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FormalismWe can rewrite the full Hamiltonian (10.1) in the local framework of the classical GS configuration as where cos (вц) is the angle between two NN spins determined classically in the previous section. Following TahirKheli and ter Haar [340], we define two doubletime GFs by The equations of motion for С,Дг, t') and F,_{y} (t, t') read
We will follow the same method as that used in Chapter 9: using the Tyablikov decoupling scheme [43] and the Fourier transforms of the retarded GFs, we have a set of equations rewritten under a matrix form
where M («) is a square matrix (2N_{z} x 2N_{Z}), g and u are the column matrices which are defined as follows:
and
where
in which, Z — 6 is the number of inplane NN, 0„,„±i the angle between two NN spins belonging to the layers n and n ± 1, в„ the angle between two inplane NN in the layer n, and
Here, for compactness we have used the following notations: (i) ]„ and /„ are the inplane interactions. In the present model is equal to ]_{s} for the two surface layers and equal to / for the interior layers. All /„ are set to be /. (ii) J_{n},n±l and /„,„±i are the interactions between a spin in the n^{th }layer and its neighbor in the (n ± l)^{w}' layer. Of course, J„,ni = f/i,n—i ^{=} 0 if n = 1, Jn,n+1 ^{=} fn,n+1 ^{=} 0 if n = N_{z}. Solving detM = 0, we obtain the spin wave spectrum w of the present system. We follow the same method in Chapter 9, we arrive at
where e is an infinitesimal positive constant and р = 1/квТ, кв being the Boltzmann constant. For spin S =1/2, the thermal average of the z component of the /th spin belonging to the nth layer is given by
In the following, we shall use the case of spin onehalf. Note that for the case of general S, the expression for (Sf) is more complicated since it involves higher quantities such as ((Sf)^{2}}. Using the GF presented above, we can calculate selfconsistently various physical quantities as functions of temperature T. The first important quantity is the temperature dependence of the angle between each spin pair. This can be calculated in a selfconsistent manner at any temperature by minimizing the free energy at each temperature to get the correct value of the angle as it has been done for a frustrated bulk spin systems [305]. In the following, we limit ourselves to the selfconsistent calculation of the layer magnetizations which allows us to establish the phase diagram as seen in the following. For numerical calculation, we used / = 0.1/ with / = 1. For positive J_{s}, we take I_{s} = 0.1 and for negative J_{s}, we use I_{s} = 0.1. A size of 80^{2} points in the first Brillouin zone is used for numerical integration. We start the selfconsistent calculation from 7 = 0 with a small step for temperature 5 x 10^{3} or 10^{1} (in units of //k_{B}). The convergence precision has been fixed at the fourth figure of the values obtained for the layer magnetizations. Phase Transition and Phase Diagram of the Quantum CaseFirst we show an example where J_{s} = —0.5 in Fig. 10.3. As seen, the surfacelayer magnetization is much smaller than the second layer one. In addition there is a strong spin contraction at 7 = 0 Figure 10.3 First two layermagnetizations obtained by the GF technique vs. 7 for J_{s} = —0.5 with / = — I_{s} = 0.1. The surfacelayer magnetization (lower curve) is much smaller than the secondlayer one. See text for comments. Figure 10.4 First two layermagnetizations obtained by the GF technique vs. T for J_{s} = 0.5 with / = /* = 0.1. for the surface layer. This is due to the antiferromagnetic nature of the inplane surface interaction J_{s}. One sees that the surface becomes disordered at a temperature 7 ~ 0.2557, while the second layer remains ordered up to T_{2} — 1.522. Therefore, the system is partially disordered for temperatures between T and T_{2}.This result is very interesting because it confirms again the existence of the partial disorder in quantum spin systems observed earlier in bulk frustrated quantum spin systems [285, 305]. Note that between Ti and T_{2}, the ordering of the second layer acts as an external field on the first layer, inducing therefore a small value of its magnetization. A further evidence of the existence of the surface transition will be provided with the surface susceptibility in the MC results shown below. Figure 10.4 shows the nonfrustrated case where J_{s} = 0.5, with / = l_{s} = 0.1. As seen, the firstlayer magnetization is smaller than the secondlayer one. There is only one transition temperature. Note the difficulty for numerical convergency when the magnetizations come close to zero. We show in Fig. 10.5 the phase diagram in the space (J_{s}, T). Phase I denotes the ordered phase with surface non collinear spin configuration, phase II indicates the collinear ordered state, and Figure 10.5 Phase diagram in the space (J_{s}, T) for the quantum Heisenberg model with N_{z} = 4,1 = /_{s} = 0.1. See text for the description of phases I to III. phase III is the paramagnetic phase. Note that the surface transition does not exist for J_{s} > //. 
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