Non-perturbatively slow spread of quantum correlations in non-resonant systems

TL;DR

The paper addresses whether strong disorder generically slows many-body quantum dynamics beyond eigenstate localization. It combines Lieb-Robinson bounds with a local Schrieffer-Wolff transformation to prove nonperturbatively slow information propagation for Hamiltonians of the form $H = H_0 + \varepsilon V$, where $H_0$ is non-resonant and composed of commuting Pauli-$Z$ terms; under a scale $r_*$ with $\Delta \sim e^{-b r_*^{\xi}}$, one finds $r_* \sim \log^{1/\xi}(\varepsilon^{-1})$ and a ballistic velocity $v \sim \varepsilon^{r_*}$, i.e., nonperturbatively slow spreading. The results establish a robust, dimension-agnostic slow-dynamics phenomenon and reveal a prethermal MBL-like regime with logarithmic entanglement growth up to non-perturbatively long times; they also provide deterministic and random non-resonant models (including incommensurate lattices) and discuss implications for classical/quantum simulation. Collectively, these findings clarify how disorder and non-resonance shape quantum dynamics, offer experimentally accessible signatures in quantum simulators, and suggest that non-resonant systems may be easier to simulate than generic many-body systems while exhibiting MBL-like phenomenology on long but finite timescales.

Abstract

Strong disorder often has drastic consequences for quantum dynamics. This is best illustrated by the phenomenon of Anderson localization in non-interacting systems, where destructive quantum wave interference leads to the complete absence of particle and information transport over macroscopic distances. In this work, we investigate the extent to which strong disorder leads to provably slow dynamics in many-body quantum lattice models. We show that in any spatial dimension, strong disorder leads to a non-perturbatively small velocity for ballistic information transport under unitary quantum dynamics, almost surely in the thermodynamic limit, in every many-body state. In these models, we also prove the existence of a "prethermal many-body localized regime", where entanglement spreads logarithmically slowly, up to non-perturbatively long time scales. More generally, these conclusions hold for all models corresponding to quantum perturbations to a classical Hamiltonian obeying a simple non-resonant condition. Deterministic non-resonant models are found, including spin systems in strong incommensurate lattice potentials. Consequently, quantum dynamics in non-resonant potentials is asymptotically easier to simulate on both classical or quantum computers, compared to a generic many-body system.

Figures (7)

• Figure 1: Left: Group velocity of each band for $h_n = n/r_\ast$ where $\varepsilon = \frac{1}{15}$ and $r_\ast = 2,3,5$. The bands become asymptotically flat as $r_\ast$ is increased. Right: Scaling of maximum group velocity with $\varepsilon$ for different system sizes $r_\ast$, which we find scales with $v_g(r_\ast, \varepsilon) \approx C(a\varepsilon)^{r_\ast}$, where $C$ and $a$ are fitting parameters.
• Figure 2: This figure illustrates the idea behind the proof on a one-dimensional chain with 10 sites; After breaking the system into regions of size $r_\ast$, time-independent perturbation theory is applied to diagonalize each, which results in the terms from $V$ that couple the regions together spreading out with tails that decay as $\sim \varepsilon^{r}$. Lieb-Robinson bounds can then be applied to the transformed system on the right to bound the coupling between sites $0$ and $9$.
• Figure 3: Left: A coarse-grained picture of an example path $\Gamma = (\Gamma_1, \dots, \Gamma_5)$. Right: Depiction of the relationship among $\mathrm S_{\mathrm{int}}^{(i-1)}$, $\mathrm S_{\mathrm{int}}^{(i)}$, and $\Gamma_i$. The darkening shades of gray show the addition of subsequent regions to $\mathrm S_{\mathrm{int}}$ (or equivalently, removal from $\mathrm S_{\mathrm{ext}}$).
• Figure 4: Depiction of the summation strategy. Left: A "small" term $\Gamma_i$. The red represents the factor of $f(u_i, v_i)$ from Lem. \ref{['lem:small_terms']}, which we can add if $B_{r_\ast}(x_i)$ is non-resonant. The dotted circle represents the condition $\mathsf d(x_i, x_{i+1}) = r_\ast+1$, reflected in $I_1$ and $I_3$. Right: If $\Gamma_i$ is big, then the summations over $u_i, v_i$ can be removed, as reflected in condition $I_2$.
• Figure 5: Figure illustrating the interpretation of the conclusion of the lemma as a sum over irreducible paths, sorted into four different types of paths. The sets $S_2$ and $S_3$ result from applying the quasilocal SW transform within $B_{r_\ast}(x_i)$ to $H_{\Gamma_i}$ and $H_{\Gamma_{i+1}}$. In (IB) and (IIB), the path ends at the boundary, but not at $\Gamma_{i+1}$, so we sum over endpoints $v$, accruing a factor of $A_{\ast}$ which will be absorbed into $\tilde{\varepsilon}^{\alpha r_\ast/2}$. The red sets illustrate where a term from $Z_{\mathrm{int}}$ occurs in the path, and because it is commuting, we can design the paths such that only one term from $Z_{\mathrm{int}}$ becomes an irreducible path component.

Theorems & Definitions (93)

• Definition 3.1: Definition \ref{['defn:noresonance']}, schematic
• Theorem 3.2: Thm. \ref{['thm:main_thm']}, schematic
• Definition 1.1: Graph distance
• Definition 1.2: $d$-dimensional graph
• Definition 1.3: $d$-dimensional square lattice
• Proposition 1.4
• proof
• Definition 1.5: Range and locality
• Definition 1.6
• Proposition 1.7