Managing singular kernels and logarithmic corrections in the staggered six-vertex model

In this paper, we investigate the spectral properties of the staggered six-vertex model with ${\mathcal{Z}}_2$ \( {\mathcal{Z}}_2 \) symmetry for arbitrary system sizes L using non-linear integral equations (NLIEs). Our study is motivated by two key questions: what is the accuracy of results based on the ODE/IQFT correspondence in the asymptotic regime of large system sizes, and what is the optimal approach based on NLIE for analyzing the staggered six-vertex model?

We demonstrate that the quantization conditions for low-lying primary and descendant states, derived from the ODE/IQFT approach in the scaling limit, are impressively accurate even for relatively small system sizes. Specifically, in the anisotropy parameter range π/4 < γ < π/2, the difference between NLIE and ODE/IQFT results for energy and quasi-momentum eigenvalues is of order $\mathcal{O}\left(L^{-2}\right)$ \( \mathcal{O}\left({L}^{-2}\right) \) .

Furthermore, we present a unifying framework for NLIEs, distinguishing between versions with singular and regular kernels. We provide a compact derivation of NLIE with a singular kernel, followed by an equivalent set with a regular kernel. We address the stability issues in numerical treatments and offer solutions to achieve high-accuracy results, validating our approach for system sizes ranging from L = 2 to L = 10²⁴.

Our findings not only validate the ODE/IQFT approach for finite system sizes but also enhance the understanding of NLIEs in the context of the staggered six-vertex model. We hope the insights gained from this study have significant implications for resolving the spectral problem of other lattice systems with emergent non-compact degrees of freedom and provide a foundation for future research in this domain.