Recover exact conjectured initial FCS slopes in interacting Bethe-ansatz models

Derive the conjectured initial slope of the full-counting statistics with respect to the ballistic ratio \(\zeta=T/X\) for interacting Bethe-ansatz-solvable models, thereby testing the BMFT–Quench Action saddle-point framework beyond the explicitly solved free-fermion and Rule 54 cases.

Background

The paper solves the saddle-point equations explicitly for free fermions and for Rule 54, but a general exact solution for interacting Bethe-ansatz models remains unavailable. The authors note that exact conjectures for the initial slope of the FCS as a function of the ballistic time-to-length ratio already exist.

Recovering those conjectured slopes from the BMFT–Quench Action equations would provide a stringent independent test of the proposed framework, particularly in the small-$\zeta$ regime where a controlled partial solution may be possible without solving the dynamics at arbitrary $\zeta$.