Dynamics of a quantum system can be described by coupled Heisenberg equations. In a generic many-body system these equations form an exponentially large hierarchy that is intractable without approximations. In contrast, in an integrable system a small subset of operators can be closed with respect to commutation with the Hamiltonian. As a result, the Heisenberg equations for these operators can form a smaller closed system amenable to an analytical treatment. We demonstrate that this indeed happens in a class of integrable models where the Hamiltonian is an element of the Onsager algebra. We explicitly solve the system of Heisenberg equations for operators from this algebra. Two specific models are considered as examples: the transverse field Ising model and the superintegrable chiral 3-state Potts model.
We construct a hierarchy of exactly solvable spin-1/2 chains with
so(N)1 critical points. Our construction is based on the framework of condensate-induced transitions between topological phases. We employ this framework to construct a Hamiltonian term that couples
N transverse field Ising chains such that the resulting theory is critical and described by the
so(N)1 conformal field theory. By employing spin duality transformations, we then cast these spin chains for arbitrary
N into translationally invariant forms that all allow exact solution by the means of a Jordan-Wigner transformation. For odd
N our models generalize the phase diagram of the transverse field Ising chain, the simplest model in our hierarchy. For even
N the models can be viewed as longer ranger generalizations of the
XY chain, the next model in the hierarchy. We also demonstrate that our method of constructing spin chains with given critical points goes beyond exactly solvable models. Applying the same strategy to the Blume-Capel model, a spin-1 generalization of the Ising chain in a generic magnetic field, we construct another critical spin-1 chain with the predicted conformal field theory (CFT) describing the criticality.