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arXiv:2012.01435 · Physical Review Letters

Entanglement and Purification Transitions in Non-Hermitian Quantum Mechanics

Sarang Gopalakrishnan, Michael J. Gullans

Published Apr 30, 2021• 145 citations
arXivHTMLPDF

Abstract

A quantum system subject to continuous measurement and postselection evolves according to a non-Hermitian Hamiltonian. We show that, as one increases the strength of postselection, this non-Hermitian Hamiltonian can undergo a spectral phase transition. On one side of this phase transition (for weak postselection), an initially mixed density matrix remains mixed at all times, and an initially unentangled state develops volume-law entanglement; on the other side, an arbitrary initial state approaches a unique pure state with low entanglement. We identify this transition with an exceptional point in the spectrum of the non-Hermitian Hamiltonian, at which PT symmetry is spontaneously broken. We characterize the transition as well as the nontrivial steady state that emerges at late times in the mixed phase using exact diagonalization and an approximate, analytically tractable mean-field theory; these methods yield consistent conclusions.

DOI
10.1103/physrevlett.126.170503

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A quantum system subject to continuous measurement and postselection evolves according to a non-Hermitian Hamiltonian. We show that, as one increases the strength of postselection, this non-Hermitian Hamiltonian can undergo a spectral phase transition. On one side of this phase transition (for weak postselection), an initially mixed density matrix remains mixed at all times, and an initially unentangled state develops volume-law entanglement; on the other side, an arbitrary initial state approaches a unique pure state with low entanglement. We identify this transition with an exceptional point in the spectrum of the non-Hermitian Hamiltonian, at which PT symmetry is spontaneously broken. We characterize the transition as well as the nontrivial steady state that emerges at late times in the mixed phase using exact diagonalization and an approximate, analytically tractable mean-field theory; these methods yield consistent conclusions.
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The authors demonstrate a mechanism whereby the Yang-Lee edge singularity in certain $P\phantom{\rule{0}{0ex}}T$-symmetric non-Hermitian Hamiltonians can trigger a first-order entanglement transition. At the transition, the entanglement entropy of the steady state jumps discontinuously from a volume-law to an area-law scaling. Physically, such non-Hermitian Hamiltonian evolutions arise when post-selecting a particular quantum trajectory of a system subject to continuous weak measurements.
The competition between unitary time evolution and quantum measurements could induce phase transitions in the entanglement characteristics of quantum many-body dynamics. In this work, we reveal such entanglement transitions in the context of non-Hermitian Floquet systems. Focusing on noninteracting fermions in a representative bipartite lattice with balanced gain/loss and under time-periodic quenches, we uncover rich patterns of entanglement transitions due to the interplay between driving and non-Hermitian effects. Specially, we find that the monotonic increase of quenched hopping amplitude could flip the system between volume-law and area-law entangled Floquet phases, yielding alternated entanglement transitions. Meanwhile, the increase of gain/loss strength could trigger area-law to volume-law reentrant transitions in the scaling behavior of steady-state entanglement entropy, which are abnormal and highly unexpected in nondriven systems. Connections between entanglement transitions and parity-time-reversal (PT) transitions in Floquet spectra are further established. Our findings not only build a foundation for exploring entanglement phase transitions in Floquet non-Hermitian setups, but also provide efficient means to engineer and control such transitions by driving fields. Published by the American Physical Society 2024
Non-Hermitian quantum many-body systems feature steady-state entanglement transitions driven by the competition between unitary dynamics and dissipation. In this work, we reveal the fundamental role of conservation laws in shaping this competition. Focusing on translation-invariant non-interacting fermionic models with U(1) symmetry, we present a theoretical framework to understand the structure of the steady-state of these models and their entanglement content based on two ingredients: the nature of the spectrum of the non-Hermitian Hamiltonian and the constraints imposed on the steady-state single-particle occupation by the conserved quantities. These emerge from an interplay between Hamiltonian symmetries and initial state, due to the non-linearity of measurement back-action. For models with complex energy spectrum, we show that the steady state is obtained by filling single-particle right eigenstates with the largest imaginary part of the eigenvalue. As a result, one can have partially filled or fully filled bands in the steady-state, leading to an entanglement entropy undergoing a filling-driven transition between critical sub-volume scaling and area-law, similar to ground-state problems. Conversely, when the spectrum is fully real, we provide evidence that local observables can be captured using a diagonal ensemble, and the entanglement entropy exhibits a volume-law scaling independently on the initial state, akin to unitary dynamics. We illustrate these principles in the Hatano-Nelson model with periodic boundary conditions and the non-Hermitian Su-Schrieffer-Heeger model, uncovering a rich interplay between the single-particle spectrum and conservation laws in determining the steady-state structure and the entanglement transitions. These conclusions are supported by exact analytical calculations and numerical calculations relying on the Faber polynomial method.
We study an entanglement phase transition in a class of chaotic non-Hermitian spin chains whose spin-spin coupling terms commute with the non-Hermitian contributions. Two representative models are investigated: the transverse-field Ising model with a complex longitudinal field and the non-Hermitian XX model with a transverse field. By analyzing their complex spectra, we find that both models undergo a dissipation-induced gapless-gapped phase transition when the transverse field exceeds a model-dependent threshold. Interestingly, the complex gap does not vary monotonically with the dissipation rate; instead, it exhibits pronounced oscillations before entering the gapped phase. By simulating their non-unitary dynamics, we show that the steady-state entanglement entropy undergoes a transition from volume-law to area-law scaling as the dissipation rate increases. Moreover, several unexpected features emerge within the volume-law regime: a larger complex gap or dissipation rate may lead to a more entangled steady state. We trace these unusual behaviors of the complex gap and the steady-state entanglement to level crossings between the maximal imaginary level and other spectral levels. Our work uncovers an exotic entanglement transition in chaotic non-Hermitian many-body systems.
The long-time state of a non-Hermitian system is determined by the eigenvalue with the largest imaginary part. In interacting many-body systems this eigenvalue usually cannot be tracked analytically, and the character of the state it selects is unknown. We construct a non-Hermitian spin ensemble of LLL spins with exactly kkk-local all-to-all interactions, in which this dominant eigenvalue can be tracked analytically from the clean limit into the disordered regime. Disorder produces a competition between an isolated spectral outlier and the edge of a many-body spectral bulk. We study three cases: purely anti-Hermitian disorder, purely Hermitian disorder, and mixed disorder of equal strength. For purely anti-Hermitian and mixed disorder, we show that when the bulk overtakes the outlier in imaginary part, the dominant eigenstate switches from an outlier state with low entanglement and magic (nonstabilizerness) to a bulk state with substantially larger entanglement and magic, with both changing at the same threshold. For k≫Lk \gg \sqrt{L}k≫L​ the bulk has a sharp spectral edge and the transition thresholds follow in closed form, while for k≪Lk \ll \sqrt{L}k≪L​ spectral tails broaden the transition into a crossover. Purely Hermitian disorder provides a contrasting case with no outlier-to-bulk switching. Finally, we map the non-Hermitian evolution exactly onto postselected trajectories of a monitored quantum system, connecting this spectral mechanism to measurement-induced transitions.