On a soluble model of an antiferromagnetic chain with Dzyaloshinsky interactions. I

The one-dimensional anisotropic XY model with antisymmetric nearest-neighbour interactions (Dzyaloshinsky interactions) in the presence of a magnetic field B in the z direction is investigated. Special attention is given to the critical behaviour of the system. In the ground state, the susceptibility χzz has a logarithmic singularity as B → 1 provided that the strength Δ of the Dzyaloshinsky interaction obeys the inequality |Δ| ≤ |γ|, where γ is the anisotropy parameter. However, if |Δ| #62; |γ|, χzz diverges with a power -12 singularity as B ↑ (1 + Δ2 - γ2)12. The specific heat is studied for low temperatures. Furthermore for a few special values of γ and B the static correlation functions 〈SαjSβk〉 are evaluated in the low-temperature limit, for α, β = x, y, z and |j - k| = 1, 2. Finally we consider the chain with Dzyaloshinsky interactions only.