Boundary emptiness formation probabilities in the six-vertex model at $\Delta = - 1/2$

https://bbb.mpim-bonn.mpg.de/b/gae-a7y-hhd

The connection between statistical mechanics and the combinatorics of alternating sign matrices is known since the work of Razumov and Stroganov on the spin-1/2 XXZ chain. One important example of this combinatorial relation occurs in the study of the emptiness formation probability $EFP_{N,m}$. This observable is defined as the sum of the squares of the ground state components of the Hamiltonian for the chain of length $N$, restricted to components where $m$ consecutive spins are aligned. At the combinatorial point $\Delta = -1/2$, it takes the form of a simple product of integers. This was shown by Cantini in 2012.

In this talk, I discuss joint work with C. Hagendorf and L. Cantini where we define a new family of overlaps $C_{N,m}$ for the spin-1/2 XXZ chain. It is equal to the linear sum of the groundstate components that have $m$ consecutive aligned spins. For reasons that will be discussed, we refer to the ratio $C_{N,m}/C_{N,0}$ as the {\it boundary emptiness formation probability}. We compute $C_{N,m}$ at the combinatorial point as a simple product of integers.