Probing phase transitions in non-Hermitian systems with quantum entanglement

TL;DR

The paper addresses how non-Hermiticity modifies quantum phase transitions in 1D spin models and how quantum-information measures can reveal these changes. It combines analytical treatment of the non-Hermitian XY model with numerical exact diagonalization of the XXZ model, computing self-normal and biorthogonal two-site densities to obtain mutual information, concurrence, negativity, and quantum coherence. The authors find that Hermitian critical points evolve into exceptional points as non-Hermiticity grows; self-normal and real-part biorthogonal measures reliably detect first-order and PT/RT transitions, while the BKT transition can be captured by concurrence and negativity at small non-Hermiticity, and an unconstrained concurrence resolves failures of the biorthogonal concurrence near exceptional points. The work provides robust entanglement-based probes of non-Hermitian QPTs and offers guidance for experimental exploration of exceptional-point physics in quantum simulators.

Abstract

We study the quantum entanglement and quantum phase transition of the non-Hermitian anisotropic spin-$\frac{1}{2}$ XY model and XXZ model with the staggered imaginary field by analytical methods and numerical exact diagonalization, respectively. Various entanglement measures, including concurrence, negativity, mutual information, and quantum coherence, and both biorthogonal and self-normal quantities are investigated. Both the biorthogonal and self-normal entanglement quantities, except the biorthogonal concurrence, are found to be capable of detecting the first-order and $\mathcal{PT}$ transitions in the XXZ model, as well as the Ising and $\mathcal{RT}$ transitions in the XY model. In addition, we introduce the unconstrained concurrence and demonstrate its effectiveness in detecting these transitions. On the other hand, the Berezinskii-Kosterlitz-Thouless (BKT) transition in the XXZ model is revealed through concurrence and negativity at small non-Hermiticity strengths. Notably, the critical points observed in the Hermitian limit evolve into exceptional points as the strength of the non-Hermiticity increases. Furthermore, we find that the first-order transition survives up to a higher non-Hermiticity strength compared to the BKT transition within the $\mathcal{PT}$-symmetric regime of the XXZ model.

Figures (9)

• Figure 1: Phase diagram of the non-Hermitian XXZ model on the $\gamma-J_z$ plane for system size $N=10$. The color scale shows the magnitude of the concurrence. The first-order transition line between the FM and XY phases is determined from the discontinuous jump in the concurrence (red squares) and in the spin-spin correlation functions (red triangles), which agree well with each other. The BKT transition between the XY and the AFM phases is determined by the maximum of the concurrence (green squares) and the crossing of the spin-spin correlation functions (green triangles), which agree with each other in small $\gamma$. The $\mathcal{PT}$ transition is determined by the discontinuity in the concurrence (blue squares) and the spin-spin correlation (blue triangles), and directly from the energy spectrum (blue circles).
• Figure 2: Concurrence (a-e) and spin-spin correlation (f-j) as a function of $J_z$ for five typical $\gamma$ values. (a-e) The red and blue dashed lines indicate the first-order and the $\mathcal{PT}$ transitions as obtained by the extrema of the concurrence's first derivative. The green dashed lines indicate the BKT transition obtained from the maximum of the concurrence. (f-g) The red and blue dashed lines are obtained by the extrema of the first derivative of $\braket{\sigma_{L/2}^z\sigma_{L/2+1}^z}_{RR}$, and the green dashed lines are obtained by the crossing points of spin-spin correlations.
• Figure 3: The ground state (a-e) and first excited state (f-j) fidelity map of five typical values of $\gamma$. The red, green, and blue dashed line denotes the first-order, BKT, and $\mathcal{PT}$ transitions, respectively, as determined from the self-normal concurrence.
• Figure 4: The negativity (first column), mutual information (second column), quantum coherence (third column), and half-chain entanglement entropy (fourth column) as a function of the imaginary field strength $\gamma$ and the anisotropy strength $J_z$ of the XXZ model. The top, middle, and bottom panel shows the self-normal, the real part of the biorthogonal, and the imaginary part of the biorthogonal quantities, respectively. The dashed lines indicate the phase boundaries determined by the extrema or the discontinuity of the concerned quantities.
• Figure 5: The self-normal entanglement measures as a function of $J_z$ for $\gamma=0$ (top panel) and $\gamma=0.15$ (bottom panel). The red, green and blue dashed lines indicate the first-order, BKT, and $\mathcal{PT}$ transition, respectively, as determined from the self-norm concurrence. All four measures capture the first-order and $\mathcal{PT}$ transitions of the system, but only the negativity detects the BKT transition by its maximum. Insets show a zoom-in of the plot in the vicinity of the BKT transition around $J_z=1$.