Statistical Mechanics - Physics at Oregon State University

184 CHAPTER 8. MEAN FIELD THEORY: CRITICAL TEMPERATURE.
Sj =
Sj =
{σ1,···,σq}
Sj =
{σ1,···,σq}
{σ1,···,σq}
σje βJ
e βh σj
σje βJσ0
σ0
q
i=1 σi e βh e β(h+h ′ )
q
i=1
q
i=1 σi e βh
q
i=0 σi
′ q
βh
e i=1 σi (8.121)
q
i=1 σi + σje −βJ
e β(J+h+h′ )σi + e −βh σj
q
i=1
q
i=1 σi e −βh e β(h+h ′ )
e β(−J+h+h′ )σi
Sj = e βh [2 cosh(β(J + h + h ′ ))] q−1 [2 sinh(β(J + h + h ′ ))]+
(8.122)
(8.123)
e −βh [2 cosh(β(−J + h + h ′ ))] q−1 [2 sinh(β(−J + h + h ′ ))] (8.124)
The last expression is independent of j, as expected.
The value for h ′ is determined by requiring that the average spin is the same
everywhere, or
which leads to S0 = Sj or
or
m = 〈σ0〉 = 〈σj〉 (8.125)
e βh [2 cosh(β(J + h + h ′ ))] q − e −βh [2 cosh(β(−J + h + h ′ ))] q =
e βh [2 cosh(β(J + h + h ′ ))] q−1 [2 sinh(β(J + h + h ′ ))]+
e −βh [2 cosh(β(−J + h + h ′ ))] q−1 [2 sinh(β(−J + h + h ′ ))] (8.126)
e βh [cosh(β(J + h + h ′ ))] q−1 [cosh(β(J + h + h ′ )) − sinh(β(J + h + h ′ ))] =
e −βh [cosh(β(−J + h + h ′ ))] q−1 [cosh(β(J − h − h ′ )) − sinh(β(J − h − h ′ ))]
(8.127)
which leads to
e βh [cosh(β(J + h + h ′
q−1
))] e −β(J+h+h′ )
=
q
i=1 σi