Non-linear integral equations for the XXX spin-1/2 quantum chain with non-diagonal boundary fields
Holger Frahm, Andreas Klümper, Dennis Wagner +1 more
The XXX spin- \frac{1}{2} 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- \frac{1}{2} 1 2 chains such NLIEs involve two functions a(x) a ( x ) and \bar{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 \log(1+a(x)) log ( 1 + a ( x ) ) and \log(1+\bar a(x)) log ( 1 + a ‾ ( x ) )
arXiv:2502.07229•SciPost Physics•Jan 19, 2026•1 citation