Exploring bulk-boundary correspondence in topological systems
Final Report Abstract
The two main objectives of the project have been the development of the general theoretical methodology for studying bound states and applying it to investigate the properties of bound states in various physical models. The latter objective has expanded beyond just the initial question of bulk-boundary correspondence; results specifically regarding bulk-boundary correspondence as well as more general ones have been obtained. A total of seven specific projects have been completed within the framework of this project. Such method is the symmetry-based approach, where models of systems with boundaries of the most general form are found, constrained only by the fundamental principles of quantum mechanics and, if present, symmetries. The main value of this approach is that such models deliver general bound-state structures. The formalism of general boundary conditions for continuum models and its adaptation to describe symmetry-constrained families of bulk systems with boundaries termed the symmetry-incorporating formalism of general continuum models with boundary conditions have been developed, which constitute this approach for continuum models. An analogous symmetry-based approach has been used to construct the lattice model of the most general form of a 1D chiral-symmetric system with a boundary. The formalism of general continuum models with boundary conditions was applied to study the bound states of the following systems: the linear-in-momentum two-component model in 1D (an insulator), 2D (a quantum anomalous Hall system in the vicinity of the topological phase transition), and 3D (a Weyl semimetal in the vicinity of its node); a Dirac semimetal in the vicinity of its node; 1D superconductors, for the study of Majorana bound states. The surface states of the Luttinger semimetal under perturbations – compressive strain and bulk-inversion asymmetry – were explored. The general lattice model of 1D chiral-symmetric systems with a boundary was used to explore the question bulkboundary correspondence and it was demonstrated that there is no "absolute” bulk-boundary correspondence is such systems. A variant of bulk-boundary correspondence for Weyl semimetals was formulated that directly characterizes the vicinity of the projected Weyl point, relating the surface-state spectrum to the Chern number of the latter. One of the main general results is the identification of the persistent "propagation effect”, whereby bound states from topologically nontrivial classes "propagate” to the related, "adjacent” in symmetry or dimension, topologically trivial classes. The presence of bound states in the linear-in-momentum two-component model of the most general form proves that this is a ubiquitous effect that is neither rare nor accidental. This is hardly possible to demonstrate conclusively with other methods. Precisely the generality of the models obtained with the employed method allows one to make definitive conclusions, which is the main value of the method. This concerns both the propagation effect and the analysis of bulk-boundary correspondence in 1D chiral-symmetric systems using general lattice models. The two main emerging overarching conclusions of our analysis is that, on the one hand, bound states appear to be more widespread and persistent than the strict topological classification suggests and, on the other hand, bulk-boundary correspondence is a more subtle and less universal phenomenon than typically believed.
Publications
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Paradoxical extension of the edge states across the topological phase transition due to emergent approximate chiral symmetry in a quantum anomalous Hall system. Physical Review B, 98(16).
Candido, Denis R.; Kharitonov, Maxim; Egues, J. Carlos & Hankiewicz, Ewelina M.
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Ever-present Majorana bound state in a generic one-dimensional superconductor with odd number of Fermi surfaces. Physical Review B, 104(13).
Kharitonov, Maxim; Hankiewicz, Ewelina M.; Trauzettel, Björn & Bergeret, F. Sebastian
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"Evolution of the surface states of the Luttinger semimetal under strain and inversion-symmetry breaking: Dirac, line-node, and Weyl semimetals”
M. Kharitonov, J.-B. Mayer, and E. M. Hankiewicz
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Formalism of general boundary conditions for continuum models
M. Kharitonov
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Formalism of general continuum models with boundary conditions, propagation of bound states from nontrivial to trivial topological classes, and the general surface-state structure near one node of a Weyl semimetal
M. Kharitonov
