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Exact results of dimerization order parameter in SU(n) antiferromagnetic chains

Yang Xian

Published Dec 1, 1994• 16 citations
OpenAlex

Abstract

After a proper definition of the dimerization order parameter for a spins system, I show that this order parameter in the SU (n) (n=2S+ 1 ) antiferromagnetic chains (or equivalently the SU(2) spins chains with Hamiltonians which project out singlet states) is, in the thermodynamic limit, directly proportional to the staggered-magnetization in the corresponding spin-½ XXZ chains which had already been mapped onto the SU (n) chains

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•May 29, 2013•0 citations
After a proper definition of the dimerization order parameter for a spins system, I show that this order parameter in the SU (n) (n=2S+ 1 ) antiferromagnetic chains (or equivalently the SU(2) spins chains with Hamiltonians which project out singlet states) is, in the thermodynamic limit, directly proportional to the staggered-magnetization in the corresponding spin-½ XXZ chains which had already been mapped onto the SU (n) chains
The author shows that SU(n) antiferromagnetic chains (or equivalently SU(2) spin-s-chains with Hamiltonians which project out singlet states) are exactly equivalent to the n 2 -state quantum Potts chain (obtained from the transfer matrix for the 2D classical Potts model), for arbitrary n (or s with n=2s+1). This implies that the models are spontaneously dimerised with a finite gap for n<2 but have a unique ground state and vanishing gap for n<or=2. The n to 0 limit may have some bearing on the quantum-Hall-effect localisation transition.
An exact correspondence between the staggered biquadratic spin-1 antiferromagnetic chain and the quantum Hamiltonian limit of the nine-state Potts model is established for finite chains with free ends. For uniform interactions this equivalence is used (via a further exact mapping to a Bethe ansatz soluble XXZ model) to calculate exactly the infinite lattice values of the ground-state energy per site and the (nonzero) gap to the lowest-energy excited state. Periodic boundary conditions and the nature of the ground state as a function of bond alternation are also discussed.
We study Heisenberg spin-1/2 and spin-1 chains with alternating ferromagnetic (J1FJ_1^FJ1F​) and antiferromagnetic (J1AJ_1^AJ1A​) nearest-neighbor interactions and ferromagnetic next-nearest-neighbor interaction (J2FJ_2^FJ2F​). In this model frustration is induced by the non-zero J2FJ_2^FJ2F​ and the effect of dimerization is realized by changing the strength of J1AJ_1^AJ1A​ for a fixed value of J1FJ_1^FJ1F​. The model with site spin sss behaves like a Haldane spin chain with site spin 2sss in the limit of vanishing J2FJ_2^FJ2F​ and large J1F/J1AJ_1^F/J_1^AJ1F​/J1A​. We show that the exact ground state of the model can be found along a line in the parameter space. The phase diagram in the space of J1A−J2FJ_1^A-J_2^FJ1A​−J2F​ is determined using numerical techniques complemented by some analytical calculations. A number of quantities, including the ground state spin, energy gap, structure factor, entanglement entropy, and zero temperature magnetization, are studied to understand the complete phase diagram. An interesting and application-wise important feature of this model is that it exhibits a macroscopic magnetization jump in the presence of a magnetic field; we study this using an effective Hamiltonian.