Logarithmic corrections arising from non-linear integral equations with singular kernels

Andreas Klumper

Wuppertal

Mon, Mar. 04th 2024, 11:00

Salle Claude Itzykson, Bât. 774, Orme des Merisiers

Andreas Kluemper, Mouhcine Azhari
We present recent results for the computational treatment of the spectra of
the integrable staggered six-vertex model and the integrable $3-\bar 3$
superspin chain.
The staggered six-vertex model has attracted the interest of several groups of
authors who derived a wealth of results (e.g.~Ikhlef, Jacobsen, Saleur 08, 12;
Frahm, Martins 12; Candu, Ikhlef 13; Frahm, Seel 14; Bazhanov, Kotousov,
Koval, Lukyanov 20). A remaining problem is how to compute the low-lying
eigenvalues for arbitrary system sizes.
We derive by proven means a set of non-linear integral equations (NLIE) with
the unpleasant property of singular terms in the kernel. Due to this fact
these equations do not lend themselves to an iterative treatment. However, we
have succeeded in deriving from the singular NLIE an equivalent set of NLIE
with purely regular kernel. This set can be solved for the lowest lying
excitations for system sizes $L = 10, 10^2 , 10^3 , ...., 10^9 ,
...10^{24}.$. Interestingly, the singular NLIE can be used to derive the CFT data
with logarithmic corrections ${\mathcal O}\left(1/(\log L)^2\right)$.
Finally, we present results for the $3-\bar 3$ superspin chain intensively
investigated by Essler, Frahm, Saleur (2005). Here we show how to derive two
sets of NLIE, a singular one and a regular one. From the singular NLIE we
derive the type of corrections to the CFT data, ${\mathcal O}\left(1/\log
L\right)$. The numerical iteration of the regular NLIE is not yet
successfully convergent: for the $3-\bar 3$ model not only is the genuine NLIE
singular, but here some of the solutions also have singular properties.
\end{document}

Contact : Vincent
PASQUIER