Application of Operator Algebras to Stochastic Dynamics and the Heisenberg Chain

Algebraic manipulations are used to reduce a new description of stochastic dynamics and quantum spin chains involving two nonlocal operators $C,D$. For the symmetric hopping of hard-core particles, and its associated Heisenberg chain, the operator algebra may be written in the reduced form $2\dot{D}=\left[\right[C,D\left]{,C}^{-1\mathrm{}}\right]$, ${2D}^2{=CDC}^{-1\mathrm{}}{D+DC}^{-1\mathrm{}}\mathrm{DC}$. These equations are shown to describe diffusive dynamics and phase change on interchange, respectively, and to lead to Bethe ansatz equations for the spectrum of the isotropic Heisenberg chain with symmetry-breaking boundary fields. This yields new exact results for the dynamics of boundary-driven systems.