Details
| Original language | English |
|---|---|
| Article number | 012 |
| Number of pages | 19 |
| Journal | SciPost Physics |
| Volume | 20 |
| Issue number | 1 |
| Early online date | 11 Feb 2025 |
| Publication status | Published - 19 Jan 2026 |
Abstract
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 \(N\).
In the case considered here the \(U (1)\) symmetry is broken by the non-diagonal boundary fields and the equations involve a novel third function \(c(x)\), which captures the inhomogeneous contributions to the T -Q equation in the ODBA. 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 steep 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.
Keywords
- Heisenberg spin chains, Integrability, integrable models, Integrable boundaries
ASJC Scopus subject areas
- Physics and Astronomy(all)
- General Physics and Astronomy
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In: SciPost Physics, Vol. 20, No. 1, 012, 19.01.2026.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Non-linear integral equations for the XXX spin-1/2 quantum chain with non-diagonal boundary fields
AU - Frahm, Holger
AU - Klümper, Andreas
AU - Wagner, Dennis
AU - Zhang, Xin
N1 - Publisher Copyright: © 2026, SciPost Foundation. All rights reserved.
PY - 2026/1/19
Y1 - 2026/1/19
N2 - 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 \(N\).In the case considered here the \(U (1)\) symmetry is broken by the non-diagonal boundary fields and the equations involve a novel third function \(c(x)\), which captures the inhomogeneous contributions to the T -Q equation in the ODBA. 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 steep 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.
AB - 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 \(N\).In the case considered here the \(U (1)\) symmetry is broken by the non-diagonal boundary fields and the equations involve a novel third function \(c(x)\), which captures the inhomogeneous contributions to the T -Q equation in the ODBA. 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 steep 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.
KW - Heisenberg spin chains
KW - Integrability
KW - integrable models
KW - Integrable boundaries
KW - Heisenberg Spinketten
KW - Integrabilität
KW - integrable Modelle
KW - integrable Randbedingungen
UR - http://www.scopus.com/inward/record.url?scp=105028990927&partnerID=8YFLogxK
U2 - 10.21468/SciPostPhys.20.1.012
DO - 10.21468/SciPostPhys.20.1.012
M3 - Article
VL - 20
JO - SciPost Physics
JF - SciPost Physics
IS - 1
M1 - 012
ER -