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arXiv:2408.07122 · Physical review. B./Physical review. B

Slow approach to adiabaticity in many-body non-Hermitian systems: The Hatano-Nelson model

Léonce Dupays, Adolfo del Campo, Balázs Dóra

Published Jan 13, 2025• 3 citations
arXivHTMLPDF

Abstract

We explore the near adiabatic dynamics in a non-Hermitian quantum many-body system by investigating a finite-time ramp of the imaginary vector potential in the interacting Hatano-Nelson model. The excess energy, the Loschmidt echo, and the density imbalance are analyzed using bosonization and exact diagonalization. The energy becomes complex valued, despite the instantaneous Hamiltonian having the same real spectrum throughout. The adiabatic limit is approached very slowly through damped oscillations. The decay scales with ${\ensuremath{\tau}}^{\ensuremath{-}1}$ with $\ensuremath{\tau}$ the ramp duration, while the oscillation period is $2L/v$ with $v$ the Fermi velocity and $L$ the system length. Yet, without the need for auxiliary controls, a shortcut to adiabaticity is found for ramp times commensurate with the period. Our work highlights the intricate interplay of adiabaticity and non-Hermitian many-body physics.

DOI
10.1103/physrevb.111.045130

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We explore the near adiabatic dynamics in a non-Hermitian quantum many-body system by investigating a finite-time ramp of the imaginary vector potential in the interacting Hatano-Nelson model. The excess energy, the Loschmidt echo, and the density imbalance are analyzed using bosonization and exact diagonalization. The energy becomes complex valued, despite the instantaneous Hamiltonian having the same real spectrum throughout. The adiabatic limit is approached very slowly through damped oscillations. The decay scales with \ensuremathτ\ensuremath−1{\ensuremath{\tau}}^{\ensuremath{-}1}\ensuremathτ\ensuremath−1 with \ensuremathτ\ensuremath{\tau}\ensuremathτ the ramp duration, while the oscillation period is 2L/v2L/v2L/v with vvv the Fermi velocity and LLL the system length. Yet, without the need for auxiliary controls, a shortcut to adiabaticity is found for ramp times commensurate with the period. Our work highlights the intricate interplay of adiabaticity and non-Hermitian many-body physics.
While non-Hermitian Hamiltonians have been experimentally realized in cold atom systems, it remains an outstanding open question of how to experimentally measure their complex energy spectra in momentum space for a realistic system with boundaries. The existence of non-Hermitian skin effects may make the question even more difficult to address given the fact that energy spectra for a system with open boundaries are dramatically different from those in momentum space; the fact may even lead to the notion that momentum-space band structures are not experimentally accessible for a system with open boundaries. Here, we generalize the widely used radio-frequency spectroscopy to measure both real and imaginary parts of complex energy spectra of a non-Hermitian quantum system for either bosonic or fermionic atoms. By weakly coupling the energy levels of a non-Hermitian system to auxiliary energy levels, we theoretically derive a formula showing that the decay of atoms on the auxiliary energy levels reflects the real and imaginary parts of energy spectra in momentum space. We further prove that measurement outcomes are independent of boundary conditions in the thermodynamic limit, providing strong evidence that the energy spectrum in momentum space is experimentally measurable. We finally apply our non-Hermitian absorption spectroscopy protocol to the Hatano-Nelson model and non-Hermitian Weyl semimetals to demonstrate its feasibility.
The entanglement dynamics in a non-Hermitian quantum system is studied numerically and analyzed from the viewpoint of quasiparticle picture. As a concrete model, we consider a one-dimensional tight-binding model with asymmetric hopping (Hatano-Nelson model) under onsite disorder and nearest-neighbor interaction. As opposed to an assertion of previous studies, the entanglement dynamics in this non-Hermitian quantum system is very different from the one in its Hermitian counterpart, especially in the delocalized regime with weak disorder; there the entanglement entropy Sent(t){S}_{\mathrm{ent}}(t)Sent​(t) shows a characteristic nonmonotonic time evolution. We have clarified and quantified the nature of this behavior in the quasiparticle picture. In the asymptotic regime of $t\ensuremath{\rightarrow}\ensuremath{\infty}$, the entanglement entropy Sent(t){S}_{\mathrm{ent}}(t)Sent​(t) in this regime saturates to a much suppressed value, which increases only logarithmically with respect to the size of the subsystem.
Shortcuts to adiabaticity (STA) reproduce adiabatic dynamics in finite time, but their counterdiabatic implementation relies on the adiabatic gauge potential (AGP), which is difficult to compute and implement in many-body systems and whose extension to open and non-Hermitian settings has remained largely model-specific. Here, we develop a general, diagonalization-free framework for engineering STA in non-Hermitian systems by representing the AGP in Krylov space. Starting from an integral representation of the counterdiabatic control, we recast the AGP as a nested-commutator series with controlled locality and generate the associated Krylov basis using the bi-Lanczos and Arnoldi algorithms. This reduces the exact or truncated AGP to a sparse tridiagonal or upper-Hessenberg matrix equation that generalizes the Hermitian construction. We demonstrate the method on a decaying two-level atom, where it recovers the exact drive and signals the exceptional point; on the interacting Hatano-Nelson model, where truncated controls rapidly suppress nonadiabatic excitations; and on a PT-symmetric Heisenberg chain, whose AGP norm detects the PT-symmetry-breaking transition. Throughout, the expansion converges with only a small fraction of the full Krylov space, offering a practical route to fast, accurate control of many-body non-Hermitian systems.
Abstract Many-body localized phases retain memory of their initial conditions in disordered interacting systems with unitary dynamics. The stability of the localized phase due to the breakdown of unitarity is of relevance to experiment in the presence of dissipation. Here we investigate the impact of non-Hermitian perturbations on many-body localization. We focus on the interacting Hatano-Nelson model which breaks unitarity via asymmetric hopping. We explore the phase diagram for the mid-spectrum eigenstates as a function of the interaction strength and the non-Hermiticity. In contrast to the non-interacting case, our findings are consistent with a two-step approach to the localized regime. We also study the dynamics of the particle imbalance. We show that the distribution of relaxation time scales differs qualitatively between the localized and ergodic phases. Our findings suggest the possibility of an intermediate dynamical regime in disordered open systems.
Simulation and analysis of multidimensional dynamics of a quantum non-Hmeritian system is a challenging problem. Gaussian wavepacket dynamics has proven to be an intuitive semiclassical approach to approximately solving the dynamics of quantum systems. A Gaussian wavepacket approach is proposed for a continuous space extension to the Hatano-Nelson model that enables transparent analysis of the dynamics in terms of complex classical trajectories. We demonstrate certain cases where the configuration space trajectory can be made fully real by transforming the initial conditions to account for the non-Hermiticity appropriately through the momentum coordinates. However, in general the complex phase space is unavoidable. For the cases where the trajectory is real, the effective force can be decomposed into that due to the potential energy surface and that due to the imaginary vector potential. The impact of the vector potential on the trajectory of the wavepacket is directly proportional to both the strength of the vector potential and the width of the wavepacket.
A quantum particle in a slowly changing potential well $V(x,t)=V(x\ensuremath{-}{x}_{0}(\ensuremath{\epsilon}t))$, periodically shaken in time at a slow frequency $\ensuremath{\epsilon}$, provides an important quantum mechanical system where the adiabatic theorem fails to predict the asymptotic dynamics over time scales longer than $\ensuremath{\sim}1/\ensuremath{\epsilon}$. Specifically, we consider a double-well potential V(x)V(x)V(x) sustaining two bound states spaced in frequency by ${\ensuremath{\omega}}_{0}$ and periodically shaken in a complex plane. Two different spatial displacements x0(t){x}_{0}(t)x0​(t) are assumed: the real spatial displacement ${x}_{0}(\ensuremath{\epsilon}t)=Asin(\ensuremath{\epsilon}t)$, corresponding to ordinary Hermitian shaking, and the complex one ${x}_{0}(\ensuremath{\epsilon}t)=A\ensuremath{-}Aexp(\ensuremath{-}i\ensuremath{\epsilon}t)$, corresponding to non-Hermitian shaking. When the particle is initially prepared in the ground state of the potential well, breakdown of adiabatic evolution is found for both Hermitian and non-Hermitian shaking whenever the oscillation frequency $\ensuremath{\epsilon}$ is close to an odd resonance of ${\ensuremath{\omega}}_{0}$. However, a different physical mechanism underlying nonadiabatic transitions is found in the two cases. For the Hermitian shaking, an avoided crossing of quasienergies is observed at odd resonances and nonadiabatic transitions between the two bound states, resulting in Rabi flopping, can be explained as a multiphoton resonance process. For the complex oscillating potential well, breakdown of adiabaticity arises from the appearance of Floquet exceptional points at exact quasienergy crossing.