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

The XXX spin- 12 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) 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) U (1) symmetric spin- 12 1 2 chains such NLIEs involve two functions a (x) a (x) and a (x) 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 N. In the case considered here the U (1) U (1) symmetry is broken by the non-diagonal boundary fields and the equations involve a novel third function c (x) c (x), which captures the inhomogeneous contributions to the T T - Q Q relation 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 (1+a (x) ) log (1 + a (x) ) and (1+ a (x) ) log (1 + a ‾ (x) )