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Solvable Topological Boundaries
Stockholm University, Faculty of Science, Department of Physics.
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: urn:nbn:se:su:diva-166135ISBN: 978-91-7797-630-1 (print)ISBN: 978-91-7797-631-8 (electronic)OAI: oai:DiVA.org:su-166135DiVA, id: diva2:1291153
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At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 7: Submitted.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: 2019-12-12Bibliographically approved
List of papers
1. Anatomy of topological surface states: Exact solutions from destructive interference on frustrated lattices
Open this publication in new window or tab >>Anatomy of topological surface states: Exact solutions from destructive interference on frustrated lattices
2017 (English)In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 96, no 8, article id 085443Article in journal (Refereed) Published
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.

Keywords
edge states, frustrated magnetism, surfaces states, topological materials
National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-147048 (URN)10.1103/PhysRevB.96.085443 (DOI)000408623700007 ()
Available from: 2017-09-20 Created: 2017-09-20 Last updated: 2019-12-12Bibliographically approved
2. Lattice models with exactly solvable topological hinge and corner states
Open this publication in new window or tab >>Lattice models with exactly solvable topological hinge and corner states
2018 (English)In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 97, no 24, article id 241405Article in journal (Refereed) Published
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.

National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-157648 (URN)10.1103/PhysRevB.97.241405 (DOI)000434762500002 ()
Available from: 2018-06-25 Created: 2018-06-25 Last updated: 2019-12-12Bibliographically approved
3. Biorthogonal Bulk-Boundary Correspondence in Non-Hermitian Systems
Open this publication in new window or tab >>Biorthogonal Bulk-Boundary Correspondence in Non-Hermitian Systems
2018 (English)In: Physical Review Letters, ISSN 0031-9007, E-ISSN 1079-7114, Vol. 121, no 2, article id 026808Article in journal (Refereed) Published
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.

National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-159108 (URN)10.1103/PhysRevLett.121.026808 (DOI)000438191700025 ()30085697 (PubMedID)
Available from: 2018-08-31 Created: 2018-08-31 Last updated: 2019-02-22Bibliographically approved
4. Boundaries of boundaries: A systematic approach to lattice models with solvable boundary states of arbitrary codimension
Open this publication in new window or tab >>Boundaries of boundaries: A systematic approach to lattice models with solvable boundary states of arbitrary codimension
2019 (English)In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 99, no 8, article id 085426Article in journal (Refereed) Published
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.

National Category
Condensed Matter Physics
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-166152 (URN)10.1103/PhysRevB.99.085426 (DOI)000459225100011 ()
Available from: 2019-02-18 Created: 2019-02-18 Last updated: 2019-12-12Bibliographically approved
5. Extended Bloch theorem for topological lattice models with open boundaries
Open this publication in new window or tab >>Extended Bloch theorem for topological lattice models with open boundaries
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.

National Category
Condensed Matter Physics
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-166153 (URN)10.1103/PhysRevB.99.085427 (DOI)000459225100012 ()
Available from: 2019-02-18 Created: 2019-02-18 Last updated: 2019-12-12Bibliographically approved
6. Symmetry-protected nodal phases in non-Hermitian systems
Open this publication in new window or tab >>Symmetry-protected nodal phases in non-Hermitian systems
2019 (English)In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 99, no 4, article id 041406Article in journal (Refereed) Published
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.

Keywords
Quantum optics, Dirac semimetal
National Category
Condensed Matter Physics
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-166133 (URN)10.1103/PhysRevB.99.041406 (DOI)000456809000002 ()
Funder
Swedish Research CouncilKnut and Alice Wallenberg Foundation
Available from: 2019-02-15 Created: 2019-02-15 Last updated: 2019-12-12Bibliographically approved
7. Non-Hermitian extensions of higher-order topological phases and their biorthogonal bulk-boundary correspondence
Open this publication in new window or tab >>Non-Hermitian extensions of higher-order topological phases and their biorthogonal bulk-boundary correspondence
2019 (English)In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 99, no 8, article id 081302Article in journal (Refereed) Published
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.

National Category
Condensed Matter Physics
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-166154 (URN)10.1103/PhysRevB.99.081302 (DOI)000459936700001 ()
Available from: 2019-02-18 Created: 2019-02-18 Last updated: 2019-12-12Bibliographically approved
8. Non-Hermitian systems and topology: A transfer-matrix perspective
Open this publication in new window or tab >>Non-Hermitian systems and topology: A transfer-matrix perspective
2019 (English)In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 99, no 24, article id 245116Article in journal (Refereed) Published
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.

National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-170798 (URN)10.1103/PhysRevB.99.245116 (DOI)000470840800003 ()
Available from: 2019-07-22 Created: 2019-07-22 Last updated: 2019-12-12Bibliographically approved

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