Kosterlitz-Thouless scaling at many-body localization phase transitions

We propose a scaling theory for the many-body localization (MBL) phase transition in one dimension, building on the idea that it proceeds via a “quantum avalanche.” We argue that the critical properties can be captured at a coarse-grained level by a Kosterlitz-Thouless (KT) renormalization group (RG) flow. On phenomenological grounds, we identify the scaling variables as the density of thermal regions and the length scale that controls the decay of typical matrix elements. Within this KT picture, the MBL phase is a line of fixed points that terminates at the delocalization transition. We discuss two possible scenarios distinguished by the distribution of rare, fractal thermal inclusions within the MBL phase. In the first scenario, these regions have a stretched exponential distribution in the MBL phase. In the second scenario, the near-critical MBL phase hosts rare thermal regions that are power-law-distributed in size. This points to the existence of a second transition within the MBL phase, at which these power laws change to the stretched exponential form expected at strong disorder. We numerically simulate two different phenomenological RGs previously proposed to describe the MBL transition. Both RGs display a universal power-law length distribution of thermal regions at the transition with a critical exponent ${\alpha }_c=2$, and continuously varying exponents in the MBL phase consistent with the KT picture.

Figure 1

Quantum avalanche [45]. A thermal inclusion initially consisting of $n_0$ spins (red region) is in contact with a set of l-bits (arrows). The inclusion thermalizes $n$ l-bits (red arrows) and thereby expands to a size $n_0+n$ (yellow region) while retaining its featureless ETH character. The effective matrix element to add the $(n+1)\text{th}$ l-bit decays exponentially from the boundary of the original inclusion.

Figure 2

Kosterlitz-Thouless RG flow obtained by integrating Eqs. (2) and (3). The MBL phase corresponds to a line of fixed points with $\rho =0$ for $\zeta <{\zeta }_c$. For $\zeta >{\zeta }_c$, even an infinitesimally small bare density of resonances grows under RG, driving the flow to the thermal phase. The dotted line denotes a schematic line of microscopic parameters, tuned, e.g., by decreasing disorder strength $W$. Note that many RG trajectories initially approach the MBL fixed line even if they eventually flow to the thermal phase; this nonmonotonicity naturally explains why numerical simulations often overestimate the extent of the MBL phase.

Figure 3

Two scenarios for the evolution of the Griffiths regions from critical to strong disorder. In scenario (i), thermal blocks have power-law distribution only at criticality, $W=W_c$ with $p_T^c\left(l_T\right)\sim l_T^{-{\alpha }_c}$. All other points in the MBL phase have stretched exponential thermal block distributions, characterized by an evolving nonzero fractal dimension $d_F>0$ of the rare regions, saturating deep in the MBL phase to $d_F=1$. In the “two-transition” scenario (ii) thermal blocks are power-law-distributed across a critical ${\mathrm{MBL}}^{*}$ phase, until a second critical disorder strength $W_c^{*}$ is reached. At the second transition, the thermal block distribution changes from power-law (${\mathrm{MBL}}^{*}$) to stretched exponential (MBL), with the fractal dimension again continuously approaching $d_F=1$.

Figure 4

DVP RG: Histograms of the number of sites (size) contained in largest cluster for each disorder realization. The system size is $L=1500$. Different lines correspond to varying disorder bandwidths $W$: $W$ is regularly spaced from 2.04 to 2.20 in units of 0.01, with three additional values $W=2.22,2.24,2.26$. The data fall in strict order from top to bottom (blue to gold) according to increasing $W$. Standard errors are estimated from the binomial distribution and indicated by the colored intervals around the histogram points; the bin size is $\delta x=25$. At larger cluster sizes, the histograms show power-law behavior with a continuously varying power as well as a thermal peak close to system size $L$. The histograms are unnormalized; at small numbers of counts, noise becomes dominant.

Figure 5

DVP RG. (a) Exponent $\alpha$ of a power-law fit to the distribution of the histograms of largest cluster size. The histograms are fit on the interval $[0.5L,0.85L]$ with a bin size of $\delta x=10$ (for $L<1500$) or $\delta x=50$ (otherwise). Mean values of $\alpha$ are obtained by resampling. The errors account for both the statistical error from resampling as well as the larger errors arising from varying the binning and fitting intervals. The value ${\alpha }_c=2$ predicted by KT theory is shown as the grey dashed line. (b) The fraction of disorder samples where the largest cluster size $x\ge 0.95L$ for various $L$. Errors are calculated from the binomial distribution, but are smaller than the marker size. Crossing points between curves of neighboring $L$ are indicated as black crosses and serve as finite-size estimates of $W_c$. Standard errors of crossings are estimated from resampling the data. (c) The value of effective critical disorder $W_c\left(L\right)$ slowly increases with the system size, illustrating the overestimation of the MBL phase on finite system sizes. The data are obtained from crossing points in panel (b) $[\tilde{L}=(L_1+L_2)/2$ is the average length of neighboring curves] and from values of $W$ when $\alpha =2$ in panel (a) $[\tilde{L}=L]$. The finite size corrections are suppressed by a logarithm of system size, consistent with KT scaling. Errors are estimates from resampling.

Figure 6

VHA RG. Distribution of the length of thermal inclusions at criticality and into the MBL phase. The disorder strength is increased from the critical value $ln{g}_0=-1.78$ (topmost curve) up to $ln{g}_0=-2.15$ corresponding to a point deep in the MBL phase. At criticality, the tail of the distribution is consistent with $1/{\ell }^2$ behavior, and the exponent increases into the MBL phase. Error bars of the histogram were calculated based on the assumption of Poisson-distributed events in each bin of the histogram. At criticality, we simulate systems with $L=3\times {10}^6$ initial blocks and $2\times {10}^3$ disorder realizations. In the MBL phase, we use $L=5\times {10}^6$ initial blocks and average over $4\times {10}^4$ disorder realizations.

Figure 7

VHA RG. Power-law scaling between ${\ell }_f$ and $\ell$ allows us to extract the fractal dimension $d_F=0.153\pm 0.005$. The fit uses the region of $\ell >{10}^4$. We note that at larger value of disorder, $ln{g}_0=-1.95$, we do not see a clear power-law scaling between $\ell$ and ${\ell }_f$ in the same region. We use systems with $L$ ranging from $5\times {10}^6$ to $7\times {10}^6$ clusters and perform averaging over ${10}^4$ to $1.2\times {10}^5$ disorder realizations. The error bars correspond to the standard deviation in each bin. Since disorder is initially introduced only in the links between the blocks, there is a region of lengths where the curves for different disorder strength coincide.

Figure 8

Finite size scaling of normalized average entanglement entropy [29] shows a crossing at a particular disorder strength, $ln{g}_0=-1.78\pm 0.005$. For stronger disorder, $lng<ln{g}_0$ entropy decreases, corresponding to the MBL phase, whereas for $lng>ln{g}_0$ normalized entropy tends to the thermal value upon increasing system size. The number of disorder realizations used is ${10}^4$ for $L=400\text{and}1000$, $2\times {10}^4$ for $L=5000\text{and}10000$, and $3\times {10}^4$ for $L=20000\text{and}40000$.

Figure 9

Distribution of normalized effective entanglement entropy $s=\langle S\rangle /S_T$ has a broad tail at the MBL transition and into the localized phase. In the delocalized phase (topmost curve, $ln{g}_0=-1.7$), the distribution is nonmonotonic, suggesting the formation of a maximum near thermal value of entanglement. In contrast, at the transition ($ln{g}_0=-1.78$) and in the MBL phase we observe a power-law-like tail that extends to volume-law values of $s\sim 0.2$. The data were obtained on small system sizes of $L={10}^4$ blocks and ${10}^6$ disorder realizations.

Figure 10

Histograms of the number of sites (size) contained in the largest cluster for each disorder distributions with different system sizes $L$ using the DVP RG. The histograms are normalized both by the number of disorder realizations and the system size $L$. Standard errors are indicated by the colored intervals; the bin size is $\delta x=25$. For each data set, the limit where there is only a single count $1/\left(N_{\mathrm{dis}}L\right)$ is shown by dashed lines.