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As mentioned in the research-interest section, we believe it important to understand the dynamics of open quantum systems from microscopic quantum mechanics. As one of such attempts, we discussed many-body localization (MBL) and universality of random matrices, which have been important concepts for thermalization of isolated quantum systems, in non-Hermitian systems and Lindblad Markovian systems.
・Universality of non-Hermitian random matrices
Dyson classified Hermitian random matrices into three classes with respect to the time-reversal symmetry (complex conjugation operation), and showed that the local spectral statistics such as the level spacing distribution has three different universality classes according to these classes. On the other hand, in non-Hermitian random matrix theory, the universality of the local spectral statistics does not change irrespective of the presence or absence of time-reversal symmetry, and thus only one type of universality is known.
Against this backdrop, we focused on the fact that the complex conjugation and transposition operations are inequivalent in non-Hermitian systems. As a result, we found that a new universal level spacing distribution emerges in the class of random matrices related to the symmetry for the transposition operation. We also argue that only three universality classes (see the right figure) exist for level spacing distributions even when we consider other possible symmetries.
Level spacing distributions, which exhibit different universality classes. In addition to the previously known class A, we found two different universality classes for the symmetry of the transposition operation.
・Lindbladian many-body localization
A typical description of an open quantum system is the Lindblad equation, which is obtained by applying, e.g., the Markov approximation. The MBL in open quantum systems under such dissipation has also attracted attention as an important problem. However, conventional studies have focused on how the structure of MBL in isolated systems (e.g., quasi-local conserved quantities) remains due to dissipation, and have not clarified whether sharp transitions unique to open systems exist.
We have discovered a new localization transition unique to disordered open quantum systems. Specifically, we have shown that, in disordered Ising spin systems described by the Lindblad equation, the statistics of the eigenvalue and eigenstate of the Lindblad super-operator undergoes a transition by disorder. In particular, for weak disorder the universal statistics of the non-Hermitian random matrix theory appears, while for strong disorder many-body localization with respect to the off-diagonal degree of freedom of the eigenstate occurs. When this "Lindbladian MBL" occurs, many-body decoherence in the random-matrix phase is prohibited, and the robustness of the decay rate for the quantum coherence appears. We also find that this transition can be characterized by the eigenvalue-spacing statistics and the operator-space entanglement of eigenstates.
The ratio of complex eigenvalues (vertical axis) as a function of the strength of disorder (horizontal axis) is shown for different sizes of the system. The intersection of the results for different sizes is regarded as the phase transition point, below which most of the eigenvalues become complex, and above which most of the eigenvalues become real.
・Non-Hermitian many-body localization
While spectral reality is guaranteed in ordinary Hermitian quantum systems, eigenvalues can be real even in non-Hermitian systems. We have found that when the disorder strength is increased in a disordered interacting system with asymmetric hopping, a novel phase transition occurs at the level of the many-body spectrum: most of the eigenvalues are complex and real for weak disorder, respectively. We showed that this dramatically changes the dynamical stability of the system under non-Hermitian time evolution. We also found that this transition is caused by a non-Hermitian extension oof the MBL transition. Namely, while the MBL in isolated systems is a phenomenon in which strong disorder prevents ETH and thermalization, in non-Hermitian systems it has a significant effect on the reality of the spectrum.
Phys. Rev. Lett. 123, 090603 (2019). [arXiv:1811.11319]
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