1205.4400 (O Foda et al.)

Partial domain wall partition functions    [PDF]

O Foda, M Wheeler

We consider six-vertex model configurations on a rectangular lattice with n (N) horizontal (vertical) lines, and "partial domain wall boundary conditions" defined as 1. all 2n arrows on the left and right boundaries point inwards, 2. n_u (n_l) arrows on the upper (lower) boundary, such that n_u + n_l = N - n, also point inwards, 3. all remaining n+N arrows on the upper and lower boundaries point outwards, and 4. all spin configurations on the upper and lower boundaries are summed over. To generate (n-by-N) "partial domain wall configurations", one can start from A. (N-by-N) configurations with domain wall boundary conditions and delete n_u (n_l) upper (lower) horizontal lines, or B. (2n-by-N) configurations that represent the scalar product of an n-magnon Bethe eigenstate and an n-magnon generic state on an N-site spin-1/2 chain, and delete the n lines that represent the Bethe eigenstate. The corresponding "partial domain wall partition function" is computed in construction {A} ({B}) as an N-by-N (n-by-n) determinant Z_{N-by-N} (Z_{n-by-n}), by starting from Izergin's (Slavnov's) determinant expression for the partition function of the original configuration, then taking the rapidity variables of the deleted lines to infinity, and normalizing appropriately. Z_{n-by-n} was computed by I Kostov. We show that 1. Z_{N-by-N} = Z_{n-by-n}, as expected from the fact that they are partition functions of the same object, 2. That each of these determinants is a discrete KP tau-function in the auxiliary space rapidities and in the inhomogeneities. Further, recalling that these determinants represent certain 0-loop structure constants in Yang-Mills theories, we show that 3. Introducing 1-loop corrections, as proposed by N Gromov and P Vieira, preserves the determinant structure.

View original: http://arxiv.org/abs/1205.4400