Strong zero modes in integrable spin-S chains

TL;DR

The paper addresses edge-localized strong zero mode physics in integrable spin-$S$ chains with open boundaries, revealing that integer spins host $2S+1$ degenerate ground states and hence cannot support a conventional SZM; nevertheless, an exact ESZM can be constructed from a family of commuting transfer matrices. By deriving an explicit matrix-product-operator form for the ESZM and analyzing its norm and locality, the authors show that the ESZM commutes with the Hamiltonian up to exponentially small corrections and induces infinite edge coherence in the appropriate observables, even though strict edge localization is weakened for $S>\tfrac12$. They connect the ESZM to boundary bound states via Bethe ansatz and discuss the role of integrability: perturbations preserving $U(1)$ symmetry can still yield long edge coherence, while breaking integrability gradually destroys it. Together with a detailed comparison to the spin-$\tfrac12$ case, the work broadens the understanding of edge modes in higher-spin integrable systems and suggests multiple ESZMs may exist for certain spins, with potential implications for robust edge coherence in quantum devices.

Abstract

We derive exact strong zero mode (ESZM) operators for integrable spin-S chains with open boundary conditions and a boundary field. Their locality properties are generally weaker than in the previously known cases, but they still imply infinite coherence times in the vicinity of the edges. We explain how such integrable chains possess multiple ground states describing a first-order quantum phase transition, and that the odd number of such states for integer S makes the weaker locality properties necessary. We make contact with more traditional approaches by showing how the ESZM for S=1/2 acts on energy eigenstates given by solutions of the Bethe equations.

Figures (18)

• Figure 1: Triple-well Landau-Ginzburg potential.
• Figure 2: Ground state phase diagram of the integrable $S=1$ chain perturbed by a single-ion anisotropy term (\ref{['HD']}).
• Figure 3: Landau-Ginzburg potential describing the four ground states for $S=3/2$. Low-lying excitations consist of domain walls between between neighbouring minima.
• Figure 4: Ground state phase diagram of the model (\ref{['HD']}) for $S=3/2$.
• Figure 5: Projected infinite-temperature autocorrelation function $f_0(t)$ for $S=\tfrac{1}{2}$, $\Delta=4.5$, $h_L=0$ and system sizes $L=10,12,14$.