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  • 1. Atteia, J.
    et al.
    Bardarson, Jens H.
    KTH, School of Engineering Sciences (SCI), Physics.
    Cayssol, J.
    Ballistic transport through irradiated graphene2017In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 96, no 24, article id 245404Article in journal (Refereed)
    Abstract [en]

    The coherent charge transport through an illuminated graphene ribbon is studied as a function of electronic doping and characteristics of the electromagnetic driving, for monochromatic circularly polarized light. We focus on the DC current carried by 2D bulk carriers which is dominant (over edge transport) for short and wide enough samples. We investigate how the ballistic conductance suppression, due to photon resonances between the valence and conduction bands, evolves when the experimentally tunable parameters are varied. The residual conductance can be associated with evanescent states and related to dynamical gaps in the Floquet quasienergy spectrum.

  • 2. Behrends, Jan
    et al.
    Bardarson, Jens H.
    KTH, School of Engineering Sciences (SCI), Physics, Condensed Matter Theory. Max Planck Institute, Germany.
    Strongly angle-dependent magnetoresistance in Weyl semimetals with long-range disorder2017In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 96, no 6, article id 060201Article in journal (Refereed)
    Abstract [en]

    The chiral anomaly in Weyl semimetals states that the left- and right-handed Weyl fermions, constituting the low energy description, are not individually conserved, resulting, for example, in a negative magnetoresistance in such materials. Recent experiments see strong indications of such an anomalous resistance response; however, with a response that at strong fields is more sharply peaked for parallel magnetic and electric fields than expected from simple theoretical considerations. Here, we uncover a mechanism, arising from the interplay between the angle-dependent Landau-level structure and long-range scalar disorder, that has the same phenomenology. In particular, we analytically show, and numerically confirm, that the internode scattering time decreases exponentially with the angle between the magnetic field and the Weyl node separation in the large field limit, while it is insensitive to this angle at weak magnetic fields. Since, in the simplest approximation, the internode scattering time is proportional to the anomaly-related conductivity, this feature may be related to the experimental observations of a sharply peaked magnetoresistance.

  • 3. Behrends, Jan
    et al.
    Rhim, Jun-Won
    Liu, Shang
    Grushin, Adolfo G.
    Bardarson, Jens H.
    KTH, School of Engineering Sciences (SCI), Physics, Condensed Matter Theory. Max-Planck-Institut für Physik Komplexer Systeme, Germany.
    Nodal-line semimetals from Weyl superlattices2017In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 96, no 24, article id 245101Article in journal (Refereed)
    Abstract [en]

    The existence and topological classification of lower-dimensional Fermi surfaces is often tied to the crystal symmetries of the underlying lattice systems. Artificially engineered lattices, such as heterostructures and other superlattices, provide promising avenues to realize desired crystal symmetries that protect lower-dimensional Fermi surfaces, such as nodal lines. In this work, we investigate a Weyl semimetal subjected to spatially periodic onsite potential, giving rise to several phases, including a nodal-line semimetal phase. In contrast to proposals that purely focus on lattice symmetries, the emergence of the nodal line in this setup does not require small spin-orbit coupling, but rather relies on its presence. We show that the stability of the nodal line is understood from reflection symmetry and a combination of a fractional lattice translation and charge-conjugation symmetry. Depending on the choice of parameters, this model exhibits drumhead surface states that are exponentially localized at the surface, or weakly localized surface states that decay into the bulk at all energies.

  • 4.
    Behrends, Jan
    et al.
    Max Planck Inst Phys Komplexer Systeme, D-01187 Dresden, Germany..
    Roy, Sthitadhi
    Max Planck Inst Phys Komplexer Systeme, D-01187 Dresden, Germany.;Univ Oxford, Phys & Theoret Chem, South Parks Rd, Oxford OX1 3QZ, England.;Univ Oxford, Rudolf Peierls Ctr Theoret Phys, Clarendon Lab, Parks Rd, Oxford OX1 3PU, England..
    Kolodrubetz, Michael H.
    Univ Calif Berkeley, Dept Phys, Berkeley, CA 94720 USA.;Lawrence Berkeley Natl Lab, Mat Sci Div, Berkeley, CA 94720 USA.;Univ Texas Dallas, Dept Phys, Richardson, TX 75080 USA..
    Bardarson, Jens H.
    KTH, School of Engineering Sciences (SCI), Physics, Condensed Matter Theory. Max Planck Inst Phys Komplexer Systeme, D-01187 Dresden, Germany.
    Grushin, Adolfo G.
    Univ Calif Berkeley, Dept Phys, Berkeley, CA 94720 USA.;CNRS, Inst Neel, F-38042 Grenoble, France.;Univ Grenoble Alpes, F-38042 Grenoble, France..
    Landau levels, Bardeen polynomials, and Fermi arcs in Weyl semimetals: Lattice-based approach to the chiral anomaly2019In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 99, no 14, article id 140201Article in journal (Refereed)
    Abstract [en]

    Condensed matter systems realizing Weyl fermions exhibit striking phenomenology derived from their topologically protected surface states as well as chiral anomalies induced by electromagnetic fields. More recently, inhomogeneous strain or magnetization were predicted to result in chiral electric E-5 and magnetic B-5 fields, which modify and enrich the chiral anomaly with additional terms. In this Rapid Communication, we develop a lattice-based approach to describe the chiral anomaly, which involves Landau and pseudo-Landau levels and treats all anomalous terms on equal footing, while naturally incorporating Fermi arcs. We exemplify its potential by physically interpreting the largely overlooked role of Fermi arcs in the covariant (Fermi level) contribution to the anomaly and revisiting the factor of 1/3 difference between the covariant and consistent (complete band) contributions to the E-5 . B-5 term in the anomaly. Our framework provides a versatile tool for the analysis of anomalies in realistic lattice models as well as a source of simple physical intuition for understanding strained and magnetized inhomogeneous Weyl semimetals.

  • 5. Bera, Soumya
    et al.
    Martynec, Thomas
    Schomerus, Henning
    Heidrich-Meisner, Fabian
    Hjörleifur Bardarson, Jens
    KTH, School of Engineering Sciences (SCI), Physics, Condensed Matter Theory. Max-Planck-Institut für Physik komplexer Systeme, Germany.
    One-particle density matrix characterization of many-body localization2017In: Annalen der Physik, ISSN 0003-3804, E-ISSN 1521-3889, Vol. 529, no 7, article id 1600356Article in journal (Refereed)
    Abstract [en]

    We study interacting fermions in one dimension subject to random, uncorrelated onsite disorder, a paradigmatic model of many-body localization (MBL). This model realizes an interaction-driven quantum phase transition between an ergodic and a many-body localized phase, with the transition occurring in the many-body eigenstates. We propose a single-particle framework to characterize these phases by the eigenstates (the natural orbitals) and the eigenvalues (the occupation spectrum) of the one-particle density matrix (OPDM) in individual many-body eigenstates. As a main result, we find that the natural orbitals are localized in the MBL phase, but delocalized in the ergodic phase. This qualitative change in these single-particle states is a many-body effect, since without interactions the single-particle energy eigenstates are all localized. The occupation spectrum in the ergodic phase is thermal in agreement with the eigenstate thermalization hypothesis, while in the MBL phase the occupations preserve a discontinuity at an emergent Fermi edge. This suggests that the MBL eigenstates are weakly dressed Slater determinants, with the eigenstates of the underlying Anderson problem as reference states. We discuss the statistical properties of the natural orbitals and of the occupation spectrum in the two phases and as the transition is approached. Our results are consistent with the existing picture of emergent integrability and localized integrals of motion, or quasiparticles, in the MBL phase. We emphasize the close analogy of the MBL phase to a zero-temperature Fermi liquid: in the studied model, the MBL phase is adiabatically connected to the Anderson insulator and the occupation-spectrum discontinuity directly indicates the presence of quasiparticles localized in real space. Finally, we show that the same picture emerges for interacting fermions in the presence of an experimentally-relevant bichromatic lattice and thereby demonstrate that our findings are not limited to a specific model.

  • 6. Dufouleur, J.
    et al.
    Xypakis, E.
    Buechner, B.
    Giraud, R.
    Bardarson, Jens Hjörleifur
    KTH, School of Engineering Sciences (SCI), Physics, Condensed Matter Theory. Max Planck Institute for Physics - Max-Planck-Gesellschaft, Germany.
    Suppression of scattering in quantum confined 2D helical Dirac systems2018In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 97, no 7, article id 075401Article in journal (Refereed)
    Abstract [en]

    Transport properties of helical Dirac fermions in disordered quantum wires are investigated in the large energy limit. In the quasiballistic regime, the conductance and the Fano factor are sensitive to disorder only when the Fermi energy is close to an opening of a transverse mode. In the limit of a large number of transverse modes, transport properties are insensitive to the geometry of the nanowire or the nature and strength of the disorder but, instead, are dominated by the properties of the interface between the ohmic contact and the nanowire. In the case of a heavily doped Dirac metallic contact, the conductance is proportional to the energy with an average transmission T = pi/4 and a Fano factor of F similar or equal to 0.13. Those results can be generalized to a much broader class of contacts, the exact values of T and F depending on the model used for the contacts. The energy dependence of Aharonov-Bohm oscillations is determined, revealing a damped oscillatory behavior and phase shifts due to the one-dimensional subband quantization and which are not the signature of the nontrivial topology.

  • 7. Erlingsson, S. I.
    et al.
    Bardarson, Jens Hjörleifur
    KTH, School of Engineering Sciences (SCI), Physics.
    Manolescu, A.
    Thermoelectric current in topological insulator nanowires with impurities2018In: Beilstein Journal of Nanotechnology, ISSN 2190-4286, Vol. 9, no 1, p. 1156-1161Article in journal (Refereed)
    Abstract [en]

    In this paper we consider charge current generated by maintaining a temperature difference over a nanowire at zero voltage bias. For topological insulator nanowires in a perpendicular magnetic field the current can change sign as the temperature of one end is increased. Here we study how this thermoelectric current sign reversal depends on the magnetic field and how impurities affect the size of the thermoelectric current. We consider both scalar and magnetic impurities and show that their influence on the current are quite similar, although the magnetic impurities seem to be more effective in reducing the effect. For moderate impurity concentration the sign reversal persists.

  • 8. Erlingsson, Sigurdur I.
    et al.
    Manolescu, Andrei
    Nemnes, George Alexandru
    Bardarson, Jens H.
    KTH, School of Engineering Sciences (SCI), Physics, Condensed Matter Theory.
    Sanchez, David
    Reversal of Thermoelectric Current in Tubular Nanowires2017In: Physical Review Letters, ISSN 0031-9007, E-ISSN 1079-7114, Vol. 119, no 3, article id 036804Article in journal (Refereed)
    Abstract [en]

    We calculate the charge current generated by a temperature bias between the two ends of a tubular nanowire. We show that in the presence of a transversal magnetic field the current can change sign; i.e., electrons can either flow from the hot to the cold reservoir, or in the opposite direction, when the temperature bias increases. This behavior occurs when the magnetic field is sufficiently strong, such that Landau and snaking states are created, and the energy dispersion is nonmonotonic with respect to the longitudinal wave vector. The sign reversal can survive in the presence of impurities. We predict this result for core-shell nanowires, for uniform nanowires with surface states due to the Fermi level pinning, and for topological insulator nanowires.

  • 9.
    Ferreiros, Yago
    et al.
    KTH, School of Engineering Sciences (SCI), Physics.
    Kedem, Yaron
    Bergholtz, Emil J.
    Bardarson, Jensen. H.
    KTH, School of Engineering Sciences (SCI), Physics.
    Mixed Axial-Torsional Anomaly in Weyl Semimetals2019In: Physical Review Letters, ISSN 0031-9007, E-ISSN 1079-7114, Vol. 122, no 5, article id 056601Article in journal (Refereed)
    Abstract [en]

    We show that Weyl semimetals exhibit a mixed axial-torsional anomaly in the presence of axial torsion, a concept exclusive of these materials with no known natural fundamental interpretation in terms of the geometry of spacetime. This anomaly implies a nonconservation of the axial current - the difference in the current of left- and right-handed chiral fermions - when the torsion of the spacetime in which the Weyl fermions move couples with opposite sign to different chiralities. The anomaly is activated by driving transverse sound waves through a Weyl semimetal with a spatially varying tilted dispersion, which can be engineered by applying strain. This leads to a sizable alternating current in the presence of a magnetic field that provides a clear-cut experimental signature of our predictions.

  • 10.
    Ferreiros, Yago
    et al.
    KTH, School of Engineering Sciences (SCI), Physics, Condensed Matter Theory.
    Zyuzin, A. A.
    KTH, School of Engineering Sciences (SCI), Physics, Condensed Matter Theory.
    Bardarson, Jens H
    KTH, School of Engineering Sciences (SCI), Physics, Condensed Matter Theory.
    Anomalous Nernst and thermal Hall effects in tilted Weyl semimetals2017In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 96, no 11, article id 115202Article in journal (Refereed)
    Abstract [en]

    We study the anomalous Nernst and thermal Hall effects in a linearized low-energy model of a tilted Weyl semimetal, with two Weyl nodes separated in momentum space. For inversion symmetric tilt, we give analytic expressions in two opposite limits: For a small tilt, corresponding to a type-I Weyl semimetal, the Nernst conductivity is finite and independent of the Fermi level; for a large tilt, corresponding to a type-II Weyl semimetal, it acquires a contribution depending logarithmically on the Fermi energy. This result is in a sharp contrast to the nontilted case, where the Nernst response is known to be zero in the linear model. The thermal Hall conductivity similarly acquires Fermi surface contributions, which add to the Fermi level-independent, zero-tilt result, and is suppressed as one over the tilt parameter at half filling in the type-II phase. In the case of inversion-breaking tilt, with the tilting vector of equal modulus in the two Weyl cones, all Fermi surface contributions to both anomalous responses cancel out, resulting in zero Nernst conductivity. We discuss two possible experimental setups, representing open and closed thermoelectric circuits.

  • 11.
    Hauschild, Johannes
    et al.
    Tech Univ Munich, Dept Phys, T42,James Franck Str 1, D-85748 Garching, Germany..
    Leviatan, Eyal
    Weizmann Inst Sci, Dept Condensed Matter Phys, IL-7610001 Rehovot, Israel..
    Bardarson, Jens H.
    KTH, School of Engineering Sciences (SCI), Physics, Condensed Matter Theory.
    Altman, Ehud
    Univ Calif Berkeley, Dept Phys, Berkeley, CA 94720 USA..
    Zaletel, Michael P.
    Princeton Univ, Dept Phys, Princeton, NJ 08540 USA..
    Pollmann, Frank
    Tech Univ Munich, Dept Phys, T42,James Franck Str 1, D-85748 Garching, Germany..
    Finding purifications with minimal entanglement2018In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 98, no 23, article id 235163Article in journal (Refereed)
    Abstract [en]

    Purification is a tool that allows to represent mixed quantum states as pure states on enlarged Hilbert spaces. A purification of a given state is not unique and its entanglement strongly depends on the particular choice made. Moreover, in one-dimensional systems, the amount of entanglement is linked to how efficiently the purified state can be represented using matrix-product states (MPS). We introduce an MPS based method that allows to find the minimally entangled representation by iteratively minimizing the second Renyi entropy. First, we consider the thermofield double purification and show that its entanglement can be strongly reduced especially at low temperatures. Second, we show that a slowdown of the entanglement growth following a quench of an infinite temperature state is possible.

  • 12.
    Herviou, Loic
    et al.
    KTH, School of Engineering Sciences (SCI), Physics, Condensed Matter Theory.
    Bardarson, Jens H.
    KTH, School of Engineering Sciences (SCI), Physics, Condensed Matter Theory. KTH Royal Inst Technol, Dept Phys, S-10691 Stockholm, Sweden..
    Regnault, Nicolas
    Univ Paris Diderot, Sorbonne Paris Cite, Sorbonne Univ, ENS,Univ PSL,CNRS,Lab Phys Ecole Normale Super, Paris, France..
    Defining a bulk-edge correspondence for non-Hermitian Hamiltonians via singular-value decomposition2019In: Physical Review A: covering atomic, molecular, and optical physics and quantum information, ISSN 2469-9926, E-ISSN 2469-9934, Vol. 99, no 5, article id 052118Article in journal (Refereed)
    Abstract [en]

    We address the breakdown of the bulk-boundary correspondence observed in non-Hermitian systems, where open and periodic systems can have distinct phase diagrams. The correspondence can be completely restored by considering the Hamiltonian's singular-value decomposition instead of its eigendecomposition. This leads to a natural topological description in terms of a flattened singular decomposition. This description is equivalent to the usual approach for Hermitian systems and coincides with a recent proposal for the classification of non-Hermitian systems. We generalize the notion of the entanglement spectrum to non-Hermitian systems, and show that the edge physics is indeed completely captured by the periodic bulk Hamiltonian. We exemplify our approach by considering the chiral non-Hermitian Su-Schrieffer-Heger and Chern insulator models. Our work advocates a different perspective on topological non-Hermitian Hamiltonians, paving the way to a better understanding of their entanglement structure.

  • 13.
    Herviou, Loic
    et al.
    KTH, School of Engineering Sciences (SCI), Physics, Condensed Matter Theory.
    Bera, Soumya
    Bardarson, Jens H.
    KTH, School of Engineering Sciences (SCI), Physics, Condensed Matter Theory.
    Multiscale entanglement clusters at the many-body localization phase transition2019In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 99, no 13, article id 134205Article in journal (Refereed)
    Abstract [en]

    We numerically study the formation of entanglement clusters across the many-body localization transition. We observe a crossover from strong many-body entanglement in the ergodic phase to weak local correlations in the localized phase, with continuous clusters throughout the phase diagram. Critical states close to the transition have a structure compatible with fractal or multiscale-entangled states, characterized by entanglement at multiple levels: small strongly entangled clusters are weakly entangled together to form larger clusters. The critical point therefore features subthermal entanglement and a power-law distributed cluster size. Upon entering the localized phase, the power-law distribution seems to persist with a varying power that crosses over into a stretched exponent before eventually becoming exponential deep in the localized phase. These results are in agreement with some of the recently proposed phenomenological renormalization-group schemes characterizing the many-body localized critical point, and serve to constrain other such schemes.

  • 14.
    Javanmard, Younes
    et al.
    Max Planck Inst Phys Komplexer Syst, D-01187 Dresden, Germany..
    Trapin, Daniele
    Max Planck Inst Phys Komplexer Syst, D-01187 Dresden, Germany..
    Bera, Soumya
    Indian Inst Technol, Dept Phys, Bombay 400076, Maharashtra, India..
    Bardarson, Jens H.
    KTH, School of Engineering Sciences (SCI), Physics, Condensed Matter Theory.
    Heyl, Markus
    Max Planck Inst Phys Komplexer Syst, D-01187 Dresden, Germany..
    Sharp entanglement thresholds in the logarithmic negativity of disjoint blocks in the transverse-field Ising chain2018In: New Journal of Physics, ISSN 1367-2630, E-ISSN 1367-2630, Vol. 20, article id 083032Article in journal (Refereed)
    Abstract [en]

    Entanglement has developed into an essential concept for the characterization of phases and phase transitions in ground states of quantum many-body systems. In this work we use the logarithmic negativity to study the spatial entanglement structure in the transverse-field Ising chain both in the ground state and at nonzero temperatures. Specifically, we investigate the entanglement between two disjoint blocks as a function of their separation, which can be viewed as the entanglement analog of a spatial correlation function. We find sharp entanglement thresholds at a critical distance beyond which the logarithmic negativity vanishes exactly and thus the two blocks become unentangled, which holds even in the presence of long-ranged quantum correlations, i.e., at the system's quantum critical point. Using time-evolving block decimation, we explore this feature as a function of temperature and size of the two blocks and present a simple model to describe our numerical observations.

  • 15.
    Lezama, Talia L. M.
    et al.
    Max Planck Inst Phys Komplexer Syst, D-01187 Dresden, Germany..
    Bera, Soumya
    Indian Inst Technol, Dept Phys, Mumbai 400076, Maharashtra, India..
    Bardarson, Jens H.
    KTH, School of Engineering Sciences (SCI), Physics, Condensed Matter Theory. Max Planck Inst Phys Komplexer Syst, D-01187 Dresden, Germany..
    Apparent slow dynamics in the ergodic phase of a driven many-body localized system without extensive conserved quantities2019In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 99, no 16, article id 161106Article in journal (Refereed)
    Abstract [en]

    We numerically study the dynamics on the ergodic side of the many-body localization transition in a periodically driven Floquet model with no global conservation laws. We describe and employ a numerical technique based on the fast Walsh-Hadamard transform that allows us to perform an exact time evolution for large systems and long times. As in models with conserved quantities (e.g., energy and/or particle number) we observe a slowing down of the dynamics as the transition into the many-body localized phase is approached. More specifically, our data are consistent with a subballistic spread of entanglement and a stretched-exponential decay of an autocorrelation function, with their associated exponents reflecting slow dynamics near the transition for a fixed system size. However, with access to larger system sizes, we observe a clear flow of the exponents towards faster dynamics and cannot rule out that the slow dynamics is a finite-size effect. Furthermore, we observe examples of nonmonotonic dependence of the exponents with time, with the dynamics initially slowing down but accelerating again at even larger times, consistent with the slow dynamics being a crossover phenomenon with a localized critical point.

  • 16. Lezama, Talia L. M.
    et al.
    Bera, Soumya
    Schomerus, Henning
    Heidrich-Meisner, Fabian
    Bardarson, Jens H.
    KTH, School of Engineering Sciences (SCI), Physics. Max-Planck-Institut für Physik komplexer Systeme, Germany.
    One-particle density matrix occupation spectrum of many-body localized states after a global quench2017In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 96, no 6, article id 060202Article in journal (Refereed)
    Abstract [en]

    The emergent integrability of the many-body localized phase is naturally understood in terms of localized quasiparticles. As a result, the occupations of the one-particle density matrix in eigenstates show a Fermi-liquid-like discontinuity. Here, we show that in the steady state reached at long times after a global quench from a perfect density-wave state, this occupation discontinuity is absent, reminiscent of a Fermi liquid at a finite temperature, while the full occupation function remains strongly nonthermal. We discuss how one can understand this as a consequence of the local structure of the density-wave state and the resulting partial occupation of quasiparticles. This partial occupation can be controlled by tuning the initial state and can be described by an effective temperature.

  • 17.
    Rhim, Jun-Won
    et al.
    Max Planck Inst Phys Komplexer Syst, D-01187 Dresden, Germany.;Inst for Basic Sci Korea, Ctr Correlated Elect Syst, Seoul 08826, South Korea.;Seoul Natl Univ, Dept Phys & Astron, Seoul 08826, South Korea..
    Bardarson, Jens H.
    KTH, School of Engineering Sciences (SCI), Physics. Max Planck Inst Phys Komplexer Syst, D-01187 Dresden, Germany..
    Slager, Robert-Jan
    Max Planck Inst Phys Komplexer Syst, D-01187 Dresden, Germany..
    Unified bulk-boundary correspondence for band insulators2018In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 97, no 11, article id 115143Article in journal (Refereed)
    Abstract [en]

    The bulk-boundary correspondence, a topic of intensive research interest over the past decades, is one of the quintessential ideas in the physics of topological quantum matter. Nevertheless, it has not been proven in all generality and has in certain scenarios even been shown to fail, depending on the boundary profiles of the terminated system. Here, we introduce bulk numbers that capture the exact number of in-gap modes, without any such subtleties in one spatial dimension. Similarly, based on these 1D bulk numbers, we define a new 2D winding number, which we call the pole winding number, that specifies the number of robust metallic surface bands in the gap as well as their topological character. The underlying general methodology relies on a simple continuous extrapolation from the bulk to the boundary, while tracking the evolution of Green's function's poles in the vicinity of the bulk band edges. As a main result we find that all the obtained numbers can be applied to the known insulating phases in a unified manner regardless of the specific symmetries. Additionally, from a computational point of view, these numbers can be effectively evaluated without any gauge fixing problems. In particular, we directly apply our bulk-boundary correspondence construction to various systems, including 1D examples without a traditional bulk-boundary correspondence, and predict the existence of boundary modes on various experimentally studied graphene edges, such as open boundaries and grain boundaries. Finally, we sketch the 3D generalization of the pole winding number by in the context of topological insulators.

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