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Journal of Physics A Mathematical and General

Ground state of XY models of interacting spins in a transverse field

Ferenc Pázmándi, Zbigniew Domański

Published Aug 7, 1993• 6 citations
Publisher

Abstract

The authors investigate the ground state of an XY model with a transverse field, characterized by an exchange matrix, whose highest eigenvalue, lambda max , is supposed to be positive and finite. They prove rigorously that for a spin S the ground state changes at the transverse field value K c =S lambda max .
DOI
10.1088/0305-4470/26/15/013

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Numerical studies by Betts and Oitmaa have led those authors to conjecture that in the XY\mathrm{XY}XY model the ground-state magnetization Mz{M}_{z}Mz​ is zero. This is a model of spins on a lattice with interactions $\ensuremath{-}J({{S}_{n}}^{x}{{S}_{m}}^{x}+{{S}_{n}}^{y}{{S}_{m}}^{y})$, that can also describe a hard-sphere boson fluid. In the present note I prove the conjecture for arbitrary spins S=12,1,32,…\mathcal{S}=\frac{1}{2}, 1, \frac{3}{2}, \dots{}S=21​,1,23​,… for positive JJJ, and treat various generalizations.
The ground state of an XY\mathrm{XY}XY model changes at a critical transverse field, above which there is a gap from the fully polarized ground state to the first excitation. We investigate the phase transition from this ``Mott insulator'' phase as the transverse field approaches its critical value. We calculate exactly some Green's functions and critical exponents for ddd-dimensional ferromagnetic lattices and for the Sherrington-Kirkpatrick spin-glass model. Finite-dimensional spin glasses are also discussed.
The effect of a transverse field Delta on the ground state in the Sherrington-Kirkpatrick model is studied numerically for systems of finite size on the basis of a mean-field approach in which fluctuations from the mean field are included in the second-order perturbation. Simple mean-field equations are solved numerically and the ground-state energy, the order parameter, and the distribution of the magnetisation for each spin are calculated as a function of Delta . Although these results show characteristic features of the spin glass, the value obtained for Delta c , the field above which the system remains paramagnetic at T=0, is much larger than the rigorous value obtained by the perturbation method with respect to the exchange couplings. To improve the mean-field approach, an attempt is made to include a reaction field in the simple mean-field equations approximately. The value of Delta c becomes considerably closer to the rigorous value.
An antiferromagnetic Ising model on the Δ chain, with a macroscopic ground state degeneracy and exponentially decaying spin correlation functions, is studied. By introducing transverse magnetic fields Γ on spins at tips of the triangles and λΓ on spins on the bottom chain, we study the ground state as a function of λ and Γ. The λ=0 problem is exactly solved and the ground state has a staggered magnetization on the bottom chain for any Γ>0. Because the transverse field on the bottom chain with finite λ destroys the staggered order, the order–disorder transition occurs as λ is increased. We calculate the transition point λ c (Γ/ J ) as a function of Γ/ J , where J denotes the magnitude of the exchange interaction, by using the series expansion method around the λ=0 limit and conclude that λ c (Γ/ J =+0)≃0.9459. It is shown that states in the degenerate ground state manifold at Γ=0 are crucial in getting a finite value of λ c (+0).