Detecting exceptional points through dynamics in non-Hermitian systems

TL;DR

This work addresses locating exceptional points in rotation-time reversal ($\mathcal{RT}$) symmetric non-Hermitian spin systems by using dynamical probes. It analyzes analytically solvable $iXY$ and $iATXY$ models and numerically treats $iXYZ$ models to show that the Loschmidt echo $\mathcal{L}(t)$ and its rate function $\lambda(t)$, along with time-averaged quantities $\eta^T$ and $\eta^S$, encode clear signatures of crossing the EP during quenches. The study demonstrates that, for quenches from the unbroken to the broken phase, $\eta^S$ exhibits a kink in $d\eta^S/dh_1$ and the steady-state $\eta^S$ saturates to a nonzero value, while unbroken-to-unbroken quenches produce distinct oscillatory or small-amplitude behavior, consistent with EP locations known from ground-state analysis and factorization surfaces. By combining analytic results with exact diagonalization, the paper establishes dynamical quantifiers as practical EP-detection criteria across both short- and long-range non-Hermitian spin models, with potential experimental relevance for quantum sensing and metrology.

Abstract

Non-Hermitian rotation-time reversal (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.

Figures (5)

• Figure 1: Parameters for the $iXY$ and the $iATXY$ model. The blue (light) shaded region represents the unbroken phase, and the broken phase is marked as red (darker), and the red dashed line represents the exceptional points. Starting with the ground state at magnetic field strength $h_0$, different quenching scenarios are depicted when the magnetic field strength is changed to $h_1<h_{ep}$ or $h_1>h_{ep}$ with $h_{ep}$ being the exceptional point.
• Figure 2: Quench for $iXY$ model with the initial state as the ground state at $h_0=2.0>h^{iXY}_{ep}$ for all $\gamma$ considered. (a) Rate function $\lambda(t)$ (ordinate) is plotted with respect to time ($t$) (abscissa) for the $iXY$ Hamiltonian in Eq. (\ref{['iXYRT']}). The sudden quench is performed by changing the magnetic fields to $h_1$ given in legends. Solid lines represent quenching from unbroken to broken phase, while dotted lines are for the quenching in the same phase. (b) Distinguishing between the broken and unbroken phases using $\eta^S$ (vertical axis) defined in Eq. (\ref{['eq:eta_s']}) against $h_1$ (horizontal axis) for different values of $\gamma$. For calculating $\eta^S$, the averaging is performed during $t \in [20, 200]$, i.e. $\tau_1=20$. (c) Derivative of $\eta^S$ with respect to $h_1$ (vertical axis) is plotted with post-quenched magnetic fields, ($h_1$) (horizontal axis). The nonanalyticity of $\frac{d\eta^S}{d h_1}$ signals the EP marked underneath by a vertical bar which can be obtained analytically with $N=1200$ (see Eq. (\ref{['eq:excep_XY_atxy']})). All the axes are dimensionless.
• Figure 3: Quench for $iATXY$ model with the initial state as the ground state at $h_a=0.5, h_0=3.0>h_{ep}^{iATXY}$. (a) Rate function, $\lambda(t)$, ($y$-axis) as a function of $t$ ($x$-axis). Dotted lines represent quenching from unbroken to broken phase, while solid lines are for the quenching in the same phase. (b) Rate of average Loschmidt echo ($\eta^S$) (ordinate) in the steady state regime against the post-quench magnetic field ($h_1$) (abscissa). From the analysis, we find that the steady state is reached at $t=100$, which is used to compute $\eta^S$ during $\tau_1=100\leq t \leq 500$ with $N=100$. The corresponding actual exceptional points are marked underneath by a vertical bar [see Eq. (\ref{['eq:excep_XY_atxy']})]. (Inset) A magnified view of the oscillations in the rate function with respect to $t$ due to the quench in the unbroken phase. All the axes are dimensionless.
• Figure 4: Evolution of the nearest-neighbor $iXYZ$ model with the magnetic field $h_0=3.0>h_{ep}^{iXYZ}$ in which the initial state is prepared, for different $\Delta$ and $\gamma$. (a) Loschidmt echo, $\mathcal{L}(t)$, (ordinate) with respect to time ($t$) (abscissa), for $\Delta=0.1$ and $\gamma=0.25$ and quenching in the corresponding broken and unbroken phases. The steady-state dynamics shows irregularity which can be due to finite size. (b) Short-time average of Loschmidt echo, $\eta^T$ (ordinate), vs post quenched magnetic field, $h_1$(abscissa) for various $\gamma$ and $\Delta$. The transient regime is taken to be $t=0$ to $\tau_0=30$. (c), (d) Behavior of $\frac{d\eta^T}{dh_1}$ (vertical axis) with respect to $h_1$ (horizontal axis) for different $\Delta$ values. The corresponding exceptional points obtained analytically are marked by a vertical bar [see Eq.( \ref{['eq:excep_XYZ']})]. Here $N=12$. All the axes are dimensionless.
• Figure 5: Predicting EP for the long-range $iXYZ$ with decaying parameter $\alpha=1$ and anisotropy parameter $\gamma=0.25$. The ground state is taken at $h_0=5.0 > h_{ep} = 3.003$ which is in the unbroken phase for all values of $\Delta$ considered. (a) $\mathcal{L}(t)$ (ordinate) with respect to $t$ (abscissa). The transient time behavior is steep when the parameters of the initial and final Hamiltonians are chosen from different phases. (b)-(e) $\eta^T$ (ordinate) against the quenching magnetic field strength, $h_1$, for different $\Delta$. To calculate $\eta^T$, $\tau_0=800$ is taken. The corresponding expected exceptional points are marked by a vertical bar [see Eq. (\ref{['eq:excep_LR']})]. All the axes are dimensionless.