Non-Hermitian Many-Body Localization

Ryusuke Hamazaki, Kohei Kawabata, and Masahito Ueda

Phys. Rev. Lett. 123, 090603 – Published 30 August 2019

Abstract

Many-body localization is shown to suppress the imaginary parts of complex eigenenergies for general non-Hermitian Hamiltonians having time-reversal symmetry. We demonstrate that a real-complex transition, which we conjecture occurs upon many-body localization, profoundly affects the dynamical stability of non-Hermitian interacting systems with asymmetric hopping that respects time-reversal symmetry. Moreover, the real-complex transition is shown to be absent in non-Hermitian many-body systems with gain and/or loss that breaks time-reversal symmetry, even though the many-body localization transition still persists.

Received 4 December 2018

DOI:https://doi.org/10.1103/PhysRevLett.123.090603

Physics Subject Headings (PhySH)

Anderson localization Cold atoms & matter waves Eigenstate thermalization Localization Open quantum systems Phase transitions

Statistical Physics & ThermodynamicsCondensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

Authors & Affiliations

Ryusuke Hamazaki¹, Kohei Kawabata¹, and Masahito Ueda^1,2

¹Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
²RIKEN Center for Emergent Matter Science, Wako 351-0198, Japan

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Vol. 123, Iss. 9 — 30 August 2019

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Images

Figure 1
(a) Weakly disordered models with asymmetric hopping. Two eigenenergies on the real axis coalesce by a non-Hermitian perturbation and become complex-conjugate pairs. (b) Strongly disordered models with asymmetric hopping. Coalescence of eigenenergies due to perturbation is prohibited by localization. (c) Disordered models with gain and/or loss. Eigenenergies acquire nonzero imaginary parts without coalescence due to the absence of TRS, irrespective of the presence of localization. (d) Eigenenergy statistics (reality and level-spacing statistics) and entanglement entropy of eigenstates for an asymmetric-hopping model [Eq. (1)] and a gain-loss model [Eq. (2)] with varying disorder strength.
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Figure 2
(a) Eigenenergies of the non-Hermitian Hamiltonian [Eq. (1)] with $L = 12$ for two values of $h$ . As $h$ increases, the number of complex eigenenergies decreases. (b) (top) Dependence of $f_{Im}$ on $h$ for $L = 6$ , 8, 10, 12, 14, and 16 [85]. As $L$ increases, $f_{Im}$ increases for $h ≲ h_{c}^{R} (≃ 8)$ and decreases for $h ≳ h_{c}^{R}$ . (bottom) Critical scaling collapse of $f_{Im}$ as a function of $(h - h_{c}^{R}) L^{1 / ν}$ , where we find $h_{c}^{R} = 8.0$ and $ν = 0.5$ . The value of $f_{Im}$ is identified to be zero if it is below the cutoff of imaginary part $C = 10^{- 13}$ ; thus complex eigenvalues ( $| Im E_{α} | ≫ C$ ) and machine errors ( $| Im E_{α} | ≪ C$ ) are clearly separated. (c) (top) Time evolution of the real part of energy of the system for $h = 2$ , 4, 6, 7, 8, 10, 12, and 14, where the initial energy decreases with increasing $h$ . The energy changes in time for $h ≲ h_{c}^{R}$ but stays almost constant for $h ≳ h_{c}^{R}$ . (bottom) Dynamics of half-chain entanglement entropy for $g = 0.1$ (solid) and $g = 0$ (dotted) with different $h$ . For $h = 2$ , $S (t)$ first exhibits a linear growth for both values of $g$ but decreases for $t ≃ 5$ only for $g = 0.1$ . For $h = 14$ , $S (t)$ exhibits a logarithmic growth for both $g = 0$ and 0.1. We take $| ψ_{0} ⟩ = | 1010 \dots ⟩$ as an initial state.
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Figure 3
(a) Nearest-level-spacing distribution of (unfolded) eigenenergies on the complex plane for $h = 2$ and that of eigenenergies on the real axis for $h = 14$ (a single disorder realization with $L = 16$ ). For $h = 2$ , the distribution is a Ginibre distribution $P_{Gin}^{C} (s)$ rather than a Poisson distribution $P_{Po}^{C} (s) = π s / 2 e^{- (π / 4) s^{2}}$ on the complex plane [88]. For $h = 14$ , the distribution is a Poisson distribution on the real axis $P_{Po}^{R} (s) = e^{- s}$ rather than the Wigner-Dyson distribution $P_{WD}^{R} (s) = π s / 2 e^{- (π / 4) s^{2}}$ . Statistics are taken from eigenenergies lying within $\pm 10 %$ of the real and imaginary parts from the middle of the spectrum. (b) (top) Half-chain entanglement entropy $S / L$ obtained by averaging $S_{α} / L$ over disorder [85] and eigenstates for which the eigenenergies lie within $\pm 2 %$ from the middle of the real part of the spectrum. With increasing $h$ , $S / L$ shows a crossover from the volume to area law. (bottom) The critical scaling collapse is confirmed as a function of $(h - h_{c}^{MBL}) L^{1 / ν}$ , where we use $h_{c}^{MBL} = 7.1$ and $ν = 1.3$ . (c) Stability of eigenstates $G$ for different values of $L$ and with ${\hat{V}}_{NH} = {\hat{b}}_{i}^{†} {\hat{b}}_{i + 1}$ [85]. With increasing $h$ , $G$ changes from $\sim α L$ to $\sim - β L$ ( $α, β > 0$ ) at $h_{c}^{MBL} ≃ 7 \pm 1$ .
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Figure 4
(a) Complex spectrum of the non-Hermitian Hamiltonian in Eq. (2) with $γ = 0.1$ for a single disorder realization. Eigenenergies have nonzero imaginary parts, irrespective of disorder strength $h$ . (b) Nearest-level-spacing distributions on the complex plane for eigenenergies having nonzero imaginary parts. The distribution exhibits random-matrix universality slightly different from the Ginibre distribution $P_{Gin}^{C} (s)$ [104] for $h = 2$ and a Poisson distribution of $P_{Po}^{C} (s)$ for $h = 14$ . Statistics are taken from eigenstates of a single disorder realization ( $L = 16$ ) for which the eigenenergies lie within $\pm 10 %$ of the real and imaginary parts from the middle of the spectrum. (c) (top) System-size dependence of $S / L$ averaged over eigenstates from the middle ( $\pm 2 %$ ) of the real part of the spectrum for different values of $L$ with $γ = 0.1$ [85]. Entanglement entropy decreases with increasing $L$ when the many-body localization sets in. (bottom) The critical scaling collapse of $S / L$ is found as a function of $(h - h_{c}^{MBL}) L^{1 / ν}$ with $h_{c}^{MBL} = 4.2$ and $ν = 1.8$ .
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