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Communications in Theoretical Physics

Ground-State Phase Diagram of Transverse Spin-2 Ising Model with Longitudinal Crystal-Field

Jie Zhao, Guozhu Wei, XU Xing-guang

Published Apr 1, 2006• 3 citations
Publisher

Abstract

The transverse spin-2 Ising ferromagnetic model with a longitudinal crystal-field is studied within the mean-field theory based on Bogoliubov inequality for the Gibbs free energy. The ground-state phase diagram and the tricritical point are obtained in the transverse field Ω/zJ-longitudinal crystal D/zJ field plane. We find that there are the first order-order phase transitions in a very small range of D/zJ besides the usual first order-disorder phase transitions and the second order-disorder phase transitions.

DOI
10.1088/0253-6102/45/4/035

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The transverse spin-2 Ising ferromagnetic model with a longitudinal crystal-field is studied within the mean-field theory based on Bogoliubov inequality for the Gibbs free energy. The ground-state phase diagram and the tricritical point are obtained in the transverse field Ω/zJ-longitudinal crystal D/zJ field plane. We find that there are the first order-order phase transitions in a very small range of D/zJ besides the usual first order-disorder phase transitions and the second order-disorder phase transitions.
We investigate the phase diagram for the spin-3/23/23/2 ferromagnetic Blume-Capel model in a transverse crystal field using the standard mean-field approximation within the framework of Bogoliubov inequality for free energy. We draw a very rich phase diagram with first- and second-order transition lines; tricritical and tetracritical points; critical endpoint of order 2 and double critical endpoint. Additionally, the behaviour of magnetisation as a function of temperature over a wide range of values of both longitudinal and transverse crystal fields is also analysed. To the best of our knowledge, this quantum spin model has only been studied employing an effective field theory, which in turn was not able to characterise completely the multicritical phenomena in its phase diagram, because that procedure is not based on free energy. Thus, our findings on the phase diagram for the present model are novel as they have been not previously reported.
Using the correlated effective-field theory,the spin-2 transverse Ising model with the crystal field for the square lattice has been investigated.The expressions of magnetization were obtained and by numerical calculation the ground-state magnetization curves were given.It was found that there were the first order-order phase transitions,the first order-disorder phase transitions and the second order phase transitions in the ground state.The second order transitions are widespread,and the first order transitions are occurred in the very narrow range of crystal field and transverse field.
The thermodynamic properties of the spin S=3/2 ferromagnetic Ising model in the presence of transverse and longitudinal crystal fields (equivalent to the Blume-Capel model with a transverse crystal field) have been studied by using two different approaches: (i) a zero-temperature mapping of the system onto a spin-1/2 quantum Ising model in longitudinal and transverse fields, together with time-independent quantum perturbation theory; and (ii) a standard mean-field approximation within the framework of the Bogoliubov inequality for the free energy. A very rich phase diagram, with different kinds of multicritical behavior, has been obtained. The results show first- and second-order transition lines, tricritical and tetracritical points, critical end points with a two-phase coexistence, double critical end points, and also double noncritical end points. Additionally, the behavior of the magnetization as a function of temperature, over a wide range of values of both longitudinal and transverse crystal fields, has also been analyzed in detail. While large magnitudes of the longitudinal crystal field select the z-spin components either in their states ±3/2 or ±1/2, it is surprising that a large transverse crystal field induces the spin component in the z direction to values ±1, which are completely different from any expected natural component. This comes indeed as a result of the zero-temperature mapping of the ground state with the superposition of the states 3/2 and -1/2, in one sector of the Hilbert space, and the states -3/2 and 1/2 on the other disjoint sector of the Hilbert space. This superposition for a large transverse crystal field prevails even for finite temperatures, implying that the exact critical points are obtained for the model on the one-dimensional lattice and the two-dimensional square lattice, and quite accurate estimates can be achieved for the three-dimensional simple cubic lattice.
The ferromagnetic spin-2 quantum Blume-Capel model in the presence of a transverse crystal field interaction is studied by employing a variational approach based on Bogoliubov inequality for the free energy and an exact mapping of the ground state. The thermodynamic properties have been obtained using a single spin cluster, which is equivalent to the usual mean-field approximation. The temperature dependence of the magnetization has been analyzed and the global phase diagram, as function of the model parameters, has been obtained for any values of the coordination number of the lattice. A variety of quantum phase transitions, multiple and multicritical points, are present in several topologies of the corresponding phase diagrams. The ground state properties of the system have also been obtained through an exact mapping of the spin-2 model onto another equivalent spin-1 Blume-Capel model in a transverse field. This mapping allows one to determine an exact asymptote formula of the phase diagram at zero temperature in the limit of large longitudinal crystal field for any lattice dimension.
The ground state properties of the two-sublattice Ising metamagnet in the transverse field are studied introducing the mean-field theory. A phase diagram is given via the h-Ω rectanguler coordinates where h is the longitudinal field and Ω is the transverse field. The ground state energy, the longitudinal total magnetization, staggered magnetization and susceptibility and the transverse total magnetization are all calculated. The results show that when the transverse field Ω is small but the longitudinal field h great, the first-order transition occurs, and when the former is great but the later small the second-order transition occurs in the system. The effect of the transverse field Ω on the ground state properties of the two-sublattice Ising metamagnet is important.
The critical properties of the one-dimensional spin-1/2 transverse Ising model in the presence of a longitudinal magnetic field were studied by the quantum fidelity method. We used exact diagonalization to obtain the ground-state energies and corresponding eigenvectors for lattice sizes up to 24 spins. The maximum of the fidelity susceptibility was used to locate the various phase boundaries present in the system. The type of dominant spin ordering for each phase was identified by examining the corresponding ground-state eigenvector. For a given antiferromagnetic nearest-neighbor interaction J_{2}, we calculated the fidelity susceptibility as a function of the transverse field B_{x} and the longitudinal field B_{z}. The phase diagram in the (B_{x},B_{z})-plane shows three phases. These findings are in contrast with the published literature that claims that the system has only two phases. For B_{x}<1, we observed an antiferromagnetic phase for small values of B_{z} and a paramagnetic phase for large values of B_{z}. For B_{x}>1 and low B_{z}, we found a disordered phase that undergoes a second-order phase transition to a paramagnetic phase for large values of B_{z}.