Spectral statistics across the many-body localization transition

The many-body localization transition (MBLT) between ergodic and many-body localized phases in disordered interacting systems is a subject of much recent interest. The statistics of eigenenergies is known to be a powerful probe of crossovers between ergodic and integrable systems in simpler examples of quantum chaos. We consider the evolution of the spectral statistics across the MBLT, starting with mapping to a Brownian motion process that analytically relates the spectral properties to the statistics of matrix elements. We demonstrate that the flow from Wigner-Dyson to Poisson statistics is a two-stage process. First, a fractal enhancement of matrix elements upon approaching the MBLT from the delocalized side produces an effective power-law interaction between energy levels, and leads to a plasma model for level statistics. At the second stage, the gas of eigenvalues has local interactions and the level statistics belongs to a semi-Poisson universality class. We verify our findings numerically on the XXZ spin chain. We provide a microscopic understanding of the level statistics across the MBLT and discuss implications for the transition that are strong constraints on possible theories.

Figure 1

(Top) Random walk in a space of Hamiltonians induces a stochastic process on the eigenenergies. The interaction between eigenlevels is set by a potential energy $U(s_i-s_j)$. (Bottom) The evolution of the interaction between levels $U\left(s\right)$ across the MBL transition determines the level statistics.

Figure 2

Averaged function $c\left(E\right)$ evolves from being almost flat at low disorder $\left(W=0.5\right)$ to a power-law decay. Note that for the intermediate values of disorder, the matrix element is enhanced at small energy differences compared to the limit of weak disorder.

Figure 3

(a) Evolution of level spacing distributions as the system is tuned towards the MBL phase. Points represent data for $L=16$, while solid lines are best fits with a two-parameter distribution (7). Red and black dashed lines correspond to Poisson and Wigner-Dyson distributions. (b) The exponent ${\gamma }_P$, controlling the tails of the level statistics, flows with $L$ for $W\lesssim 2.5$, but is constant in the vicinity of MBLT $W_c\approx 3.6$. (c) In contrast, $\beta$, controlling the level repulsion, remains constant for $W\lesssim 2$, and starts to flow closer to the MBLT.