Static and dynamic properties of the XXZ chain with long-range interactions

Abstract

The one-dimensional XXZ model (s=1/2) in a transverse field, with uniform longrange 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,χ zz T , are presented for the different critical regimes. The two-spin static and dynamic correlation functions, < S z j S z l >, are also presented, and the dynamic susceptibility, χ zz q (ω),is obtained in closed form. Explicit results are presented at T = 0, and it is shown that the isothermal susceptibility, χ zz T , is different from the static one, χ zz q (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.

Critical properties of the XXZ model with long-range interactions on the double chain

“…Finally, we would like to point out that, for J 1 ¼ J 2 ; the quantum critical behaviour is identical to the one obtained for the model when we have a single chain [1,4], and as already mentioned we do not have multiple transitions.…”

“…with j m ¼ arccosðÀðh þ 4IM z Þ=J m Þ: It should be noted that for J 1 ¼ J 2 the results obtained so far reproduce the known ones for the single chain [1]. By minimizing the free energy functional, given in Eqs.…”

“…In this case, it is not possible to obtain an explicit equation for the critical surface and it has been determined numerically. The interception of the critical surfaces of first and second orders occurs at I ¼ 0, and corresponds to a bicritical line since it contains the known bicritical point present for J 1 ¼ J 2 [1,4], when the results reduce to the ones for the single chain.…”

“…It is also shown that, differently from the results obtained for the single chain, the system can present multiple quantum transitions. The critical behaviour of the XXZ model with longrange interactions among the spins along the z direction, in a single chain, has been recently considered by the authors [1]. In that paper they present a comprehensive study of the quantum and classical transitions undergone by the model.…”

A study of the spin- XZ chain with long-range interactions in the external magnetic field

“…Furthermore, the Jordan-Wigner transformation and integral Gaussian transformation are also frequently introduced to the system [6,7], and the phase transition property at finite temperature is obtained. Similarly, in this paper we investigate the thermodynamic property of the one-dimensional spin- 1 2 XZ chain in external magnetic field and uniform long-range interactions by means of Jordan-Wigner transformation, Gaussian transformation and Bogoliubov transformation.…”