Random two-body dissipation induced non-Hermitian many-body localization
Authors: Liu Jing-Hu , Xu Zhi-Hao ,   State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China,   Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China
Published: January 2024

<sec>Recent researches on disorder-driven many-body localization (MBL) in non-Hermitian quantum systems have aroused great interest. In this work, we investigate the non-Hermitian MBL in a one-dimensional hard-core Bose model induced by random two-body dissipation, which is described by</sec><sec>　　　　　　　　　　　<inline-formula><tex-math id="M1">\begin{document}$ \hat{H}=\displaystyle\sum\limits_{j}^{L-1}\left[ -J\left( \hat{b}_{j}^{\dagger}\hat{b}_{j+1}+\hat {b}_{j+1}^{\dagger}\hat{b}_{j}\right) +\frac{1}{2}\left( U-{\mathrm{i}}\gamma_{j}\right) \hat{n}_{j}\hat{n}_{j+1}\right] \notag,$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231987_M1.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231987_M1.png"/></alternatives></inline-formula> </sec><sec>with the random two-body loss <inline-formula><tex-math id="M2">\begin{document}$\gamma_j\in\left[0,W\right]$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231987_M2.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231987_M2.png"/></alternatives></inline-formula>. By the level statistics, the system undergoes a transition from the AI<inline-formula><tex-math id="M3">\begin{document}$^{\dagger}$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231987_M3.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231987_M3.png"/></alternatives></inline-formula> symmetry class to a two-dimensional Poisson ensemble with the increase of disorder strength. This transition is accompanied by the changing of the average magnitude (argument) <inline-formula><tex-math id="M4">\begin{document}$\overline{\left\langle {r}\right\rangle}$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231987_M4.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231987_M4.png"/></alternatives></inline-formula> (<inline-formula><tex-math id="M5">\begin{document}$\overline{-\left\langle \cos {\theta}\right\rangle }$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231987_M5.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231987_M5.png"/></alternatives></inline-formula>) of the complex spacing ratio, shifting from approximately 0.722 (0.193) to about 2/3 (0). The normalized participation ratios of the majority of eigenstates exhibit finite values in the ergodic phase, gradually approaching zero in the non-Hermitian MBL phase, which quantifies the degree of localization for the eigenstates. For weak disorder, one can see that average half-chain entanglement entropy <inline-formula><tex-math id="M8">\begin{document}$\overline{\langle S \rangle}$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231987_M8.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231987_M8.png"/></alternatives></inline-formula> follows a volume law in the ergodic phase. However, it decreases to a constant independent of <i>L</i> in the deep non-Hermitian MBL phase, adhering to an area law. These results indicate that the ergodic phase and non-Hermitian MBL phase can be distinguished by the half-chain entanglement entropy, even in non-Hermitian system, which is similar to the scenario in Hermitian system. Finally, for a short time, the dynamic evolution of the entanglement entropy exhibits linear growth with the weak disorder. In strong disorder case, the short-time evolution of <inline-formula><tex-math id="M9">\begin{document}$\overline{S(t)}$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231987_M9.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231987_M9.png"/></alternatives></inline-formula> shows logarithmic growth. However, when <inline-formula><tex-math id="M10">\begin{document}$t\geqslant10^2$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231987_M10.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231987_M10.png"/></alternatives></inline-formula>, <inline-formula><tex-math id="M11">\begin{document}$\overline{S(t)}$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231987_M11.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231987_M11.png"/></alternatives></inline-formula> can stabilize and tend to the steady-state half-chain entanglement entropy <inline-formula><tex-math id="M12">\begin{document}$\overline{ S_0 }$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231987_M12.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231987_M12.png"/></alternatives></inline-formula>. The results of the dynamical evolution of <inline-formula><tex-math id="M13">\begin{document}$\overline{S(t)}$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231987_M13.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231987_M13.png"/></alternatives></inline-formula> imply that one can detect the occurrence of the non-Hermitian MBL by the short-time evolution of <inline-formula><tex-math id="M14">\begin{document}$\overline{S(t)}$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231987_M14.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231987_M14.png"/></alternatives></inline-formula>, and the long-time behavior of <inline-formula><tex-math id="M15">\begin{document}$\overline{S(t)}$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231987_M15.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231987_M15.png"/></alternatives></inline-formula> signifies the steady-state information.</sec>