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  • 1.
    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.

  • 2.
    Edvardsson, Elisabet
    et al.
    Stockholm University, Faculty of Science, Department of Physics.
    Mossberg, Eva
    The Q-toothpick Cellular Automaton2019In: Journal of Cellular Automata, ISSN 1557-5969, Vol. 14, no 1-2, p. 51-68Article in journal (Refereed)
    Abstract [en]

    In a toothpick-type cellular automaton, a shape is drawn, and then at each time-step copies of the same shape are attached at certain predetermined places. The resulting pattern exhibits unexpected growth properties. We investigate the fractal-like large-scale behavior of the Q-toothpick cellular automaton, which is built from quarter circles, with starting configurations consisting of an arbitrary number of quarter circles. In this paper, we prove that infinitely long barriers of quarter circles arise in the pattern, and divide it into non-interacting triangular parts. Furthermore, we show that the behavior of these triangular parts is described by the one-dimensional elementary cellular automaton rule 18 and is related to the Sierpinski triangle.

  • 3.
    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.

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