The one-dimensional XXZ model (s=12) in a transverse field, with uniform long-range interactions among the transverse components of the spins, is studied. The model is exactly solved by introducing the Jordan–Wigner transformation and the integral Gaussian transformation. The complete critical behaviour and the critical surface for the quantum and classical transitions, in the space generated by the transverse field and the interaction parameters, are presented. The crossover lines for the various classical/quantum regimes are also determined exactly. It is shown that, besides the tricritical point associated with the classical transition, there are also two quantum critical points: a bicritical point where the classical second-order critical line meets the quantum critical line, and a first-order transition point at zero field. It is also shown that the phase diagram for the first-order classical/quantum transitions presents the same structure as for the second-order classical/quantum transitions. The critical classical and quantum exponents are determined, and the internal energy, the specific heat and the isothermal susceptibility, χTzz, are presented for the different critical regimes. The two-spin static and dynamic correlation functions, 〈SjzSlz〉, are also presented, and the dynamic susceptibility, χqzz(ω), is obtained in closed form. Explicit results are presented at T=0, and it is shown that the isothermal susceptibility, χTzz, is different from the static one, χqzz(0). Finally, it is shown that, at T=0, the internal energy close to the first-order quantum transition satisfies the scaling form recently proposed by Continentino and Ferreira.
