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  • 1. Behrends, Jan
    et al.
    Kunst, Flore K.
    Stockholm University, Faculty of Science, Department of Physics.
    Sbierski, Björn
    Transversal magnetotransport in Weyl semimetals: Exact numerical approach2018In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 97, no 6, article id 064203Article in journal (Refereed)
    Abstract [en]

    Magnetotransport experiments on Weyl semimetals are essential for investigating the intriguing topological and low-energy properties of Weyl nodes. If the transport direction is perpendicular to the applied magnetic field, experiments have shown a large positive magnetoresistance. In this work we present a theoretical scattering matrix approach to transversal magnetotransport in a Weyl node. Our numerical method confirms and goes beyond the existing perturbative analytical approach by treating disorder exactly. It is formulated in real space and is applicable to mesoscopic samples as well as in the bulk limit. In particular, we study the case of clean and strongly disordered samples.

  • 2. Budich, Jan Carl
    et al.
    Carlström, Johan
    Stockholm University, Faculty of Science, Department of Physics.
    Kunst, Flore K.
    Stockholm University, Faculty of Science, Department of Physics.
    Bergholtz, Emil J.
    Stockholm University, Faculty of Science, Department of Physics.
    Symmetry-protected nodal phases in non-Hermitian systems2019In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 99, no 4, article id 041406Article in journal (Refereed)
    Abstract [en]

    Non-Hermitian (NH) Hamiltonians have become an important asset for the effective description of various physical systems that are subject to dissipation. Motivated by recent experimental progress on realizing the NH counterparts of gapless phases such as Weyl semimetals, here we investigate how NH symmetries affect the occurrence of exceptional points (EPs), that generalize the notion of nodal points in the spectrum beyond the Hermitian realm. Remarkably, we find that the dimension of the manifold of EPs is generically increased by one as compared to the case without symmetry. This leads to nodal surfaces formed by EPs that are stable as long as a protecting symmetry is preserved, and that are connected by open Fermi volumes. We illustrate our findings with analytically solvable two-band lattice models in one and two spatial dimensions, and show how they are readily generalized to generic NH crystalline systems.

  • 3.
    Edvardsson, Elisabet
    et al.
    Stockholm University, Faculty of Science, Department of Physics.
    Kunst, Flore K.
    Stockholm University, Faculty of Science, Department of Physics.
    Bergholtz, Emil J.
    Stockholm University, Faculty of Science, Department of Physics.
    Non-Hermitian extensions of higher-order topological phases and their biorthogonal bulk-boundary correspondence2019In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 99, no 8, article id 081302Article in journal (Refereed)
    Abstract [en]

    Non-Hermitian Hamiltonians, which describe a wide range of dissipative systems, and higher-order topological phases, which exhibit novel boundary states on corners and hinges, comprise two areas of intense current research. Here we investigate systems where these frontiers merge and formulate a generalized biorthogonal bulk-boundary correspondence, which dictates the appearance of boundary modes at parameter values that are, in general, radically different from those that mark phase transitions in periodic systems. By analyzing the interplay between corner/hinge, edge/surface and bulk degrees of freedom we establish that the non-Hermitian extensions of higher-order topological phases exhibit an even richer phenomenology than their Hermitian counterparts and that this can be understood in a unifying way within our biorthogonal framework. Saliently this works in the presence of the non-Hermitian skin effect, and also naturally encompasses genuinely non-Hermitian phenomena in the absence thereof.

  • 4.
    Kunst, Flore
    Stockholm University, Faculty of Science, Department of Physics.
    Topology Meets Frustration: Exact Solutions for Topological Surface States on Geometrically Frustrated Lattices2017Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    One of the main features of topological phases is the presence of robust boundary states that are protected by a topological invariant. Famous examples of such states are the chiral edge states of a Chern insulator, the helical edge states of a two-dimensional Z2 insulator, and the Fermi arcs of Weyl semimetals. Despite their omnipresence, these topological boundary states can typically only be theoretically investigated through numerical studies due to the lack of analytical solutions for their wave functions. In the rare cases that wave-function solutions are available, they only exist for simple fine-tuned systems or for semi-infinite systems. Exact solutions are, however, common in the field of flat bands physics, where they lead to an understanding of the bulk bands rather than the boundary physics. It is well known that fully-periodic lattices with a frustrated geometry host localized modes that have a constant energy throughout the Brillouin zone. These localized modes appear due to a mechanism referred to as destructive interference, which leads to the disappearance of the wave-function amplitude on certain lattice sites. Making use of this mechanism, it is shown in this licentiate thesis that exact wave-function solutions can also be found on d-dimensional geometrically frustrated lattices that feature (d − 1)-dimensional boundaries. These exact solutions localize to the boundaries when the frustrated lattice hosts a topological phase and correspond to the robust, topological boundary states.

    This licentiate thesis revolves around the publication, which describes the method to finding these exact, analytical solutions for the topological boundary states on geometrically frustrated lattices, which was authored by the author of this licentiate thesis together with Maximilian Trescher and Emil J. Bergholtz and published in Physical Review B on August 30, 2017 with the title Anatomy of topological surface states: Exact solutions from destructive interference on frustrated lattices. An introduction is given on topological phases in condensed matter systems focussing on those models of which explicit examples are given in the paper: two-dimensional Chern insulators and three-dimensional Weyl semimetals. Moreover, by making use of the kagome lattice as an example the appearance of localized and semi-localized modes on geometrically frustrated lattices is elaborated upon. The chapters in this licentiate thesis thus endeavor to provide the reader with the proper background to comfortably read, understand, place into context and judge the relevance of the work in the accompanying publication. The licentiate thesis finishes with an outlook where it is discussed that the method presented in the paper can be generalized to an even larger class of lattices and can also be applied to find exact solutions for higher-order topological phases such as corner and hinge states. 

  • 5.
    Kunst, Flore K.
    et al.
    Stockholm University, Faculty of Science, Department of Physics.
    Dwivedi, Vatsal
    Non-Hermitian systems and topology: A transfer matrix perspectiveManuscript (preprint) (Other academic)
    Abstract [en]

    Non-Hermitian topological systems are known to exhibit features strikingly different from their Hermitian counterparts. We study these systems using a generalized transfer matrix approach, which provides a unifying analytical framework for tight-binding models with periodic as well as open boundary conditions. This leads to an analytical and intuitive understanding of many of the unusual properties of non-Hermitian systems, such as the non-Hermitian skin effect, the breakdown of the conventional bulk-boundary correspondence, and the appearance of exceptional points with an order scaling with system size.

  • 6.
    Kunst, Flore K.
    et al.
    Stockholm University, Faculty of Science, Department of Physics.
    Dwivedi, Vatsal
    Non-Hermitian systems and topology: A transfer-matrix perspective2019In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 99, no 24, article id 245116Article in journal (Refereed)
    Abstract [en]

    Topological phases of Hermitian systems are known to exhibit intriguing properties such as the presence of robust boundary states and the famed bulk-boundary correspondence. These features can change drastically for their non-Hermitian generalizations, as exemplified by a general breakdown of bulk-boundary correspondence and a localization of all states at the boundary, termed the non-Hermitian skin effect. In this paper, we present a completely analytical unifying framework for studying these systems using generalized transfer matrices, a real-space approach suitable for systems with periodic as well as open boundary conditions. We show that various qualitative properties of these systems can be easily deduced from the transfer matrix. For instance, the connection between the breakdown of the conventional bulk-boundary correspondence and the existence of a non-Hermitian skin effect, previously observed numerically, is traced back to the transfer matrix having a determinant not equal to unity. The vanishing of this determinant signals real-space exceptional points, whose order scales with the system size. We also derive previously proposed topological invariants such as the biorthogonal polarization and the Chern number computed on a complexified Brillouin zone. Finally, we define an invariant for and thereby clarify the meaning of topologically protected boundary modes for non-Hermitian systems.

  • 7.
    Kunst, Flore K.
    et al.
    Stockholm University, Faculty of Science, Department of Physics.
    Edvardsson, Elisabet
    Stockholm University, Faculty of Science, Department of Physics.
    Budich, Jan Carl
    Bergholtz, Emil J.
    Stockholm University, Faculty of Science, Department of Physics.
    Biorthogonal Bulk-Boundary Correspondence in Non-Hermitian Systems2018In: Physical Review Letters, ISSN 0031-9007, E-ISSN 1079-7114, Vol. 121, no 2, article id 026808Article in journal (Refereed)
    Abstract [en]

    Non-Hermitian systems exhibit striking exceptions from the paradigmatic bulk-boundary correspondence, including the failure of bulk Bloch band invariants in predicting boundary states and the (dis) appearance of boundary states at parameter values far from those corresponding to gap closings in periodic systems without boundaries. Here, we provide a comprehensive framework to unravel this disparity based on the notion of biorthogonal quantum mechanics: While the properties of the left and right eigenstates corresponding to boundary modes are individually decoupled from the bulk physics in non-Hermitian systems, their combined biorthogonal density penetrates the bulk precisely when phase transitions occur. This leads to generalized bulk-boundary correspondence and a quantized biorthogonal polarization that is formulated directly in systems with open boundaries. We illustrate our general insights by deriving the phase diagram for several microscopic open boundary models, including exactly solvable non-Hermitian extensions of the Su-Schrieffer-Heeger model and Chern insulators.

  • 8.
    Kunst, Flore K.
    et al.
    Stockholm University, Faculty of Science, Department of Physics. Freie Universität Berlin, Germany.
    Trescher, Maximilian
    Bergholtz, Emil J.
    Stockholm University, Faculty of Science, Department of Physics. Freie Universität Berlin, Germany.
    Anatomy of topological surface states: Exact solutions from destructive interference on frustrated lattices2017In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 96, no 8, article id 085443Article in journal (Refereed)
    Abstract [en]

    The hallmark of topological phases is their robust boundary signature whose intriguing properties-such as the one-way transport on the chiral edge of a Chern insulator and the sudden disappearance of surface states forming open Fermi arcs on the surfaces of Weyl semimetals-are impossible to realize on the surface alone. Yet, despite the glaring simplicity of noninteracting topological bulk Hamiltonians and their concomitant energy spectrum, the detailed study of the corresponding surface states has essentially been restricted to numerical simulation. In this work, however, we show that exact analytical solutions of both topological and trivial surface states can be obtained for generic tight-binding models on a large class of geometrically frustrated lattices in any dimension without the need for fine-tuning of hopping amplitudes. Our solutions derive from local constraints tantamount to destructive interference between neighboring layer lattices perpendicular to the surface and provide microscopic insights into the structure of the surface states that enable analytical calculation of many desired properties including correlation functions, surface dispersion, Berry curvature, and the system size dependent gap closing, which necessarily occurs when the spatial localization switches surface. This further provides a deepened understanding of the bulkboundary correspondence. We illustrate our general findings on a large number of examples in two and three spatial dimensions. Notably, we derive exact chiral Chern insulator edge states on the spin-orbit-coupled kagome lattice, and Fermi arcs relevant for recently synthesized slabs of pyrochlore-based Eu2Ir2O7 and Nd2Ir2O7, which realize an all-in-all-out spin configuration, as well as for spin-ice-like two-in-two-out and one-in-three-out configurations, which are both relevant for Pr2Ir2O7. Remarkably, each of the pyrochlore examples exhibit clearly resolved Fermi arcs although only the one-in-three-out configuration features bulk Weyl nodes in realistic parameter regimes. Our approach generalizes to symmetry protected phases, e.g., quantum spin Hall systems and Dirac semimetals with time-reversal symmetry, and can furthermore signal the absence of topological surface states, which we illustrate for a class of models akin to the trivial surface of Hourglass materials KHgX where the exact solutions apply but, independently of Hamiltonian details, yield eigenstates delocalized over the entire sample.

  • 9.
    Kunst, Flore K.
    et al.
    Stockholm University, Faculty of Science, Department of Physics.
    van Miert, Guido
    Bergholtz, Emil J.
    Stockholm University, Faculty of Science, Department of Physics.
    Boundaries of boundaries: A systematic approach to lattice models with solvable boundary states of arbitrary codimension2019In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 99, no 8, article id 085426Article in journal (Refereed)
    Abstract [en]

    We present a generic and systematic approach for constructing D−dimensional lattice models with exactly solvable d−dimensional boundary states localized to corners, edges, hinges, and surfaces. These solvable models represent a class of “sweet spots” in the space of possible tight-binding models—the exact solutions remain valid for any tight-binding parameters as long as they obey simple locality conditions that are manifest in the underlying lattice structure. Consequently, our models capture the physics of both (higher order) topological and nontopological phases as well as the transitions between them in a particularly illuminating and transparent manner.

  • 10.
    Kunst, Flore K.
    et al.
    Stockholm University, Faculty of Science, Department of Physics.
    van Miert, Guido
    Bergholtz, Emil J.
    Stockholm University, Faculty of Science, Department of Physics.
    Extended Bloch theorem for topological lattice models with open boundaries2019In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 99, no 8, article id 085427Article in journal (Refereed)
    Abstract [en]

    While the Bloch spectrum of translationally invariant noninteracting lattice models is trivially obtained by a Fourier transformation, diagonalizing the same problem in the presence of open boundary conditions is typically only possible numerically or in idealized limits. Here we present exact analytic solutions for the boundary states in a number of lattice models of current interest, including nodal-line semimetals on a hyperhoneycomb lattice, spin-orbit coupled graphene, and three-dimensional topological insulators on a diamond lattice, for which no previous exact finite-size solutions are available in the literature. Furthermore, we identify spectral mirror symmetry as the key criterium for analytically obtaining the entire (bulk and boundary) spectrum as well as the concomitant eigenstates, and exemplify this for Chern and Z2 insulators with open boundaries of codimension one. In the case of the two-dimensional Lieb lattice, we extend this further and show how to analytically obtain the entire spectrum in the presence of open boundaries in both directions, where it has a clear interpretation in terms of bulk, edge, and corner states.

  • 11.
    Kunst, Flore K.
    et al.
    Stockholm University, Faculty of Science, Department of Physics.
    van Miert, Guido
    Bergholtz, Emil J.
    Stockholm University, Faculty of Science, Department of Physics.
    Lattice models with exactly solvable topological hinge and corner states2018In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 97, no 24, article id 241405Article in journal (Refereed)
    Abstract [en]

    We devise a generic recipe for constructing D-dimensional lattice models whose d-dimensional boundary states, located on surfaces, hinges, corners, and so forth, can be obtained exactly. The solvability is rooted in the underlying lattice structure and as such does not depend on fine tuning, allowing us to track their evolution throughout various phases and across phase transitions. Most saliently, our models provide boundary solvable examples of the recently introduced higher-order topological phases. We apply our general approach to breathing and anisotropic kagome and pyrochlore lattices for which we obtain exact corner eigenstates, and to periodically driven two-dimensional models as well as to three-dimensional lattices where we present exact solutions corresponding to one-dimensional chiral states at the hinges of the lattice. We relate the higher-order topological nature of these models to reflection symmetries in combination with their provenance from lower-dimensional conventional topological phases.

  • 12.
    Kunst, Flore Kiki
    Stockholm University, Faculty of Science, Department of Physics.
    Solvable Topological Boundaries2019Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    The hallmark of topological phases of matter is the presence of robust boundary states. In this dissertation, a formalism is developed with which analytical solutions for these states can be straightforwardly obtained by making use of destructive interference, which is naturally present in a large family of lattice models. The validity of the solutions is independent of tight-binding parameters, and as such these lattices can be seen as a subset of solvable systems in the landscape of tight-binding models. The approach allows for a full control of the topological phase of the system as well as the dispersion and localization of the boundary states, which makes it possible to design lattice models possessing the desired topological phase from the bottom up. Further applications of this formalism can be found in the fields of higher-order topological phases—where boundary states localize to boundaries with a codimension larger than one—and of non-Hermitian Hamiltonians—which is a fruitful approach to describe dissipation, and feature many exotic features, such as the possible breakdown of bulk-boundary correspondence—where the access to exact solutions has led to new insights.

  • 13.
    Mohamed El Hassan, Ashraf
    et al.
    Stockholm University, Faculty of Science, Department of Physics.
    Kunst, Flore
    Stockholm University, Faculty of Science, Department of Physics.
    Moritz, Alexander
    Stockholm University, Faculty of Science, Department of Physics.
    Andler, Guillermo
    Stockholm University, Faculty of Science, Department of Physics.
    Bergholtz, Emil
    Stockholm University, Faculty of Science, Department of Physics.
    Bourennane, Mohamed
    Stockholm University, Faculty of Science, Department of Physics.
    Corner state of light in photonic waveguides2018In: Article in journal (Refereed)
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