We 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 Tahir-Kheli and ter Haar , 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  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 (2Nz x 2NZ), g and u are the column matrices which are defined as follows:
in which, Z — 6 is the number of in-plane NN, 0„,„±i the angle between two NN spins belonging to the layers n and n ± 1, в„ the angle between two in-plane NN in the layer n, and
Here, for compactness we have used the following notations:
(i) ]„ and /„ are the in-plane 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) Jn,n±l and /„,„±i are the interactions between a spin in the nth layer and its neighbor in the (n ± l)w' layer. Of course, J„,n-i = f/i,n—i = 0 if n = 1, Jn,n+1 = fn,n+1 = 0 if n = Nz.
Solving det|M| = 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 n-th layer is given by
In the following, we shall use the case of spin one-half. 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 self-consistently 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 self-consistent 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 . In the following, we limit ourselves to the self-consistent 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 Js, we take Is = 0.1 and for negative Js, we use Is = -0.1. A size of 802 points in the first Brillouin zone is used for numerical integration. We start the self-consistent calculation from 7 = 0 with a small step for temperature 5 x 10-3 or 10-1 (in units of //kB). 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 Case
First we show an example where Js = —0.5 in Fig. 10.3. As seen, the surface-layer 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 layer-magnetizations obtained by the GF technique vs. 7 for Js = —0.5 with / = — Is = 0.1. The surface-layer magnetization (lower curve) is much smaller than the second-layer one. See text for comments.
Figure 10.4 First two layer-magnetizations obtained by the GF technique vs. T for Js = 0.5 with / = /* = 0.1.
for the surface layer. This is due to the antiferromagnetic nature of the in-plane surface interaction Js. One sees that the surface becomes disordered at a temperature 7 ~ 0.2557, while the second layer remains ordered up to T2 — 1.522. Therefore, the system is partially disordered for temperatures between T and T2.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 T2, 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 non-frustrated case where Js = 0.5, with / = ls = 0.1. As seen, the first-layer magnetization is smaller than the second-layer 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 (Js, 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 (Js, T) for the quantum Heisenberg model with Nz = 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 Js > //.