Simple formulas of directional amplification from non-Bloch band theory

Xue, Wen-Tan; Li, Ming-Rui; Hu, Yu-Min; Song, Fei; Zhong, Wang

doi:10.1103/physrevb.103.l241408

Cited by 78 publications

(13 citation statements)

References 55 publications

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“…Finally, we discuss the consistency between our theorem and previous works. In [82], the authors consider the frequency space Green's function G(x, y, ω) for non-Hermitian Hamiltonians in one dimension with a single site per unit cell. It is found that for certain hopping parameters, G(x, y, ω) is sensitive to the boundary condition.…”

Section: General Resultsmentioning

confidence: 99%

Boundary condition independence of non-Hermitian Hamiltonian dynamics

2021

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The non-Hermitian skin effect, namely, that the eigenvalues and eigenstates of a non-Hermitian tight-binding Hamiltonian have significant differences under open or periodic boundary conditions, is a remarkable phenomenon of non-Hermitian systems. Inspired by the presence of the non-Hermitian skin effect, we study the evolution of wave packets in non-Hermitian systems, which can be determined using the single-particle Green's function. Surprisingly, we find that in the thermodynamic limit, the Green's function does not depend on boundary conditions, despite the presence of skin effect. We provide a general proof for this statement in arbitrary dimension with finite hopping range, with an explicit illustration in the non-Hermitian Su-Schrieffer-Heeger model. We also explore its applications in noninteracting open quantum systems described by the master equation. We demonstrate that the evolution of the density matrix is independent of the boundary condition.

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Section: General Resultsmentioning

confidence: 99%

Boundary condition independence of non-Hermitian Hamiltonian dynamics

2021

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“…As we show in the plot, these remain delocalized across all three regimes, and thus the bulk is left intact, with the exception of one zero singular mode in the non-trivial regime (here ν = −1), which is exponentially localized at the left boundary of the finite-size system, see Eqs. (32) and (33). In contrast, the left eigenvectors display NHSE, both for trivial and non-trivial configurations alike.…”

Section: Singular Value Spectrummentioning

confidence: 94%

“…As the NH gap closes (C = 1) and we enter the non-trivial phase for C > 1, an exponentially small singular value appears under OBC that corresponds to a pair of exponentially localized right and left singular vectors, Eqs. ( 32) and (33), while all other singular vectors remain plane waves, i.e., the bulk is left intact by the change in the boundary term. Left (e) and right (f) singular vectors localize at opposite ends.…”

Section: Explicit Calculation Of the Zero Singular Modesmentioning

confidence: 99%

“…[29,32] investigated the same kind of amplification in photonic lattices, also making use of the SVD, but the topological characterization is there given in terms of standard topological band theory, with no connection to NH Hamiltonians. Directional amplification in NH lattices has also been studied in the context of non-Bloch band theory [33]. Amplification is made possible by the bosonic implementation of the NH Hamiltonian and the presence of active reservoirs (incoherent pumps), and it would be otherwise forbidden for fermions.…”

Section: Introductionmentioning

confidence: 99%

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Restoration of the non-Hermitian bulk-boundary correspondence via topological amplification

Brunelli¹,

Wanjura²,

Nunnenkamp³

2022

Preprint

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Non-Hermitian (NH) lattice Hamiltonians display a unique kind of energy gap and extreme sensitivity to boundary conditions. Due to the NH skin effect, the separation between edge and bulk states is blurred and the (conventional) bulk-boundary correspondence is lost. Here, we restore the bulk-boundary correspondence for the most paradigmatic class of NH Hamiltonians, namely those with one complex band and without symmetries. We obtain the desired NH Hamiltonian from the (mean-field) unconditional evolution of driven-dissipative cavity arrays, in which NH terms-in the form of non-reciprocal hopping amplitudes, gain and loss-are explicitly modelled via coupling to (engineered and non-engineered) reservoirs. This approach removes the arbitrariness in the definition of the topological invariant, as point-gapped spectra differing by a complex-energy shift are not treated as equivalent; the origin of the complex plane provides a common reference (base point) for the evaluation of the topological invariant. This implies that topologically non-trivial Hamiltonians are only a strict subset of those with a point gap and that the NH skin effect does not have a topological origin. We analyze the NH Hamiltonians so obtained via the singular value decomposition, which allows to express the NH bulk-boundary correspondence in the following simple form: an integer value ν of the topological invariant defined in the bulk corresponds to |ν| singular vectors exponentially localized at the system edge under open boundary conditions, in which the sign of ν determines which edge. Non-trivial topology manifests as directional amplification of a coherent input with gain exponential in system size. Our work solves an outstanding problem in the theory of NH topological phases and opens up new avenues in topological photonics.

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“…These phenomena have been experimentally observed in various physical systems [12][13][14][15][16][17][18]. It is significant to investigate some phenomena induced by the non-Hermitian skin effect because this effect can lead to novel non-Hermitian phenomena [19][20][21][22][23][24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

Scaling rule in critical non-Hermitian skin effect

Yokomizo,

Murakami

2021

Preprint

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Non-Hermitian systems show a non-Hermitian skin effect, where the bulk states are localized at a boundary of the systems with open boundary conditions. In this paper, we study dependence of the localization length of the eigenstates on a system size in a specific non-Hermitian model with a critical non-Hermitian skin effect, where the energy spectrum undergoes discontinuous transition in the thermodynamic limit. We analytically show that the eigenstates exhibit remarkable localization, known as scale-free localization, where the localization length is proportional to a system size. Our result gives a theoretical support for the scale-free localization, which has been proposed only numerically in previous works.

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Simple formulas of directional amplification from non-Bloch band theory

Cited by 78 publications

References 55 publications

Boundary condition independence of non-Hermitian Hamiltonian dynamics

Boundary condition independence of non-Hermitian Hamiltonian dynamics

Restoration of the non-Hermitian bulk-boundary correspondence via topological amplification

Scaling rule in critical non-Hermitian skin effect

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