Magnetic susceptibility of the two-dimensional Ising model ... - LMPT

a 1 (y) = 1 3
a 2 (y) = 2 3
a 3 (y) = 1
64
×
sinh γ(0)+γ(2π/3) sinh γ(π)+γ(2π/3) sinh 2 γ(2π/3)+γ(π/3)
2 2 2
sinh γ(0)+γ(π/3) sinh γ(π)+γ(π/3) sinh γ(π/3) sinh γ(2π/3) , (3.13)
2 2
sinh γ(0)+γ(π/3) sinh γ(0)+γ(π)
2 2
cos(2πy/3), (3.14)
sinh γ(π/3) sinh γ(π/3)+γ(π)
2
1
sinh γ(0)+γ(π/3) sinh γ(π)+γ(π/3) sinh γ(0)+γ(2π/3) sinh γ(π)+γ(2π/3)
2 2 2 2
1
sinh γ(π/3) sinh γ(2π/3) sinh 2 γ(π/3)+γ(2π/3)
2
.
× (3.15)
The transfer matrix 2 3 × 2 3 has 8 eigenvalues, some of them are equal. Besides that,
some eigenvectors have zero components. As result, the expression for the correlation
function (3.3) contains only three (not seven) independent terms. If we take into account
the definition (2.11), (2.12) of the function γ(q) for particular values of quasimomentum
q = 0, π/3, 2π/3, π , we get exact correspondence between this three terms and (3.10)–
(3.15).
4 Momentum representation of the correlation
function
Since we have the expression (3.2) for g n (r), which depends on both components of r , we
can make the Fourier transform. Let us write the momentum representation of (3.1) in the
form similar to (2.16)–(2.17)
∑
˜G(p) = ξξ T ˜g n (p), (4.1)
n
˜g n (p) = ∑ r
e −|x|/Λ g n (r)e ipr , (4.2)
where
∑ ∞∑ N∑
= . (4.3)
r x=−∞ y=1
After performing the summation in (4.2) we have
˜g n (p) =
en/Λ ∑
( (b) ∏ n
n!N n−1
[q]
) sinh
e −η j
(
sinh γ
j=1 j
cosh
(
Λ −1 + n ∑
γ j
)F n[q]
2 (
j=1
) δ p
∑
Λ −1 + n y −
γ j − cos p x
j=1
n∑
q j
). (4.4)
j=1
8