Statistical Mechanics - Physics at Oregon State University

8.6. BETHE APPROXIMATION. 185
or
e −βh [cosh(β(−J + h + h ′
q−1
))] e −β(J−h−h′ )
cosh(β(J + h + h ′ ))
cosh(β(J − h − h ′ 2
= e q−1
)) βh′
(8.128)
(8.129)
The self-consistency equations can be solved numerically. The interesting
question, as usual, pertains to spontaneous magnetic order. Hence we now
focus on the case h = 0. A trivial solution is h ′ = 0. This gives S0 = 0 and
hence m = 0, and corresponds to a non-magnetic state.
In order to see if there are other solutions one can consider both sides of
8.129 as a function of h ′ . The right hand side is a simple exponentially increasing
function, which has a positive curvature. The left hand side is also an increasing
function, as can be seen from
∂
∂h ′
cosh(β(J + h ′ ))
cosh(β(J − h ′ )) =
β sinh(β(J + h′ )) cosh(β(J − h ′ )) + cosh(β(J + h ′ )) sinh(β(J − h ′ ))
cosh 2 (β(J − h ′ ))
sinh(2βJ)
β
cosh 2 (β(J − h ′ ))
=
(8.130)
This is positive indeed. Since the hyperbolic cosine is minimal at zero, we
see that this derivative is maximal for h ′ = J. Hence the left hand side of
8.129 is one for h ′ = 0, increases with a positive curvature until h ′ = J, and
then increases with a negative curvature until it approaches e 2βJ in the limit
h ′ → ∞. Plots for two cases are shown in the following figures.
−3
−2
x
−1
10.0
7.5
y 5.0
2.5
0.0
0
Figure 8.3: β = 1 , J = 1 , q = 3
1
2
3