Scrambling Dynamics across a Thermalization-Localization Quantum Phase Transition

We study quantum information scrambling, specifically the growth of Heisenberg operators, in large disordered spin chains using matrix product operator dynamics to scan across the thermalization-localization quantum phase transition. We observe ballistic operator growth for weak disorder, and a sharp transition to a phase with subballistic operator spreading. The critical disorder strength for the ballistic to subballistic transition is well below the many body localization phase transition, as determined from finite size scaling of energy eigenstate entanglement entropy in small chains. In contrast, we find that the transition from subballistic to logarithmic behavior at the actual eigenstate localization transition is not resolved in our finite numerics. These data are discussed in the context of a universal form for the growing operator shape and substantiated with a simple phenomenological model of rare regions.

Figure 1

Phase diagram of operator spreading in disordered interacting spin systems with different disorder models. The Heisenberg Hamiltonian is defined using Pauli operators instead of spin-$1/2$ operators, so the $W$ normalization is twice as large relative to the spin-$1/2$ convention.

Figure 2

Plot of the contours of the averaged $\log (C)$, for the mixed field Ising model with Gaussian disorder. Averaged over $\sim 200$ disorder realizations, for three disorders, $W=0.2$ (ballistic), $W=1.0$ (intermediate), and $W=3.8$ (logarithmic). Bond dimension is 32. Convergence with bond dimension is discussed in the Supplemental Material [68]. Fluctuations away from the disorder averaging are discussed in Fig. 4 and in the corresponding section.)

Figure 3

(a) The extracted broadening coefficient $p$ and butterfly velocity $v_B$ are plotted for different sized systems, versus disorder. Note, $v_B$ goes to zero and $p$ has a peak at around disorder $W\sim 0.5$ with small finite-size effect. Error bars obtained from the 95% confidence interval of fitting, are shown for the largest system size. (b) Finite-size scaling on half-chain entanglement entropy estimates that the localization transition occurs at $W_c\sim 2.21$. The data collapse to the degree 3 polynomial ansatz $g[(W-W_c)L^{1/n}]$ with $n\sim 0.95$ is shown in the inset. The shaded region is the intermediate region.

Figure 4

(a) The bold black lines are single realizations of $-15$ contour lines of $\log (C)$ at disorder $W=0.8$ for the mixed field Ising model with Gaussian disorder. Note the colored patch is given by the SD of the $x$ positions for 180 realizations at a given time. Note that the two disorder realizations have distinct behaviors after $t=25$, with one being significantly slower because of a local bottleneck of large disorder. (b) SD of $x$ cuts at times $t=25$ and $t=50$, for 180 realizations for different disorders are plotted, which peaks at $W\sim 0.5$ and coincides with the critical disorder where $v_B$ vanishes.