|
|
| Fig. 1: A rare thermal bubble in MBL bulk phase. (Source: J. Ding, after De Roeck et al. [1]) |
Recent discovery of many-body localization (MBL) in disordered systems have sparked intense excitement. Unlike typical non-integrable many-body quantum systems, MBL systems do not reach thermal equilibrium under intrinsic dynamics, even at energy densities corresponding to infinite effective temperatures. In one spatial dimension, the existence of a stable MBL phase has been essentially confirmed, but the stability of MBL in higher dimensions remains controversial. This paper examines the "avalanche" argument [1] against the existence of MBL in two dimensions and higher.
We imagine a disordered system which is hypothetically globally in a MBL phase. It is possible for the system to have rare regions in which all site energies are close to each other. These regions with atypically weak disorder are locally thermalizing. For simplicity, we imagine a spherical thermal bubble coupled to localized spins with coupling strength exponentially decaying with distance to the bubble, and study the evolution of the thermal bubble in a larger MBL system. A schematic of this is shown in Fig. 1.
We assume the thermal bubble can be described by random matrix theory, and assume the localized spins do not interact with each other. The ratio (interaction strength)/(level spacing) allows us to heuristically determine if localized spins hybridize with delocalized states in the thermal bubble. We find that for large enough initial bubble size, a localized spin closest to the bubble will be absorbed and become fully delocalized, forming a larger combined thermal system with twice the Hilbert space dimension and half the level spacing. Assuming the new combined thermal system is also described by random matrix theory, we can iterate this process. Because the bubble-spin interactions decay exponentially with distance, but (in two dimensions and higher) the Hilbert space dimension grows superexponentially with the radius of the thermal system, the hybridization condition will always be satisfied, even for spins arbitrarily far away from the initial thermal bubble.
Thus, MBL is unstable with respect to rare thermal regions in two dimensions and higher. One single large enough thermal inclusion will create an avalanche that thermalizes the entire system.
© Jixun Ding. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] W. De Roeck and F. M. C. Huveneers, "Stability and Instability Towards Delocalization in Many-body Localization Systems," Phys. Rev. B 95, 155129 (2017).