Validity of the Lieb-Mattis theorem in the J₁-J₂ Heisenberg model

The Lieb-Mattis theorem shows that, for the nonfrustrated spin-S Heisenberg antiferromagnet, the ground-state total spin is equal to N_A-N_BS, where N_A and N_B are the site numbers of two sublattices, respectively. For several J₁-J₂ clusters, we calculate their ground-state total spins by exact diagonalization, and find that the conclusion of Lieb and Mattis is valid as long as Néel order exists. Frustrations which are not able to destroy Néel order cannot change the value of the ground-state total spin. Also, our calculations show that if the ground-state total spin does not abide by the conclusion of Lieb and Mattis, the ground state has no Néel order. For a J₁-J₂ system of large enough size, those states which have total spins between NA -N_BS and the lowest possible total spin are not able to become the ground state for arbitrary strength of frustration. To study the phase transition of the J₁-J₂ model from Néel order to spin disorder, in some respects, the cluster with the geometric shape shown in this paper may be a better choice than usual n x n clusters since it can directly give the rather accurate critical ratio J₂/J₁.