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arXiv:2212.12403 · Physical Review A

Detecting exceptional points through dynamics in non-Hermitian systems

Keshav Das Agarwal, Tanoy Kanti Konar, Leela Ganesh Chandra Lakkaraju, Aditi Sen

Published Jul 25, 2024• 14 citations
arXivHTMLPDF

Abstract

Non-Hermitian rotation-time reversal $(\mathcal{RT})$-symmetric spin models possess two distinct phases, the unbroken phase in which the entire spectrum is real and the broken phase which contains complex eigenspectra, thereby indicating a transition point, referred to as an exceptional point. We report that the dynamical quantities, namely, the short- and long-time average of the Loschmidt echo, which is the overlap between the initial and the final states, and the corresponding rate function can faithfully predict the exceptional point. In particular, when the initial state is prepared as the ground state in the unbroken phase of the non-Hermitian Hamiltonian and the system is quenched to either the broken or unbroken phase, we analytically demonstrate that the rate function and the average Loschmidt echo can distinguish between the quench that occurred in the broken or the unbroken phase for the nearest-neighbor non-Hermitian $XY$ model with uniform and alternating magnetic fields, thereby indicating the exceptional point. Furthermore, we exhibit that such quantities are capable of identifying the exceptional point even in models like the non-Hermitian short- and long-range $XYZ$ model with magnetic field, which can only be solved numerically, thereby establishing it as detection criteria for recognizing exceptional points.

DOI
10.1103/physreva.110.012226

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Non-Hermitian rotation-time reversal (RT)(\mathcal{RT})(RT)-symmetric spin models possess two distinct phases, the unbroken phase in which the entire spectrum is real and the broken phase which contains complex eigenspectra, thereby indicating a transition point, referred to as an exceptional point. We report that the dynamical quantities, namely, the short- and long-time average of the Loschmidt echo, which is the overlap between the initial and the final states, and the corresponding rate function can faithfully predict the exceptional point. In particular, when the initial state is prepared as the ground state in the unbroken phase of the non-Hermitian Hamiltonian and the system is quenched to either the broken or unbroken phase, we analytically demonstrate that the rate function and the average Loschmidt echo can distinguish between the quench that occurred in the broken or the unbroken phase for the nearest-neighbor non-Hermitian XYXYXY model with uniform and alternating magnetic fields, thereby indicating the exceptional point. Furthermore, we exhibit that such quantities are capable of identifying the exceptional point even in models like the non-Hermitian short- and long-range XYZXYZXYZ model with magnetic field, which can only be solved numerically, thereby establishing it as detection criteria for recognizing exceptional points.
In the traditional quantum theory, one-dimensional quantum spin models possess a factorization surface where the ground states are fully separable and have vanishing bipartite as well as multipartite entanglement. We report that in the non-Hermitian counterpart of these models, these factorization surfaces either can predict the exceptional points where the unbroken-to-broken transition occurs or can guarantee the spectrum to be real, thereby suggesting a procedure to reveal the unbroken phase. We first analytically demonstrate it for the nearest-neighbor rotation-time- (RT)(\mathcal{RT})(RT) symmetric XYXYXY model with uniform and alternating transverse magnetic fields, referred to as the iATXYiATXYiATXY model. Exact diagonalization techniques are then employed to establish this fact for the RT\mathcal{RT}RT-symmetric XYZXYZXYZ model with short- and long-range interactions as well as for the long-range iATXYiATXYiATXY model. Moreover, we show that although the factorization surface prescribes the unbroken phase of the non-Hermitian model, the bipartite nearest-neighbor entanglement at the exceptional point is nonvanishing.
Abstract Non-Hermitian systems with exceptional points lead to many intriguing phenomena due to the coalescence of both eigenvalues and corresponding eigenvectors, in comparison to Hermitian systems where only eigenvalues degenerate. In this paper, we propose an alternative and accurate proposal based on the entropy uncertainty relation (EUR) to detect the exceptional points and identify different phases of the non-Hermitian systems. In particular, we reveal a general connection between the EUR and the exceptional points of non-Hermitian system. Compared to the unitary Hermitian dynamics, the behaviors of EUR in the non-Hermitian system are well defined into two different ways depending on whether the system is located in unbroken or broken phase regimes. In the unbroken phase regime where EUR undergoes an oscillatory behavior, while in the broken phase regime where the oscillation of EUR breaks down. Moreover, we identify the critical phenomena of non-Hermitian systems in terms of the EUR in the dynamical limit. It is found that the EUR can detect exactly the critical points of non-Hermitian systems beyond (anti-)PT symmetric systems. Finally, we comment on the prospective experimental situation.
Non-Hermitian quantum systems exhibit qualitatively distinct physical behavior compared to Hermitian systems, a prime example being spectral singularities known as exceptional points. Their relevance in, e.g., quantum sensing, unidirectional transport, and robust lasing makes it important to be able to identify exceptional points through observable features of a many-body system. Here, using as an example a one-dimensional complex XY spin chain realizing both rotation-time RT- and parity-time PT-symmetric regimes, we develop a framework for detecting exceptional points based on the distinction between left and right eigenvectors of the Hamiltonian, which in a non-Hermitian system are no longer the adjoint of each other. We first show that a global measure constructed from the difference between the Hamiltonian and its adjoint locates exceptional points via distinct non-analytic behavior. At the level of observables, differences in local spin correlations evaluated on the right and left eigenstates provide a reliable static detection scheme. In contrast, static bipartite entanglement measures fail to capture this distinction, urging us to study the quantum dynamics of the model. Following a sudden quench, we demonstrate that the time-averaged right-left entanglement entropy difference directly encodes signatures of the exceptional point. In the RT-symmetric regime, it exhibits a pronounced peak at the exceptional point, whereas in the PT-symmetric regime it behaves as an order-parameter-like quantity, remaining finite in one phase and vanishing at the transition. Our results establish a direct link between the structure of non-Hermitian eigenstates and observable signatures of exceptional points, providing a practical route to identify them in existing quantum simulators.
Non-Hermitian spin systems have attracted extensive interest due to their unconventional magnetic properties, rich entanglement resources, and unusual quantum criticality phenomena. In this paper, based on the exact solution of a one-dimensional non-Hermitian spin-1/2 XY model with rotation-time-reversal (RT\mathcal{RT}RT) symmetry [X. Z. Zhang et al., Phys. Rev. A 87, 012114 (2013)], we study the ground-state energy density, magnetization, correlation functions, quantum entanglement, and correlation in RT\mathcal{RT}RT-symmetric and RT\mathcal{RT}RT-broken phases as well as their characteristics at the exceptional point. We find that the energy density rises faster with the non-Hermitian parameter $|\ensuremath{\gamma}|$ in the symmetric region than in the broken one and is elevated rapidly at the exceptional point, and the decay of magnetization has similar results. We analyze the effect of $\ensuremath{\gamma}$ on the system and find that the energy density decreases linearly with the external magnetic field hhh for $\ensuremath{\gamma}>0,$ while it is bifurcated when $\ensuremath{\gamma}=0$. In addition, the behaviors of the energy density indicate that the phase transition caused by symmetry breaking is second order, which is further demonstrated by the magnetization, quantum entanglement, etc. The numerical results of the correlation functions surprisingly indicate that the RT\mathcal{RT}RT-broken phase has quasi-long-range order, which is quite different from the Hermitian XY model. Especially, the crossover behavior of the ground-state entanglement shows that it increases with $\ensuremath{\gamma}$ in the symmetric region, which is opposite to the case of the broken one, and its maximum always appears at the exceptional point. The above behaviors at the phase boundary are actually ascribed to the fierce competition between $\ensuremath{\gamma}$ and hhh, which results in a rapid decline of magnetization and the appearance of the maximum of entanglement.
PT\mathcal{PT}PT-Symmetric Spin--Boson Model with a Continuous Bosonic Spectrum: Exceptional Points and Dynamics
This work studies a PT\mathcal{PT}PT-symmetric non-Hermitian spin--boson model, consisting of a non-Hermitian two-level system coupled to a continuous bosonic bath. The static properties of the system are analyzed through a projection method derived from the displacement operator. We find that only a single exceptional point (EP) emerges, in contrast to non-Hermitian spin--boson models with finite modes, which typically exhibit multiple EPs. Notably, only a single real eigenvalue is found before the EP, which differs markedly from typical non-Hermitian systems where a pair of real eigenvalues precedes the EP. The time evolution of observables is further investigated via the Dirac--Frenkel time-dependent variational principle. Compared to its Hermitian counterpart, the non-Hermitian model exhibits distinct dynamical signatures, most notably the emergence of oscillations with periodic amplified amplitude. In the PT\mathcal{PT}PT-unbroken phase, the system exhibits sustained oscillatory dynamics with suppressed decoherence, whereas in the PT\mathcal{PT}PT-broken phase, additional dissipative channels accelerate decoherence and drive rapid convergence toward a stable steady state. These results shed light on how PT\mathcal{PT}PT symmetry protects coherent light--matter interactions in non-Hermitian quantum systems.
We study the exceptional-point (EP) structure and the associated quantum dynamics in a system consisting of a non-Hermitian qubit and a Hermitian qubit. We find that the system possesses two sets of EPs, which divide the system-parameter space into P T -symmetry unbroken, partially broken and fully broken regimes, each with distinct quantum-dynamics characteristics. Particularly, in the partially broken regime, while the P T -symmetry is generally broken in the whole four-dimensional Hilbert space, it is preserved in a two-dimensional subspace such that the quantum dynamics in the subspace are similar to those in the P T -symmetry unbroken regime. In addition, we reveal that the competition between the inter-qubit coupling and the intra-qubit driving gives rise to a complex pattern in the EP variation with system parameters.