Strong zero modes in integrable spin-$S$ chains

<jats:p>
We derive exact strong zero mode (ESZM) operators for integrable spin-
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>S</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mi>S</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
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
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>S</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mi>S</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
makes the weaker locality properties necessary. We make contact with more traditional approaches by showing how the ESZM for
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>S=\tfrac12</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mo>=</mml:mo>
<mml:mstyle displaystyle="false">
<mml:mfrac>
<mml:mn>1</mml:mn>
<mml:mn>2</mml:mn>
</mml:mfrac>
</mml:mstyle>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
acts on energy eigenstates given by solutions of the Bethe equations.
</jats:p>