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arXiv:1705.03408 · Communications in Mathematical Physics

Six-Vertex Model and Non-linear Differential Equations. I: Spectral Problem

W. Galléas

Published Aug 23, 2018• 2 citations
arXivHTMLPDF
DOI
10.1007/s00220-018-3232-7

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A bstract In this paper, we investigate the spectral properties of the staggered six-vertex model with $$ {\mathcal{Z}}_2 $$ 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 $$ \mathcal{O}\left({L}^{-2}\right) $$ O L − 2 . 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 = 10 24 . 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.

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This paper is a continuation of our previous work "Six-vertex model and non-linear differential equations I. Spectral problem" in which we have put forward a method for studying the spectrum of the six-vertex model based on non-linear differential equations. Here we intend to elaborate on that approach and also discuss properties of the spectrum unveiled by the aforementioned differential formulation of the transfer matrix's eigenvalue problem. In particular, we intend to demonstrate how this differential approach allows one to study continuous symmetries of the transfer matrix's spectrum through the Lie groups method.
A bstract In this paper, we investigate the spectral properties of the staggered six-vertex model with Z2 {\mathcal{Z}}_2 Z2​ 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 O(L−2) \mathcal{O}\left({L}^{-2}\right) O(L−2) O L − 2 . 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 = 10 24 . 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.