Can Majorana Fermions correctly describe the ordered state of an antiferromagnetic Heisenberg chain?

To check the reliability of the Majorana representation of quantum spin systems, we use it to compute the ground state energy of an antiferromagnetic spin 1/2 chain to one loop order. We find a very small one loop correction of the mean field energy and a discrepancy of >100% compared to the Bethe Ansatz result. We conclude that a careful handling of the gauge degrees of freedom of this representation is crucial to get correct results.