Universal Properties of Many-Body Delocalization Transitions

We study the dynamical melting of “hot” one-dimensional many-body localized systems. As disorder is weakened below a critical value, these nonthermal quantum glasses melt via a continuous dynamical phase transition into classical thermal liquids. By accounting for collective resonant tunneling processes, we derive and numerically solve an effective model for such quantum-to-classical transitions and compute their universal critical properties. Notably, the classical thermal liquid exhibits a broad regime of anomalously slow subdiffusive equilibration dynamics and energy transport. The subdiffusive regime is characterized by a continuously evolving dynamical critical exponent that diverges with a universal power at the transition. Our approach elucidates the universal long-distance, low-energy scaling structure of many-body delocalization transitions in one dimension, in a way that is transparently connected to the underlying microscopic physics. We discuss experimentally testable signatures of the predicted scaling properties.

Popular Summary

Exotic quantum phenomena like coherent quantum memories and topological edge states can occur near absolute zero temperature, but they are typically destroyed by moderate temperatures. Recently proposed many-body localized quantum glasses offer an exception to this limitation and can exhibit quantum coherent behavior in “hot” systems without the need for cooling.

We develop a new numerical technique to investigate phase transition between such hot but coherent quantum glasses and classical fluids. These transitions represent an entirely new class of phase transitions in which the laws of thermodynamics break down sharply at a glass-freezing transition. Working in one dimension, we find that close to the melting transition, the particles in the fluid phase do not obey the familiar rules of Einstein’s Brownian motion. Rather, they move much more slowly and exhibit subdiffusive dynamics since they must quantum tunnel through puddles of frozen glass. We propose ways to experimentally verify our theoretical predictions using ensembles of cold atoms. We also note that our model can be applied to large systems in which the computational costs scale polynomially with system size.

Extensions of our techniques will enable efficient numerical simulations of a variety of quantum-to-classical transitions, such as the many-body localized transitions in higher dimensions, which were previously inaccessible using conventional numerical methods.

Figure 1

Phase diagram and finite-size crossovers for one-dimensional many-body delocalization transitions. $W$ parametrizes disorder strength. For strong disorder $W>W_c$, a nonergodic many-body localized glass is obtained, which lacks thermal transport but exhibits slow dephasing and entanglement dynamics with length-time scaling $L\sim \log {\tau }_{\mathrm{deph}}$ (where the subscript “deph” emphasizes that this is the dephasing time associated with virtual quantum fluctuations rather than the energy transport time scale, which is strictly infinite in the strong-disorder glass). Below a critical disorder strength $W_c$, the MBL glass melts into a thermal, energy-conducting liquid. The melting transition is continuous (second order), characterized by a single diverging length scale $\xi \sim |W-W_c{|}^{-\nu }$, where $\nu \approx 3.5\pm 0.3$. Energy transport (entanglement spreading) in the thermal liquid is characterized by a subdiffusive (sub-ballistic) power-law scaling between time and length $\tau \sim L^z$, with dynamical exponent $z\sim \xi$ that diverges continuously upon approaching the transition.

Figure 2

Finite-size scaling for the disorder-averaged localization length ${\xi }_{\mathrm{loc}}$ (defined as the size of the longest resonant cluster for a given disorder realization). ${\xi }_{\mathrm{loc}}/L$ collapse to a universal form (main panel) when plotted against the scaling variable $(W-W_c)L^{1/\nu }$, with correlation length exponent $\nu \approx 3.5\pm 0.3$. Insets show the unscaled ${\xi }_{\mathrm{loc}}/L$ curves. Results are averaged over $\sim {10}^4$ disorder realizations.

Figure 3

Energy transport near the transition. In the delocalized phase ($W<W_c$), excess energy initially localized near the origin spreads to a distance $\delta r\sim t^{1/z}$ in time $t$ (bottom inset). The dynamical critical exponent $z$ diverges continuously from the delocalized side of the transition and vanishes inside the quantum critical glass phase (main panel). The numerical results are consistent with $z$ diverging as $|W-W_c{|}^{-\zeta }$ with $\zeta =\nu$ (top inset).

Figure 4

Two cases for merging resonances. The energy cost for changing the state of one cluster (blue circle to open circle connected by black arrow) can be partially compensated by changing the state of another cluster. The reduced energy penalty, $\delta E_{12}$, for such “flip-flop” transitions is either ${\delta }_1-{\mathrm{\Lambda }}_2$ or ${\delta }_1{\delta }_2/{\mathrm{\Lambda }}_2$ depending on whether ${\delta }_1\gtrless {\mathrm{\Lambda }}_2$, as shown in panels (a) and (b), respectively.

Figure 5

Finite-size scaling for the disorder-averaged localization length ${\xi }_{\mathrm{loc}}$ (see Fig. 2) using a simplified model that ignores the renormalization of the couplings. Results are averaged over $\sim {10}^4$ disorder realizations.

Figure 6

Transport path through resonant cluster ($W<W_c$). Schematic depiction of the sequence of resonance-merging steps ($i,i+1,i+2$) that produces a gap in the transport path through the resonant cluster. See text for a detailed explanation. Blue rectangles are resonantly linked clusters of spins. Black and gray lines indicate resonant links between previously formed clusters. The dashed crossed-out red line indicates a spurious connection that should not be added to the transport path of the cluster.