Renormalization-group study of the many-body localization transition in one dimension

Using a new approximate strong-randomness renormalization group (RG), we study the many-body localized (MBL) phase and phase transition in one-dimensional quantum systems with short-range interactions and quenched disorder. Our RG is built on those of Zhang et al. [Phys. Rev. B 93, 224201 (2016)] and Goremykina et al. [Phys. Rev. Lett. 122, 040601 (2019)], which are based on thermal and insulating blocks. Our main addition is to characterize each insulating block with two lengths: a physical length and an internal decay length $\zeta$ for its effective interactions. In this approach, the MBL phase is governed by a RG fixed line that is parametrized by a global decay length $\tilde{\zeta }$, and the rare large thermal inclusions within the MBL phase have a fractal geometry. As the phase transition is approached from within the MBL phase, $\tilde{\zeta }$ approaches the finite critical value corresponding to the avalanche instability, and the fractal dimension of large thermal inclusions approaches zero. Our analysis is consistent with a Kosterlitz-Thouless-like RG flow, with no intermediate critical MBL phase.

Figure 1

A summary of the RG rules. The rules are formulated in terms of the lengths $L^I$ and $d$ for $I$ blocks and $L^T$ for $T$ blocks. (a) An $ITI\mapsto I$ move where a $T$ block at the cutoff $\left(L_n^T=\mathrm{\Lambda }\right)$ fails to thermalize the neighboring $I$ blocks. (b) A $TIT\mapsto T$ move where an $I$ block at the cutoff $\left(d_n=\mathrm{\Lambda }\right)$ thermalizes via interactions with neighboring $T$ blocks.

Figure 2

(a) The probability of a sample being thermal. The curves are for $N$ from $8\times {10}^3$ to $512\times {10}^3$ by powers of four. The number of samples used to generate each data point is ${10}^9/N$. Standard errors are smaller than the markers. (b) $logQ$, an indicator of the quality of the one-parameter scaling ansatz for different values of $\nu$ and ${\zeta }_{\mathrm{bare},\mathrm{c}}$. Smaller values indicate a better fit, with $logQ=0$ indicating the best possible result.

Figure 3

Results of numerical simulations of single samples with $N={10}^7$. Each thick line consists of three [(b) and (c)] to five [(a)] thin lines representing single samples: When these individual samples can be distinguished, that gives a crude estimate of the statistical errors. The data shown was recorded during the final 10% of RG steps so as to isolate the large-$\mathrm{\Lambda }$ behavior. (a) The RG flow is plotted on the same plane as that of Ref. [39] for direct comparison. Simulations were done with ${\zeta }_{\mathrm{bare}}\in [0.330,0.370]$ (right to left) by steps of 0.005. (b) The distribution of the physical lengths of decimated thermal blocks at criticality and in each phase for ${\zeta }_{\mathrm{bare}}=[0.300,0.375]$ by steps of 0.025. (c) The scaling of ${\rho }^T\left(\mathrm{\Lambda }\right)$ with $\mathrm{\Lambda }$ for ${\zeta }_{\mathrm{bare}}\in [0.300,0.375]$ again (top to bottom). ${\rho }^T\left(\mathrm{\Lambda }\right)$ was calculated by numerically estimating the derivative of the cumulative distribution function of $L^T$ at $L^T=\mathrm{\Lambda }$ with a quadratic fit to the first ${10}^3$ data points. This was done up to the scale where ${10}^{4}T$ blocks remained. The raw data was smoothed using a Savitzky-Golay filter with a window containing 1% of the data.

Figure 4

Numerical estimates of the parameters $\epsilon$ and $\kappa$, and the fractal dimension $d_f$ deep in the MBL phase and at criticality (from right to left). $\epsilon$ and $\kappa$ were estimated by fitting numerical values of ${\rho }^T\left(\mathrm{\Lambda }\right)$ to the form $(1+\kappa )/{\mathrm{\Lambda }}^{1-\epsilon }$. $d_{f,\mathrm{ana}}$ is the expression given in Eq. (11). $d_{f,\mathrm{num}}$ was computed by fitting finite-size RG data to the form $\langle L_{\mathrm{frac}}^T\rangle \propto {\langle L^T\rangle }^{d_{f,\mathrm{num}}}$. The points are plotted using the values of $\tilde{\zeta }$ obtained at the end of the finite-size simulations of the RG flows, which do not reach $\tilde{\zeta }=1$ for the critical systems. The discrepancy between the estimates $d_{f,\mathrm{num}}$ and $d_{f,\mathrm{ana}}$ can be accounted for by well-understood finite-size effects; this is discussed in the main text of Sec. 4b.