Dynamical Scar States in Driven Fracton Systems

One-dimensional fracton systems can exhibit perfect localization, failing to reach thermal equilibrium under arbitrary local unitary time evolution. We investigate how this nonergodic behavior manifests in the dynamics of a driven fracton system, specifically a one-dimensional Floquet quantum circuit model featuring conservation of a $U(1)$ charge and its dipole moment. For a typical basis of initial conditions, a majority of states heat up to a thermal state at near-infinite temperature. In contrast, a small number of states flow to a localized steady state under the Floquet time evolution. We refer to these athermal steady states as “dynamical scars,” in analogy with the scar states observed in the spectra of certain many-body Hamiltonians. Despite their small number, these dynamical scars are experimentally relevant due to their high overlap with easily prepared product states. Each scar state displays a single agglomerated fracton peak, in agreement with the steady-state configurations of fractonic random circuits. The details of these scars are insensitive to the precise form of the Floquet operator, which is constructed from random unitary matrices. Rather, dynamical scar states arise directly from fracton conservation laws, providing a concrete mechanism for the appearance of scars in systems with constrained quantum dynamics.

Figure 1

Floquet fractonic circuit (period 3): each site is a three-state qudit. Each gate (colored box) conserves $S_z^{\text{total}}$ and ${\overrightarrow{P}}_{\text{total}}$ of the three qudits it acts upon. The block diagonal Haar-random unitary with its nontrivial blocks is also shown. All gates of a particular color are identical.

Figure 2

Bipartite entropy $S$ of (a) pure eigenstates, exhibiting Hilbert space fragmentation [34, 35], (b) steady states of initial conditions slightly different from eigenstates ($\mathrm{\Delta }ϵ={10}^{-3}$), and (c) steady states of random initial conditions.

Figure 3

(a) For typical initial conditions, there is one scar state per sector, as diagnosed by entropy (shown for $Q=1$, $P=1$). (b) A typical scar state (red) features a single fracton peak, while a typical thermal state (blue) has a mostly flat $⟨S_z⟩$ profile. (c) Scar states have high fidelity with minimal product density matrices ($L=9$).

Figure 4

(a) Frequency (normalized by total number of scar states) of the number of dipole sectors per $S_z$ eigenvalue $Q$. The quadratic best fit (dashed line) agrees well with the analytic prediction ($L=15$). (b) The number of scar states scale as $L^a$, where $a\approx 3$ to high accuracy, while the total number of states scales exponentially, i.e., $\sim 3^L$.