We consider a unitary circuit where the underlying gates are chosen
to be \check{R}Ř-matrices
satisfying the Yang-Baxter equation and correlation functions can be
expressed through a transfer matrix formalism. These transfer matrices
are no longer Hermitian and differ from the ones guaranteeing local
conservation laws, but remain mutually commuting at different values of
the spectral parameter defining the circuit. Exact eigenstates can still
be constructed as a Bethe ansatz, but while these transfer matrices are
diagonalizable in the inhomogeneous case, the homogeneous limit
corresponds to an exceptional point where multiple eigenstates coalesce
and Jordan blocks appear. Remarkably, the complete set of (generalized)
eigenstates is only obtained when taking into account a combinatorial
number of nontrivial vacuum states. In all cases, the Bethe equations
reduce to those of the integrable spin-1
chain and exhibit a global SU(2)
symmetry, significantly reducing the total number of eigenstates
required in the calculation of correlation functions. A similar
construction is shown to hold for the calculation of out-of-time-order
correlations.
Positive One–Point Commuting Difference Operators∗†