Physics 212: Statistical mechanics II
Lecture XIII
The first part of this lecture explains further the behavior of correlation functions near a secondorder critical point. The second part returns to the 1D Ising model in order to understand how
rescaling transformations tell us something about this solvable model. Then we will discuss the
approximate behavior of the Ising model in higher dimensions, where there is a phase transition at
some nonzero temperature Tc between an ordered state and a disordered one.
The concept of “anomalous dimensions” is one of the more fundamental in the field of statistical
mechanics and field theory. Simply put, it means that quantities near a critical point, where the
correlation length is very large, can depend in a rather nontrivial way on both the correlation length
ξ and the lattice spacing or short-distance cutoff a. The term “anomalous” refers to the breakdown
of simple dimensional analysis.
As an example, consider the correlation function of spins at the critical point of the Ising
model in three dimensions. One might guess that the spin-spin correlator should be independent
of microscopic details in some sense, but taking this literally will be shown to lead to meanfield theory, which is quantitatively wrong for this transition. Really a fundamental breakthrough
in the understanding of field theory and quantum phase transitions was in understanding how
anomalous dimensions, i.e., violations of the scaling predicted by mean field theory, occur for
almost all interesting problems.
We found in mean-field theory that the correlation fell off as r−(d−2) up to the correlation
length, then fell off exponentially. This happened because the Fourier transform of the correlation,
for ξ = ∞, went as 1/k 2 , which in d dimensions corresponds to the Coulomb potential. That is,
the correlations satisfy the homogeneous Laplacian equation ∇2 m = 0 at criticality, except for a
source at the origin. Hence G(r) falls off logarithmically in 2D and as 1/r in 3D.
The correlation function is related to the susceptibility: χ ∼ dd rG(r), so χ ∼ ξ 2 in MFT.
This accords with our naive expectation that χ should depend only on the diverging length scale
ξ. However, this is wrong for the 3D Ising model. In fact, χ does have a dependence on the short
length scale a:
χ ∼ aη ξ 2−η .
(1)
R
This η is an example of an anomalous dimension: the mean-field guess is incorrect and there is in
some sense a dependence on the short-distance physics, but this dependence can still be universal
(η is equal for all problems in the same universality class). Understanding how this occurs is one
of the major contributions of the RG to statistical physics and field theory.
(We can also go back and understand this from the Landau free energy. I won’t be talking
about functional integration at this point, but in Fourier space the quadratic part of the exponent
looks like
(k 2 + r02 )|m(k)|2 ,
(2)
so the correlator hm(k)m(−k)i ∼ (k 2 + r02 )−2 , and at criticality this gives the mean-field result
G(r) ∼ r−1 in three dimensions. The model neglecting the fourth and higher terms is known as the
Gaussian model. Incidentally, mean-field exponents are also referred to as “classical” because
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they correspond to a saddle-point of the Landau free energy, in the same way as classical mechanics
corresponds to a saddle-point (small-h̄ limit) of the Feynman path integral.)
We mentioned before a decimation or block-spin transformation on the 1D Ising model. There
the conclusion was that we could sum over half the spins of the chain to obtain a new model for
just the remaining half of the spins; then the new coupling satisfies
0
e2K = cosh(2K).
(3)
This was derived, for three spins, via
!
X
e
βJ(σ1 σ2 +σ2 σ3 )
X
=
σ1 ,σ2 ,σ3
X
X σ2
βJ(σ1 σ2 +σ2 σ3 )
e
σ1 ,σ3
0
eK σ1 σ3 .
= C
(4)
σ1 ,σ3
The overall constant C follows from
0
0
CeK = e2K + e−2K ⇒ C = 2eK .
(5)
The rescaling relation (3) implied (here K ≡ βJ)
tanh(K 0 ) = (tanh K)2
(6)
Adding a magnetic field H, the rescaled Boltzmann factor should now be
eβJ
0σ
1 σ3 +βH
0 (σ +σ )/2
1
3
=
X
1
eβJ(σ1 σ2 +σ2 σ3 )+βH(σ2 + 2 (σ1 +σ3 ))
(7)
σ2
Here the factor of 21 in the magnetic terms for spin 1 and 3 arises because, when we pass to a chain
of many spins, we want to avoid double counting. There are eight terms in the sum, which group
into four terms of the new problem: (let H̃ be the reduced magnetic field βH)
0
0
CeK +H̃
0
Ce−K
0
0
CeK −H̃
= e2K+2H̃ + e−2K
= eH̃ + e−H̃
= e2K−2H̃ + e−2K .
(8)
This can be simplified by introducing u = e−K , v = e−H̃ . Then
v
0
0
u
!1/2
=
u2 + v 2 u−2
v −2 u−2 + u2
=
(v + v −1 )2
u4 + u−4 + v 2 + v −2
!1/4
C = (v + v −1 )1/2 u4 + u−4 + v 2 + v −2
1/4
.
(9)
We interpret the new quantities K 0 , H̃, C as the spin-spin coupling and magnetic field of the rescaled
problem, plus an addition to the free energy reflecting the degrees of freedom that were integrated
out.
2
Note that u = 1 and any v is a fixed point of the above (zero coupling), and that u = 0 and
v = 1 is also a fixed point. At low temperature or large K, and small field, consider the rescaling
of the magnetic field.
This means that the “RG flows” are away from the T = 0, H = 0 fixed point in both directions:
this is a completely unstable fixed point.
Some things to remember about the RG, which will become more clear in the next few lectures:
1. It is a one-way process: many initial configurations are combined into a single configuration
of the rescaled problem. Hence it is not a group, strictly speaking.
2. It is not a uniquely defined process: for example, the book by Cardy also carries out an exact
RG of the 1D Ising model, but groups spins by threes rather than by twos. Similarly, in higher
dimensions there are often different approximate RG methods applied to the same model.
3. Only in one-dimensional models, in general, is the space of possible couplings finite-dimensional.
The main approximation in higher dimensions is in truncating the problem to a few most relevant
couplings.
4. Critical exponents and other interesting scaling properties near a transition are related to
RG flows near a fixed point of the RG.
We can represent the 1D RG flows at H = 0 on a line from K = 0 to K = ∞. However, for
this simple problem the flows are always from K = ∞ (strong coupling) down to K = 0 (weak
coupling), or from low temperature to high temperature. In two dimensions at H = 0 we can
approximately draw the RG flows on a line, but now this is only an approximate truncation. Now
there is a fixed point at Tc , and both T = 0 and T = ∞ are stable fixed points. The next step
is to understand point (4): how the RG flows lead to important information about experimental
quantities.
The simplest example is the correlation length: after a rescaling by a factor b, we expect bξ 0 = ξ,
where ξ 0 is the correlation length in the rescaled system. This was verified explicitly for the 1D
case with b = 2. Since ξ is reduced under rescaling, we flow from the vicinity of a critical point
with ξ = ∞ to points of smaller correlation length. Let K ∗ be the value of the coupling at a fixed
point, and let the rescaling equation be K → R(K) so that R(K ∗ ) = K ∗ . Starting from a K near
K ∗ , so K − K ∗ is small, we have
K 0 = R(K) = R(K ∗ ) +
where we have defined
y=
dR
(K − K ∗ ) = K ∗ + by (K − K ∗ ),
dK
dR
log dK
dR
= logb
.
log b
dK
(10)
(11)
and the derivative is evaluated at the fixed point K = K ∗ . We expect ξ(K) = A(K − K 0 )−ν from
the definition of the critical exponent ν, where A is some constant. Since ξ(K) = bξ(K 0 ) as a simple
consequence of rescaling,
A(K − K ∗ )−ν = bA(K 0 − K)−ν = bA (by (K − K ∗ ))−ν .
(12)
This implies νy = 1 or ν = 1/y. Hence the critical exponent ν is simply related to the action
of the RG flows near the fixed point, through the definition of y (11). In general, eigenvalues of
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the linearized RG flows near a fixed point will turn out to determine critical exponents and other
properties of the fixed point. Note that the above also shows that, for smooth RG flows, ν is equal
on both the ordered and disordered sides of a critical point.
Let’s call the above y the thermal eigenvalue yth and ask what happens in the presence of a
magnetic field. In the process we will learn more about the idea of “scaling forms.” In general we
might have some large number of couplings K1 , K2 , K3 , . . ., and want to understand the renormalization group transformation in this large-dimensional space. Linearizing around a fixed point K∗ ,
we can write
Ka0 − Ka∗ = R(K)a − Ka∗ = Tab (Kb − Kb∗ ).
(13)
where here a, b are indices running over the different couplings and T is the linearized transformation
Tab =
∂Ka0
.
∂Kb0
(14)
Let the eigenvalues of the matrix Tab be λi , and let φi be its left eigenvectors. Since T is not
in general symmetric, its left eigenvectors may be different from its right eigenvectors. In fact its
eigenvalues need not be real, but for the physical cases discussed here the eigenvalues are real.
Then
X
φia Tab = λi φib .
(15)
a
Now define “scaling variables” ui through
ui =
X
φia (Ka − Ka∗ ).
(16)
a
This gives a new coordinate system near the fixed point which behaves simply under the rescaling:
u0i =
=
=
X
a
X
a,b
X
φia (Ka0 − Ka∗ )
φia Tab (Ka0 − Ka∗ )
λi φib (Kb − Kb∗ ) = λi ui .
(17)
b
For the Ising model example, the two directions in this coordinate system will just be the thermal
direction and the magnetic direction, as suggested by symmetry. For more complicated problems,
the transformation to these scaling/orthogonal coordinates simplifies the problem considerably.
Now define the quantities yi through byi = λi , where b is the rescaling factor. These yi are
called renormalization group eigenvalues. One useful property of the yi is that they allow us to
determine whether a direction in coupling space is a “relevant” or “irrelevant” direction.
If one of the yi is positive, then that ui gets driven away from its fixed point value under
rescaling transformations. This is called a “relevant” coupling. If one of the yi is negative, then
that ui gets driven toward its fixed point value. This is an “irrelevant” coupling. Addition of
irrelevant couplings does not modify universal properties such as critical exponents, as we’ll see,
and this explains much of the power of RG methods. Finally, if one of the yi is zero, then that
indicates a “marginal” coupling: some additional information is required to learn whether the
coupling is “marginally relevant”, “marginally irrelevant“, or “exactly marginal”.
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To make clear how this is useful, consider the scaling transformation of the free energy. For
the singular part of the free energy per site, we can ignore the additive constant generated upon
rescaling (this can be done since that additive term contains short-ranged physics, is therefore
smooth near the transition, and thus contributes to the nonsingular part of the free energy). The
singular part satisfies
fs (K) = b−d fs (K0 ),
(18)
in order that the singular part of the total free energy (number of sites times the above) be constant
under rescaling. This implies
fs (ut , uh ) = b−d fs (byt ut , byh uh ) = b−nd fs (bnyt ut , bnyh uh ),
(19)
where in the second equation we have rescaled n times.
More precisely, the full rescaling equation for the free energy per site is
f (K) = g(K) + b−d f (K0 ),
(20)
where g is the additive constant generated from the sum over short-distance degrees of freedom.
(This corresponds to the overall constant C 6= 1 in our 1D Ising model RG.) This inhomogeneous
equation for the full free energy becomes a homogeneous equation (18) for the singular part fs , since
g is nonsingular by the argument above that any contribution from short distances is regular at
the transition. This is the reason why we separated f into fs and fn above: the rescaling equation
for f is inhomogeneous, while that for fs is homogeneous.
The singular free energy rescaling (18) is analogous to what we did before with the correlation
length: the physical equivalence of the systems requires that the correlation length in real units and
the free energy per volume in real units remain constant. These impose strong constraints on the
rescaling behavior (these are known as “renormalization conditions” in quantum field theory).
We can now use the rescaling equation for fs to obtain the scaling function used in the previous
lecture near a critical point. We would like to work close to the critical point so that the linearization
above is valid. Pick some positive ut0 close enough to the critical point that the linear equations
are valid, and suppose that an initial point ut is even closer to the critical point (on either side),
and requires n rescalings to be ±ut0 : then
|bnyt ut | = ut0 ,
or
n=
1
logb (|ut0 /ut |)
yt
(21)
(22)
Then the free energy equation
fs (ut , uh ) = b−nd fs (bnyt ut , bnyh uh )
(23)
fs (ut , uh ) = |ut0 /ut |−d/yt fs (±ut0 , uh |ut0 /ut |yh /yt )
(24)
becomes
This is exactly the scaling form we conjectured before, with
2−α=
d
,
yt
κ = yh /yt .
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(25)
Note that 2 − α = dν, one of the scaling relations that was derived before. While the value of
fs at the reference point ut0 may be different depending on whether the positive or negative sign
appears in the above, the exponents are the same on both sides. You can check for yourself that the
mean-field exponents are obtained for yt = 2 and yh = 3. The exact exponents from the solution
of the 2D Ising model satisfy the scaling relations with yt = 1 and yh = 15/8.
To summarize, under the assumption that there are only two relevant directions for the Ising
model critical point in the infinite-dimensional space of couplings, we get a scaling form immediately.
Our current understanding is that all the differences between two members of the Ising “universality
class” correspond to irrelevant operators, which do not modify critical exponents (you are asked to
confirm this on the problem set). There are critical points with more than two relevant directions,
and we will talk about such a “tricritical point” soon; note however that each relevant direction
corresponds to one experimental parameter or “knob” that must be tuned, so that tricritical points
are relatively rare, and higher-order multicritical points almost nonexistent.
Most of what we do in future lectures will not depend on having a lattice model, but first let’s
say a bit more about an approximation to the 2D Ising model that we mentioned briefly before. This
approximation is known as the “Migdal-Kadanoff approximation”: its advantage is its simplicity,
but it is not much used in practice since it is an uncontrolled approximation.
Recall that the problem with real-space renormalization on the 2D Ising model is that we
generate lots of nonlocal couplings. One advantage of continuum momentum-space methods to be
discussed shortly is that, by working in the basis of scaling variables, it is much more clear which
perturbations are relevant and which are not. Migdal-Kadanoff is a trick to redesign the 2D lattice
so that the RG equations are closed, without generating nonlocal couplings.
More precisely, to make the Migdal-Kadanoff approximation correct, one should work on a
“hierarchical lattice” that is not continuously connected to normal lattices like the square lattice.
While there are still papers using Migdal-Kadanoff, a much preferable set of approaches depend on
controlled expansions where the small parameter may be related to dimensionality (−expansion)
or the symmetry of the order parameter (large-N expansion), to cite two famous examples.
The rescaling procedure in this approximation consists of two steps: 1. bond moving. 2.
decimation. (A picture of this, in case you’ve missed class, is in Huang problem 18.2, although
there are some typos in that discussion.) The rescaling equation is
0
e2K = cosh(4K),
(26)
where before, for the 1D model, it was,
0
e2K = cosh(2K),
(27)
In terms of x = e2K , the Migdal-Kadanoff RG equation is
x0 =
x2 + x−2
.
2
(28)
In addition to the x = 1 and x = ∞ solutions that were present before, this equation has an
intermediate fixed point. Finding that point x∗ exactly requires solution of a quartic, which we
won’t bother to do (can be done in Mathematica), but it is useful to understand a bit more about
this fixed point. The easiest way to show that there must be an unstable fixed point between 1 and
6
∞ is to plot the function f (x) = −x + (x2 + x−2 )/2. As an exercise, you will be asked to linearize
the RG near this point and find ν.
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