Abstract
In search for 1D quantum chains with a free spectrum or simple 1D models with nice properties, we were inspired by Paul Fendley’s work on ℤ𝑛-invariant interacting chain (i.e. Baxter’s clock model) with a “free” spectrum on open boundary condition (OBC)1. He showed that the repeated action of the Lie bracket of the Hamiltonian (i.e. [𝐻𝑜𝑏𝑐 ,⋅ ]) on a seed operator is linear in terms of a finite number of linearly independent higher Hamiltonians obtained from the integrability of the model, which is proven using a set of conserved charges. This means that he can find shifting operators (analogous to raising & lowering operators) and rewrite the Hamiltonian in terms of it, thus recovering the “free” spectrum (analogous to the spectrum of free fermions). We tried to follow a similar procedure to see if the Baxter’s clock model in closed boundary conditions (CBCs) would also result in a “free” spectrum. We were able to find a set of conserved charges to prove the integrability of the clock model in CBCs. Also, we determined that the finite dimensional Krylov space for [𝐻𝑐𝑏𝑐 ,⋅ ] with some local operator must be decomposable by a direct sum of certain matrices up to similar transformations if the spectrum were to be “free” as in Fendley’s case. However, this is not the case for the clock model in CBCs. Along with other computational evidence, it suggests that the model we have considered does not have a “free” spectrum.
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