Integrable Stochastic Processes Associated with the $D_2$ Algebra

Abstract

We introduce novel integrable stochastic processes associated with the $D_2$ quantum group, which can be decomposed into two XXX spin chains (or two symmetric simple exclusion processes). We establish the integrability of the model under three types of boundary conditions (periodic, twisted, and open boundaries), and present its exact solution, including the spectrum, eigenstates, and some observables.

Figures (8)

• Figure 1: Transition rates of the particles in the model.
• Figure 2: Decomposition of the stochastic process. Here, the solid and empty circles denote a particle and a hole, respectively.
• Figure 3: Illustration of the transition process. As an example, we consider the transition process from $|-2\rangle \otimes |+2\rangle$ to $|-1\rangle \otimes |+1\rangle$.
• Figure 4: Left panel: The evolution of $|-2\rangle\otimes|-2\rangle\otimes|-1\rangle$ with $N=3$. We observe that $c_{-1,-2,-2}=c_{-2,-1,-2}$ and $c_{j,k,l}\to \frac{1}{3}$ in the limit $t\to\infty$. Right panel: The evolution of $|-2\rangle\otimes|-1\rangle\otimes|+1\rangle$ with $N=3$. Here, $c_{-1,+1,-2}=c_{+1,-2,-1}=c_{-2,+1,-1}=c_{+2,-2,-2}$, $c_{+1,-1,-2}=c_{-1,-2,+1}=c_{-2,+2,-2}=c_{-2,-2,+2}$ and $c_{j,k,l}\to\frac{1}{9}$ in the limit $t\to\infty$.
• Figure 5: Dependence of the expansion coefficient on time $t$ in the regime $t\ll 1$. The initial state is $|-2\rangle\otimes|-1\rangle\otimes|+1\rangle$ with $N=3$. Left panel: Time evolution of the coefficient $c_{-2,-2,+2}$; the dashed line corresponds to $y = t$. Right panel: Time evolution of the coefficient $c_{-1,+1,-2}$; the dashed line corresponds to $y = t^2$.

Theorems & Definitions (3)

• Remark
• Remark
• Remark