Phase transitions and generalized biorthogonal polarization in non-Hermitian systems

Edvardsson, Elisabet; Kunst, Flore K.; Yoshida, Tsuneya; Bergholtz, Emil J.

doi:10.1103/physrevresearch.2.043046

Cited by 43 publications

(30 citation statements)

References 64 publications

(84 reference statements)

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“…We note that the biorthogonal polarization P is equal for models that are related to each other via unitary transformations acting locally, e.g., P SSH for the nonreciprocal SSH model equals P Lee for Lee's model discussed in Sect. III.A.1 (Edvardsson et al, 2019b).…”

Section: Biorthogonal Bulk-boundary Correspondencementioning

confidence: 99%

Exceptional topology of non-Hermitian systems

2021

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We review the current understanding of the role of topology in non-Hermitian (NH) systems, and its far-reaching physical consequences observable in a range of dissipative settings. In particular, we elucidate how the paramount and genuinely NH concept of exceptional degeneracies, at which both eigenvalues and eigenvectors coalesce, leads to phenomena drastically distinct from the familiar Hermitian realm. An immediate consequence is the ubiquitous occurrence of nodal NH topological phases with concomitant open Fermi-Seifert surfaces, where conventional band-touching points are replaced by the aforementioned exceptional degeneracies. We furthermore discuss new notions of gapped phases including topological phases in single-band systems, and clarify how a given physical context may affect the symmetry-based topological classification. A unique property of NH systems with relevance beyond the field of topological phases consists in the anomalous relation between bulk-and boundary-physics, stemming from the striking sensitivity of NH matrices to boundary conditions. Unifying several complementary insights recently reported in this context, we put together a clear picture of intriguing phenomena such as the NH bulk-boundary correspondence, and the NH skin effect. Finally, we review applications of NH topology in both classical systems including optical setups with gain and loss, electric circuits, mechanical systems, and genuine quantum systems such as electronic transport settings at material junctions, and dissipative cold-atom setups.

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Section: Biorthogonal Bulk-boundary Correspondencementioning

confidence: 99%

Exceptional topology of non-Hermitian systems

2021

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“…This correspondence is based on the tacit assumption that, as long as the system is large, boundary conditions do not affect bulk properties. Natural questions which have been investigated are related to whether non-Hermiticity disrupts topological properties [22,23], whether new topological invariants can be introduced [24][25][26], and whether BBC holds true and in which sense [27][28][29][30]. A major issue regarding the restoration a non-Hermitian BBC is that non-Hermitian 1D tight-binding Hamiltonians with point gapped spectrum under periodic boundary conditions (PBCs) always yield the non-Hermitian skin effect [31][32][33][34][35][36][37][38][39][40], that is the unusual accumulation of bulk eigenstates at the ends of the same lattice under open 9).…”

Section: Introductionmentioning

confidence: 99%

Non-Hermitian skin effect as an impurity problem

Roccati

2021

Preprint

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A striking feature of non-Hermitian tight-binding Hamiltonians is the high sensitivity of both spectrum and eigenstates to boundary conditions. Indeed, if the spectrum under periodic boundary conditions is point gapped, by opening the lattice the non-Hermitian skin effect will necessarily occur. Finding the exact skin eigenstates may be demanding in general, and many methods in the literature are based on ansatzes and on recurrence equations for the eigenstates' components. Here we devise a general procedure based on the Green's function method to calculate the eigenstates of non-Hermitian tight-binding Hamiltonians under open boundary conditions. We apply it to the Hatano-Nelson and non-Hermitian SSH models and finally we contrast the edge states localization with that of bulk states.

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“…[56] and for the generalized case in ref. [57]

Θ = 1 - \lim_{L \to \infty} \frac{1}{L} 〈ψ_{L} (||, (0false \sum_{n = 1}^{N} n {\prod^{̂}}_{n})) ψ_{R}〉

where

{\prod^{̂}}_{n}

represents the projections onto the n th unit cell under OBC, L is system size, and

ψ_{L}

and

ψ_{R}

are the left and right eigenstates for the system, respectively. For the case of

Θ = 1

, it hints that there is one pair of the protected boundary states.…”

Section: Energy Spectra and Winding Numbermentioning

confidence: 99%

“…For example, in ref. [57], two pairs, one pair, and zero pair of the protected boundary states exist.…”

Section: Energy Spectra and Winding Numbermentioning

confidence: 99%

Topological Phase Transition and Eigenstates Localization in a Generalized Non‐Hermitian Su–Schrieffer–Heeger Model

Zhang

Huang

et al. 2020

Annalen der Physik

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The topological properties of a generalized non‐Hermitian Su–Schrieffer–Heeger model are investigated and it is demonstrated that the non‐Hermitian phase transition and the non‐Hermitian skin effect can be induced by intra‐cell asymmetric coupling under open boundary conditions. Through investigating and calculating the non‐Hermitian winding number with generalized Brillouin zone theory, it is found that the present non‐Hermitian system has an exact bulk‐boundary correspondence relationship. Meanwhile, the non‐Hermitian winding number is used to characterize the non‐Hermitian phase transition and determine the phase transition boundary, and it is found that the non‐Hermitian phase transition is not completely induced by the asymmetric coupling strength. By means of the mean inverse participation ratio, the factors that affect the eigenstates localization are shown and it is revealed that large system size or large asymmetric coupling strength can leave the system in the localized state. Additionally, it is found that for the asymmetric coupling strength and the system size, the eigenstates localization is much more sensitive to the asymmetric coupling strength.

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Phase transitions and generalized biorthogonal polarization in non-Hermitian systems

Cited by 43 publications

References 64 publications

Exceptional topology of non-Hermitian systems

Exceptional topology of non-Hermitian systems

Non-Hermitian skin effect as an impurity problem

Topological Phase Transition and Eigenstates Localization in a Generalized Non‐Hermitian Su–Schrieffer–Heeger Model

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