Six-vertex model on a finite lattice: integral representations for nonlocal correlation functions

Abstract

We consider the problem of calculation of correlation functions in the six-vertex model with domain wall boundary conditions. To this aim, we formulate the model as a scalar product of off-shell Bethe states, and, by applying the quantum inverse scattering method, we derive three different integral representations for these states. By suitably combining such representations, and using certain antisymmetrization relation in two sets of variables, it is possible to derive integral representations for various correlation functions. In particular, focusing on the emptiness formation probability, besides reproducing the known result, obtained by other means elsewhere, we provide a new one. By construction, the two representations differ in the number of integrations and their equivalence is related to a hierarchy of highly nontrivial identities.

Figures (4)

• Figure 1: The model: (a) The six possible vertex configurations, and their weights; (b) The $N\times N$ square lattice with domain wall boundary conditions; also shown is the assigment of parameters $\lambda_1,\ldots,\lambda_N$ and $\nu_1,\ldots,\nu_N$ to the vertical and horizontal lines, respectively.
• Figure 2: (a) A possible $s$th-row configuration, with $s$ up arrows at positions $r_1,\dots,r_s$ (here, $N=7$, $s=2$, $r_1=2$, $r_2=4$); (b) The corresponding top and bottom portions resulting from splitting the original lattice in correspondence of the $s$th row.
• Figure 3: Emptiness formation probability: (a) Basic definition, as the probability of having on all edges within a rectangular region of size $(N-r)\times s$ in the top-left corner of the lattice, arrows pointing down or left (here is shown the case $s=2$, $r=4$ and $N=7$); (b) Equivalent definition, as the probability of having, between the $s$th and $(s+1)$th horizontal lines, $N-r$ down arrows at positions $N-r+1,\ldots,N$; (c) In terms of the row configuration probability, as a sum over $1\leqslant r_1 <\ldots< r_s\leqslant r$, where the dashed line shows the border which cannot be passed by the positions $r_1,\ldots,r_s$ of the up arrows in the summation.
• Figure 4: One more definition of emptiness formation probability: (a) The same as in figure \ref{['fig-EFP-NW']}a, but reflected with respect to the SW-NE diagonal and with all arrows reversed; (b) The same as in figure \ref{['fig-EFP-NW']}b, but now, between the $r$th and $(r+1)$th horizontal lines, among the $r$ up arrows, $s$ of them must be at positions $1,\ldots,s$; (c) In terms of row configuration probability, as a sum over the positions of the remaining $n=r-s$ up arrows $s+1\leqslant r_{s+1}<\ldots<r_{n+s}\leqslant N$, where the dashed line now shows that all positions at $1,\ldots,s$ are occupied.