The many-body localized phase of the quantum random energy model

The random energy model (REM) provides a solvable mean-field description of the equilibrium spin-glass transition. Its quantum sibling (the QREM), obtained by adding a transverse field to the REM, has similar properties and shows a spin-glass phase for sufficiently small transverse field and temperature. In a recent work, some of us have shown that the QREM further exhibits a many-body localization-delocalization (MBLD) transition when viewed as a closed quantum system, evolving according to the quantum dynamics. This phase encloses the familiar equilibrium spin-glass phase. In this paper, we study in detail the MBLD transition within the forward-scattering approximation and replica techniques. The predictions for the transition line are in good agreement with the exact diagonalization numerics. We also observe that the structure of the eigenstates at the MBLD critical point changes continuously with the energy density, raising the possibility of a family of critical theories for the MBLD transition.

Figure 1

The phase diagram of the QREM, in the $\mathrm{\Gamma }\text{−}\epsilon$ plane (top) and the $\mathrm{\Gamma }\text{−}T$ plane (bottom). The red shaded region contains localized eigenstates and the blue shaded region contains ergodic eigenstates. The green shaded regions indicate the numerically observed boundary region: dark green circles from exact diagonalization (Sec. 3), and light green diamonds from the numerical FSA (Sec. 5c). The black dashed lines are the conjectured limiting boundaries of ${\epsilon }_c=\pm \mathrm{\Gamma }$, the green dashed lines are the analytic estimate of ${\epsilon }_c$ within the single-resonance approximation (Sec. 5b), and (bottom only) the blue dashed lines indicate the thermodynamic phase boundaries as predicted by canonical calculations [30].

Figure 2

Probability density function for the eigenstate single-spin magnetization $(y$ axis), for small bins over a range of energy densities $(x$ axis). Each vertical slice is a separate probability distribution. These distributions are at $\mathrm{\Gamma }=0.20$ and $N=14$, with energy density windows $\delta \epsilon =0.02$.

Figure 3

Eigenstate autocorrelation function ${\langle {\sigma }_1^z\left(t\right){\sigma }_1^z\left(0\right)\rangle }_C$ [see Eq. (13)] as a function of time (vertical axis) and eigenstate energy density (horizontal axis). The vertical dashed lines indicate the location of the MBLD transition as determined by the QREM's spectral statistics. These results are taken at $\mathrm{\Gamma }=0.20$ and $N=13$ with energy bins $\delta \epsilon =0.02$.

Figure 4

Properties of the QREM (probabilistic) spectrum. All data are at $\mathrm{\Gamma }=0.20$, and $N=8$ (blue) to 14 (red). Eigenstates are binned within energy density windows of size $0.02$. These curves have been collapsed using the scaling form $y=y_c+f\left[N^{\frac{1}{\tilde{\nu }}}(\epsilon -{\epsilon }_c)\right]$. (a) The disorder-averaged level spacing ratio. ${\epsilon }_c=-0.34\left(2\right),\tilde{\nu }=0.39\left(9\right)$, and $r_c=0.44\left(1\right)$. (b) The deviation of the frequency of smaller-than-average energy gaps from GOE statistics. ${\epsilon }_c=-0.33\left(2\right),\tilde{\nu }=0.41\left(9\right)$, and $J_{1c}=0.73\left(1\right)$. (c) The deviation of the frequency of larger-than-average energy gaps from GOE statistics. ${\epsilon }_c=-0.34\left(2\right),\tilde{\nu }=0.46\left(9\right)$, and $J_{2c}=0.80\left(1\right)$.

Figure 5

Finite-size scaling parameters for the level spacing statistics as a function of $\mathrm{\Gamma }$. The shaded bands indicate the range of parameter values over which the data collapse well. Red corresponds to $\left[\overline{r}\right]$, green corresponds to $J_1$, and blue corresponds to $J_2$. (a) The critical energy density, overlaid on a portion of the phase diagram of Fig. 1. (b) The critical exponent $\tilde{\nu }$. (c) The value of the level spacing statistics at criticality. We plot $(\left[\overline{r}\right]-r^{\left(\text{WD}\right)})/(r^{\left(\text{P}\right)}-r^{\left(\text{WD}\right)})$ and denote it by $J_{\left[\overline{r}\right]}$, in analogy with $J_1$ and $J_2$.

Figure 6

The disorder-averaged magnitude of the eigenstate single-spin magnetization, as determined via exact diagonalization, as a function of the eigenstate energy density. These results are taken at $\mathrm{\Gamma }=0.20$, from $N=8$ (blue) to $N=14$ (red). Eigenstates are binned in energy density windows of size $0.02$. The background shading corresponds to the predicted phase at that energy density: red is localized, blue is delocalized, and green is the transition region as determined by the QREM's spectral statistics.

Figure 7

Probability density function for the difference between single-spin magnetizations of spectrally adjacent eigenstates $(y$ axis), over a range of energy densities $(x$ axis). Each vertical slice is a separate probability distribution. These distributions are at $\mathrm{\Gamma }=0.20$ and $N=14$. We used energy density windows of $0.02$ for each distribution.

Figure 8

The decay time of the eigenstate connected autocorrelation function $\left[\overline{{〈{\sigma }_1^z\left(t\right){\sigma }_1^z\left(0\right)〉}_C}\right]$, as a function of energy density. These results are taken at $\mathrm{\Gamma }=0.20,N=8$ (blue) to $N=13$ (red). The binning in energy density is $\mathrm{\Delta }\epsilon =0.025$. (a) The unscaled decay times. The vertical dashed line marks the critical energy density as determined by the QREM's spectral statistics. (b) Collapsed curves (omitting $N=8)$, using the form $\tau =(a+bN)f\left[N^{\frac{1}{\tilde{\nu }}}(\epsilon -{\epsilon }_c)\right]$. ${\epsilon }_c=-0.32\left(2\right),\tilde{\nu }=0.40\left(7\right),a=-2.9\left(1\right)\times {10}^2$, and $b=38\left(2\right)$.

Figure 9

The disorder-averaged order of magnitude of the eigenstate IPR from exact diagonalization, divided by system size. These results are taken at $\mathrm{\Gamma }=0.20$ for sizes $N=8$ (blue) to $N=14$ (red). Eigenstates are binned in energy density windows of size $\delta \epsilon =0.02$. The background shading corresponds to the predicted phase at that energy density: red is localized, blue is delocalized, and green is the transition region as determined by the spectral statistics $\left[\overline{r}\right]$.

Figure 10

Semi-log plot of probability overlaps between eigenstates separated by energy $\omega$, in the delocalized phase $\left(\epsilon =-0.10,\mathrm{\Gamma }=0.40\right)$ for $N=8$ (blue) to $N=14$ (red). These data have been averaged over states within an energy window $\delta \epsilon =0.05$ centered on $\epsilon =-0.10$, and averaged over samples. The lines have been shifted vertically to illustrate their dependence on $N$. Although the slopes of the lines depend slightly on $N$, the rough value is $-0.50$.

Figure 11

Log-log plot of probability overlaps between eigenstates separated by energy $\omega$, in the localized phase $\left(\epsilon =-0.20,\mathrm{\Gamma }=0.10\right)$ for $N=8$ (blue) to $N=14$ (red). These data have been averaged over states within an energy window $\delta \epsilon =0.05$ centered on $\epsilon =-0.20$, and averaged over samples. The lines have been shifted vertically to illustrate their dependence on $N$. The lines all have slopes of $-2$, and are well described by perturbation theory [see Eq. (23)].

Figure 12

Eigenstate properties as determined within the forward-scattering approximation, as a function of energy density. All data are at $\mathrm{\Gamma }=0.20$, and $N=10$ (blue) to 20 (red). These curves have been collapsed using the scaling form $y=f\left[N^{\frac{1}{\tilde{\nu }}}(\epsilon -{\epsilon }_c)\right]$. The hatched regions in the bottom two plots are a reminder: the FSA's predictions for observables in the delocalized phase are not meaningful. (a) The probability of a sample containing a resonance. The critical parameters are ${\epsilon }_c=-0.32\left(1\right)$ and $\tilde{\nu }=0.75\left(6\right)$. (b) The disorder-averaged IPR within the FSA. The critical parameters are ${\epsilon }_c=-0.31\left(1\right)$ and $\tilde{\nu }=0.83\left(7\right)$. (c) The disorder-averaged single-spin magnetization of the FSA wave functions. The critical parameters are ${\epsilon }_c=-0.30\left(1\right)$ and $\tilde{\nu }=0.80\left(9\right)$.

Figure 13

A comparison of the disorder-averaged IPR obtained from ED (dashed line) and from the FSA (solid line), at $\mathrm{\Gamma }=0.20$ and with $N$ ranging from 8 (blue) to 14 (red). The background shading indicates the phase at that energy density as determined by the spectral statistics in ED: red is localized, blue is delocalized, and green is the transition region. Note that the FSA results are a good approximation in the localized region $\left(\epsilon <{\epsilon }_c\right)$ but differ qualitatively in the delocalized region $\left(\epsilon >{\epsilon }_c\right)$. This is not surprising since the FSA is not valid in this region.