University of California at San Diego – Department of Physics – Prof. John McGreevy
Physics 217 Winter 2016
Assignment 2 –
Solutions
Due 2pm Thursday, January 28, 2016
This homework begins with two simple problems to warm up your brain.
1. Reminder. Check from the ‘bridge’ equation F = −T log Z that F = E − T S where
E is the average energy and S = −∂T F is the entropy.
2. What is the relationship between the spin-spin correlation function hsi sj i and the
probability that si and sj are pointing in the same direction? (Hint: (1 + si sj )/2 is a
projector.)
3. Real-space RG for the SAW. Consider again the problem of a self-avoiding walk
on the square lattice. Construct an RG scheme with zoom factor λ = 3 (so that nine
sites of the fine lattice are represented by one site of the coarse lattice). Find the RG
map K 0 (K), find its fixed points and estimate the critical exponent at the nontrivial
fixed point. Is it closer to the numerical result than the λ = 2 schemes discussed in
lecture and by Creswick?
In the following problems we consider a closed (periodic) chain of N classical spins si = ±1
with Hamiltonian
X
X
H = −J
si si+1 − h
si + const, sN +1 = s1
i
The partition function is Z(βJ, βh) =
i.e. set β = 1.
i
P
{s}
e−βH ; let’s measure J, h in units of temperature,
4. Ising model in 1d by transfer matrix.
(a) Show that the partition function can be written as
Z = tr2 T N
where T is the 2 × 2 matrix
T =
1
eJ+h e−J
e−J eJ−h
(called the transfer matrix) and tr2 M = M11 + M22 denotes trace in this twodimensional space. Express Z in terms of the eigenvalues of T and find the free
T
energy density f = − N
log Z in the thermodynamic (N → ∞) limit. Plot the
free energy for h = 0 as a function of x = e−4J for 0 ≤ x ≤ 1.
(b) Find an expression for the correlation function
G(m) ≡ hsi sm+i i − hsi ihsi+m i
using the transfer matrix. Show that as N → ∞,
G(m) ∼ e−m/ξ
where ξ =
log
1
λ1
λ2
where λ1 > λ2 are eigenvalues of T . Note that ξ → ∞ when
λ1 → λ2 . For what values of β, h, J does this happen?
5. Decimation of 1d Ising model in a field.
Now suppose that N is even.
(a) Decimate the even sites:
X
e−H(s) ≡ ∆eHeff (sodd ) .
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More explicitly, identify the terms in H(s) that depend on any one even site,
H2 (s) and define its contribution to Heff by
X
e−H2 (s) ≡ ∆e−∆Heff (s1 ,s3 )
s2
P
P
Rewriting Heff (sodd ) = −J 0 ss − h0 s − const in the usual form, find J 0 , h0
and the constant in terms of the microscopic parameters J, h.
I find
cosh β(2J + h) cosh β(2J − h)
cosh2 βh
1
cosh 2β(2J + h)
0
h = h+
log
2β
cosh 2β(2J − h)
p
∆ = cosh βh cosh β(2J + h) cosh β(2J − h).
1
J =
log
4β
0
(b) Let w ≡ tanh βJ, v ≡ tanh βh. Plot some RG trajectories in the v, w plane.
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(c) Find all the fixed points and compute the exponents near each of the fixed points.
The fixed points are J ? = ∞, h? = 0: the ferromagnetic, strong coupling fixed
point, and J ? = 0, any h, the paramagnetic, high-temperature fixed point. Near
the paramagnetic fixed point,
w0 = w, v 0 = v 2 (1 − w2 ).
The two eigenvalues of the 2 × 2 matrix
R=
∂(v 0 , w0 )
∂(v, w)
are
1
log ∂v v 0 |v? =0 = −∞
very irrelevant.
log 2
1
yh =
log ∂w w0 |v? =0 = 0 marginal.
log 2
yv =
(1)
Near the ferromagnetic fixed point, w0 = 2w, t0 = 2t, where t ≡ 1 − v. Here we
have yv = yt = 1, two relevant perturbations.
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