Glassy quantum dynamics in translation invariant fracton models

TL;DR

The paper demonstrates that translation-invariant three-dimensional fracton models can exhibit glassy quantum dynamics without quenched disorder. Type I fractons show mobility suppressed by a factor $e^{-W/T}$ and, when coupled to a finite-temperature bath, approach equilibrium via a logarithmic-in-time relaxation over an exponentially wide time window, while Type II fractons display subdiffusive motion with a super-exponential relaxation time $t_{\text{relax}} \sim \exp\left(c' W^{2}/T^{2}\right)$ at low $T$. The analysis reveals two distinct dynamical regimes: slow, activated-like transport for the charge sector and unusually slow bath-mediated equilibration in Type I, and extremely slow, barrier-limited diffusion in Type II, both in translation-invariant 3D settings. The work suggests potential phases where thermal conductivity and charge transport decouple (thermal metal but charge insulator) for generalized $U(1)$ fracton models, linking fracton physics to broader questions in MBL, glassy dynamics, and three-dimensional topological order. These findings open avenues for exploring localization-like dynamics in higher dimensions and for leveraging fracton dynamics in quantum information contexts.

Abstract

We investigate relaxation in the recently discovered "fracton" models and discover that these models naturally host glassy quantum dynamics in the absence of quenched disorder. We begin with a discussion of "type I" fracton models, in the taxonomy of Vijay, Haah, and Fu. We demonstrate that in these systems, the mobility of charges is suppressed exponentially in the inverse temperature. We further demonstrate that when a zero temperature type I fracton model is placed in contact with a finite temperature heat bath, the approach to equilibrium is a logarithmic function of time over an exponentially wide window of time scales. Generalizing to the more complex "type II" fracton models, we find that the charges exhibit subdiffusion upto a relaxation time that diverges at low temperatures as a super-exponential function of inverse temperature. This behaviour is reminiscent of "nearly localized" disordered systems, but occurs with a translation invariant three-dimensional Hamiltonian. We also conjecture that fracton models with conserved charge may support a phase which is a thermal metal but a charge insulator.

Figures (14)

• Figure 1: The X-Cube model is represented by spins $\sigma$ placed on the links of a cubic lattice and is given by the sum of a twelve-spin $\sigma_x$ operator at each cube $c$ and planar four-spin $\sigma_z$ operators at each vertex $v$.
• Figure 2: Topological excitations of the X-Cube model are depicted in (i) and (ii). Fractons $e^{(0)}$ are created at corners by acting on the ground state by a membrane operator $\mathcal{M}$ that is the product of $\sigma_z$ operators along red links. Wilson line operators create a composite topological excitation $e^{(2)}$.
• Figure 3: A single fracton hop. Starting from a single isolated fracton, we can move it over by one site by the action of a $\sigma_z$ operator, shown in red, in step a). However, this creates two additional fractons which together form a dimension-1 excitation that can move along a line without creating any further excitations. As shown in b), this pair can then be moved off to infinity by the action of a Wilson line of $\sigma_z$ operators.
• Figure 4: A fracton hop mediated by an $e^{(2)}$ excitation. a) Starting from a single isolated fracton and an $e^{(2)}$, we first act by a $\sigma_z$ operator (in red) resulting in three fractons. b) Acting with another $\sigma_z$ takes us to a configuration with an $e^{(2)}$ and a fracton that has moved over by two sites. In this manner, fractons can hop while remaining on-shell.
• Figure 5: Dominant second order on-shell processes between charged fracton sector and thermal bath. (i) Two bosons convert into four fractons. (ii) Two bosons convert into a boson and two fractons. (iii) A boson and a fracton convert into three fractons.