L.L. Gonçalves

The XY model on the one-dimensional superlattice: static properties

Abstract The XY model ( s= 1 2 ) on the one-dimensional alternating superlattice (closed chain) is solved exactly by using a generalized Jordan–Wigner transformation and the Green function method. Closed expressions are obtained for the excitation spectrum, the internal energy, the specific heat, the average magnetization per site, the static susceptibility, χ zz , and the two-spin correlation function in the field direction at arbitrary temperature. At T =0, it is shown that the system presents multiple second-order phase transitions induced by the transverse field, which are associated to the zero energy mode with wave number equal to 0 or π. It is also shown that the average magnetization as a function of the field presents, alternately, regions of plateaux (disordered phases) and regions of variable magnetization (ordered phases). The static correlation function presents an oscillating behavior in the ordered phase and its period goes to infinity at the critical point.

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

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.

The anisotropic XY model on the 1D alternating superlattice

Abstract The anisotropic XY-model in a transverse field ( s= 1 2 ) on the one-dimensional alternating superlattice (closed chain) is considered. The solution of the model is obtained by introducing a generalized Jordan–Wigner transformation which maps the system onto a non-interacting fermion gas. The exact excitation spectrum is determined by reducing the problem to a diagonalization of a block matrix, and it is shown that it is numerically identical to the one obtained by using the approximate transfer matrix method. The induced magnetization and the susceptibility χ zz are determined as a function of the transverse field, and it is shown that, at T =0, the susceptibility presents multiple singularities. It is also shown, as expected, that this critical behaviour driven by transverse field belongs to the same universality class of the model on the alternating chain.

The XY-model on the one-dimensional superlattice

Abstract The XY-model ( S = 1 2 ) on the one-dimensional superlattice (closed chain) is considered. The exact solution of the model is obtained by using a generalized Jordan-Wigner transformation and the Green function technique. Special attention is given to the study of the excitation spectrum and the results are compared with those obtained by using the transfer matrix technique for the ferromagnetic case only. The thermodynamic properties are also studied and an explicit solution for the internal energy is presented.

SEPARATION OF THE INDUCED MAGNETIZATION IN THE ONE-DIMENSIONAL XY MODEL

The one-dimensional (s = 1/2, open chain) isotropic ferromagnetic XY model in an inhomogeneous transverse field is considered. The exact profile of the induced magnetization is obtained by using the Green function technique. Results are presented for the magnetization profile at different finite temperatures and various strengths of the field. In the scaling region, from the analytical results obtained for the profile of the interface at T = 0, and by a suitable renormalization of the magnetization, it is shown that near the critical point it satisfies a scaling relation identical to the one obtained by van der Waals theory for fluids, provided the correlation length is adequately defined.

Quantum transitions of the isotropic X Y model with long-range interactions on the inhomogeneous periodic chain

The isotropic

The longitudinal dynamic correlation and dynamic susceptibility of the isotropic XY-model on the 1D alternating superlattice☆

XY

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

model

Surface Tension of the Induced Magnetization in the 1D Isotropic XY-Model

(s=1/2)

CRITICAL DYNAMICS OF THE XY-MODEL ON THE ONE-DIMENSIONAL SUPERLATTICE BY POSITION SPACE RENORMALIZATION GROUP

in a transverse field, with uniform long-range interactions among the transverse components of the spins, on the inhomogeneous periodic chain, is studied. The model, composed of