Fracton systems exhibit restricted mobility of their excitations due to the presence of higher-order conservation laws. Here we study the time evolution of a one-dimensional fracton system with charge and dipole moment conservation using a random unitary circuit description. Previous work has shown that when the random unitary operators act on four or more sites, an arbitrary initial state eventually thermalizes via a universal subdiffusive dynamics. In contrast, a system evolving under three-site gates fails to thermalize due to strong ``fragmentation'' of the Hilbert space. Here we show that three-site gate dynamics causes a given initial state to evolve toward a highly nonthermal state on a timescale consistent with Brownian diffusion. Strikingly, the dynamics produces an effective attraction between isolated fractons or between a single fracton and the boundaries of the system, as in the Casimir effect of quantum electrodynamics. We show how this attraction can be understood by exact mapping to a simple classical statistical mechanics problem, which we solve exactly for the case of an initial state with either one or two fractons.
Fracton systems exhibit restricted mobility of their excitations due to the presence of higher-order conservation laws. Here we study the time evolution of a one-dimensional fracton system with charge and dipole moment conservation using a random unitary circuit description. Previous work has shown that when the random unitary operators act on four or more sites, an arbitrary initial state eventually thermalizes via a universal subdiffusive dynamics. In contrast, a system evolving under three-site gates fails to thermalize due to strong ``fragmentation'' of the Hilbert space. Here we show that three-site gate dynamics causes a given initial state to evolve toward a highly nonthermal state on a timescale consistent with Brownian diffusion. Strikingly, the dynamics produces an effective attraction between isolated fractons or between a single fracton and the boundaries of the system, as in the Casimir effect of quantum electrodynamics. We show how this attraction can be understood by exact mapping to a simple classical statistical mechanics problem, which we solve exactly for the case of an initial state with either one or two fractons.
We show that the combination of charge and dipole conservation-characteristic of fracton systems-leads to an extensive fragmentation of the Hilbert space, which, in turn, can lead to a breakdown of thermalization. As a concrete example, we investigate the out-of-equilibrium dynamics of one-dimensional spin-1 models that conserve charge (total S-z) and its associated dipole moment. First, we consider a minimal model including only three-site terms and find that the infinite temperature autocorrelation saturates to a finite value-showcasing nonthermal behavior. The absence of thermalization is identified as a consequence of the strong fragmentation of the Hilbert space into exponentially many invariant subspaces in the local S-z basis, arising from the interplay of dipole conservation and local interactions. Second, we extend the model by including four-site terms and find that this perturbation leads to a weak fragmentation: The system still has exponentially many invariant subspaces, but they are no longer sufficient to avoid thermalization for typical initial states. More generally, for any finite range of interactions, the system still exhibits nonthermal eigenstates appearing throughout the entire spectrum. We compare our results to charge and dipole moment-conserving random unitary circuit models for which we reach identical conclusions.
We study a random circuit model of constrained fracton dynamics, in which particles on a one-dimensional lattice undergo random local motion subject to both charge and dipole moment conservation. The configuration space of this system exhibits a continuous phase transition between a weakly fragmented (``thermalizing'') phase and a strongly fragmented (``nonthermalizing'') phase as a function of the number density of particles. Here, by mapping to two different problems in combinatorics, we identify an exact solution for the critical density nc. Specifically, when evolution proceeds by operators that act on $\ensuremath{\ell}$ contiguous sites, the critical density is given by ${n}_{c}=1/(\ensuremath{\ell}\ensuremath{-}2)$. We identify the critical scaling near the transition, and we show that there is a universal value of the correlation length exponent $\ensuremath{\nu}=2$. We confirm our theoretical results with numeric simulations. In the thermalizing phase the dynamical exponent is subdiffusive, z=4, while at the critical point it increases to ${z}_{c}\ensuremath{\gtrsim}6$.
We show that the combination of charge and dipole conservation-characteristic of fracton systemsleads to an extensive fragmentation of the Hilbert space, which, in turn, can lead to a breakdown of thermalization.As a concrete example, we investigate the out-of-equilibrium dynamics of one-dimensional spin-1 models that conserve charge (total S z ) and its associated dipole moment.First, we consider a minimal model including only three-site terms and find that the infinite temperature autocorrelation saturates to a finite value-showcasing nonthermal behavior.The absence of thermalization is identified as a consequence of the strong fragmentation of the Hilbert space into exponentially many invariant subspaces in the local S z basis, arising from the interplay of dipole conservation and local interactions.Second, we extend the model by including four-site terms and find that this perturbation leads to a weak fragmentation: The system still has exponentially many invariant subspaces, but they are no longer sufficient to avoid thermalization for typical initial states.More generally, for any finite range of interactions, the system still exhibits nonthermal eigenstates appearing throughout the entire spectrum.We compare our results to charge and dipole moment-conserving random unitary circuit models for which we reach identical conclusions.
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.
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.
We study the spreading of initially local operators under unitary time evolution in a one-dimensional random quantum circuit model that is constrained to conserve a Uð1Þ charge and also the dipole moment of this charge.These constraints are motivated by the quantum dynamics of fracton phases.We discover that the charge remains localized at its initial position, providing a crisp example of a nonergodic dynamical phase of random circuit dynamics.This localization can be understood as a consequence of the return properties of low-dimensional random walks, through a mechanism reminiscent of weak localization, but insensitive to dephasing.The charge dynamics is well described by a system of coupled hydrodynamic equations, which makes several nontrivial predictions that are all in good agreement with numerics in one dimension.Importantly, these equations also predict localization in two-dimensional fractonic random circuits.We further find that the immobile fractonic charge emits nonconserved operators, whose spreading is governed by exponents that are distinct from those observed in nonfractonic circuits.These nonstandard exponents are also explained by our coupled hydrodynamic equations.Where entanglement properties are concerned, we find that fractonic operators exhibit a short time linear growth of observable entanglement with saturation to an area law, as well as a subthermal volume law for operator entanglement.The entanglement spectrum is found to follow semi-Poisson statistics, similar to eigenstates of many-body localized systems.The nonergodic phenomenology is found to persist to initial conditions containing nonzero density of dipolar or fractonic charge, including states near the sector of maximal charge.Our work implies that low-dimensional fracton systems should preserve forever a memory of their initial conditions in local observables under noisy quantum dynamics, thereby constituting ideal memories.It also implies that one-and two-dimensional fracton systems should realize true many-body localization (MBL) under Hamiltonian dynamics, even in the absence of disorder, with the obstructions to MBL in translationinvariant systems and in spatial dimensions greater than one being evaded by the nature of the mechanism responsible for localization.We also suggest a possible route to new nonergodic phases in high dimensions.