Recognizing critical lines via entanglement in non-Hermitian systems

TL;DR

The paper investigates a RT-symmetric non-Hermitian spin chain that combines an imaginary iXY interaction with a Hermitian KSEA term under a transverse field, showing it can describe an XX+KSEA chain coupled to local and nonlocal baths. Through a Jordan–Wigner mapping to a quadratic free-fermion model, the authors identify two regimes: Region I with \gamma>K features an exceptional point at $h_{EP}=\sqrt{1+\gamma^2-K^2}$, while Region II with \gamma\le K is RT-symmetry protected and exhibits a critical point at $h_c=1$ and a factorization surface at $h_f=\sqrt{1+\gamma^2-K^2}$. Nearest-neighbor bipartite entanglement detects both exceptional points and factorization, with its derivative signaling critical lines; dynamical signatures via Loschmidt echo and entanglement fluctuations further reveal equilibrium and dynamical phase transitions, though rate-function signals can be misleading in some parameter ranges. The work demonstrates entanglement as a powerful probe of non-Hermitian many-body physics and its relation to Hermitian counterparts, with potential experimental realization via reservoir engineering and quantum trajectories.

Abstract

The non-Hermitian model exhibits counterintuitive phenomena that are not observed in the Hermitian counterparts. To probe the competition between non-Hermitian and Hermitian interacting components of the Hamiltonian, we focus on a system containing non-Hermitian $XY$ spin chain and Hermitian Kaplan-Shekhtman-Entin-Aharony (KSEA) interactions along with the transverse magnetic field. We show that the non-Hermitian model can be an effective Hamiltonian of a Hermitian $XX$ spin-$\frac{1}{2}$ with KSEA interaction and a local magnetic field that interacts with local and nonlocal reservoirs. The analytical expression of the energy spectrum divides the system parameters into two regimes: in one region, the strength of Hermitian KSEA interactions dominates over the imaginary non-Hermiticity parameter, while in the other, the opposite is true. In the former situation, we demonstrate that the nearest-neighbor entanglement and its derivative can identify quantum critical lines with the variation of the magnetic field. In this domain, we determine a surface where the entanglement vanishes, similar to the factorization surface, known in the Hermitian case. On the other hand, when non-Hermiticity parameters dominate, we report the exceptional and critical points where the energy gap vanishes and illustrate that bipartite entanglement is capable of detecting these transitions as well. Going beyond this scenario, when the ground state evolves after a sudden quench with the transverse magnetic field, both the rate function and the fluctuation of bipartite entanglement quantified via its second moment can detect critical lines generated without quenching dynamics.

Figures (9)

• Figure 1: Schematic representation of the effective non-Hermitian Hamiltonian. When the $XX$ model with KSEA interactions between neighboring sites and transverse magnetic fields are in contact with local (LB) and non-local baths (GB), the dynamics can be governed by the effective non-Hermitian model, studied in this work. It leads to a possible method to realize this Hamiltonian in laboratories.
• Figure 2: Energy, $\pm \epsilon_p$ (ordinate) of the iKSEA model against $\phi_p$. (a) and (b). $\gamma \leq K$ , i.e., Hermitian KSEA interactions dominates over non-Hermiticity parameter while in (c) and (d), the opposite picture is considered with $\gamma >k$. Different lines represent different strength of the magnetic field, $0.6$ (lightest)$\leq h\leq 1.4$ (darkest). Solid and dotted lines indicate real and imaginary energies respectively. All axes are dimensionless.
• Figure 3: Contour plot of nearest-neighbor entanglement against external magnetic field, $h$ (vertical axis) and non-Hermiticity parameter, $\gamma$ (horizontal axis) of the spin chain with both non-Hermitian XY and Hermitian KSEA interactions, given in Eq. (\ref{['eq:ksea_xy']}). Here we set $K=0.75$ and $N=5000$. Lines with triangles and circles represent the exceptional points and factorization surface respectively, while the dashed line represents the critical line. All axes are dimensionless.
• Figure 4: Variation of nearest-neighbor entanglement, $\mathcal{E}$ (ordinate) as a function of external magnetic field, $h$ (abscissa). (a) Bipartite entanglement between the nearest neighbor spins for $\gamma<K$ (solid lines) and $\gamma>K$ (dashed and dashed-dotted lines). Other system parameters are $K = 0.75$ and $N=5000$. (b) Derivative of nearest-neighbor entanglement with $h$ for $\gamma<K$. $\frac{d\mathcal{E}}{dh}$ is non-analytic at the factorization points (gray vertical lines) and at the critical point $h_c=1$ (dotted vertical line). (c) $\frac{d\mathcal{E}}{dh}$ vs $h$ for $\gamma>K$ which is non-analytic at the exceptional points (gray vertical lines) and at the critical point, $h_c=1$ (dotted vertical line). All the axes are dimensionless.
• Figure 5: Rate function, $\lambda(t)$ (vertical axis) as a function of time, $t$ (horizontal axis). (a) Critical times are mentioned in the graph as $t^{*}_{h_1}$. The system size is $N=5000$ and other parameters are $\gamma=0.1, K=0.2$, $h_0=0.4$. Different lines represents $\lambda(t)$ for quenching in different $h_1$ values, with $h_1=0.1, 0.9$ and $2.0$ showing DQPT. (b) DQPT condition on quenched parameter, $h_1$ (solid blue curve) according to Eq.(\ref{['eq:h1_cond']}). No DQPT occurs when the value of $h_1$ lies in the green (dark shaded) region $h_1\in (h_a, h_b)$, while when it crosses the yellow (light shaded) region $h_1\in[0, h_a]\cup[h_b,1]$, it shows DQPT without crossing the critical lines. Here $h_a\sim0.1628, h_b\sim0.8072$ and all the axes are dimensionless.