Skin-Anderson localization transitions in disordered hybrid-nonreciprocal systems

Abstract

Anderson (localization) transition is a universal wave phenomenon characterized by a disorder-induced quantum phase transition from extended to localized states, whereas the non-Hermitian skin effect is a generic feature of non-Hermitian systems that causes bulk states to localize at the boundaries. Here, we report an unexpected skin-Anderson localization transition arising from the interplay between these two phenomena in hybrid-nonreciprocal systems that exhibit both reciprocity and nonreciprocity in different spatial directions. In the weak-disorder regime, the states are boundary-extended, meaning they are extended in reciprocal spatial dimensions but localized at the boundaries in nonreciprocal dimensions due to the non-Hermitian skin effect. As disorder increases, these boundary-extended states transition to boundary-localized states at a critical disorder strength. Remarkably, the corresponding critical points exhibit universal characteristics akin to those of the Anderson transition in its Hermitian counterpart, including identical critical exponents within numerical errors. When disorder exceeds a higher critical threshold, a second transition occurs in which boundary-localized states become bulk-localized, thereby eliminating the non-Hermitian skin effect. Thus, the skin-Anderson localization transition establishes a new framework for controlling state localization by unifying the physics of Anderson transitions with non-Hermitian topology.

Figures (3)

• Figure 1: (a) A schematic diagram illustrates the reciprocal, hybrid-nonreciprocal, and nonreciprocal systems in two dimensions (2D). The hybrid-nonreciprocal system exhibits both symmetric (denoted as $t$) and asymmetric (denoted as $r_{\pm}$) hopping, whose transfer matrices can be mapped to its reciprocal counterpart under certain conditions. (b) A generalized phase diagram for hybrid-nonreciprocal systems indicates two critical transitions: a skin-Anderson transition and a boundary-to-bulk localization transition at $W_{c,1}$ and $W_{c,2}$, respectively. (c), (d) and (f) depict schematic representations of boundary-extended, boundary-localized, and bulk-localized states, three typical phases in disordered hybrid-nonreciprocal systems in 2D.
• Figure 2: (a) $\bar{\lambda}_M(W)$ for the hybrid-nonreciprocal Rashba model with $\alpha=0.1$, $\tilde{t}=0.1$, $E=0$, and various $M=16,32,\cdots,160$. (b) The scaling function $\bar{\lambda}_M=f(\ln{[M|W-W_{c,1}|^{\nu}]})$ of data in (a), where the effect of the irrelevant scaling variable is eliminated. The upper and lower branches stand for the boundary-extended and boundary-localized phases, respectively. (c) $\ln{p_2}$ for various $L$. The black dashed lines are the scaling function \ref{['eq_8']} obtained through a $\chi^2$ fit. (d) Critical exponent $\nu$ v.s $\tilde{t}$ of the hybrid-nonreciprocal Rashba model obtained through the scaling analysis of $\bar{\lambda}_M$ (empty circle) and $p_2$ (empty square). The black solid line guides the critical exponent of its reciprocal counterpart.
• Figure 3: (a) $\eta_m(W)$ of the hybrid-nonreciprocal Rashba model under the same parameters in Fig. \ref{['fig2']}(a). Inset: The scaling function near $W_{c,2}$. (b) A phase diagram of the hybrid-nonreciprocal Rashba model of $\alpha=0.1$ and $E=0$, determined through the scaling analysis of $W_{c,1}$ and $W_{c,2}$, from which one can identify the emergence of boundary-extended, boundary-localized, and bulk-localized phases. For $\tilde{t}=0$ (reciprocal limit), $W_{c,1}=W_{c,2}$, standing for the critical point of the conventional Anderson transition (the pentagram) separating extended and localized states.