Statistical Mechanics - Physics at Oregon State University

9.7. RENORMALIZATION GROUP THEORY. 225
where M is the Jacobian matrix of the renormalization group transformation.
The eigenvalues of this matrix determine the stability of the critical point. If an
eigenvalue has an absolute value less than one, the direction of the corresponding
eigenvector corresponds to a direction in which the fixed point is stable. The
deviations will become smaller in this direction. Eigenvalues larger than one
correspond to unstable directions. A stable fixed point has all eigenvalues less
than one.
For the one-dimensional Ising chain we have
∂ ˜ J ′
∂ ˜
=
J
1 sinh(2
2
˜ J + ˜ h)
cosh(2 ˜ J + ˜ 1 sinh(2
+
h) 2
˜ J − ˜ h)
cosh(2 ˜ J − ˜ (9.175)
h)
∂ ˜ J ′
∂˜
=
h
1 sinh(2
4
˜ J + ˜ h)
cosh(2 ˜ J + ˜ 1 sinh(2
−
h) 4
˜ J − ˜ h)
cosh(2 ˜ J − ˜ 1 sinh(
−
h) 2
˜ h)
cosh( ˜ h)
∂˜ h ′
∂ ˜ J
= sinh(2 ˜ J + ˜ h)
cosh(2 ˜ J + ˜ h) − sinh(2 ˜ J − ˜ h)
cosh(2 ˜ J − ˜ h)
∂˜ h ′
∂˜
= 1 +
h
1 sinh(2
2
˜ J + ˜ h)
cosh(2 ˜ J + ˜ 1 sinh(2
+
h) 2
˜ J − ˜ h)
cosh(2 ˜ J − ˜ h)
(9.176)
(9.177)
(9.178)
and the matrix M for the one-dimensional Ising model at the point ( ˜ J, 0) is
simple
tanh(2 ˜ J) 0
0 1 + tanh(2 ˜ J)
At the critical point (0, 0) the form is simply
0 0
0 1
(9.179)
(9.180)
This means that along the ˜ J direction the critical point (0, 0) is very stable;
first order deviations disappear and only higher order terms remain. In the ˜ h
direction the deviations remain constant.
Suppose that we have found a fixed point ˜ J ∗ for a more-dimensional Ising
model at h = 0. Since the problem is symmetric around h = 0 the direction
corresponding to ˜ J must be the direction of an eigenvector of the matrix M.
Suppose the corresponding eigenvalue is λ. For small deviations at constant
field h = 0 we can therefore write
δ ˜ J ′ = λδ ˜ J (9.181)
Now assume that we constructed this small deviation in ˜ J by changing the
temperature slightly. Hence we assume that δ ˜ J = α(T − Tc) where α is a