Localization in Fractonic Random Circuits

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 $U(1)$ charge and also the dipole moment of this charge. These constraints are motivated by the quantum dynamics of fracton phases. We discover that 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 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 nonergodic phases in high dimensions.

Popular Summary

Many-body localization (MBL) is a quantum phenomenon in which an isolated system fails to reach thermodynamic equilibrium and permanently retains memory of its initial condition. While useful for exploring exotic physics, the theoretical understanding of MBL is limited by a small number of available techniques, which apply only in very restricted settings. Meanwhile, a complementary set of techniques involving random quantum circuits has been developed to understand many-body quantum chaos. Here, we demonstrate how these techniques can provide a new window into MBL and related phases of quantum matter.

To connect random quantum circuits with MBL, we borrow ideas from a third field of research known as fractons—phases of matter whose elementary excitations have restricted mobility (they either cannot move or can move only in certain directions). One might think that random circuits with fractonlike constraints would thermalize, just like any other random circuit. However, we find that this is not the case. Fractonic charges fail to spread even though the circuit allows such charges to move. This localization mechanism is distinct from the conventional understanding of MBL.

Our work greatly expands the range of systems to which notions of MBL may be applied. It impacts our understanding of quantum dynamics of fracton phases and also opens a new direction for the application of random circuit techniques.

Figure 1

Random unitary circuit: Each site is a three-state qudit. Each gate (blue box) locally conserves $S_z^{\text{total}}$, the total $z$ component of the three qudits it acts upon, and ${\overrightarrow{P}}_{\text{total}}$, the total dipole moment of the three qudits. It is a block-diagonal unitary of the form shown on the right, with each block (red boxes) of each gate independently Haar random. The smaller $1\times 1$ blocks do not flip the qudits, while the larger $2\times 2$ blocks produce charge- and dipole-conserving qudit flips as indicated.

Figure 2

Right-weight profile (averaged over 10 runs) of the spreading dipole operator in a system of $L=21$ sites with open boundary conditions, at a range of times.

Figure 3

Right-weight profile (averaged over 10 runs) of the spreading fracton charge operator in a system of $L=21$ sites with open boundary conditions.

Figure 4

Fluctuations in the right weights of the fracton charge operator. (a) Circuit to circuit fluctuations in the right-weights of the fracton charge operator for six different runs computed at late times. $i$ in ${\rho }_R^i$ labels typical runs. (b) Right-weight profile of the fracton operator averaged over different number of disorder realizations. The right-weight profile for sites 4 through 21 is shown in the plot and that for sites 1-3 is shown in an inset. This demonstrates that averaging over $\mathcal{O}(10)$ configurations is sufficient.

Figure 5

Right-weight profile (averaged over 10 runs) of the spreading fracton charge operator in a system of $L=21$ sites with gates of different sizes (with open boundary conditions). (a) $L=21$ spin chain with four-qudit gates. (b) $L=21$ spin chain with five-qudit gates.

Figure 6

Log-log plot of the integrated area under the fracton peak in the right-weight plots (${\omega }_{\text{peak}}$) as a function of time for three different gate sizes (averaged over 10 runs). On the $y$ axis, we plot the logarithm of the area under the central peak at time $t$ minus the area under the central peak at $t\to \infty$.

Figure 7

Integrated weight of fracton peak in the right weight as a function of system size at large $t(=200)$ (averaged over 10 runs). The uncertainties in the fitting parameters are given below the plot.

Figure 8

The $S_z$ expectation values (averaged over 10 runs) in the spin-1 random unitary circuit model, initialized with a single fracton. (a) $⟨S_z⟩$ vs position for different values of $t$in an $L=18$ spin chain. (b) $⟨S_z⟩$ vs position for different system sizes at large $t(=200)$.

Figure 9

Area under the $⟨S_z⟩$ peak as a function of system size at large $t(=200)$ (data averaged over 10 runs). The peak is defined as the region bounded by the central site $\pm 2$. The uncertainty in the fitting parameter is given in the legend.

Figure 10

The $⟨S_z⟩$ in the numerically obtained thermal or maximal entropy state of a classical 3-state model with fixed charge $Q$ and dipole moment $P$. The above plot is for periodic boundary conditions, with the dipole moment defined mod $L$ (the choice of origin is arbitrary, and the anticlockwise direction is taken to be positive). The results for open boundary conditions are similar: There is no localization of charge in the “classical” maximum entropy state. (a) $Q=1$, $P=2$, $L=8$. (b) $Q=2$, $P=2$, $L=8$.

Figure 11

The fracton charge probability distribution function is given by the Boltzmann distribution: The tails of $⟨S_z⟩$ have been plotted on a semilog scale for different system sizes ($L$) as a function of position at large $t$ (averaged over 10 runs).

Figure 12

Plot showing how the diffusion constant of dipoles scales with gate size. The diffusion constant is extracted by preparing an initial condition with a single dipole, tracking how dipole charge density at the origin decays as a function of time and fitting to $1/\sqrt{Dt}$. We obtain a behavior $D\sim (\text{gate size}{)}^2$, consistent with our expectations.

Figure 13

Plot showing how the charge localization length ${\xi }_{\text{fracton}}$ scales with gate size. The localization length is obtained by looking at $⟨S^z(r)⟩$ in the long-time steady state and is defined simply as the half width at half maximum of the $S^z$ peak. We find a scaling ${\xi }_{\text{fracton}}\sim (\text{gate size}{)}^2$, consistent with the predictions of our analysis.

Figure 14

A fracton initialized at site 10 in an $L=21$ system (under periodic boundary conditions) with a strength $-4$ dipole on its right and a $+2$ dipole on its left shifts to the right by two sites, i.e., $\widetilde{{\eta }_0}=-2$ in this case (data from a typical run).

Figure 15

Position of the propagating front as a function of time in the spin-1 model for the fracton and dipole operators ($L=21$) for a typical run.

Figure 16

Front broadening in the right-weight profiles (averaged over 10 runs) of the fracton and dipole operators ($L=21$). The uncertainties in the fitting parameters are given in the legend.

Figure 17

Power-law behavior of diffusive tails behind propagating fronts ($L=21$). The right-weight profiles (averaged over 10 runs) have been rescaled before fitting. The dipole operator has a power-law exponent $-3/2$, while the fracton operator has an anomalous exponent of $-5/2$. Fitting parameter uncertainties are specified in the plot.

Figure 18

Local observable entanglement entropy $S_x^c(t)$ (averaged over 10 runs) as a function of time in the spin-1 system ($L=18$) with dipole conservation (solid lines) and without (dashed lines).

Figure 19

Operator entanglement (averaged over 10 runs) for the fracton charge operator in a spin chain with $L=9$. The late-time operator entanglement of the corresponding thermal state and for the state evolved without the dipole constraint are also plotted, and these are well above the corresponding value for the fracton charge operator. The “thermal” state here is the equal-weight superposition over all $S^z$ product states with a given charge and dipole moment.

Figure 20

The level spacings of the entanglement Hamiltonian for the late-time state obey semi-Poisson statistics, in contrast with an integrable Anderson-like phase, which would obey Poisson statistics.

Figure 21

Fracton diffusion in the random circuit system initialized with multiple fracton peaks (averaged over 10 runs). (a) Diusion constant of fractons in a two-fracton system is lower than that of the dipoles by a factor of $1/l^2$ where $l$ is the initial separation between the fractons. The uncertainties in the fitting parameters are given in the legend. (b) Right-weight profile in a system with a finite density of fractons (we look at the superposition of the following strings: $\mathbb{I}$ everywhere except at site 2, $\mathbb{I}$ everywhere except at site 5, $\mathbb{I}$ everywhere except at site 13, $\mathbb{I}$ everywhere except at site 16). Note that the fractons end up agglomerating at their center of mass.

Figure 22

The $⟨S_z(t)⟩$ close to the maximal charge sector. The late-time profile shows a peak, indicating localization even at finite fracton density.

Figure 23

The $⟨S_z(t)⟩$ for an initial condition in the single charge sector but packed with dipoles. The late-time profile shows a peak indicating localization even in this limit.

Figure 24

Right-weight profile (for a typical run) of the spreading fracton charge operator in a large $S$ system with open boundary conditions. (a) $L=18$ spin-10 chain. (b) $L=15$ spin-100 chain.

Figure 25

Power-law behavior of tails lagging behind propagating fronts (for a typical run) in a large $S$ ($S=10$, $L=18$) system with open boundary conditions. Uncertainties in fitting parameters are given in the plot.

Figure 26

Integrated weight in the fracton peak in the right-weight profile for large $S$ at late times ($t\sim 200$).