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arXiv:2102.01097 · Physical Review A

Eigenstate entanglement entropy in a PT -invariant non-Hermitian system

Ranjan Modak, Bhabani Prasad Mandal

Published Jun 24, 2021• 27 citations
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Abstract

Much has been learned about universal properties of the eigenstate entanglement entropy for one-dimensional lattice models, which is described by a Hermitian Hamiltonian, while much less has been understood for non-Hermitian systems. In the present work we study a non-Hermitian, noninteracting model of fermions which is invariant under combined $PT$ transformation. Our models show a phase transition from a $PT$ unbroken phase to broken phase as we tune the Hermiticity-breaking parameter. Entanglement entropy of such systems can be defined in two different ways, depending on whether we consider only right (or equivalently, only left) eigenstates or a combination of both left and right eigenstates which form a complete set of biorthonormal eigenstates. We demonstrate that the entanglement entropy of the ground state and also of the typical excited states shows some unique features in both of these phases of the system. Most strikingly, entanglement entropy obtained taking a combination of both left and right eigenstates shows an exponential divergence with system size at the transition point. While in the $PT$-unbroken phase, the entanglement entropy obtained from only the right (or equivalently, left) eigenstates shows identical behavior to an equivalent Hermitian system which is connected to the non-Hermitian system by a similarity transformation.

DOI
10.1103/physreva.103.062416

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Much has been learned about universal properties of the eigenstate entanglement entropy for one-dimensional lattice models, which is described by a Hermitian Hamiltonian, while much less has been understood for non-Hermitian systems. In the present work we study a non-Hermitian, noninteracting model of fermions which is invariant under combined PTPTPT transformation. Our models show a phase transition from a PTPTPT unbroken phase to broken phase as we tune the Hermiticity-breaking parameter. Entanglement entropy of such systems can be defined in two different ways, depending on whether we consider only right (or equivalently, only left) eigenstates or a combination of both left and right eigenstates which form a complete set of biorthonormal eigenstates. We demonstrate that the entanglement entropy of the ground state and also of the typical excited states shows some unique features in both of these phases of the system. Most strikingly, entanglement entropy obtained taking a combination of both left and right eigenstates shows an exponential divergence with system size at the transition point. While in the PTPTPT-unbroken phase, the entanglement entropy obtained from only the right (or equivalently, left) eigenstates shows identical behavior to an equivalent Hermitian system which is connected to the non-Hermitian system by a similarity transformation.
We study the properties of the entanglement spectrum in gapped non-interacting non-Hermitian systems, and its relation to the topological properties of the system Hamiltonian. Two different families of entanglement Hamiltonians can be defined in non-Hermitian systems, depending on whether we consider only right (or equivalently only left) eigenstates or a combination of both left and right eigenstates. We show that their entanglement spectra can still be computed efficiently, as in the Hermitian limit. We discuss how symmetries of the Hamiltonian map into symmetries of the entanglement spectrum depending on the choice of the many-body state. Through several examples in one and two dimensions, we show that the biorthogonal entanglement Hamiltonian directly inherits the topological properties of the Hamiltonian for line gapped phases, with characteristic singular and energy zero modes. The right (left) density matrix carries distinct information on the topological properties of the many-body right (left) eigenstates themselves. In purely point gapped phases, when the energy bands are not separable, the relation between the entanglement Hamiltonian and the system Hamiltonian breaks down.
Non-Hermitian degeneracies are usually discussed through spectral coalescence, whereas entanglement is a property of eigenvectors and need not be fixed by the eigenvalues alone. We formulate a compact bipartite model that separates these two notions. A Hermitian operator with a degenerate eigenspace is transformed by an invertible non-unitary similarity map. The resulting Hamiltonian is non-Hermitian and retains a non-defective degeneracy at every finite value of the non-Hermiticity parameter. For an exactly solvable two-qubit realization, the right eigenstates evolve continuously from product states to maximally entangled states although the spectrum is unchanged. We distinguish the positive right-state reduced density matrix from the generally non-positive biorthogonal reduction, for which entropy may become complex. The same two-qubit solution gives a closed partial-transpose negativity and a Schmidt-gauged non-local magic. Entanglement grows monotonically with the non-Hermiticity parameter, whereas the non-local magic vanishes for both the product and maximally entangled limits and is largest at an intermediate coupling. In larger bipartite spaces, the Page entropy and Haar-averaged purity provide reference values for eigenstate typicality. These diagnostics separate non-defective degeneracy, exceptional-point sensitivity, Haar-typical entanglement, and non-stabilizer correlations without relying on a proliferation of basis-dependent spectral quantities.
The scaling law of entanglement entropy could undergo qualitative changes during the nonunitary evolution of a quantum many-body system. In this work, we uncover such entanglement phase transitions for free fermions in one-dimensional non-Hermitian quasicrystals (NHQCs). We identify two types of entanglement transitions with different scaling laws and critical behaviors due to the interplay between non-Hermitian effects and quasiperiodic potentials. The first type represents a typical volume-law to area-law transition, which happens together with a PT-symmetry breaking and a localization transition. The second type features an abnormal log-law to area-law transition, which is mediated by a critical phase with a volume-law scaling in the steady-state entanglement entropy. These entangling phases and transitions are demonstrated in two representative models of noninteracting NHQCs. Our results thus advance the study of entanglement transitions in non-Hermitian disordered systems and further disclose the rich entanglement patterns in NHQCs.
We consider a PT-symmetric Fermi gas with an exceptional point, representing the critical point between PT-symmetric and symmetry broken phases. The low energy spectrum remains linear in momentum and is identical to that of a Hermitian Fermi gas. The fermionic Green's function decays in a power law fashion for large distances, as expected from gapless excitations, although the exponent is reduced from -1 due to the quantum Zeno effect. In spite of the gapless nature of the excitations, the ground state entanglement entropy saturates to a finite value, independent of the subsystem size due to the non-Hermitian correlation length intrinsic to the system. Attractive or repulsive interaction drives the system into the PT-symmetry broken regime or opens up a gap and protects PT symmetry, respectively. Our results challenge the concept of universality in non-Hermitian systems, where quantum criticality can be masked due to non-Hermiticity.
Entanglement entropy is a powerful tool in characterizing universal features in quantum many-body systems. In quantum chaotic Hermitian systems, typical eigenstates have near maximal entanglement with very small fluctuations. Here, we show that for Hamiltonians displaying non-Hermitian many-body quantum chaos, modeled by the Ginibre ensemble, the entanglement entropy of typical eigenstates is greatly suppressed. The entropy does not grow with the Hilbert space dimension for sufficiently large systems, and the fluctuations are of equal order. We derive the novel entanglement spectrum that has infinite support in the complex plane and strong energy dependence. We provide evidence of universality, and similar behavior is found in the non-Hermitian Sachdev-Ye-Kitaev model, indicating the general applicability of the Ginibre ensemble to dissipative many-body quantum chaos.
The PT -symmetric non-Hermitian systems have been widely studied and explored both in theory and in experiment due to their various interesting features.In this work we focus on the dynamical features of a triplequbit system, one of which evolves under a local PT -symmetric Hamiltonian.An abnormal dynamic pattern in the entropy evolution process is identified that presents a parameter-dependent stable state, determined by the non-Hermiticity of Hamiltonian in the broken phase of PT symmetry.The entanglement and mutual information of a two-body subsystem can increase beyond the initial values, which do not exist in the Hermitian and two-qubit PT -symmetric systems.Moreover, an experimental demonstration of the stable states in non-Hermitian systems with nonzero entropy and entanglement is realized on a four-qubit quantum simulator with nuclear spins.Our work reveals the distinctive dynamic features in the triple-qubit PT -symmetric system and paves the way for practical quantum simulation of multiparty non-Hermitian system on quantum computers.