Anatomy of topological surface states: Exact solutions from destructive interference on frustrated lattices

Kunst, Flore K.; Trescher, Maximilian; Bergholtz, Emil J.

doi:10.1103/physrevb.96.085443

Cited by 34 publications

(49 citation statements)

References 92 publications

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“…Finally, our formalism extends the real-space biorthogonal polarization [46] to more general lattice topologies than those considered in Refs. [46,78,79].…”

Section: Introductionmentioning

confidence: 99%

Non-Hermitian systems and topology: A transfer-matrix perspective

2019

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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. arXiv:1812.02186v3 [cond-mat.mes-hall]

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“…Finally, our formalism extends the real-space biorthogonal polarization [46] to more general lattice topologies than those considered in Refs. [46,78,79].…”

Section: Introductionmentioning

confidence: 99%

Non-Hermitian systems and topology: A transfer-matrix perspective

2019

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“…There has been a lot of progress on models in higher dimensions, that exhibit exact zero modes, see for instance [39]. It would be interesting to investigate if it is possible to construct models, that exhibit 'one-sided' zero modes along the lines of the ones described in this paper, even in those higher-dimensional systems.…”

Section: Discussionmentioning

confidence: 91%

Zero modes of the Kitaev chain with phase-gradients and longer range couplings

Mahyaeh

Ardonne

2018

J. Phys. Commun.

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We present an analytical solution for the full spectrum of Kitaevʼs one-dimensional p-wave superconductor with arbitrary hopping, pairing amplitude and chemical potential in the case of an open chain. We also discuss the structure of the zero-modes in the presence of both phase gradients and next nearest neighbor hopping and pairing terms. As observed by Sticlet et al, one feature of such models is that in a part of the phase diagram, zero-modes are present at one end of the system, while there are none on the other side. We explain the presence of this feature analytically, and show that it requires some fine-tuning of the parameters in the model. Thus as expected, these 'one-sided' zeromodes are neither protected by topology, nor by symmetry. IntroductionOne of the characteristic features of many topological phases is the presence of gapless boundary modes. The (fractional) quantum Hall states are a prime example [1-3], and their boundary modes provide strong evidence of the topological nature of these states. Another prime example is the Kitaev chain, whose topological p-wave superconducting phase features so-called 'Majorana zero modes' at its edges [4]. Trying to establish the existence of the topological phase is often done by trying to establish the presence of the boundary modes. This has led to strong evidence for the topological phase in for instance strongly spin-orbit coupled nano-wires that are proximity coupled to an s-wave superconductor in the presence of a magnetic field [5-9], or in chains of magnetic ad-atoms [10][11][12][13]. It has been proposed that the zero energy Majorana bound states can be used as topologically protected q-bits, for quantum information processing purposes [14,15]. By now, there exist various proposals to manipulate these q-bits, either in T-junction systems, in which the Majorana bound states can be braided explicitly [16], or in Josephson coupled Kitaev chains, in which the coupling of the various chains allows operation on the q-bits [17].Despite the intense research on the Kitaev chain models, there are still interesting features that deserve attention. In this paper, we look into one of them. It was observed by Sticlet et al [18], that the zero-modes of Kitaev chains carrying a current, i.e., in the presence of a gradient in the phase of the order parameter, have interesting properties. The most striking feature is that is it possible that at one edge of the chain, there is pair of Majorana bound states (or better, one 'ordinary' Dirac zero mode), while there is no zero mode at the other end of the chain. Clearly, from a topological point of view, this means that the chain is in a trivial phase, but it is nevertheless worthwhile to investigate these zero-modes further. In this paper, we explain the presence of these zero-modes, via an exact solution of the zero modes of an extended Kitaev chain, i.e., in the presence of both complex and next nearest-neighbor hopping an pairing terms. We show that it is necessary to fine tune the couplings in order that these 'one-sid...

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“…In this section, we study lattices with open boundary conditions in one direction of the form shown in Fig. 1(c), and show that in addition to finding exact solutions for the boundary modes [28,29,33], we can also fully diagonalize the entire system if the spectral symmetry E( k || , k ⊥ ) = E( k || , −k ⊥ ) is present in the spectrum of the periodic Bloch Hamiltonian. Moreover, we show that this method still works in the presence of SOC by satisfying weak constraints.…”

Section: Exact Solutionsmentioning

confidence: 99%

“…Indeed, by setting the dimension D = 2 and 3 and implementing the relevant lattice Hamiltonian, we showed in Ref. 33 that these solutions correspond to the chiral, edge modes of a Chern insulator and the Fermi arcs in Weyl semimetals, respectively. To showcase this formalism explicitly, we show an explicit example in Sec.…”

Section: A Boundary Modesmentioning

confidence: 99%

Extended Bloch theorem for topological lattice models with open boundaries

2019

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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 co-dimension 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. arXiv:1812.03099v2 [cond-mat.mes-hall]

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Anatomy of topological surface states: Exact solutions from destructive interference on frustrated lattices

Cited by 34 publications

References 92 publications

Non-Hermitian systems and topology: A transfer-matrix perspective

Non-Hermitian systems and topology: A transfer-matrix perspective

Zero modes of the Kitaev chain with phase-gradients and longer range couplings

Extended Bloch theorem for topological lattice models with open boundaries

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