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

Abstract

The XXZ model ðs ¼ 1 2 Þ in a transverse field on a double chain with a uniform long-range interaction among the z components of the spins is considered. The nearest-neighbour interactions are restricted to the components in the xy plane and to the spins within the same chain leg, such that the Hamiltonian is given by Hwhere N is the number of sites of the lattice and m; n ðm; n ¼ 1; 2Þ label the chain legs. The model is solved exactly by introducing the Jordan-Wigner and integral Gaussian transformations, which map the Hamiltonian in a non-interacting fermion system and corresponds to an extension of the model recently studied by the authors for a single chain. The equation of state is obtained in closed form, and the critical classical (at T40) and quantum (at T ¼ 0) behaviours can be determined exactly. The quantum critical surface is determined in the space generated by the transverse field and interaction parameters, and the crossover lines separating the different critical regimes are also obtained. It is also shown that, differently from the results obtained for the single chain, the system can present multiple quantum transitions. rThe 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. Although the model presents first-and secondorder quantum and classical transitions, which is a very important feature, no multiple transitions are present. However, multiple quantum transitions can occur in magnetic systems as it has been shown in the experimental results recently obtained for ZrZn 2 [2]. This result has been the main motivation for studying spin models which are exactly soluble and can present multiple quantum transitions. Therefore, we will consider the XXZ model ðs ¼ 1 2 Þ in a transverse field on a double chain with long-range uniform interaction among z components of spins. The nearest-neighbour interactions are assumed isotropic, and restricted to the spin components in the XY plane and to the spins within the same chain leg. It should be noted that due to the isotropy of the nearest-neighbour interactions the total magnetization along the z direction is conserved. Assuming periodic boundary conditions the Hamiltonian is given byx m;j S x m;jþ1 þ S y m;j S y m;jþ1 Þ À X 2 m;n¼1 I N X N j;k¼1 S z m;j S z n;kwhere N is the number of sites of one leg and m (m ¼ 1; 2) labels the chain legs. By introducing the Jordan-Wigner transformation [3] and a Gaussian transformation, we can ARTICLE IN PRESS www.elsevier.com/locate/jmmm 0304-8853/$ -see front matter r

Thermodynamic properties of the XXZ double chain with long-range interactions

“…(5) and (6) in Ref. [6]. That is to say, our cut-off approximation is consistent with the Gaussian approximation of Ref.…”

“…That is to say, our cut-off approximation is consistent with the Gaussian approximation of Ref. [6] in case of T → 0. However, different from Ref.…”

“…The quantum phase transition and the thermodynamic properties of the system at finite temperature are investigated. It is remarkable that our numerical results in the degenerate condition are in full accord with those of de Lima and Goncalves [6]. A concise physical image and numerical results of the system are obtained under the Jordan-Wigner transformation and cut-off approximation framework.…”

“…The quantum phase transition of one-dimensional s = 1/2 XXZ double chain with z-directional uniform long-range interactions in the external magnetic field is studied through the Jordan-Wigner transformation and integral Gaussian transformation [6]. It is found that the double chain system presents multiple quantum transitions, which is consistent with the experimental findings in the quantum phase transition property in the itinerant Ferromagnet ZrZn 2 [7].…”

“…To this end, we generalize the double-chain model in Ref. [6] to a uniform long range interaction within the two chains in the z direction and study its effect on the thermodynamic properties of the system. Meanwhile, the low temperature thermodynamic behavior and the quantum phase transition driven by quantum fluctuation at zero temperature of quantum spin chains have been a subject of active research over several decades.…”