Spectral deformations in non-Hermitian lattices with disorder and skin effect: A solvable model

Stefano Longhi

Phys. Rev. B 103, 144202 – Published 8 April 2021

Abstract

We derive analytical results on energy spectral phase transitions and deformations in the simplest model of a one-dimensional lattice displaying the non-Hermitian skin effect, namely, the Hatano-Nelson model with unidirectional hopping under on-site potential uncorrelated disorder on the complex energy plane. Although the energy spectrum under open boundary conditions (OBCs) exactly reproduces the distribution of on-site potential disorder, the energy spectrum under periodic boundary conditions (PBC) undergoes spectral deformations, from one or more closed loops in the fully delocalized phase with no overlap with the OBC spectrum to a mixed spectrum (closed loops and some OBC energies) in the mobility edge phase to a complete collapse toward the OBC spectrum in the bulk localized phase. Such transitions are observed as the strength of disorder is increased. Depending on the kind of disorder, different interesting behaviors are found. In particular, for continuous disorder with a radial distribution on the complex energy plane it is shown that in the delocalized phase the energy spectrum under PBC is locked and fully insensitive to disorder, whereas transition to the bulk localized phase is signaled by the change in a topological winding number. When the disorder is described by a discrete distribution, the bulk localization transition never occurs, whereas topological phase transitions associated with PBC energy spectral splittings can be observed.

Received 17 February 2021
Revised 22 March 2021
Accepted 31 March 2021

DOI:https://doi.org/10.1103/PhysRevB.103.144202

Physics Subject Headings (PhySH)

Anderson localization Mathematical physics Skin effect

Disordered systems Random & disordered media

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & OpticalGeneral Physics

Authors & Affiliations

Stefano Longhi ^*

Dipartimento di Fisica, Politecnico di Milano, Piazza L. da Vinci 32, I-20133 Milano, Italy and IFISC (UIB-CSIC), Instituto de Fisica Interdisciplinar y Sistemas Complejos, E-07122 Palma de Mallorca, Spain

^*stefano.longhi@polimi.it

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Vol. 103, Iss. 14 — 1 April 2021

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Images

Figure 1
Schematic of the non-Hermitian lattice with unidirectional hopping under (a) the OBC (linear chain geometry) and (b) the PBC (ring geometry with a magnetic flux $Φ$ ).
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Figure 2
PBC energy spectra in the random dimer model and in the thermodynamic limit $N \to \infty$ as predicted by Eq. (16). The on-site potential $V_{n}$ can take only the two values $- W$ and $W$ with probabilities $p$ and $1 - p$ . In (a) $p = 1 / 2$ , whereas in (b) $p = 1 / 3$ . The various curves, labeled by 1–4, refer to increasing values of disorder $W / J$ . In (a), $W / J = 0$ (curve 1), $W / J = 0.5$ (curve 2), $W / J = W_{c} / J = 1$ (critical point, curve 3), and $W / J = 1.3$ (curve 4). In (b), $W / J = 0$ (curve 1), $W / J = 0.6$ (curve 2), $W / J = W_{c} / J = 0.921$ (critical point, curve 3), and $W / J = 1.3$ (curve 4).
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Figure 3
PBC energy spectra in the random trimer model. The on-site potential $V_{n}$ can take only the three values $- W, 0$ , and $W$ with probabilities $p_{1}, p_{2}$ , and $p_{3} = 1 - p_{1} - p_{2}$ . In (a) $p_{1} = p_{2} = p_{3} = 1 / 3$ , whereas in (b) $p_{1} = 1 / 4, p_{2} = 1 / 3$ , and $p_{3} = 5 / 12$ . The various curves, labeled by 1–4, refer to increasing values of disorder $W / J$ . In (a), $W / J = 0$ (curve 1), $W / J = 1$ (curve 2), $W / J = W_{c} / J = \sqrt[3]{13 / 5} = 1.3751$ (critical point, curve 3), and $W / J = 1.6$ (curve 4). In (b), $W / J = 0$ (curve 1), $W / J = 1$ (curve 2), $W / J = 1.3$ (curve 3), and $W / J = 1.8$ (curve 4). The energy spectra are computed in the thermodynamic limit $N \to \infty$ from the transcendental equation Eq. (12).
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Figure 4
Energy spectra, under the OBC and PBC, for a continuous random disorder with radial symmetry, illustrating the existence of three different phases for $H_{OBC}$ and $H_{PBC}$ depending on the strength of disorder $R_{0}$ . Dark points correspond to eigenenergies of $H_{OBC}$ solely with eigenmodes squeezed on the right edge of the lattice (skin states), light red points correspond to eigenenergies of both $H_{OBC}$ and $H_{PBC}$ with bulk localized eigenstates, whereas the red circles of radius $J$ and $R$ in the left and central panels correspond to eigenenergies of $H_{PBC}$ solely with extended states. The critical disorder strength $R_{m}$ , above which the system enters in the bulk localized phase, is implicitly defined by Eq. (23) given in the text.
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Figure 5
Numerically computed eigenvalues (energy spectra) of $H_{OBC}$ (open black circles) and $H_{PBC}$ (bold red circles) in a lattice comprising $N = 600$ sites for a single realization of on-site potential disorder with radial symmetry. The radial probability distribution of disorder $g (R)$ is given by Eq. (24) of the main text. In (a) $R_{0} / J = 0.5$ , in (b) $R_{0} / J = 1$ , in (c) $R_{0} / J = 1.5$ , in (d) $R_{0} / J = 2$ , and in (e) $R_{0} / J = 2.8$ . The critical disorder strength $R_{m}$ , above which the system enters into the bulk localized phase, is $R_{m} = e J ≃ 2.7813 J$ . Note that in (a) $H_{PBC}$ is in the delocalized phase and $H_{OBC}$ in the skin phase, the eigenvalues of $H_{PBC}$ lie on the circumference of radius $J$ with the OBC and PBC energy spectra fully disjoint. In (c) and (d) $H_{PBC}$ is in the mobility edge phase and $H_{OBC}$ in the mixed phase with eigenenergies of $H_{PBC}$ partly distributed along a circle of radius $R < J$ , corresponding to extended states, and partly coincident with eigenenergies of $H_{OBC}$ (the points in the outer of the circle of radius $R$ ), corresponding to bulk localized eigenstates. The eigenvectors of $H_{OBC}$ in the interior of the circle of radius $R$ are skin states. In (b) the system is at the critical point separating delocalized and nobility edges phases for $H_{PBC}$ (or likewise skin and mixed phases for $H_{OBC}$ ). In (e) the system is in the bulk localized phase with equal PBC/OBC energy spectra and all eigenfunctions localized in the bulk.
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Figure 6
Behavior of the radius $R$ versus disorder strength $R_{0}$ for the on-site potential disorder with radial distribution given by Eq. (24) in the mobility edge phase. Vertical circles refer to numerical simulations for 20 different realizations of disorder in a lattice comprising $N = 600$ sites under the PBC, whereas the solid curve is the theoretical prediction of Eq. (26).
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