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physica status solidi (b)

Study of Surface Effects in Quantum Heisenberg and XY Antiferromagnets

J. Cabral Neto, J. Ricardo de Sousa

Published Apr 1, 1999• 6 citations
Publisher

Abstract

Using a finite-cluster mean-field approximation, we study the characteristics of phase transitions for the semi-infinite simple cubic quantum spin-1/2 anisotropic Heisenberg model with ferromagnetic and antiferromagnetic interactions. The dependence of the critical temperature on the parameter Δ is obtained, where Δ = Js/Jb – 1, Js andJb are the nearest-neighbour coupling constants at the surface and bulk, respectively. Particular attention has been drawn to the difference between the criticality for the F (ferromagnetic) and AF (antiferromagnetic) models. It is shown that quantum influence modifies the criticality of the anisotropic Heisenberg model with Tc (F case) < TN (AF case) and Tc > TN in the bulk-paramagnetic and surface paramagnetic critical lines, respectively. On the other hand, the quantum XY model presents the same criticality for the two types of interactions (F and AF).

DOI
10.1002/(sici)1521-3951(199904)212:2<343::aid-pssb343>3.0.co;2-z

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Using a finite-cluster mean-field approximation, we study the characteristics of phase transitions for the semi-infinite simple cubic quantum spin-1/2 anisotropic Heisenberg model with ferromagnetic and antiferromagnetic interactions. The dependence of the critical temperature on the parameter Δ is obtained, where Δ = Js/Jb – 1, Js andJb are the nearest-neighbour coupling constants at the surface and bulk, respectively. Particular attention has been drawn to the difference between the criticality for the F (ferromagnetic) and AF (antiferromagnetic) models. It is shown that quantum influence modifies the criticality of the anisotropic Heisenberg model with Tc (F case) < TN (AF case) and Tc > TN in the bulk-paramagnetic and surface paramagnetic critical lines, respectively. On the other hand, the quantum XY model presents the same criticality for the two types of interactions (F and AF).
The phase diagram of the anisotropic quantum spin-1/2 Heisenberg antiferromagnet in the presence of competitive nearest (J1)({J}_{1})(J1​)- and next-nearest $({J}_{2}=\ensuremath{\alpha}{J}_{1})$-neighbor interactions on a simple cubic lattice is studied within the framework of the differential operator technique. The Hamiltonian is solved by using an effective-field theory in a two-spin cluster. We propose a function for the free energy, which is similar to the Landau expansion, in order to find the ground state (T=0)(T=0)(T=0) and the finite-TTT phase diagram. At zero temperature, we find a first-order quantum phase transition in the point ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta})$ between the antiferromagnetic (AF) and superantiferromagnetic (SAF) phases. This is dependent on the anisotropy parameter $\ensuremath{\Delta}$ ($\ensuremath{\Delta}=0$ and 1 correspond to the isotropic Heisenberg and Ising models, respectively). The quantum critical values of ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}\ensuremath{\ne}1)={\ensuremath{\alpha}}_{c}^{q}(\ensuremath{\Delta})$ are smaller than those of the classical model ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}=1)={\ensuremath{\alpha}}_{c}^{c}=1/4$ (for example, for $\ensuremath{\Delta}=0$, we find ${\ensuremath{\alpha}}_{c}^{q}(0)=0.21$). The former monotonically increases with the anisotropy parameter $\ensuremath{\Delta}$. The phase diagram in $T\text{\ensuremath{-}}\ensuremath{\alpha}$ plane presents a second-order transition at ${T}_{2}(\ensuremath{\alpha})$ between the AF and paramagnetic (P) phases. The phase transition between the SAF and P phases is of first-order ${T}_{1}(\ensuremath{\alpha})$ for all values of $\ensuremath{\Delta}∊[0,1]$. The phase transition lines ${T}_{1}(\ensuremath{\alpha})$ and ${T}_{2}(\ensuremath{\alpha})$ merge in a critical end point (CEP) that is dependent on $\ensuremath{\Delta}$ [i.e., ${T}_{\text{CEP}}(\ensuremath{\Delta})$ and ${\ensuremath{\alpha}}_{\text{CEP}}(\ensuremath{\Delta})$ are the temperature and frustration parameters at the CEP]. The ordered phases (AF and SAF) are separated by first-order transitions and exhibit a nontrivial re-entrant behavior at low temperatures.
We use a Migdal-Kadanoff-like renormalization group approach to study the critical behaviour of a semi-infinite simple cubic Ising ferromagnet whose (1, 0, 0) free surface contains anisotropic (in spin space) Heisenberg ferromagnetic interactions. The phase diagram presents three phases (namely the paramagnetic, the bulk ferromagnetic and the surface ferromagnetic ones) which join on a multicritical point The location of this point is calculated as a function of the anisotropy. The various universality classes of the problem are exhibited.
The anisotropic nearest-neighbour Heisenberg model for the simple cubic lattice has been investigated by interpolating the anisotropy between the Ising and isotropic Heisenberg limits via general spin high-temperature series expansions of the zero-field suspectibility. This is done by estimating the critical temperature (Tc(3)) and the susceptibility exponent γ from the analysis of the series by the Ratio and Pade approximants methods. It is noted that Tc(3) varies with anisotropy while γ is almost the same for the anisotropic system, and a jump in it occurs for the isotropic case in agreement with the universality hypothesis. The effect of anisotropy on the susceptibility is also shown. Further, it is seen that estimates of γ for the two extreme limits agree well with those of previous theoretical as well as experimental investigations. In addition, critical temperatures have been summarised in a relation, and expressions for the magnetisation have been derived.