Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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approximate calculation of the flow of coupling
constants is given in the next section). The
existence of a global trajectory for such coordi-
nates helps us to control the limit for moments of
the probability measure (correlation functions).
The question of coordinates and the representation
of the irrelevant terms will be taken up in the
section‘‘RigorousRGanalysis.’’
Ultraviolet Cutoff Removal
The next issue is ultraviolet cutoff removal in field
theory. This problem can be put into the earlier
framework as follows. Let
N
be a sequence of
positive numbers which tend to 0 as N !1.
Following the remark after [3], we replace the unit
cutoff covariance C by the covariance C
N
defined
by taking
N
(instead of 1) as the lower end point in
the integral [3]. Thus,
N
acts as a short-distance or
ultraviolet cutoff. It is easy to see that
C
N
ðx yÞ¼S
N
Cðx yÞ½28
Consider the partition function Z
N
() in a cube
=[R=2, R=2]
d
:
Z
N
ðÞ¼
Z
d
C
N
ðÞe
V
0
ð;;
~
N
;
~
g
N
;
~
N
Þ
½29
where V
0
is given by [23] with g
0
,
0
replaced by
~
g
N
,
~
N
, respectively. By dimensional analysis we can
write
~
N
¼
ð2½dþ2Þ
N
;
~
g
N
¼
ð4½dÞ
N
g;
~
N
¼
ð2½dÞ
N
½30
where g, , are dimensionless parameters. Now
distributed according to C
N
equals in distribution
S
N
distributed according to C. Therefore, choosing
N
= L
N
, we get
Z
N
ðÞ¼
Z
d
C
ðÞe
V
0
ð;S
N
;
~
N
;
~
g
N
;
~
N
Þ
¼
Z
d
C
ðÞe
V
0
ð
N
;;g; Þ
½31
where
N
= [L
N
R=2, L
N
R=2]
d
. Thus, the field
theory problem of removing the ultraviolet cutoff,
that is, taking the limit
N
!0, has been reduced to
the study of a statistical mechanical model in a very
large volume. The latter has to be analyzed via RG
iterations as before.
Critical Field Theories
As mentioned earlier, we have to study the flow of
local interactions as well as that of irrelevant terms.
Together they constitute the RG trajectory and we
have to prove that it exists globally. In general, the
trajectory will tend to explode after a large number
of iterations due to growing relevant terms (char-
acterized in terms of the expanding Wick monomials
mentioned earlier). Wilson pointed out that the
saving factor is to exploit fixed points and their
invariant manifolds by tuning the initial interaction
so that the RG has a global trajectory. This leads to
the notion of a critical manifold which can be
defined as follows. A fixed point will have contract-
ing and/or marginal attractive directions besides the
expanding ones. In the language of dynamical
systems, the critical manifold is the stable or center
stable manifold of the fixed point in question. This
is determined by a detailed study of the discrete
flow. In the examples above, it amounts to fixing
the initial ‘‘mass’’ parameter
0
=
c
(g
0
) with a
suitable function
c
such that the flow remains
bounded in an invariant set. The critical manifold is
then the graph of a function from the space of
contracting and marginal variables to the space of
’s which remains invariant under the flow.
Restricted to it the flow will now converge to a
fixed point. All references to initial coupling
constants have disappeared. The result is known as
a critical theory.
Critical theories have been rigorously constructed
in a number of cases. Take the standard
4
in d
dimensions. Then [] = (d 2)=2. For d > 5 the
4
interaction is irrelevant and the Gaussian fixed point
is attractive with one unstable direction (corre-
sponding to ). In this case one can prove that the
interactions converge exponentially fast to the
Gaussian fixed point on the critical manifold. For
d = 4 the interaction is marginal and the Gaussian
fixed point attractive for g > 0. The critical theory
has been constructed by Gawedzki and Kupainen
(1984) starting with a sufficiently small coupling
constant. The fixed point is Gaussian (interactions
vanish in the limit) and the convergence rate is
logarithmic. This is thus a mean-field theory with
logarithmic corrections, as expected on heuristic
grounds. The mathematical construction of the
critical theory in d = 3 is an open problem. (It is
expected to exist with a non-Gaussian fixed point,
and this is indicated by the perturbative expansion
of Wilson and Fisher in 4 dimensions.) However,
the critical theory for d = 3 for [] = (3 )=4
for >0 held very small has been rigorously
constructed by Brydges et al. (2003). This theory
has a nontrivial hyperbolic fixed point of O(). The
stable manifold is constructed in a small neighbor-
hood of the fixed point. Note that the covariance
without cutoff is Osterwalder–Schrader positive and
thus this is a candidate for a nontrivial EQFT. For
= 1 we have the standard situation in d = 3, and
276 Exact Renormalization Group
this remains open, as mentioned earlier. A very
simplified picture of the above is furnished by the
perturbative computation in the next section.
Unstable Fixed Points
We may attempt to construct field theories around
unstable fixed points. In this case the initial
parameters have to be adjusted as functions of the
cutoff in such a way as to stabilize the flow in the
neighborhood of the fixed point. This may be called
a genuine renormalization. A famous example of
this is pure Yang–Mills theory in d = 4, where the
Gaussian fixed point has only marginal unstable
directions. Bałaban in a series of papers ending in
Bałaban (1989) considered Wilson’s lattice cutoff
version of Yang–Mills theory in d = 4 with initial
coupling fixed by the two-loop asymptotic freedom
formula. He proved, by lattice RG iterations, that in
the weak-coupling regime the free energy per unit
volume is bounded above and below by constants
independent of the lattice spacing. Instability of the
flow is expected to lead to mass generation for
observables but this is a famous open problem.
Another example is the standard nonlinear sigma
model for d = 2. Here too the flow is unstable
around the Gaussian fixed point and we can set the
initial coupling constant by the two-loop asymptotic
freedom formula. Although much is known via
approximation methods (as well as by methods
based on integrable systems) this theory remains to
be rigorously constructed as an EQFT.
Let us now consider a relatively simpler
example, that of constructing a massive super-
renormalizable scalar field theory. This has been
studied in d = 3, with [] = (d 2)=2 = 1=2. We
get =
~
, g = L
N
~
g, = L
2N
~
,and
~
g is taken to be
small. is marginal, whereas g, are relevant
parameters and grow with the iterations. After N
iterations, they are brought up to
~
g,
~
together
with remainders. This realizes the so-called
massive continuum
4
theory in d = 3, and this
has been mathematically controlledintheexact
RG framework. This was proved by Brydges,
Dimock, and Hurd and earlier by Benfatto,
Cassandro, Gallavotti, and others, (see the refer-
ences in Brydges et al. (1998) and Benfatto and
Gallavotti (1995)).
The Exact RG as a Continuous Semigroup
The discrete semigroup defined in [13] of the previous
section has a natural continuous counterpart. Just take
L to be a continuous parameter, L = e
t
, t 0, and
write by abuse of notation T
t
, S
t
,
t
instead of T
e
t
,etc.
The continuous transformations T
t
,
T
t
F ¼ S
t
t
F ½32
give a semigroup
T
t
T
s
¼ T
tþs
½33
of contractions on L
2
(d
C
) with
C
as invariant
measure. One can show that T
t
is strongly contin-
uous and, therefore, has a generator which we will
call L. This is defined by
LF ¼ lim
t!0
þ
T
t
1
t
F ½34
whenever this limit exists. This restricts F to a
suitable subspace D(L) L
2
(d
C
). D(L) contains,
for example, polynomials in fields as well as twice-
differentiable bounded cylindrical functions. The
generator L can be easily computed. To state it, we
need some definitions. Define (D
n
F)(; f
1
, ..., f
n
)as
the nth tangent map at along directions f
1
, ..., f
n
.
The functional Laplacian
_
is defined by
_
FðÞ¼
Z
d
_
ðÞðD
2
FÞð ; ; Þ½35
where
_
=u. Define an infinitesimal dilatation
operator
DðxÞ¼x rð xÞ½36
and a vector field X,
XF ¼½ð DFÞð; ÞðDFÞð; DÞ½37
Then, an easy computation gives
L¼
1
2
_
þX ½38
T
t
is a semigroup with L as generator. Therefore,
T
t
= e
tL
. Let F
t
() = T
t
F(). Then F
t
satisfies the
linear PDE
@F
t
@t
¼LF
t
½39
with the initial condition F
0
= F. This evolution
equation assumes a more familiar form if we write
F
t
= e
V
t
, V
t
being known as the effective potential.
We get
@V
t
@t
¼LV
t
1
2
ðV
t
Þ
ðV
t
Þ
½40
where
ðV
t
ðÞÞ
ðV
t
ðÞÞ
¼
Z
d
_
ðÞðð DV
t
Þð; ÞÞ
2
½41
and V
0
= V. This infinite-dimensional nonlinear
PDE is a version of Wilson’s flow equation.
Note that the linear semigroup T
t
acting on
Exact Renormalization Group 277
functions induces a semigroup R
t
acting non-
linearly on effective potentials giving a trajectory
V
t
= R
t
V
0
.
Equations like the above are notoriously difficult
to control rigorously, especially for large times.
However, they may be solved in formal perturbation
theory when the initial V
0
is small via the presence
of small parameters. In particular, they give rise
easily to perturbative flow equations for coupling
constants. They can be obtained to any order but
then there is the remainder. It is hard to control the
remainder from the flow equation for effective
potentials in bosonic field theories. They require
other methods based on the discrete RG. Never-
theless, these approximate perturbative flows are
very useful for getting a preliminary view of the
flow. Moreover, their discrete versions figure as an
input in further nonperturbative analysis.
Perturbative Flow
It is instructive to see this in second-order perturba-
tion theory. We will simplify by working in infinite
volume (no infrared divergences can arise because
_
(x y) is of fast decrease). Now suppose that we
are in standard
4
theory with [] = (d 2)=2 and
d > 2. We want to show that
V
t
¼
Z
dx
t
:jrð xÞj
2
: þg
t
:ðxÞ
4
:
þ
t
:ðxÞ
2
:
½42
satisfies the flow equation in second order modulo
irrelevant terms provided the parameters flow
correctly. We will ignore field-independent terms.
The Wick ordering is with respect to the covariance
C of the invariant measure. The reader will notice
that we have ignored a
6
term which is actually
relevant in d = 3 for the above choice of []. This is
because we will only discuss the d = 3 case for the
model discussed at the end of this section and for
this case the
6
term is irrelevant. We will assume
that
t
,
t
are of order O(g
2
). Plug in the above in
the flow equation. The quantity
n, m
t
:P
n, m
: repre-
sents one of the terms above with m fields and n
derivatives. Because L is the generator of the
semigroup T
t
we have
@
@t
L
n;m
t
: P
n;m
:
¼
d
n;m
t
dt
ðd m½nÞ
n;m
t
: P
n;m
: ½43
Next turn to the nonlinear term in the flow equation
and insert the
4
term (the others are already of
order O(g
2
)). This produces a double integral of
_
(x y):(x)
3
::(y)
3
:, which after complete Wick
ordering, gives
g
2
t
2
16
Z
dx dy
_
ðx yÞ :ðxÞ
3
ðyÞ
3
:
þ9Cðx y Þ: ðxÞ
2
ðyÞ
2
: þ36Cðx yÞ :ðxÞðyÞ:
þ6Cðx y Þ
2
½44
Consider the nonlocal
4
term. We can localize it by
writing
:ðxÞ
2
ðyÞ
2
: ¼
1
2
: ðxÞ
4
þ ðyÞ
4
ððxÞ
2
ðyÞ
2
Þ
2
: ½45
The local part gives a
4
contribution and the last
term above gives rise to an irrelevant contribution
because it produces additional derivatives. The
coefficients are well defined because C,
_
are smooth
and
_
(x y) is of fast decrease. Now the nonlocal
2
term is similarly localized. It gives a relevant local
2
contribution as well as a marginal jrj
2
contribu-
tion. Finally, the same principle applies to the
nonlocal
6
contribution and gives rise to further
irrelevant terms. Then it is easy to see by matching
that the flow equation is satisfied in second order up
to irrelevant terms (these would have to be compen-
sated by adding additional terms in V
t
) provided
dg
t
dt
¼ð4 dÞg
t
ag
2
t
þ Og
3
t
d
t
dt
¼ 2
t
bg
2
t
þ Og
3
t
d
t
dt
¼ cg
2
t
þ Og
3
t
½46
where a, b, c are positive constants. We see from the
above formulas that, up to second order in g
2
,as
t !1, g
t
! 0 for d 4. In fact, for d 5 the decay
rate is O(e
t
) and for d = 4 the rate is O (t
1
).
However, to see if V
t
converges, we also have to
discuss the
t
,
t
flows. It is clear that in general the
t
flow will diverge. This is fixed by choosing the
initial
0
to be the bare critical mass. This is
obtained by integrating up to time t and then
expressing
0
as a function of the entire g trajectory
up to time t. Assume that
t
is uniformly bounded
and take t !1. This gives the critical mass as
0
¼ b
Z
1
0
ds e
2s
g
2
s
¼
c
ðg
0
Þ½47
This integral converges for all cases discussed above.
With this choice of
0
we get
t
¼ b
Z
1
0
ds e
2s
g
2
sþt
½48
278 Exact Renormalization Group
and this exists for all t and converges as t !1.
Now consider the perturbative flow. It is easy to
see from the above that for d 4,
t
converges as
t !1.
We have not discussed the d = 3 case because the
perturbative g fixed point is of order O(1). But
suppose we take, in the d = 3 case, [] = (3 )=4
with >0 held small as in Brydges et al. (2003).
Then the above perturbative flow equations are
easily modified (by taking account of [43]) and we
get, to second order, an attractive fixed point
g
= O() of the g flow. The critical bare mass
0
can be determined as before and the
t
flow
converges. The qualitative picture obtained above
has a rigorous justification.
Rigorous RG Analysis
We will give a brief introduction to rigorous RG
analysisinthediscretesetupinthesection‘‘TheRG
asaDiscreteSemigroup’’concentratingonthe
principal problems encountered and how one
attemptstosolvethem.Ourapproachisborrowed
from Brydges et al. (2003). It is a simplification
of the methods initiated by Brydges and Yau in
(1990) and developed further by Brydges et al.
(1998). The reader will find other approaches to
rigorous RG methods in the selected references, such
as those of Bałaban, Gawedzki and Kupiainen,
Gallavotti, and others. We will take as a concrete
example the scalar field model introduced earlier.
At the core of the analysis is the choice of good
coordinates for the partition function density, z,of
thesection‘‘TheRGasaDiscreteSemigroup’’.This
is provided by a polymer representation (defined
below) which parametrizes z by a couple (V, K),
where V isalocalpotentialand K isasetfunction
also depending on the fields. Then the RG transfor-
mation T
L
maps (V, K)toanew(V, K). (V, K)
remain good coordinates as the volume tends to 1,
whereas z(volume) diverges. There exist norms
which are suited to the fixed-point analysis of (V, K)
to new (V, K). Now comes the important point: z
does not uniquely specify the representation (V, K).
Therefore, we can take advantage of this nonunique-
ness to keep K small in norm and let most of the
action of T
L
reside in V. This process is called
extraction in Brydges et al. (2003). It makes sure that
K is an irrelevant term, whereas the local flow of V
gives rise to discrete flow equations in coupling
constants. We will not discuss extraction any further.
In the following, we introduce the polymer represen-
tation and explain how the RG transformation acts
on it.
To proceed further, we first introduce a simplifi-
cationinthesetupusedinthesection‘‘TheRGasa
DiscreteSemigroup.’’Recallthatthefunction u
introduced in [3] was smooth, positive definite, and
of rapid decrease. We will simplify further by
imposing the stronger property that it is actually of
finite range: u(x) = 0 for jxj1. We say that u is of
finite range 1. It is easy to construct such functions.
For example, if g is any smooth function of finite
range 1/2, then u = g g is a smooth positive-definite
function of finite range 1. This implies that the
fluctuation covariance
L
of [7] has finite range L.
As a result,
n
in [10] has finite range L
nþ1
and the
corresponding fluctuation fields
n
(x) and
n
(y) are
independent when jx yjL
nþ1
.
Polymer Representation
Pave R
d
with closed cubes of side length 1 called
1-blocks or unit blocks denoted by , and suppose
that is a large cube consisting of unit blocks. A
connected polymer X is a closed connected
subset of these unit blocks. A polymer
activity K(X, ) is a map X, ! R where the fields
depend only on the points of X. We will set
K(X, ) = 0ifX is not connected. A generic form of
the partition function density z(, ) after a certain
number of RG iterations is
zðÞ¼
X
1
N¼0
1
N!
X
X
1
;...;X
N
e
VðX
c
Þ
Y
N
j¼1
KðX
j
Þ½49
Here X
j
are disjoint polymers, X =
S
X
j
,and
X
c
=nX. V is a local potential of the form [23] with
parameters , g, . We have suppressed the -depen-
dence. Initially, the activities K
j
= 0, but they will arise
under RG iterations and the form [49] remains stable,
as we will see. The partition function density is thus
parametrized as a couple (V, K).
Norms for Polymer Activities
Polymer activities K(X, ) are endowed with a norm
kK(X)k, which must satisfy two properties:
_
X \
_
Y ¼ 0 )kK
1
ðXÞK
2
ðYÞk kK
1
ðXÞkkK
1
ðYÞk
kT
L
KðXÞk c
jXj
kKðXÞk ½50
where
_
X is the interior of X and jXj is the number
of blocks in X. c is a constant of order O(1). The
norm measures (Fre´che´t) differentiability proper-
ties of the activity K(X, ) with respect to the field
as well as its admissible growth in .The
growth is admissible if it is
C
integrable. The
second property above ensures the stability of
the norm under RG iteration. For a fixed polymer
X, the norm is such that it gives rise to a Banach
Exact Renormalization Group 279
space of activities K(X). The final norm kk
A
incorporates the previous one and washes out the
set dependence,
kKk
A
¼ sup
X
X
AðXÞkKðXÞk ½51
where A(X) = L
(dþ2)jXj
. This norm essentially ensures
that large polymers have small activities. The details
of the above norms can be found in Brydges et al.
(2003).
The RG operation map f is a composition of two
maps. The RG iteration map z ! T
L
z induces a map
V !
~
V
0
L
and a nonlinear map
~
T
L
: K !
~
K =
~
T
L
(K).
We then compose this with a (nonlinear) extraction
map E which takes out the expanding (relevant)
parts of
~
K !E(
~
K) = K
0
and compensates the local
potential
~
V
0
L
! V
0
such that T
L
z remains invariant.
We denote by f the composition of these two maps
with
V ! V
0
¼ f
V
ðV; KÞ; K ! K
0
¼ f
K
ðV; KÞ½52
The Map
˜
T
L
Consider applying the RG map T
L
to [49]. The map
consists of a convolution
L
followed by the
rescaling S
L
. In the integration over the fluctuation
field , we will exploit the independence of (x) and
(y) when jx yjL. To do this, we pave by
closed blocks of side L, called L-blocks, so that
each L-block is a union of 1-blocks. Let
X
L
be
the L-closure of a set X, namely the smallest union
of L-blocks containing X. The polymers will be
combined into L-polymers which are, by definition,
connected unions of L blocks. The combination is
performed in such a way that the new polymers are
associated to independent functionals of .
Let
~
V(X, )), to be chosen later, be a local
potential independent of . For a coupling constant
sufficiently small, there is a bound
ke
VðYÞ
k2
jYj
½53
We assume that
~
V is so chosen that the same bound
holds when V is replaced by
~
V. Define
Pð;;Þ¼e
Vð;þ Þ
e
~
Vð;Þ
½54
Then we have
e
VðX
c
;þÞ
¼ e
Vð
X
c
;þÞ
¼
Y
X
c
ðe
~
Vð;Þ
þ Pð;;ÞÞ ½ 55
where
X
c
is the closure of X
c
. Expand out the
product and insert into the representation [49] for
z(, þ )). We then rewrite the resulting sum in
terms of L-polymers. The sum splits into a sum over
connected components. Define, for every connected
L-polymer Y,
BK ðYÞ¼
X
NþM1
1
N!M!
X
ðX
j
Þ;ð
i
Þ!Y
e
~
VðX
0
Þ
Y
N
j¼1
KðX
j
Þ
Y
M
i¼1
Pð
i
Þ½56
where X
0
= Y n( [ X
j
) [ ( [
j
) and the sum over the
distinct
i
, and disjoint 1-polymers X
j
is such that
their L-closure is Y. Equation [49] now becomes
zðÞ¼
X
1
N!
X
Y
1
;...;Y
N
e
~
VðY
c
Þ
Y
N
j¼1
BKðY
j
Þ½57
where the sum is over disjoint, connected closed
L-polymers. We now perform the fluctuation inte-
gration over followed by the rescaling. Now
~
V(Y
c
)
is independent of . The -integration sails through
and then factorizes because the Y
j
, being disjoint
closed L-polymers, are separated from each other by
a distance L. The rescaling brings us back to 1-
polymers and reduces the volume from to L
1
.
Therefore,
z
0
ðL
1
Þ¼T
L
zðÞ
¼
X
1
N!
X
X
1
;...;X
N
e
~
V
L
ðX
c
Þ
Y
N
j¼1
ðT
L
BKÞðX
j
Þ½58
where the sum is over disjoint 1-polymers,
X
c
=L
1
nX. By definition
~
V
L
()=S
L
~
V(L) and
(T
L
BK )(Z)=S
L
L
BK(LZ). This shows that the
representation [49] is stable under iteration and,
furthermore, gives us the map
V !
~
V
L
K !
~
K ¼
~
T
L
ðKÞ¼T
L
BK
½59
The norm boundedness of K implies that
~
T
L
(K)is
norm bounded. We see from the above that a
variation in the choice of
~
V is reflected in the
corresponding variation of
~
K. The extraction map E
now takes out from
~
K the expanding parts and then
compensates it by a change of
~
V
L
in such a way that
the representation [58] is left invariant by the
simultaneous replacement
~
V
L
! V
0
,
~
K ! K
0
= E(
~
K).
The extraction map is nonlinear. Its linearization is a
subtraction operation and this dominates in norm the
nonlinearities, (Brydges et al. 1998).
The map V !
~
V
L
! V
0
leads to a discrete flow of
the coupling constants in V. It is convenient to write
K = K
pert
þ R,whereR is the remainder. Then the
coupling constant flow is a discrete version of the
280 Exact Renormalization Group
continuous flows encountered in the last section,
together with remainders which are controlled by the
size of R. In addition, we have the flow of K.The
discrete flow of the pair (coupling constants, K)canbe
studied in a Banach space norm. Once one proves that
the nonlinear parts satisfy a Lipshitz property, the
discrete flow can be analyzed by the methods of stable-
manifold theory of dynamical systems in a Banach
space context. The reader is referred to the article by
Brydges et al. (2003) for details of the extraction map
and the application of stable-manifold theory in the
construction of a global RG trajectory.
Further Topics
Lattice RG Methods
Statistical mechanical systems are often defined on a
lattice. Moreover, the lattice provides an ultraviolet
cutoff for Euclidean field theory compatible with
Osterwalder–Schrader positivity. The standard lat-
tice RG is based on Kadanoff–Wilson block spins.
Its mathematical theory and applications have
been developed by Bałaban, and Gawedzki and
Kupiainen (see Gawedzki and Kupiainen (1986) and
references therein). This leads to multiscale decom-
positions of the Gaussian lattice field as a sum of
independent fluctuation fields on increasing length
scales. Brydges et al. (2004) have shown that
standard Gaussian lattice fields have multiscale
decompositions as a sum of independent fluctuation
fields with the finite-range property introduced in
the last section. This permits the development of
rigorous lattice RG theory in the spirit of the
continuum framework of the previous section.
Fermionic Field Theories
Field theories of interacting fermions are often
simpler to handle than bosonic field theories. Because
of statistics, fermion fields are bounded and pertur-
bation series converges in finite volume in the
presence of an ultraviolet cutoff. The notion of
studying the RG flow at the level of effective
potentials makes sense. At any given scale, there is
always an ultraviolet cutoff and the fluctuation
covariance being of fast decrease provides an infrared
cutoff. This is illustrated by the work of Gawedzki
and Kupiainen (1985), who gave a nonperturbative
construction in the weak effective coupling regime of
the RG trajectory for the Gross–Neveu model in two
dimensions. This is an example of a model with an
unstable Gaussian fixed point where the initial
coupling has to be adjusted as a function of the
ultraviolet cutoff consistent with ultraviolet asympto-
tic freedom so as to stabilize the flow.
Acknowledgments
The author is grateful to David Brydges for very
helpful comments on a preliminary version of this
article.
See also: Operator Product Expansion in Quantum Field
Theory; Renormalization: Statistical Mechanics and
Condensed Matter.
Further Reading
Bałaban T (1982) (Higgs)
2,3
quantum fields in a finite volume II,
an upper bound. Communications in Mathematical Physics
86: 555–594.
Bałaban T (1989) Large field renormalization. 2: Localization,
exponentiation and bounds for the R operation. Communica-
tions in Mathematical Physics 122: 355–392.
Benfatto G and Gallavotti G (1995) Renormalization Group.
Princeton, NJ: Princeton University Press.
Brydges DC and Yau HT (1990) Grad perturbations of massless
Gaussian fields. Communication in Mathematical Physics
129: 351–392.
Brydges D, Dimock J, and Hurd T (1998) Estimates on
renormalization group transformations. Canadian Journal of
Mathematics 50(4): 756–793.
Brydges DC, Mitter PK, and Scoppola B (2003) Critical (F
4
)
3,
.
Communications in Mathematical Physics 240: 281–327.
Brydges D, Guadagni G, and Mitter PK (2004) Finite range
decomposition of Gaussian processes. Journal of Statistical
Physics 115: 415–449.
Gawedzki K and Kupiainen A (1984) Massless lattice
4
theory in
four dimensions: a nonperturbative control of a renormaliz-
able model. Communications in Mathematical Physics
99: 197–252.
Gawedzki K and Kupiainen A (1985) Gross–Neveu models
through convergent perturbation expansions. Communica-
tions in Mathematical Physics 102: 1–30.
Gawedzki K and Kupiainen A (1986) Asymptotic freedom beyond
perturbation theory. In: Osterwalder K and Stora R (eds.)
Critical Phenomena, Random Systems, Gauge Theories.
North-Holland: Les Houches.
Gelfand IM and Vilenkin NYa (1964) Generalized Functions,vol.4.
New York: Academic Press.
Kopper C and Muller V (2000) Renormalization proof for
spontaneously broken Yang–Mills theory with flow equations.
Communications in Mathematical Physics 209: 477–516.
Polchinski J (1984) Renormalization and effective Lagrangians.
Nuclear Physics B 231: 269–295.
Rivasseau V (1991) From Perturbative to Constructive Renorma-
lization. Princeton NJ: Princeton University Press.
Wilson KG (1983) Renormalization group and critical phenom-
ena. Reviews of Modern Physics 55: 583–599.
Wilson KG and Kogut J (1974) Renormalization group and the
expansion. Physics Reports 12C: 75–200.
Exact Renormalization Group 281
F
Falicov–Kimball Model
Ch Gruber, Ecole Polytechnique Fe
´
de
´
rale
de Lausanne, Lausanne, Switzerland
D Ueltschi, University of Arizona, Tucson, AZ, USA
ª 2006 Elsevier Ltd. All rights reserved.
A Brief History
The ‘ ‘Falicov–Kimball model’ ’ was first considered by
Hubbard and Gutzwiller during 1963–65 as a simpli-
fication of the Hubbard model. In 1969, Falicov and
Kimball introduced a model that included a few extra
complications, in order to investigate metal–insulator
phase transitions in rare-earth materials and transition-
metal compounds (Falicov and Kimball 1969). Experi-
mental data suggested that this transition is due to the
interactions between electrons in two electronic states:
nonlocalized states (itinerant electrons), and states that
are localized around the sites corresponding to the
metallic ions of the crystal (static electrons).
A tight-binding approximation leads to a model
defined on a lattice (the crystal) and two species of
particles are considered. The first species consists of
spinless quantum fermions (we refer to them as
‘ ‘electrons’’), and the second species consists of localized
holes or electrons (‘‘classical particles’ ’). Electrons hop
between nearest-neighbor sites but classical particles do
not. Both species obey Fermi statistics (in particular, the
Pauli exclusion principle prevents more than one
particle of a given species to occupy the same site).
Interactions are on-site and thus involve particles of
different species; they can be repulsive or attractive.
The very simplicity of the model allows for a
broad range of applications. It was studied in the
context of mixed valence systems, binary alloys, and
crystal form ation. Adding a magnetic field yields the
flux phase problem. The Falicov–Kimball model can
also be viewed as the simplest model where
quantum particles interact with classical fields.
The fifteen years following the introduction of the
model saw studies based on approximate methods,
such as Green’s function techniques, that gave rise to
a lot of confusion. A breakthrough occurred in 1986
when Brandt and Schmidt, and Kennedy and Lieb,
proposed the first rigorous results. In particular,
Kennedy and Lieb showed in their beautiful paper
that the electrons create an effective interaction
between the classical particles and that a phase
transition takes place for any value of the coupling
constant, provided the temperature is low enough.
Many studies by mathematical physicists fol-
lowed and several results are presented in this
short survey. Recent years have seen an increasing
interest from condensed matter physicists. We
encourage interested readers to consult the reviews
by Freericks and Zlatic
´
(2003), Gruber and Macris
(1996), and Je¸ drzejewski and Lema´nski (2001).
Mathematical Setting
Definitions
Let Z
d
denote a finite cubic box. The config-
uration space for the classical particles is
¼f0; 1g
¼ ! ¼ð!
x
Þ : x 2 ; and !
x
¼ 0; 1
fg
where !
x
= 0 or 1 denotes the absence or presence of
a classical particle at the sit e x. The total number of
classical particles is N
c
(!) =
P
x2
!
x
. The Hilbert
space for the spinless quantum particles (‘‘elec-
trons’’) is the usual fermionic Fock space
F
¼
M
jj
N¼0
H
;N
where H
, N
is the Hilbert space of square summable,
antisymmetric, complex functions =(x
1
, ..., x
N
)
of N variables x
i
2 .Leta
y
x
and a
x
denote the
standard creation and annihilation operators of an
electron at x; recall that they satisfy the antic-
ommutation relations
fa
x
; a
y
g¼0; fa
y
x
; a
y
y
g¼0; fa
x
; a
y
y
g¼
xy
The Hamiltonian for the Falicov–Kimball model is an
operator on F
that depends on the configurations of
classical particles. Namely, for ! 2
,wedefine
H
ð!Þ¼
X
x;y2
jxyj¼1
a
y
x
a
y
U
X
x2
!
x
a
y
x
a
x
The first term represents the kinetic energy of the
electrons. The second term represents the on-site
attraction (U > 0) or repulsion (U < 0) between
electrons and classical particles.
The Falicov–Kimball Hamiltonian can be written
with the help of a one-body Hamiltonian h
, which
is an operator on the Hilbert space for a single
electron ‘
2
(). Indeed, we have
H
ð!Þ¼
X
x;y2
h
xy
ð!Þa
y
x
a
y
The matrix h
(!) = (h
xy
(!)) is the sum of a hopping
matrix (adjacency matrix) t
, and of a matrix v
(!)
that represents an external potential due to the
classical particles. Namely, we have
h
xy
ð!Þ¼t
xy
U!
x
xy
where t
xy
is one if x and y are nearest neighbors, and is
zero otherwise. The spectrum of t
lies in (2d,2d),
and the eigenvalues of v
(!)areU (with degeneracy
N
c
(!)) and 0 (with degeneracy jjN
c
(!)). Denoting
j
(A) the eigenvalues of a matrix A, it follows from the
minimax principle that
j
ðAÞkBk
j
ðA þ BÞ
j
ðAÞþkBk
Let
1
(!)
2
(!)
jj
(!) be the eigenvalues
of h
(!). Choosing A = v
(!)andB = t
in the
inequality above, we find that for U > 0,
U 2d <
j
ð!Þ < U þ 2d for j ¼ 1; ...; N
c
ð!Þ
2d <
j
ð!Þ < 2d for j ¼ N
c
ð!Þþ1; ...; jj
In particular, for any configuration ! and any ,
Spec h
ð!ÞðU 2d; U þ 2dÞ[ð2d; 2dÞ
Thus, for U > 4d, the spectrum of h
(!) has the
‘‘universal’’ gap (U þ 2d, 2d). A similar property
holds for U < 4d.
Canonical Ensemble
A fruitful approach towards understanding the
behavior of the Falicov–Kimball model is to first
fix the configuration of the classical particles, and
then to introduce the groun d-state energy E
(N
e
, !)
as the lowest eigenvalue of H
(!) in the subspace
H
, N
e
:
E
ðN
e
;!Þ¼ inf
2H
;N
e
;
kk¼1
hjH
ð!Þji¼
X
N
e
j¼1
j
ð!Þ
A typical problem is to find the set of ground-
state configurations, that is, the set of configura-
tions that minimize E
(N
e
, !)forgivenN
e
and
N
c
= N
c
(!).
In the case U > 4d and N
e
= N
c
(!), the ground-
state energy E
(N
c
(!), !) has a convergent expan-
sion in powers of U
1
:
E
ðN
c
ð!Þ;!Þ
¼UN
c
ð!Þþ
X
k2
1
kU
k1
X
x
1
; ...; x
k
2
jx
i
x
i1
j¼1
0 < mðfx
i
gÞ < k
ð1Þ
mðfx
i
gÞ
k 2
mðfx
i
gÞ 1
½1
where m(x
1
, ..., x
k
) is the number of sites x
i
with
!
x
i
= 0. The last sum also includes the condition
jx
k
x
1
j= 1. Sim ple estimates show that the series is
less than (2d = (U 4d))N
c
(!). The lowest-order term
is a nearest-neighbor interact ion,
1
U
X
fx;yg:jxyj¼1
w
x
;1w
y
that favors pairs with different occupation numbers.
Formula [1] is the starting point for most studies of
the phase diagram for large U. A similar expansion
holds for U < 4d and N
e
= jjN
c
(!).
A simple derivation of expansion [1] using Cauchy
formula can be found in Gruber and Macris (1996).It
can be extended to positive temperatures with the
help of Lie–Schwinger series (Datta et al. 1999).
Phase diagrams are better discussed in the limit of
infinite volumes where boundary effects can be
discarded. Let
per
be the set of configurations on Z
d
that are periodic in all d directions, and
per
(
c
)
per
be the set of periodic configurations with density
c
.
For ! 2
per
and
e
2 [0, 1], we introduce the energy
per site in the infinite volume limit by
eð
e
;!Þ¼ lim
%Z
d
1
jj
E
ðN
e
;!Þ½2
Here, the limit is taken over any sequence of
increasing cubes, and N
e
= b
e
jjc is the integer
part of
e
jj. Existence of this limit follows from
standard arguments.
In the case of the empty configuration !
x
0,
we get the well-known energy per site of free lattice
electrons: for k 2 [, ]
d
, let "(k) =
P
d
= 1
cos k
;
then
eð
e
;! 0Þ¼
1
ð2Þ
d
Z
"ðkÞ<"
F
ð
e
Þ
"ðkÞ dk
where "
F
(
e
) is the Fermi energy, defined by
e
¼
1
ð2Þ
d
Z
"ðkÞ<"
F
ð
e
Þ
dk
284 Falicov–Kimball Model
The other simple situation is the full configu-
ration !
x
1, whose energy is e(
e
, ! 1) =
e(
e
, ! 0) U
e
.
Let e(
e
,
c
) denote the absolute ground-state
energy density, namely,
eð
e
;
c
Þ¼ inf
!2
per
ð
c
Þ
eð
e
;!Þ
Notice that e(
e
, !) is convex in
e
, and that e(
e
,
c
)
is the convex envelope of {e(
e
, !):! 2
per
(
c
)}. It
may be locally linear around some (
e
,
c
). This is
the case if the infimum is not realized by a periodic
configuration. The nonperiodic ground states can be
expressed as linear combinations of two or more
periodic ground states (‘‘mixtures’’). That is, for 1
i n there are
i
0 with
P
i
i
= 1, !
(i)
2
per
,
and
(i)
e
, such that
e
¼
X
i
i
ðiÞ
e
;
c
¼
X
i
i
c
ð!
ðiÞ
Þ
and
eð
e
;
c
Þ¼
X
i
i
eð
ðiÞ
e
;!
ðiÞ
Þ
The simplest mixture is the ‘‘segregated state’’ for
densities
e
<
c
:take!
(1)
to be the empty
configuration, !
(2)
to be the full c onfiguration,
(1)
e
= 0,
(2)
e
=
e
=
c
,and
2
= 1
1
=
c
.
If d 2, a mixture between configurations !
(i)
can be realized as follows. First, partition Z
d
into
domains D
1
[[D
n
such that jD
i
j=jj!
i
and
j@D
i
j=jj!0as % Z
d
. Then, define a nonperiodic
configuration ! by setting !
x
= !
(i)
x
for x 2 D
i
(see the
illustration in Figure 1). The canonical energy can be
computed from [2], and it is equal to
eð
e
;!Þ¼ inf
ð
ðiÞ
e
Þ:
P
i
i
ðiÞ
e
¼
e
X
n
i¼1
i
eð
ðiÞ
e
;!
ðiÞ
Þ
Furthermore, the infimum is realized by densities
(i)
e
such that there exists
e
with
e
(
e
, !
(i)
) =
(i)
e
for all
i (see [4] below for the definition of
e
(
e
, !)).
We define the canonical ground-state phase
diagram as the set of ground states ! (either a
periodic configuration or a mixture) that minimize
the ground-state energy for given densities
e
,
c
:
G
can
ð
e
;
c
Þ¼ ! : eð
e
;!Þ¼eð
e
;
c
Þ and
c
ð!Þ¼
c
fg
Grand-Canonical Ensemble
Properties of the system at finite temperatures
are usually investigated within the grand-canonical
formalism. The equilibrium state is characterized by
an inverse temperature = 1=k
B
T, and by chemical
potentials
e
,
c
, for the electrons and for the
classical particles, respectively. In this formalism,
the thermodynamic properties are derived from the
partition functions
Z
ð;
e
;!Þ¼tr
F
e
½H
ð!Þ
e
N
Z
ð;
e
;
c
Þ¼
X
!2
e
c
N
c
ð!Þ
Z
ð;
e
;!Þ
½3
Here, N
=
P
x2
a
y
x
a
x
is the operator for the total
number of electrons. We then define the free energy by
F
ð;
e
;
c
Þ¼
1
log Z
ð;
e
;
c
Þ
The first partition function in [3] allows us to
introduce an effective interaction for the classical
particles, mediated by the electrons, by
F
ð;
e
;
c
;!Þ¼
c
N
c
ð!Þ
1
log Z
ð;
e
;!Þ
It depends on the inverse temperature . Taking the
limit of zero temperature gives the corresponding
ground-state energy of the electrons in the classical
configuration !:
E
ð
e
;
c
;!Þ¼lim
!1
F
ð;
e
;
c
;!Þ
¼
c
N
c
ð!Þþ
X
j:
j
ð!Þ<
e
ð
j
ð!Þ
e
Þ
Notice that F
and E
are strictly decreasing and
concave in
e
,
c
(E
is actually linear in
c
). We
also define the energy density in the infinite volume
limit by considering a sequence of increasing cubes.
For ! 2
per
,
eð
e
;
c
;!Þ¼ lim
%Z
d
1
jj
E
ð
e
;
c
;!Þ
The corresponding electronic density is
e
ð
e
;!Þ¼ lim
%Z
d
1
jj
#fj :
j
ð!Þ <
e
g
¼
@
@
e
eð
e
;
c
;!Þ½4
Figure 1 A two-dimensional mixed configuration formed by
periodic configurations of densities 0, 1=5, and 1=2.
Falicov–Kimball Model 285