Non-linear integral equations for the XXX spin-1/2 quantum chain with non-diagonal boundary fields

verfasst von: Holger Frahm, Andreas Klümper, Dennis Wagner, Xin Zhang
Abstract: The XXX spin-\(\frac{1}{2}\) Heisenberg chain with non-diagonal boundary fields represents a cornerstone model in the study of integrable systems with open boundaries. Despite its significance, solving this model exactly has remained a formidable challenge due to the breaking of \(U(1)\) symmetry. Building on the off-diagonal Bethe Ansatz (ODBA), we derive a set of nonlinear integral equations (NLIEs) that encapsulate the exact spectrum of the model.
For \(U(1)\) symmetric spin-\(\frac{1}{2}\) chains such NLIEs involve two functions \(a(x)\) and \(\bar{a}(x)\) coupled by an integration kernel with short-ranged elements. The solution functions show characteristic features for arguments at some length scale which grows logarithmically with system size \(N\).
For the non \(U(1)\) symmetric case, the equations involve a novel third function \(c(x)\), which captures the inhomogeneous contributions of the \(T\)-\(Q\) relation. The kernel elements coupling this function to the standard ones are long-ranged and lead for the ground-state to a winding phenomenon. In \(\log(1+a(x))\) and \(\log(1+\bar a(x))\) we observe a sudden change by \(2\pi\)i at a characteristic scale \(x_1\) of the argument. Other features appear at a value \(x_0\) which is of order \(\log N\). These two length scales, \(x_1\) and \(x_0\), are independent: their ratio \(x_1/x_0\) is large for small \(N\) and small for large \(N\). Explicit solutions to the NLIEs are obtained numerically for these limiting cases, though intermediate cases (\(x_1/x_0 \sim 1\)) present computational challenges.
This work lays the foundation for studying finite-size corrections and conformal properties of other integrable spin chains with non-diagonal boundaries, opening new avenues for exploring boundary effects in quantum integrable systems.
Organisationseinheit(en): Institut für Theoretische Physik
Externe Organisation(en): Bergische Universität Wuppertal
CAS - Institute of Physics
Typ: Preprint
Anzahl der Seiten: 15
Publikationsdatum: 11.02.2025
Publikationsstatus: Elektronisch veröffentlicht (E-Pub)
Elektronische Version(en): https://arxiv.org/abs/2502.07229 (Zugang: Offen)

BibTeX

@techreport{c0ea8949850e43a382b604ea353a204b,
title = "Non-linear integral equations for the XXX spin-1/2 quantum chain with non-diagonal boundary fields",
abstract = "The XXX spin-\(\frac{1}{2}\) Heisenberg chain with non-diagonal boundary fields represents a cornerstone model in the study of integrable systems with open boundaries. Despite its significance, solving this model exactly has remained a formidable challenge due to the breaking of \(U(1)\) symmetry. Building on the off-diagonal Bethe Ansatz (ODBA), we derive a set of nonlinear integral equations (NLIEs) that encapsulate the exact spectrum of the model.For \(U(1)\) symmetric spin-\(\frac{1}{2}\) chains such NLIEs involve two functions \(a(x)\) and \(\bar{a}(x)\) coupled by an integration kernel with short-ranged elements. The solution functions show characteristic features for arguments at some length scale which grows logarithmically with system size \(N\).For the non \(U(1)\) symmetric case, the equations involve a novel third function \(c(x)\), which captures the inhomogeneous contributions of the \(T\)-\(Q\) relation. The kernel elements coupling this function to the standard ones are long-ranged and lead for the ground-state to a winding phenomenon. In \(\log(1+a(x))\) and \(\log(1+\bar a(x))\) we observe a sudden change by \(2\pi\)i at a characteristic scale \(x_1\) of the argument. Other features appear at a value \(x_0\) which is of order \(\log N\). These two length scales, \(x_1\) and \(x_0\), are independent: their ratio \(x_1/x_0\) is large for small \(N\) and small for large \(N\). Explicit solutions to the NLIEs are obtained numerically for these limiting cases, though intermediate cases (\(x_1/x_0 \sim 1\)) present computational challenges.This work lays the foundation for studying finite-size corrections and conformal properties of other integrable spin chains with non-diagonal boundaries, opening new avenues for exploring boundary effects in quantum integrable systems.",
author = "Holger Frahm and Andreas Kl{\"u}mper and Dennis Wagner and Xin Zhang",
year = "2025",
month = feb,
day = "11",
language = "English",
type = "WorkingPaper",
}