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Physical review. B, Condensed matter

Spin-1 Ising model in a transverse crystal field

J. Oitmaa, A. M. A. von Brasch

Published May 8, 2003• 33 citations
Publisher

Abstract

We investigate a ferromagnetic spin-1 Ising model with a transverse single-ion crystal-field term. By means of a mapping to the usual spin-1/2 transverse Ising model, we determine the location of the $T=0$ quantum critical point exactly in one dimension, and to high precision in higher dimensions. The nature of the excited states is also explored. We show how this approach can be generalized to higher spin values.

DOI
10.1103/physrevb.67.172402

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Physical review. B, Solid state•Jun 1, 1975•35 citations
We investigate a ferromagnetic spin-1 Ising model with a transverse single-ion crystal-field term. By means of a mapping to the usual spin-1/2 transverse Ising model, we determine the location of the T=0T=0T=0 quantum critical point exactly in one dimension, and to high precision in higher dimensions. The nature of the excited states is also explored. We show how this approach can be generalized to higher spin values.
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.
We solve the spin-1 quantum Ising model with single-ion anisotropy by mapping it onto a series of segmented spin-1/2 transverse Ising chains, separated by the S(z)=0 states called holes. A recursion formula is derived for the partition function to simplify the summation of hole configurations. This allows the thermodynamic quantities of this model to be rigorously determined in the thermodynamic limit. The low temperature behavior is governed by the interplay between the hole excitations and the fermionic excitations within each spin-1/2 Ising segment. The quantum critical fluctuations around the Ising critical point of the transverse Ising model are strongly suppressed by the hole excitations.
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 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.
The correlation functions ${\ensuremath{\rho}}_{\mathrm{xx}}(R,t)$ and ${\ensuremath{\rho}}_{\mathrm{yy}}(R,t)$ are calculated for the one-dimensional Ising model with a transverse field hhh at T=0T=0T=0. This model corresponds to the XYX YXY model with $\ensuremath{\gamma}=1$, and is equivalent to the two-dimensional anisotropic (${J}_{1}\ensuremath{\rightarrow}\ensuremath{\infty}$, ${J}_{2}\ensuremath{\rightarrow}0$) Ising model. The additional dimension in the classical model is the imaginary time of the quantum model. For all values of hhh, ${\ensuremath{\rho}}_{\mathrm{yy}}(R,t)=\ensuremath{-}(\frac{1}{{h}^{2}})(\frac{{\ensuremath{\partial}}^{2}}{\ensuremath{\partial}{t}^{2}}){\ensuremath{\rho}}_{\mathrm{xx}}(R,t)$. At the critical field h=hc=1h={h}_{c}=1h=hc​=1, ${\ensuremath{\rho}}_{\mathrm{xx}}(R,t)\ensuremath{\sim}{({R}^{2}\ensuremath{-}{t}^{2})}^{\frac{\ensuremath{-}1}{8}}$. For h<1,h>1h<1,h>1h<1,h>1 the results already obtained for the XY\mathrm{XY}XY model are recovered. We give some consequences from this equivalence in higher dimension, concerning the behavior of the correlation function at the critical field at T=0T=0T=0.