Non-invertible symmetry breaking in a frustration-free spin chain

TL;DR

We address spontaneous breaking of non-invertible symmetries in a frustration-free spin-$\tfrac{1}{2}$ chain constructed from rank-1 projectors with $H_{af}=\sum_j A_j F_{j+1}$ and ground states $|g_a\rangle$ and $|g_f\rangle$, which are annihilated by $H_{af}$ and become orthogonal in the thermodynamic limit. Bravyi–Gosset criteria show the model is gapped since the eigenvalues of $T_\psi$ are $\{0,\frac{f-a}{(1+a^2)(1+f^2)}\}$, while non-invertible symmetry generators $S_f$ and $S_a$ commute with the Hamiltonian and act on the ground states with a finite scaling factor, enabling an algebraic-quantum-theory proof that the two ground-state sectors belong to inequivalent representations of the local observable algebra ${\cal A}$. The order operator ${\cal O}$ reveals a nonzero, opposite action in the two representations, $\pi_a({\cal O})=\sigma\,\mathrm{Id}$ and $\pi_f({\cal O})=-\sigma\,\mathrm{Id}$ with $\sigma=\frac{(a-f)^2}{(1+a^2)(1+f^2)}$, and the absence of an intertwiner confirms spontaneous breaking of the non-invertible symmetry. This work demonstrates a non-duality-based origin for non-invertible symmetry breaking and provides a framework for exploring similar phenomena in higher-rank, frustration-free models within the language of algebraic quantum theory.

Abstract

A nearest-neighbor, frustration-free spin $\frac{1}{2}$ chain can be constructed {\it via} projectors of various ranks á la Bravyi-Gosset. We show that in the rank 1 case this system is gapped and has two ground states resembling ferromagnetic states. These states spontaneously break the non-invertible symmetry connecting them. The latter is proved using the machinery of algebraic quantum theory. The non-invertible symmetries of this system do not come from a duality.