Random Matrix Ensemble for the Level Statistics of Many-Body Localization

We numerically study the level statistics of the Gaussian $\beta$ ensemble. These statistics generalize Wigner-Dyson level statistics from the discrete set of Dyson indices $\beta =1$, 2, 4 to the continuous range $0<\beta <\infty$. The Gaussian $\beta$ ensemble covers Poissonian level statistics for $\beta \to 0$, and provides a smooth interpolation between Poissonian and Wigner-Dyson level statistics. We establish the physical relevance of the level statistics of the Gaussian $\beta$ ensemble by showing near-perfect agreement with the level statistics of a paradigmatic model in studies on many-body localization over the entire crossover range from the thermal to the many-body localized phase. In addition, we show similar agreement for a related Hamiltonian with broken time-reversal symmetry.

Figure 1

Numerically obtained distributions of $r$ (top) and $s$ (bottom) for the Gaussian $\beta$ ensemble at various $\beta$. The curves for $\beta =0$ are obtained analytically from the expressions given in the main text.

Figure 2

Numerically obtained distributions of $r$ and $s$ for the Hamiltonian at various $W$ (solid lines) and the corresponding distributions for the Gaussian $\beta$ ensemble (dashed lines, identical color scheme). The top and bottom left plots are for $\mathrm{\Delta }=1$, the bottom right one for $\mathrm{\Delta }=2$.

Figure 3

Numerically obtained distributions of $r$ for the Hamiltonian at various $W$ (solid lines) and the corresponding distributions for the Gaussian $\beta$ ensemble (dashed lines, identical color scheme) for $L=12$, 14 (top panels) and the estimated value of $\beta$ for the spectra of the Hamiltonian as a function of $L$ and $W$ (bottom panels).

Figure 4

Numerically obtained distribution of $r$ for $H+H^{\prime }$ at various $W$ (solid lines) and the corresponding distributions for the Gaussian $\beta$ ensemble (dashed lines, identical color scheme).

Figure 5

Numerically obtained distributions of $r^{(n)}$ with $n=2$, 3, 4, 5 for the Hamiltonian at various $W$ (solid lines) and the corresponding distributions for the Gaussian $\beta$ ensemble (dashed lines, identical color scheme).