Non-Hermiticity induced thermal entanglement phase transition

Abstract

Theoretical analysis of a prototypical two-qubit effective non-Hermitian system characterized by asymmetric Heisenberg $XY$ interactions in the absence of external magnetic fields demonstrates that maximal bipartite entanglement and quantum phase transitions can be induced exclusively through non-Hermiticity. At thermal equilibrium as $T\rightarrow 0$, the system attains maximal entanglement ${C}=1$ for values of the non-Hermiticity parameter greater than a critical value $γ>γ_c=J\sqrt{(1-δ^2)}$, where $J$ denotes the exchange interaction and $δ$ represents the anisotropy of the system; conversely, for $γ< γ_c$, entanglement is nonmaximal and given by ${C} = \sqrt{(1 - (γ/J)^2)}$. The entanglement undergoes a discontinuous transition to zero precisely at $γ= γ_c$. This phase transition originates from the closing of the energy gap at a non-Hermiticity-driven ground state degeneracy, which is fundamentally different from an exceptional point. This work suggests the use of singular-value-decomposition generalized density matrix for the computation of entanglement in bi-orthogonal systems.

Figures (5)

• Figure 1: Non-Hermiticity induced Hermitian degeneracy. Energy spectra of $H$ for two distinct values of $\delta$, illustrate the appearance of a Hermitian degeneracy induced by non-Hermiticity, which differs from an exceptional point (EP). When the parameter $\gamma$ is varied, the ground state and the first excited state interchange their positions. It is also observed that ground-state-interchange takes place at smaller values of $\gamma$ when $\delta$ is larger. This state switching plays a crucial role in the thermal entanglement transition described in the main text. Here, $J=1$ is chosen.
• Figure 2: Thermal entanglement in the non-Hermitian Heisenberg-Ising model $(\delta=1)$. (a) The concurrence $C$, calculated from the thermal mixed state $\rho$ as defined in Eq. \ref{['eq-rho']}, is presented as a function of $\gamma$ at three distinct temperatures. At temperature near zero, entanglement reaches its maximum for non-zero values of $\gamma$, whereas at finite temperatures, a stronger non-Hermiticity is required to achieve comparable entanglement. In panel (b), the concurrence ${C}$ is plotted against temperature for three different values of $\gamma$, illustrating an exponential decay of entanglement with increasing temperature. Panel (c) depicts ${C}$ within the entire parameter range $0 \leq T \leq J/3$ and $0 \leq \gamma < J$. Region below the dashed line represents concurrence ${C}>0.9$, follows from the Eq. \ref{['eq-7']}. Maximal entanglement observed at $T=0$ originates from the non-Hermiticity-assisted non-degenerate ground state, which corresponds to a Bell state, as shown in panel (d). This behavior contrasts sharply with that of the Hermitian system subjected to an external transverse field $B \hat{z}$, described by $H = H_{XY} + B(S_1^z + S_2^z)$; its ground state is non-maximally entangled. Corresponding energy spectrum, $\{\pm\sqrt{J^2+B^2},\pm J\}$, and zero temperature concurrence are provided in panel (e) for comparison, where $\alpha_{\pm} = B \pm \sqrt{J^2 + B^2}$.
• Figure 3: Non-Hermiticity induced thermal entanglement phase transition. Thermal concurrence at absolute zero temperature ($T=0$) and at $T=0.1J$ is shown in panels (a) and (b), respectively, across the full range of the anisotropy parameter $0 \leq \delta \leq1$ and non-Hermiticity $0 \leq \gamma < J$. The results indicate that entanglement experiences a discontinuous transition from $C=1$ when $\gamma > \gamma_c$ to $C=0$ exactly at the critical point $\gamma_c = J(1-\delta^2)^{1/2}$, followed by an asymptotic increase for values of $\gamma < \gamma_c$. This sharp discontinuity in zero temperature entanglement at the critical non-Hermiticity $\gamma_c$ serves as a hallmark of quantum phase transition. Panel (c) shows that the discontinuity shifts toward lower values of $\gamma$ as $\delta$ increases. Panel (d) shows that the entanglement phase transition coincides with the occurrence of non-Hermiticity-assisted Hermitian degeneracy where $E_0 = E_1$ (marked by circles), where a change in the ground state from a non-maximally entangled to a maximally entangled state as a function of $\gamma$ occurs.
• Figure 4: The contour lines defined by the equation $(J+\gamma_1)(J+\gamma_2) = J^2 \delta^2$ in the $(\gamma_1, \gamma_2)$ parameter plane indicate the transition in ground-state entanglement from maximal to nonmaximal values for different anisotropy $\delta$. For each specific $\delta$, the entanglement remains maximal below the corresponding contour line, while above it, the entanglement becomes nonmaximal. The special case of $\gamma_1=-\gamma_2$ is shown by the dashed line.
• Figure 5: Various entanglement measures, such as entropy ($S$), concurrence ($C$), and entanglement of formation ($\xi$), are presented for the bi-orthogonal pure state $\lvert R_0\rangle\langle L_0\rvert$. It is observed that the conventional density matrix $\rho$ in Eq. \ref{['Eq-B1']} yields inconsistent entanglement values, whereas measures derived from the SVD density matrix $\rho^{SVD}$ in Eq. \ref{['Eq-B3']} provide consistent results.