Localization in Fractonic Random Circuits

Abstract

We study the spreading of initially-local operators under unitary time evolution in a onedimensional random quantum circuit model which is constrained to conserve a U (1) charge and also the dipole moment of this charge. These constraints are motivated by the quantum dynamics of fracton phases. We discover that charge remains localized at its initial position, providing a crisp example of a non-ergodic 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 non-trivial predictions which 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 non-conserved operators, whose spreading is governed by exponents distinct from those observed in non-fractonic circuits. These non-standard 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 non-ergodic phenomenology is found to persist to initial conditions containing non-zero 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 translation invariant 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 non-ergodic phases in high dimensions.

Fracton phases of matter

“…While dipole absorption can lead to thermalization at the longest timescales in generic three-dimensional fracton models, the behavior of one-dimensional fracton systems can be a bit more complicated, leading in some cases to truly non-ergodic behavior. Such a failure of a one-dimensional fracton system to fail to thermalize at any timescale was first encountered in the context of random unitary circuit dynamics [13]. Random unitary circuits provide, in a certain sense, the most generic form of unitary time evolution, without the presence of additional constraining conservation laws, such as energy conservation.…”

“…In Reference [13], Pai et al studied random unitary circuits subject to the two conservation laws of charge and dipole conservation, thereby implementing fracton behavior. (This model takes fracton behavior as a starting point, in order to study its physical consequences, as opposed to deriving fracton conservation laws from some underyling microscopic interactions.)…”

“…Thus, this system exhibits non-ergodic behavior, never forgetting the initial position of the fracton. Various other aspects of the dynamics were studied in [13], such as an anomalous exponent in the "tail" of the ballistically spreading peak. It was also shown that fractons could attract each other under random unitary evolution, consistent with the gravitational analysis of Reference [14].…”

Quantum butterfly effect in polarized Floquet systems

“…Inspired by this work, a series of RUCs with different symmetries have been proposed [6,7,11], which introduce extra conservation laws in the quantum dynamics and give rise to diffusive transport on top of ballistic information propagation in conventional models. With the right conservation laws [12][13][14], localization is also possible.…”

Quantum robustness of fracton phases

“…The physical origin of the thermal fragility in two-dimensional stabilizer codes is the finite energy barrier between different topological ground states, e.g., the toric code becomes thermally stable only for dimensions larger three [14]. The recently discovered fracton phases reach beyond such topological codes and have interesting crosslinks to other domains in physics like elasticity [38][39][40][41][42][43], localization [15,44,45], gravity [46][47][48], Majorana fermions [49,50], and deconfined quantum criticality [51]. * matthias.muehlhauser@fau.de † matthias.walther@fau.de ‡ david.reiss@fu-berlin.de § kai.phillip.schmidt@fau.de…”