Exponentially slow thermalization in 1D fragmented dynamics

TL;DR

The paper investigates slow thermalization in one-dimensional constrained quantum and classical dynamics that fragment the Hilbert space into many Krylov sectors. By modeling boundary-induced decoherence as a depolarizing bath and projecting intra-sector dynamics to a maximally mixed state, the authors map thermalization to a Markov process on the Krylov graph and connect the thermalization time to the graph's conductance. They conjecture and provide evidence that exponentially fragmented dynamics yield thermalization times that are either infinite or exponentially large in the system size, with proofs established for broad classes linked to hyperbolic groups and heat-kernel conductance. Through concrete models (dipole-conserving, tJ_z, spin-1 breakdown, and pair-flip-like constructions) and fragmentation-transitions in East and higher-range dipole models, the work reveals multiple mechanisms—bottlenecks, large diameters, and real-space constraints—that slow thermalization and highlights connections to expander graphs and Benjamini-type conjectures. The results offer a unified framework to predict slow relaxation from structural constraints and suggest avenues for extending the approach to quantum dynamics and higher dimensions, with implications for understanding constrained dynamics in many-body systems.

Abstract

We investigate the thermalization dynamics of 1D systems with local constraints coupled to an infinite temperature bath at one boundary. The coupling to the bath eventually erases the effects of the constraints, causing the system to tend towards a maximally mixed state at long times. We show that for a large class of local constraints, the time at which thermalization occurs can be extremely long. In particular, we present evidence for the following conjecture: when the constrained dynamics displays strong Hilbert space fragmentation, the thermalization time diverges exponentially with system size. We show that this conjecture holds for a wide range of dynamical constraints, including dipole-conserving dynamics, the $tJ_z$ model, and a large class of group-based dynamics, and relate a general proof of our conjecture to a different conjecture about the existence of certain expander graphs.

Figures (13)

• Figure 1: A schematic of the general setup. In a), an open 1d chain is coupled to a thermal bath at one of its boundaries, which induces depolarizing noise on one of its boundary sites. We conjecture that in all exponentially fragmented dynamics, the thermalization time $t_{\rm th}$ is either infinite, or scales exponentially with system size. In b), a finite chain is bipartitioned into $A$ and $A^c$, with the state on $A^c$ now playing the role of the bath. This dynamics generically thermalizes at least as slowly as the former type, although thermalization may only be possible when $|A^c| / |A|$ diverges sufficiently quickly as $|A|\rightarrow\infty$.
• Figure 2: Coarse-graining and the Krylov graph. Each vertex represents a basis state in $\mathcal{H}$, with a solid line drawn between two vertices if the corresponding states are connected by the constraint-preserving part of the dynamics, and a dashed line drawn if they are connected by the constraint-breaking part; the resulting graph is $\mathcal{G}_\mathcal{H}$. Each Krylov sector (purple circles) defines a vertex of the Krylov graph $\mathcal{G}_\mathcal{H}$, with an edge drawn between two sectors if states in each sector are connected to one another under the constraint-breaking part.
• Figure 3: Different types of Hilbert space connectivity that give rise to exponentially slow thermalization. In class 0, Hilbert space remains disconnected even in the presence of the bath, and $t_{\rm th} = \infty$. In class I, the diameter of the Krylov graph $\mathcal{G}_\mathcal{K}$ is $\Omega(\exp(L))$, meaning that it takes the bath at least $\sim\exp(L)$ time steps to move the system across Hilbert space. In class II, the bath can connect any two states in only ${\rm poly}(L)$ time steps, but Hilbert space possesses strong bottlenecks that render the thermalization dynamics exponentially slow. In this class, $\mathcal{G}_\mathcal{K}$ typically has a tree-like structure as in the figure, where $\mathcal{G}_\mathcal{K}$ can become disconnected into two thermodynamically large pieces after only a small number of edges are cut.
• Figure 4: (a) Allowed dynamical moves of the spin-1 breakdown model. (b) Sizes of each Krylov sector $\mathcal{K}_Q$ with charge $Q$. Notice that the sector sizes are symmetric about $Q_{\rm max}/2$ (particle-hole symmetric). The sizes also exhibit a self-similar structure, which can be understood from an underlying recursion relation of $|\mathcal{K}_Q|$ (see App. \ref{['app:breakdown']})
• Figure 5: Numerical results for the total charge relaxation dynamics in the spin-1 breakdown model with boundary depolarizing noise. The blue line shows results obtained from a direct simulation of the stochastic dynamics of the spin-1 chain under deep RU circuits in the bulk + depolarizing channel at the endpoint. The orange line is obtained by simulating the effective random walk process on the corresponding Krylov graph. Both results are consistent with a diffusive charge relaxation.

Theorems & Definitions (57)

• Definition 1: Hyperbolic groups
• Theorem 1: Gromov
• Definition 2: Amenability
• Proposition 1
• Lemma 1
• proof
• Proposition 2
• proof
• Proposition 3
• Proposition 4