Ground-state phase diagram of the spin-1/2 ferromagnetic-antiferromagnetic alternating Heisenberg chain with anisotropy.
Abstract
The ground state of the ferromagnetic-antiferromagnetic alternating spin-t Heisenberg chain is stud ied with Ising-type anisotropy on the ferromagnetic bond. This model tends to the spin-l antiferromag netic Heisenberg chain with on-site anisotropy in the limit of strong ferromagnetic bond. It is found that the Haldane-gap phase, the large-D phase, and the two types of Neel phases appear as the ground state of this model. The duality relations between these four phases are proved for several limiting cases. The string order parameter characterizing the large-D phase is introduced based on these duality relations. The ground-state phase diagram is determined by the numerical diagonalization of the finite-size system. I. INTRODUCfION Haldane's conjecture 1 ,2 opened up an exciting field in the study of quantum spin systems. 3 - I8 The concept of the string order introduced by den Nijs and Rommelse 4 and later by Tasaki 5 has succeeded in clarifying the phys ical nature of this peculiar quantum state which possesses a hidden long-range "string" order accompanied by the breakdown of the hidden symmetry. 6, 7 The presence of this order is also checked numerically. 8, 9 The exactly solvable spin... } model l2 ,13 also has this order and the same symmetry breakdown. 6,8 The experimentally ob served edge state with spin t can be regarded as the evi dence of the hidden symmetry breakdown. 10, II These observations suggest that further physical in sight into this state might be obtained by regarding the spin-I operator as the strongly coupled two spin- + operators. The author thus introduced the spin-+ ferromagnetic-antiferromagnetic alternating Heisenberg chain which tends to the spin-I Heisenberg antiferromag net in the limit of the infinite ferromagnetic coupling. Through the investigation of this model, it is found that the Haldane phase can be regarded as the special case of the dimer phase of the spin-+ model which also possesses the extended string order. 16 The argument based on the nonlocal unitary transform by Kohmoto and Tasaki 17 and Takada 18 also supports this conclusion. On the other hand, in the spin-I Heisenberg antiferro magnetic chain, the on-site easy plane anisotropy is known to destabilize the Haldane phase, leading to the so-called large-D phase, while easy axis anisotropy leads to the transition to the Neel phase. 4,5,14,15 It is the purpose of the present work to investigate the effect of this type of anisotropy in the spin-+ alternating Heisenberg chain. We consider the Hamiltonian