Obstruction to Ergodicity from Locality and $U(1)$ Higher Symmetries on the Lattice

Abstract

We argue that the presence of \emph{any} exact $U(1)$ higher-form symmetry, under mild assumptions, presents a fundamental obstruction to ergodicity under unitary dynamics in lattice systems with local interactions and finite on-site Hilbert space dimension. Focusing on the two-dimensional case, we show that such systems necessarily exhibit Hilbert space fragmentation and explicitly construct Krylov sectors whose number scales exponentially with system size. While these sectors cannot be distinguished by symmetry quantum numbers, we identify the emergent integrals of motion which characterize them. Our symmetry-based approach is insensitive to details of the Hamiltonian and the lattice, providing a systematic explanation for ergodicity-breaking in a range of systems, including quantum link models.

Figures (4)

• Figure 1: Warm-up example. (a) Oriented square lattice with top/bottom and left/right edges identified. Orange lines denote the 1-form generators $Q_{\gamma_y}$ and the blue dashed line the charged operator $V_{\Gamma_y}$. Green loops denote $Q_\gamma$ on contractible loops and $Q_{L/R}$ the left/right edges of the corresponding $Q_\gamma$. (b) Two fragmented states with identical symmetry numbers under $Q_{\gamma_{x,y}}$ and all $Q_{\square}$, but lying in distinct Krylov sectors. The blue dashed lines indicate where $V_{\Gamma_y}$ were applied relative to the state with all spins up on the vertical bonds.
• Figure 2: Generalizing the warm-up example. (a) Example of bulk fragmentation: we place the qubits along the orange lines in extremal states of $Q_{\gamma}$. The green lines indicate the emergent integrals of motion. The dashed blue line is a charged operator toggling between the Krylov sectors. (b) Fragmentation on a general lattice. Orange lines indicate strings of qubits in extremal eigenstates of $Q_\gamma$ and green lines indicate the emergent integrals of motion.
• Figure 3: Fragmentation for non-onsite $U(1)$ generators. (a) Examples of states in distinct Krylov sectors in a square lattice system with the symmetry Eq. \ref{['eq:ising-symmetry']}. The shaded blue ribbons indicate where the charged operators $V_\gamma$ were applied. (b) Fragmentation from general 1-form symmetry on the square lattice. The orange, green, and blue strips indicate the symmetry generators, emergent integrals of motion, and charged operators, respectively.
• Figure 4: Applications. (a) Quantum link model on the triangular lattice with $U_\triangle = \prod_{l \in \triangle} U_l$ (left) and Gauss' law operator (right). (b) Examples of qubit configurations on a square plaquette indicating the presence of a $U(1)$ 1-form charge only in the presence of a CZ 1-form charge and configurations on octagonal plaquettes with trivial CZ charge but non-trivial $U(1)$ 1-form charge.