Hilbert-Space Fragmentation from Strict Confinement

We study one-dimensional spin-$1/2$ models in which strict confinement of Ising domain walls leads to the fragmentation of Hilbert space into exponentially many disconnected subspaces. Whereas most previous works emphasize dipole moment conservation as an essential ingredient for such fragmentation, we instead require two commuting U(1) conserved quantities associated with the total domain-wall number and the total magnetization. The latter arises naturally from the confinement of domain walls. Remarkably, while some connected components of the Hilbert space thermalize, others are integrable by Bethe ansatz. We further demonstrate how this Hilbert-space fragmentation pattern arises perturbatively in the confining limit of ${\mathbb{Z}}_2$ gauge theory coupled to fermionic matter, leading to a hierarchy of timescales for motion of the fermions. This model can be realized experimentally in two complementary settings.

Figure 1

(a) Connectivity within the sector $(n_{\mathrm{DW}}=8,S^z=-2)$ for $L=18$. This sector has a total Hilbert-space dimension of 4410. (b) Ratio of the size of the largest emergent subsector within the largest $(n_{\mathrm{DW}},S^z)$ sector to that of the entire sector, for different system sizes.

Figure 2

(a) Entanglement entropy of the eigenstates within the sector $(n_{\mathrm{DW}}=8,S^z=-2)$ under an equibipartitioning of the system. Red line, Page value of the $(n_{\mathrm{DW}},S^z)$ sector; green line, Page value of the largest connected subsector. (b) Entanglement entropy growth (normalized by the Page value) after a quantum quench starting from random product states, averaged over 200 initial states.

Figure 3

(a) Entanglement entropy of eigenstates within an emergent subsector built from the root configuration $0\boxed{111111000000}\boxed{010101010101}0$ for system size $L=26$. This subsector has dimension 12 376 and is nonintegrable. (b) Entanglement entropy of eigenstates within an emergent subsector built from the root configuration $0\boxed{000000000000}\boxed{010101010101}0$ for system size $L=26$. This subsector has dimension 27 132 and is integrable. Red lines mark the Page value of the corresponding subsector.

Figure 4

(a) Expectation value of the spin in the middle of the chain under time evolution with Hamiltonian (3), starting from initial configurations with two pairs of domain walls: $00\cdots 011\dots 100\cdots 011\dots 100\cdots 0$. The dashed lines mark the diagonal ensemble average values: $⟨{\sigma }_{13}^z{⟩}_{\text{diag}}={\sum }_n⟨n|{\sigma }_{13}^z|n⟩|c_n{|}^2$, where $|n⟩$ denotes the eigenstate of the Hamiltonian and $c_n=⟨n|{\psi }_0⟩$ is the overlap between the initial state and each eigenstate. (b) Scaling of the saturation timescale $t_{*}$ as a function of $d_{\mathrm{DW}}$.