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Renormalization Group in physics
Renormalization Group (RG) is very sophisticated, beautiful and powerful method
for analyzing physical phenomena (especially so called critical phenomena).
RG is based on the invariance of Hamiltonian under some scale transformations.
e.g.) 1 dim Ising model
The Renormalization Group in Real Space
his chapter, we develop the real-space RG for the one-dimensional
ing model. The Ising model is especially useful for illustrating the
ion of Kadanoff averaging and its effect on the partition function. This
was solved exactly in Chapter 3, so comparisons with those results will
the efficacy of the RG method to be assessed. In the next chapter, we
ply the RG method to the two-dimensional (2D) Ising model.
The One-Dimensional Ising Model
ergy E of the 1D Ising model is
H = −J
i
si
si+1
, (7.1)
> 0 is the coupling constant. The partition function,
Z =
{si=±1}
e−H/kBT =
{si=±1} i
eKsisi+1 , (7.2)
K = J/kB
T, kB
is Boltzmann’s constant, and T is the absolute tem-
e, determines the thermodynamic properties of the Ising model in
f the Helmholtz free energy
F = −kB
T ln Z . (7.3)
ution of this model was derived in Sec. 3.2.
Renormalization by Decimation
are several ways of implementing the real-space RG because of the
ty in choosing the block spins. This can rarely be performed exactly,
tematic approximations are available to make the calculation man-
. The 1D Ising model is an exception in that the RG calculations can
ed out exactly.
In this chapter, we develop the real-space RG for the one-dimensional
(1D) Ising model. The Ising model is especially useful for illustrating the
realization of Kadanoff averaging and its effect on the partition function. This
model was solved exactly in Chapter 3, so comparisons with those results will
enable the efficacy of the RG method to be assessed. In the next chapter, we
will apply the RG method to the two-dimensional (2D) Ising model.
7.1 The One-Dimensional Ising Model
The energy E of the 1D Ising model is
H = −J
i
si
si+1
, (7.1)
where J > 0 is the coupling constant. The partition function,
Z =
{si=±1}
e−H/kBT =
{si=±1} i
eKsisi+1 , (7.2)
where K = J/kB
T, kB
is Boltzmann’s constant, and T is the absolute tem-
perature, determines the thermodynamic properties of the Ising model in
terms of the Helmholtz free energy
F = −kB
T ln Z . (7.3)
The solution of this model was derived in Sec. 3.2.
7.2 Renormalization by Decimation
There are several ways of implementing the real-space RG because of the
flexibility in choosing the block spins. This can rarely be performed exactly,
but systematic approximations are available to make the calculation man-
ageable. The 1D Ising model is an exception in that the RG calculations can
be carried out exactly.
(1D) Ising model. The Ising model is especially useful for illustr
realization of Kadanoff averaging and its effect on the partition func
model was solved exactly in Chapter 3, so comparisons with those re
enable the efficacy of the RG method to be assessed. In the next ch
will apply the RG method to the two-dimensional (2D) Ising mode
7.1 The One-Dimensional Ising Model
The energy E of the 1D Ising model is
H = −J
i
si
si+1
,
where J > 0 is the coupling constant. The partition function,
Z =
{si=±1}
e−H/kBT =
{si=±1} i
eKsisi+1 ,
where K = J/kB
T, kB
is Boltzmann’s constant, and T is the abso
perature, determines the thermodynamic properties of the Ising
terms of the Helmholtz free energy
F = −kB
T ln Z .
The solution of this model was derived in Sec. 3.2.
7.2 Renormalization by Decimation
There are several ways of implementing the real-space RG becau
flexibility in choosing the block spins. This can rarely be performe
but systematic approximations are available to make the calculat
ageable. The 1D Ising model is an exception in that the RG calcula
be carried out exactly.
We will perform a decimation, whereby a partial evaluation of the
function (7.2) is carried out by to summing over alternate spins on th
fficacy of the RG method to be assessed. In the next chapter, we
e RG method to the two-dimensional (2D) Ising model.
he One-Dimensional Ising Model
E of the 1D Ising model is
H = −J
i
si
si+1
, (7.1)
is the coupling constant. The partition function,
Z =
{si=±1}
e−H/kBT =
{si=±1} i
eKsisi+1 , (7.2)
J/kB
T, kB
is Boltzmann’s constant, and T is the absolute tem-
termines the thermodynamic properties of the Ising model in
Helmholtz free energy
F = −kB
T ln Z . (7.3)
of this model was derived in Sec. 3.2.
enormalization by Decimation
veral ways of implementing the real-space RG because of the
choosing the block spins. This can rarely be performed exactly,
tic approximations are available to make the calculation man-
1D Ising model is an exception in that the RG calculations can
ut exactly.
erform a decimation, whereby a partial evaluation of the partition
) is carried out by to summing over alternate spins on the lattice.
where J > 0 is the coupling constant. The partition function,
Z =
{si=±1}
e−H/kBT =
{si=±1} i
eKsisi+1 , (7.2)
where K = J/kB
T, kB
is Boltzmann’s constant, and T is the absolute tem-
perature, determines the thermodynamic properties of the Ising model in
terms of the Helmholtz free energy
F = −kB
T ln Z . (7.3)
The solution of this model was derived in Sec. 3.2.
7.2 Renormalization by Decimation
There are several ways of implementing the real-space RG because of the
flexibility in choosing the block spins. This can rarely be performed exactly,
but systematic approximations are available to make the calculation man-
ageable. The 1D Ising model is an exception in that the RG calculations can
be carried out exactly.
We will perform a decimation, whereby a partial evaluation of the partition
function (7.2) is carried out by to summing over alternate spins on the lattice.
The decimation proceeds by summing over the spins on, say, the odd sites.
This is shown schematically in Fig. 7.1. The first step is to write the partition
function in a form where the spins on the even and odd sites are separated:
Z =
{s2i=±1} 2i {s2i+1=±1} 2i+1
eK(s2is2i+1+s2i+1s2i+2 . (7.4)
K K K K K K K K
Figure 7.1: Decimation of the 1D Ising model. The spins on the sites marke
by filled circles can be summed over independently, leaving a partition fun
tion expressed only in terms of spins on sites marked by open circles.
As indicated in Fig. 7.1, in a model that has interactions only between ad
jacent spins,the spins on odd sites do not interact with one another. Thu
the sums in the partition function over alternate sites may be evaluated in
dependently of one another. The contributions from the odd sites can b
determined by focussing on the contribution from a typical odd spin s2n+
Performing the partition sum over odd spins leaves only even spins:
s2n+1=±1
eK(s2ns2n+1+s2n+1s2n+2)
= eK(s2n+s2n+2) + e−K(s2n+s2n+2)
= 2 cosh K(s2n
+ s2n+2
) . (7.5
in terms of which the partition function becomes
Z =
{s2i=±1} 2i
2 cosh K(s2i
+ s2i+2
) . (7.6
As noted above, this is an exact expression.
7.3 Recursion Relations
The expression in (7.6) represents the partial evaluation of the partitio
function obtained by summing over the spins on odd sites. This term ha
two types of contributions: a spin-independent term from the removed (odd
spins and a spin-dependent term from the remaining (even) spins. Notice tha
the resulting interaction between spins does not appear to have the form
the original Ising energy in (7.1). However, we now assert that the effect
2i 2i+2
for every configuration of neighboring spins s2i
and s2i+1
. Ther
configurations obtained from s2i
= ±1 and s2i+2
= ±1, so we m
s2i
= +1 , s2i+2
= +1 , z(K)eK′
= 2 cosh 2K
s2i
= −1 , s2i+2
= −1 , z(K)eK′
= 2 cosh 2K
s2i
= +1 , s2i+2
= −1 , z(K)e−K′
= 2 ,
s2i
= −1 , s2i+2
= +1 , z(K)e−K′
= 2 .
Thus, there are only two distinct conditions to satisfy, one
to aligned spins and one corresponding to opposed spins. T
of pairs of configurations results from the up-down symmet
model in the absence of a magnetic field. We can express the
above in terms of K′ and z(K) separately by dividing and m
two distinct equations, which yields, respectively,
e2K′
= cosh 2K ,
z2(K) = 4 cosh 2K .
The first of these equations can be solved for K′ in terms of K
K′ = 1
2
ln(cosh 2K) .
rmalization Group in Real Space 105
recursion relation for the coupling constant. This recursion relation
ed entirely in terms of the original coupling constant, i.e. no new
ns are generated by the decimation. This is a special property of
ng model; in higher dimensions the RG generates additional inter-
tween spins that are consistent with the symmetry of the original
on (7.11) can be written in a more useful form by using the fact
Helmholtz free energy F = −kB
T ln Z is an extensive quantity, so
is also extensive: ln Z = Nf(K), where N is the number of sites
tem. Hence, Eq. (7.8) may be written as
Z(K, N) = [z(K)]N/2Z(K′, N/2) , (7.13)
This is the recursion relation for the coupling constant. This recursion relation
is expressed entirely in terms of the original coupling constant, i.e. no new
interactions are generated by the decimation. This is a special property of
the 1D Ising model; in higher dimensions the RG generates additional inter-
actions between spins that are consistent with the symmetry of the original
model.
Equation (7.11) can be written in a more useful form by using the fact
that the Helmholtz free energy F = −kB
T ln Z is an extensive quantity, so
that ln Z is also extensive: ln Z = Nf(K), where N is the number of sites
in the system. Hence, Eq. (7.8) may be written as
Z(K, N) = [z(K)]N/2Z(K′, N/2) , (7.13)
where, on account of Eq. (7.10), Z is the same function of K′ and 1
2
N on the
right-hand side of this equation as it is of K and N on the left-hand side.
Additionally, since the form of the energy is preserved after each decimation,
the functional form of z(K) is preserved as well. Taking logarithms, we have
ln Z(K, N) = Nf(K) = 1
2
N ln z(K) + 1
2
Nf(K′) , (7.14)
which can be rearranged as
f(K′) = 2f(K) − ln z(K) . (7.15)
By invoking Eq. (7.11), we obtain
f(K′) = 2f(K) − ln 2(cosh 2K)1/2 . (7.16)
This is the second recursion relation of the 1D Ising model.
Figure 7.2 summarizes the repeated application of the decimation proce-
106 The Renormalization Group in Real
K K K K K K K K K K K K K K K K
K K K K K K K K
K K K K
K K
Figure 7.2: Schematic illustration of the recursive decimation of the 1
model. At each step, every second spin spin, indicated by the filled
are incorporated into the partition function, which results in an Ising
with the same energy, but with a renormalized coupling constant.
7.4 Fixed Points
The behavior of the coupling constant under repeated decimation
examined by first writing the recursion relation in Eq. (7.10) as K′ =
Thus, beginning with an initial value K0
of the coupling constant, deci
yields K1
= R(K0
), a second decimation produces K2
= R(K1
), and
This can be represented diagrammatically as in Fig. 7.3.
Suppose we have some initial value of the coupling constant K0
,
Hamiltonian
Partition func
Free energy
( )
Sum over odd spins, →
We can get the parameter relation between the original and coarse grained system :
104 The Renormalization Group in Real Spa
this assertion. For the moment, we write
Z =
{s2i=±1} 2i
2 cosh K(s2i
+ s2i+2
)
=
{s2i=±1} 2i
z(K)eK′s2is2i+2 (7
= [z(K)]N/2
{s2i=±1} 2i
eK′s2is2i+2 , (7
in which z(K) is the spin-independent part of the partition function a
K′ is the new (i.e. renormalized) coupling constant. Consistency betw
Eqs. (7.2) and (7.7) requires that
2 cosh K(s2i
+ s2i+2
) = z(K)eK′s2is2i+2 (7
for every configuration of neighboring spins s2i
and s2i+1
. There are four s
configurations obtained from s2i
= ±1 and s2i+2
= ±1, so we must have t
K′
If we denote , we can derive
For fixed point ( ), we get
108 The Renormalization Group in Real Space
and the recursion relation reduces to
f(K∗) ≈ ln 2(cosh 2K∗)1/2
= ln 2 + 1
2
ln(cosh 2K∗) . (7.18)
As K → 0, cosh K → 1, so
f(K∗) → ln 2 , (7.19)
and the free energy
F = −NkB
Tf(K∗) = −NkB
T ln 2 , (7.20)
(For 2 dim, we get non-trivial result : )
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