Nonergodic extended states in the $\beta$ ensemble

Matrix models showing a chaotic-integrable transition in the spectral statistics are important for understanding many-body localization (MBL) in physical systems. One such example is the $\beta$ ensemble, known for its structural simplicity. However, eigenvector properties of the $\beta$ ensemble remain largely unexplored, despite energy level correlations being thoroughly studied. In this work we numerically study the eigenvector properties of the $\beta$ ensemble and find that the Anderson transition occurs at $\gamma =1$ and ergodicity breaks down at $\gamma =0$ if we express the repulsion parameter as $\beta =N^{-\gamma }$. Thus other than the Rosenzweig-Porter ensemble (RPE), the $\beta$ ensemble is another example where nonergodic extended (NEE) states are observed over a finite interval of parameter values $(0<\gamma <1)$. We find that the chaotic-integrable transition coincides with the breaking of ergodicity in the $\beta$ ensemble but with the localization transition in the RPE or the 1D disordered spin-1/2 Heisenberg model. As a result, the dynamical timescales in the NEE regime of the $\beta$ ensemble behave differently than the latter models.

Figure 1

Density of states (DOS) of the $\beta$ ensemble (a) averaged over 500 disordered realizations for various $\gamma$ where $N=8192$. (b) $\gamma =0.6$ and (c) $\gamma =1.1$ for various $N$. We also show Wigner's semicircle law (solid bold) and Gaussian distribution $\mathcal{N}(0,1/4)$ (dashed bold).

Figure 2

Short-range spectral correlations for $\beta$ ensemble: PDF of (a) NNS and (b) modified RNNS varying $\beta$ for $N=8192$, and varying $N$ for $\beta =0.2$ and 2.5 (c), $\gamma =0.3$ (d), $\gamma =0$ (e), and $\gamma =-0.3$ (f). The markers indicate numerical data, while solid lines denote empirical analytical forms [Eqs. (12) and (13) in [35] for NNS and Eq. (1) in [65] for RNNS]. We also show analytical expressions for the Poisson ensemble (dashed bold) and GOE (solid bold).

Figure 3

Ensemble average of $\tilde{r}$ for $\beta$ ensemble vs (a) $\beta$ and (b) $\gamma$, and (c) for RPE vs $\tilde{\gamma }$ and (d) Heisenberg model vs disorder strength, $h$. For (a), (b), and (c) system size, $N$, and for (d) chain length, $L$, is varied. In (a) we show the $\langle \tilde{r}\rangle$ for $N=3$ [Eq. (7) in 63] with a dashed line. The insets of (b), (c), and (d) show collapsed data following the ansatz in Eq. (7), where critical parameter and exponents are also given.

Figure 4

Power spectrum of noise for (a) the $\beta$ ensemble and (d) RPE for $N=8192$ as a function of dimensionless frequency, $\omega =2\pi k/N$. In (b), the power spectrum is shown for various $\gamma$ [denoted with markers similar to (a)] and $N$ (denoted by different colors as in the legend). We see that $\langle P_{\omega }^{\delta }\rangle$ shifts downwards with increasing $N$ for $\gamma =-0.3$, where the dashed line $\propto 1/\omega$ is placed as a guide to the eye. (b) $\langle P_{\omega }^{\delta }\rangle$ for $\gamma =0.3$ while varying $N$ (data shifted in the $Y$ direction for clarity). Dashed and solid lines indicate $1/\omega$ and $1/{\omega }^2$ behaviors, respectively. In (a), (c), and (d), stars denote the critical frequencies, ${\omega }_c$, separating heterogeneous behavior. The inset of (c) shows numerically obtained ${\omega }_c$, where the solid line denotes the linear fit in log-log scale. We also show the analytical $\langle P_{\omega }^{\delta }\rangle$ for the Poisson ensemble (dashed bold) and GOE [solid bold; Eq. (10) in 69].

Figure 5

Eigenstate statistics: Shannon entropy, $S$, and relative Rényi entropy of two types [${\mathcal{R}}_{1,2}$ in Eq. (9)] for the $\beta$ ensemble, RPE, and 1D disordered spin-1/2 Heisenberg model, as a function of system parameters for different matrix size, $N$, and chain length, $L$ (values given in the legends). The critical values of parameters indicating ergodic and localization transitions are marked with a dashed line in all plots. The inset shows collapsed data following the ansatz in Eq. (7), where numerically obtained critical parameters and exponents are also given. The inset of (c) shows collapsed data of $S^{\prime }=S/ln\left(0.48N\right)$.

Figure 6

Inverse participation ratio: for $\beta$ ensemble as a function of $\gamma$ for various system sizes, $N$. Inset shows collapsed data following the ansatz in Eq. (7) along with critical parameter and exponent.

Figure 7

Entropic localization length for (a) the $\beta$ ensemble, (b) RPE, and (c) 1D disordered spin-1/2 Heisenberg model for various system sizes and chain lengths as a function of the repulsion parameter $\beta$. We show the line $d_N/N=\beta$ with a dashed curve. The inset of (a) shows ${\beta }^{★}$ below which $d_N/N$ becomes constant.

Figure 8

Density of Shannon entropy: for (a) the $\beta$ ensemble, $\gamma =0.6$ (b) RPE, $\tilde{\gamma }=1.5$, and (c) the Heisenberg model, $h=2.4$, while varying system sizes, $N$, and chain lengths, $L$, where $\mathcal{S}$ and $S$ are the scaled and original Shannon entropies, respectively ($D_1$ is the fractal dimension obtained from system size scaling of $S$.)

Figure 9

(a) Phase diagram: The three distinct phases observed in the $\beta$ ensemble, RPE, and 1D disordered spin-1/2 Heisenberg model are demarcated against the critical parameter values. Markers indicate fractal exponents $D_{1,2}$ as explained in the legend. (b) Fraction of localized states: as a function of $\gamma$ for the $\beta$ ensemble for various system sizes. The inset shows ${\alpha }_{\xi }$, the system size scaling exponent of $\xi$ vs $\gamma$ along with a linear fit, ${\alpha }_{\xi }=a\gamma +b$, where $a=0.9656\pm 0.0676$ and $b=-1.1618\pm 0.0370$.

Figure 10

Survival probability for $\beta$ ensemble: (a) time evolution for $\gamma =-0.3$ and various system size, $N$. We show Thouless ($t_{\text{Th}}$) and relaxation time ($t_{\text{R}}$) with markers and the correlation hole ($t_{\text{hole}}$) with a dashed line in each case. The inset shows $t_{\text{Th}}$ and $t_{\text{R}}$ as a function of $N$ along with linear fit in log-log scale with a solid line, indicating a $\sqrt{N}$ dependence. (b) $N=1024$ varying $\gamma$ (c) correlation hole vs $\gamma$ for various $N$. The inset shows ${\gamma }^{★}$ vs $N$ in log-log scale ($t_{\text{hole}}\approx 0$ for $\gamma \ge {\gamma }^{★}$). (d) Asymptotic value of survival probability vs $\gamma$ for different $N$. The inset shows system size scaling ($\overline{R}\propto N^{{\alpha }_{\overline{R}}}$) where ${\alpha }_{\overline{R}}\approx \gamma -1$ for $\gamma \le 1$ and 0 for $\gamma >1$.