Localization from Hilbert space shattering: From theory to physical realizations

We show how a finite number of conservation laws can globally “shatter” Hilbert space into exponentially many dynamically disconnected subsectors, leading to an unexpected dynamics with features reminiscent of both many-body localization and quantum scars. A crisp example of this phenomenon is provided by a “fractonic” model of quantum dynamics constrained to conserve both charge and dipole moment. We show how the Hilbert space of the fractonic model dynamically fractures into disconnected emergent subsectors within a particular charge and dipole symmetry sector. This shattering can occur in arbitrary spatial dimensions. A large number of the emergent subsectors, exponentially many in system volume, have dimension one and exhibit strictly localized quantum dynamics—even in the absence of spatial disorder and in the presence of temporal noise. Other emergent subsectors display nontrivial dynamics and may be constructed by embedding finite-sized nontrivial blocks into the localized subspace. While “fractonic” models provide a particularly clean realization, the shattering phenomenon is more general, as we discuss. We also discuss how the key phenomena may be readily observed in near term ultracold atom experiments. In experimental realizations, the conservation laws are approximate rather than exact, so the localization only survives up to a prethermal timescale that we estimate. We comment on the implications of these results for recent predictions of Bloch/Stark many-body localization.

Figure 1

Fractonic random unitary circuit: each site is a three-state qudit. Each gate (blue box) locally conserves charge $Q={\sum }_j{S}_j^z$ and dipole moment $P={\sum }_{j}j{S}_j^z$ of the three qudits it acts upon. The block diagonal Haar-random unitary with its nontrivial blocks is also shown. Figure taken from Ref. [35].

Figure 2

Entanglement entropy of eigenstates as a function of dipole moment for a system with $L=13$ sites, in the symmetry sector with total charge $Q=0$. The color-bar denotes the number of eigenstates with entanglement entropy $S$ and a given $(Q,P)$. For each symmetry sector $(Q,P)$, there is a co-existence of low and high entanglement eigenstates, in sharp contrast to the usual behavior expected from the eigenstate thermalization hypothesis.

Figure 3

(a) Exponentially many strictly inert states in a model with range $\ell$ can be constructing by dividing the system into size $\ell$ blocks, and randomly picking each block to be of extremal positive or negative charge. A range $\ell$ gate (blue rectangles) acting on such a state locally sees either a configuration of maximal charge, or a configuration of maximal dipole moment for a given charge—and hence is forbidden from making any local rearrangements. (b) Dynamical subspaces of varying sizes can be constructed by embedding “active,” i.e., noninert, blocks into inert backgrounds. As long as the active block has a finite size, it can be prevented from melting the inert regions by surrounding it with “shielding” regions of equal or greater size. (c) Dynamics of charge $\langle S_x^z\left(t\right)\rangle$ starting from an initial state with a central active region surrounded by shielding regions. We see that the central region thermalizes, but is not able to melt the boundary spins which remain inert.

Figure 4

(a) Scaling of dimension of inert subspace as a function of system size, for $\ell$ site gates $\ell =3,4,\text{and}5$. For $\ell =3\text{and}4,$ the results are consistent with the analytic predictions. For $\ell =5,$ we have not worked out an analytic prediction, but the results are consistent with the lower bound established in Sec. 3a. (b) Plot showing subsector size distribution. For three site gates, the frequency of subsectors of a particular size decreases polynomially with the Hilbert space dimension of the subsector. For four site gates there is an initial polynomial decrease followed by a saturation, in that beyond a certain subsector size, further increases in subsector size do not seem to translate into a decrease in frequency. Notably, the maximum size of the emergent subsectors for four site gates is much larger than that for three site gates. (c) Plot showing the relative sizes of the largest emergent subsector and largest $(Q,P)$ symmetry sector, which corresponds to $Q=0\text{and}P=0$. For three site gates, the size of the largest emergent subsector is a vanishing fraction of the size of the largest symmetry sector, in the thermodynamic limit, whereas these sizes scale similarly for four and five site gates, showing that the fracturing is more severe for three site gates.

Figure 5

(a) Breakup of the Hilbert space into symmetry sectors labeled by charge $Q$ and dipole moment $P$ (b) Further shattering of each symmetry sector into emergent subsectors of various size, here shown for the symmetry sector with $(Q,P)=(0,0)$.

Figure 6

Inert states in one and two dimensions. Thick green lines indicate domain walls. (a) The state is divided into blocks of length $\ell$, and each block is chosen randomly to be $+$ or $-$. Each such pattern is an eigenstate of the dynamics when $Q$ and $P_x$ are conserved because a range $\ell$ gate (gray blocks) either sees a pattern of extremal local charge or extremal local dipole moment. (b) Translating one dimensional inert states along $\widehat{y}$ still gives inert states (c) When $Q$, $P_x$ and $P_y$ are conserved in 2d, the domain walls can be allowed to “roughen,” while the state still remains an eigenstate of the dynamics. (d) When charge, dipole and quadrupole are conserved, inert states are obtained by dividing the system into blocks of size $\ell \times \ell$, and picking each block to be $+$ or $-$.

Figure 7

How to obtain dipole and quadrupole conservation. (a) A one dimensional system in a “tilted potential” conserves dipole moment upto an exponentially long prethermal timescale. (b) When a tilted potential is applied at an angle in two dimensions, dipole moment in the direction of the tilt is conserved but one has to worry about hopping in an equipotential shell perpendicular to the tilt direction (shown). The color scale denotes the potential energy due to a tilt at ${60}^{\circ }$ relative to the $\widehat{x}$ axis. Sites within the equipotential shell see a quasiperiodic potential set by the distance from the equipotential line. (c) Quadrupole conservation in two dimensions may be obtained by placing the system in a harmonic trap, with inequivalent trap frequencies along two orthogonal directions, and with the trap rotated with respect to lattice axes.

Figure 8

(a) A plot showing the average bipartite entanglement entropy of eigenstates as a function of subsector size. The Page value would be the thermal entanglement entropy for a subsector of this size. Data are for $L=15$, three site gates, and open boundary conditions, and all eigenstates with $Q=0,1$ are considered. Note that while the eigenstate entanglement broadly tracks the appropriate Page value for the subsector, there is still a wide distribution, with many eigenstates having significantly subthermal half-chain entanglement, including eigenstates with strictly zero entanglement (perfect scars) in subsectors that do not exhibit trivial dynamics. This is related to the physics of bottlenecks described in Sec. A. (b) Entanglement entropy of individual eigenstates within the largest emergent subsector, plotted as a function of Floquet quasienergy $\varphi$. Now the eigenstates do have entanglement close to the thermal (Page) value, and this agreement gets better as system size is increased.

Figure 9

Figure showing the dynamics of entanglement starting from a random product state (not in the $z$ basis). Entanglement entropy is given as a ratio of the Page value for a random product state in a Hilbert space of dimension $3^L$. For three site gates, the entanglement entropy saturates to well below its Page value, consistent with expectations given the strong fracturing of Hilbert space. For four site gates, the saturating value of entanglement entropy is much closer to the Page value, with a slow upward drift with increasing system size.