Categorical Symmetries in Spin Models with Atom Arrays

Abstract

Categorical symmetries have recently been shown to generalize the classification of phases of matter, significantly broadening the traditional Landau paradigm. To test these predictions, we propose a simple spin chain model that encompasses all gapped phases and second-order phase transitions governed by the categorical symmetry $\mathsf{Rep}(D_8)$. This model not only captures the essential features of non-invertible phases but is also straightforward enough to enable practical realization. Specifically, we outline an implementation using neutral atoms trapped in optical tweezer arrays. Employing a dual-species setup and Rydberg blockade, we propose a digital simulation approach that can efficiently implement the many-body evolution in several nontrivial quantum phases.

Figures (3)

• Figure 1: (a) The anyon chain for a categorical symmetry (left) is used to construct lattice models, generically on a constrained Hilbert space. Here, we realize this on a spin-chain, realized on an unconstrained 3-qubit tensor product Hilbert space (right). (b) Lattice configuration and atomic level diagram used in our proposal for realizing the spin chain in (a) with categorical symmetry. The silver and the golden circles represent the data and the ancillary atomic qubits, respectively. (c) Trotterization scheme of the quantum circuit for simulating the many-body dynamics. On-site and inter-site terms can be realized by only driving data atoms or ancillary atoms, respectively.
• Figure 2: (a) and (b) show the evolution of the local and string order parameters $\mathcal{O}_\mathrm{loc}$ and $\mathcal{O}_\mathrm{str}$ for the annealing across the phase transition $\mathrm{Trivial}\rightarrow \mathsf{Rep}(D_8)/({\mathbb Z}_2\times{\mathbb Z}_2)$ SSB and $\mathrm{Trivial}\rightarrow\mathrm{SPT}$, respectively. The results are obtained by time-evolving block decimation (TEBD) with $L=18$ sites ($54$ data qubits) and $\tau=1$. (c) and (d) show the errors in (a) and (b) as a function of the step size $\tau$ with $L=6$. The dashed curves are polynomial fits ($\propto\tau^2$) to the Trotter errors. In (a), symmetry-breaking defects are included at the boundaries to select one of the degenerate ground states.
• Figure 3: Characterization of the quantum phase transition $({\mathbb Z}_2\times {\mathbb Z}_2) \textrm{ SSB } \rightarrow\mathsf{Rep}(D_8) \textrm{ SSB}$. (a) and (b) show the spectrum (lowest five eigenenergies) and the order parameter $\mathcal{O}_\mathrm{loc}=\sum_iX_i^\mathrm{I}/L$ as a function of $\lambda$. The results are obtained by exact diagonalization with $L=6$ sites ($18$ qubits). In (b), a symmetry-breaking perturbation $-0.01\sum_i X_i^\mathrm{I}$ is applied.