Quantum spin chains from Onsager algebras and reflection K-matrices

Abstract

We present a representation of the generalized p-Onsager algebras O p (An ) in which the generators are expressed as local Hamiltonians of XXZ type spin chains with various boundary terms reflecting the Dynkin diagrams. Their symmetry is described by the reflection K matrices which are obtained recently by a q-boson matrix product construction related to the 3D integrability and characterized by Onsager coideals of quantum affine algebras. The spectral decomposition of the K matrices with respect to the classical part of the Onsager algebra is described conjecturally. We also include a proof of a certain invariance property of boundary vectors in the q-boson Fock space playing a key role in the matrix product construction.

Integrable boundaries for the q-Hahn process

“…Further details will be reported elsewhere. Other approaches to the construction of K-matrices that may be useful for the derivation of the full K-matrix for non-compact spin chains can be found in [2,[29][30][31][32][33]. One may also ask whether the boundary conditions we introduced are the most general stochastic boundaries.…”

Generalised Onsager Algebra in Quantum Lattice Models

“…However, the method in Sec. 4.2 only works for the homogenous case (31). Instead, we can add inhomogeneities in the transfer matrix (64) constructed in Sec.…”

“…Similar to the free fermionic eight-vertex model, we construct the Lax operator for the Fendley model with r = 2, cf. (31). Since we would like the Hamiltonian to be the second conserved charges beside momentum, which has local density acting on three consecutive sites, it is natural to consider a four-dimensional auxiliary space (i.e.…”

“…a Lie subalgebra of the affine Lie algebra A (1) n−1 . Results for other generalisations can be found in [30], even in the presence of the q-deformation [25,31]. Recently, a systematic construction of this type of generalisations was presented in [1], dubbed "generalised Onsager algebras", classifying all the Lie subalgebras of different Kac-Moody algebras with Dolan-Grady-like relations.…”