theory of Chern polarization via crystalline defect charge
, Journal of Physics: Condensed Matter, Vol: 38, ISSN: 0953-8984The 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.
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Journal articleQian Y, Kim Y, Kim DH, et al., 2026,
Correlated singular flat bands on the surface pentagonal lattice of ferromagnetic CoS2.
, Nat Commun, Vol: 17Flat bands have been proposed in various geometrically frustrated lattices, yet their experimental realization has been limited only to the two-dimensional (2D) kagome and three-dimensional (3D) pyrochlore lattices. While synthesizing bulk crystals with desired lattice geometry is generally challenging, the surfaces of 3D bulk materials sometimes naturally host 2D lattice structures hardly achievable in 2D bulk crystals. Thus, the electrons localized on such 2D surfaces can be a new platform for exotic flat bands residing on the unexplored lattice geometry. Here, we report the first experimental observation of singular flat bands with band crossings in the 2D pentagonal lattice on the surface of the 3D ferromagnetic topological semimetal CoS2. We demonstrate that the coupling between localized surface flat bands and extended bulk topological bands gives rise to the surface non-Fermi liquid behavior characterized by the quasiparticle scattering rate with linear temperature dependence and pronounced quasiparticle broadening. This study of CoS2 highlights metallic pyrite materials as an ideal platform for exploring the exotic properties of spin-polarized singular flat bands localized on the surface pentagonal lattice, which provides a fresh perspective for investigating correlated electron phenomena in pentagon-based materials.
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Journal articleSchindler F, 2026,
A pedestrian’s guide to the topological phases of free fermions
, SciPost Physics Lecture Notes, Vol: 2026, ISSN: 2590-1990These 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 manyof the models we work with are non-interacting. We start out with symmetry protectedtopological (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 super conductors, and explain how their classification changesin 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 articleTerakawa S, Liu B, Schindler F, et al., 2026,
Proximity induced magnetic anisotropy and trefoil fermiology in monolayer FeCl₂/Bi(111)
, Advanced Materials, Vol: 38, ISSN: 0935-9648Interfaces 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.
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Journal articleDavenport H, Schindler F, Knolle J, 2026,
Berry curvature of low-energy excitons in rhombohedral graphene
, Physical Review B, Vol: 113, ISSN: 2469-9950We 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.
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Journal articleHwang Y, Zhu P, Hughes TL, 2026,
Spin-momentum locking and topological vector charge response with conserved spin
, Physical Review B, Vol: 113, ISSN: 2469-9950 -
Journal articleDavenport H, Knolle J, Schindler F, 2026,
Exciton berryology
, Physical Review B, Vol: 113, ISSN: 2469-9950In 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.
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Journal articleChaduteau A, Lee DKK, Schindler F, 2026,
Lindbladian versus postselected non-hermitian topology
, Physical Review Letters, Vol: 136, ISSN: 0031-9007The 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.
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Journal articleBanerjee N, Bell C, Ciccarelli C, et al., 2025,
Materials for quantum technologies: a roadmap for spin and topology
, Applied Physics Reviews, Vol: 12, ISSN: 1931-9401In 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 articleVelury S, Hwang Y, Hughes TL, 2025,
Global and local topological crystalline markers for rotation-symmetric insulators
, Physical Review B, Vol: 112, ISSN: 2469-9950 -
Journal articleMorales-Pérez A, Devescovi C, Hwang Y, et al., 2025,
Transversality-enforced tight-binding models for three-dimensional photonic crystals aided by topological quantum chemistry
, Physical Review B, Vol: 111, ISSN: 2469-9950 -
Journal articleYang K, Liu Y, Schindler F, et al., 2025,
Engineering miniband topology via band folding in moiré superlattice materials
, Physical Review B, Vol: 111, ISSN: 2469-9950The 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.
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Journal articleChaduteau A, Raess N, Davenport H, et al., 2025,
Momentum-space modulated symmetries in the Luttinger liquid
, Physical Review B, Vol: 111, ISSN: 2469-9950The 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.
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Journal articleSchindler F, Bulchandani VB, Benalcazar WA, 2025,
Nonlinear breathers with crystalline symmetries
, Physical Review B: Condensed Matter and Materials Physics, Vol: 111, ISSN: 1098-0121Nonlinear 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.
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Journal articleDavenport H, Knolle J, Schindler F, 2024,
Interaction-induced crystalline topology of excitons
, Physical Review Letters, Vol: 133, ISSN: 0031-9007We 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.
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Journal articleDevescovi C, Morales-Pérez A, Hwang Y, et al., 2024,
Author Correction: Axion topology in photonic crystal domain walls.
, Nat Commun, Vol: 15 -
Journal articleDevescovi C, Morales-Pérez A, Hwang Y, et al., 2024,
Axion topology in photonic crystal domain walls.
, Nat Commun, Vol: 15Axion insulators are 3D magnetic topological insulators supporting hinge states and quantized magnetoelectric effects, recently proposed for detecting dark-matter axionic particles via their axionic excitations. Beyond theoretical interest, obtaining a photonic counterpart of axion insulators offers potential for advancing magnetically-tunable photonic devices and axion haloscopes based on axion-photon conversion. This work proposes an axionic 3D phase within a photonic setup. By building inversion-symmetric domain-walls in gyrotropic photonic crystals, we bind chiral modes on inversion-related hinges, ultimately leading to the realization of an axionic channel of light. These states propagate embedded in a 3D structure, thus protected from radiation in the continuum. Employing a small external gyromagnetic bias, we transition across different axionic mode configurations, enabling effective topological switching of chiral photonic fibers. While demonstrating the possibility of realizing axion photonic crystals within state-of-the-art gyrotropic setups, we propose a general scheme for rendering axion topology at domain walls of Weyl semimetals.
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