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

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.