Details
| Original language | English |
|---|---|
| Number of pages | 15 |
| Publication status | E-pub ahead of print - 11 Feb 2025 |
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 \(N\).
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.
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2025.
Research output: Working paper/Preprint › Preprint
}
TY - UNPB
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
PY - 2025/2/11
Y1 - 2025/2/11
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 \(N\).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.
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 \(N\).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.
M3 - Preprint
BT - Non-linear integral equations for the XXX spin-1/2 quantum chain with non-diagonal boundary fields
ER -