Random two-body dissipation induced non-Hermitian many-body localization

Abstract

<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>