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Extended Bloch theorem for topological lattice models with open boundaries
Stockholm University, Faculty of Science, Department of Physics.
Stockholm University, Faculty of Science, Department of Physics.
2019 (English)In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 99, no 8, article id 085427Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
2019. Vol. 99, no 8, article id 085427
National Category
Condensed Matter Physics
Research subject
Theoretical Physics
Identifiers
URN: urn:nbn:se:su:diva-166153DOI: 10.1103/PhysRevB.99.085427ISI: 000459225100012OAI: oai:DiVA.org:su-166153DiVA, id: diva2:1289430
Available from: 2019-02-18 Created: 2019-02-18 Last updated: 2019-03-04Bibliographically approved
In thesis
1. Solvable Topological Boundaries
Open this publication in new window or tab >>Solvable Topological Boundaries
2019 (English)Doctoral 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.

Place, publisher, year, edition, pages
Stockholm: Department of Physics, Stockholm University, 2019. p. 91
National Category
Condensed Matter Physics
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-166135 (URN)978-91-7797-630-1 (ISBN)978-91-7797-631-8 (ISBN)
Public defence
2019-04-12, sal FB52, AlbaNova universitetscentrum, Roslagstullsbacken 21, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 7: Submitted. Paper 8: Manuscript.

Available from: 2019-03-20 Created: 2019-02-22 Last updated: 2019-03-15Bibliographically approved

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