# Formalism

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 [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 (2*N _{z}* x

*2N*g and u are the column matrices which are defined as follows:

_{Z}),

and

where

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) *J _{n},n*±l and /„,„±i are the interactions between a spin in the

*n*layer and its neighbor in the

^{th }*(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 = N*

_{z}.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 [305]. 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 *J _{s},* we take

*I*= 0.1 and for negative

_{s}*J*we use

_{s},*I*= -0.1. A size of 80

_{s}^{2}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 /

*/k*The convergence precision has been fixed at the fourth figure of the values obtained for the layer magnetizations.

_{B}).# Phase Transition and Phase Diagram of the Quantum Case

First we show an example where *J _{s}* = —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 *J _{s}* = —0.5 with / = —

*I*= 0.1. The surface-layer magnetization (lower curve) is much smaller than the second-layer one. See text for comments.

_{s}Figure 10.4 First two layer-magnetizations 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 in-plane 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*1.522. Therefore, the system is partially disordered for temperatures between

_{2}—*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

*T*i and

*T*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.

_{2},Figure 10.4 shows the non-frustrated case where *J _{s}* = 0.5, with / =

*l*= 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.

_{s}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*= 4

_{z}*,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}* > //.