Symmetry Fragmentation

Abstract

In quantum many-body systems with kinetically constrained dynamics, the Hilbert space can split into exponentially many disconnected subsectors, a phenomenon known as Hilbert-space fragmentation. We study the interplay of such fragmentation with symmetries, focusing on charge conserving systems with charge conjugation and translation symmetries as a concrete example. The non-Abelian algebra of these symmetries and the projectors onto the fragmented subsectors leads to the emergence of exponentially many logical qubits encoded in degenerate pairs of eigenstates, which can be highly entangled. This algebra also provides necessary conditions for experimental signatures of Hilbert-space fragmentation, such as the persistence of density imbalances at late times.

Figures (3)

• Figure 1: Symmetry fragmentation. Top: Interaction graph of the Hamiltonian \ref{['eq:xnor']} showing four Krylov sectors. Charge conjugation symmetry (pink bonds) pairs the two largest Krylov sectors together while leaving the smaller two invariant. Translation symmetry (orange arrows) collapses both pairs of sectors, since neither is translation invariant.
• Figure 2: Dynamics of encoded qubits under perturbed evolution at $L=18$. The time-evolved charge conjugation operator $\braket{X(t)}$ (see text) exhibits oscillations with period $8/\delta$, where $\delta$ is the qubit splitting induced by the perturbation. An initial state formed by superposing two frozen states of the unperturbed dynamics (green circles) exhibits coherent oscillations under the perturbed dynamics, while an initial superposition of two complex eigenstates (orange squares) exhibits damped oscillations.
• Figure 3: Site-resolved dynamics of an initial computational basis state on $L=24$ sites. The local expectation value $\braket{Z_i(t)}$ is represented by a color scale with light (dark) corresponding to the value $+1$ ($-1$). The initial state belongs to a Krylov sector of dimension 1456 that is neither charge-conjugation nor translation invariant, resulting in relaxation to a staggered magnetization profile at late times.