What can be learnt from the nonperturbative renormalization group?
We point out some limits of the perturbative renormalization group used in statistical mechanics both at and out of equilibrium. We argue that the non-perturbative renormalization group formalism is a promising candidate to overcome some of them. We present some results recently obtained in the...
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | What can be learnt from the nonperturbative renormalization group? / B. Delamotte, L. Canet // Condensed Matter Physics. — 2005. — Т. 8, № 1(41). — С. 163–179. — Бібліогр.: 51 назв. — англ. |
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irk-123456789-1194812017-06-08T03:03:56Z What can be learnt from the nonperturbative renormalization group? Delamotte, B. Canet, L. We point out some limits of the perturbative renormalization group used in statistical mechanics both at and out of equilibrium. We argue that the non-perturbative renormalization group formalism is a promising candidate to overcome some of them. We present some results recently obtained in the literature that substantiate our claims. We finally list some open issues for which this formalism could be useful and also review some of its drawbacks. Ми звертаємо увагу на певні обмеження пертурбативної ренормалізаційної групи, що використовується в рівноважній та нерівноважній статистичній механіці. Ми стверджуємо, що формалізм непертурбативної ренормалізаційної групи є добрим кандидатом для того, щоб обійти деякі з них. Ми приводимо деякі недавно отримані в літературі результати, що підтверджують наші думки. В кінці ми перелічуємо деякі відкриті питання, для яких може бути корисний цей формалізм, а також робимо огляд його недоліків. 2005 Article What can be learnt from the nonperturbative renormalization group? / B. Delamotte, L. Canet // Condensed Matter Physics. — 2005. — Т. 8, № 1(41). — С. 163–179. — Бібліогр.: 51 назв. — англ. 1607-324X PACS: 64.60.Ak, 11.10.Hi, 05.10.Cc DOI:10.5488/CMP.8.1.163 http://dspace.nbuv.gov.ua/handle/123456789/119481 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We point out some limits of the perturbative renormalization group used
in statistical mechanics both at and out of equilibrium. We argue that the
non-perturbative renormalization group formalism is a promising candidate
to overcome some of them. We present some results recently obtained in
the literature that substantiate our claims. We finally list some open issues
for which this formalism could be useful and also review some of its drawbacks. |
| format |
Article |
| author |
Delamotte, B. Canet, L. |
| spellingShingle |
Delamotte, B. Canet, L. What can be learnt from the nonperturbative renormalization group? Condensed Matter Physics |
| author_facet |
Delamotte, B. Canet, L. |
| author_sort |
Delamotte, B. |
| title |
What can be learnt from the nonperturbative renormalization group? |
| title_short |
What can be learnt from the nonperturbative renormalization group? |
| title_full |
What can be learnt from the nonperturbative renormalization group? |
| title_fullStr |
What can be learnt from the nonperturbative renormalization group? |
| title_full_unstemmed |
What can be learnt from the nonperturbative renormalization group? |
| title_sort |
what can be learnt from the nonperturbative renormalization group? |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119481 |
| citation_txt |
What can be learnt from the nonperturbative renormalization group? / B. Delamotte, L. Canet // Condensed Matter Physics. — 2005. — Т. 8, № 1(41). — С. 163–179. — Бібліогр.: 51 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT delamotteb whatcanbelearntfromthenonperturbativerenormalizationgroup AT canetl whatcanbelearntfromthenonperturbativerenormalizationgroup |
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2025-07-08T15:57:00Z |
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2025-07-08T15:57:00Z |
| _version_ |
1837094899430719488 |
| fulltext |
Condensed Matter Physics, 2005, Vol. 8, No. 1(41), pp. 163–179
What can be learnt from the
nonperturbative renormalization
group?
B.Delamotte∗1 ,L.Canet† 2
1 Laboratoire de Physique Théorique des Liquides,
Université Paris VI-Pierre et Marie Curie,
75252 Paris Cedex 05, France
2 School of Physics and Astronomy,
University of Manchester,
Manchester M13 9PL, United Kingdom
Received December 2, 2004
We point out some limits of the perturbative renormalization group used
in statistical mechanics both at and out of equilibrium. We argue that the
non-perturbative renormalization group formalism is a promising candidate
to overcome some of them. We present some results recently obtained in
the literature that substantiate our claims. We finally list some open issues
for which this formalism could be useful and also review some of its draw-
backs.
Key words: renormalization group, nonperturbative field theory,
dynamical critical phenomena
PACS: 64.60.Ak, 11.10.Hi, 05.10.Cc
1. Introduction
Field theory together with renormalization group (RG) have provided very pow-
erful means to investigate continuous phase transitions in equilibrium and non-
equilibrium statistical physics. The need for field theoretical techniques is now well
understood: when spatial (and/or time) fluctuations are large in a N -body system,
the mean field approach is inadequate and it becomes necessary to keep track of
the space (and/or time) dependence of the order parameter. In the continuum, sta-
tistical fluctuations are summed over through a functional integral so that theories
to be dealt with are (euclidean) field theories. The difficulty with these theories is
that close to a continuous phase transition, fluctuations on all length (and/or time)
∗E-mail: delamotte@lptl.jussieu.fr
†E-mail: leonie@theory.ph.man.ac.uk
c© B.Delamotte, L.Canet 163
B.Delamotte, L.Canet
scales contribute and render the perturbative approach problematic. The origin of
the problem is well known and is twofold. First, at each order of the perturba-
tion expansion (of, say, a correlation function) all fluctuations of all wavelengths
(and/or frequencies) contributing to this order are summed over (Feynman graphs).
Since they all contribute algebraically, the resulting sums (integrals) diverge at very
short (ultraviolet) or very long (infrared) wavelengths. Getting rid of these ultra-
violet divergences is the subject of perturbative renormalization and is now well
under control [1–3]. Second, even after renormalization, the perturbation series are
in general non-convergent and, as such, difficult to use to obtain reliable quanti-
tative results. This is, for instance, the case for the perturbation expansion of the
β-functions of the RG flow and of the critical exponents. This is all the more se-
vere that the dimensions of interest are far below the critical dimension since, then,
the field theories are in their strong coupling regime. Fortunately, in some cases,
the perturbation series turn out to be Borel summable so that efficient resummati-
on techniques (Padé-Borel, conformal mappings, etc) allow us to compute physical
quantities reliably and with high accuracy. This is the case for the O(N) models in
three dimensions for which five (ε-expansion) and six (fixed dimension expansion)
loops have been computed and where all available methods lead to consistent and
accurate results for critical exponents [4]. Some of the most celebrated equilibrium
systems for which resummation techniques also work well are ferromagnetic systems
with cubic anisotropy or with quenched disorder [5–7]. However, and contrary to
the common belief, this ideal picture turns out to be the exception rather than the
rule. In most cases, the situation is more complicated for at least two reasons. First,
in many cases, only the two first orders of perturbation series are known and are
insufficient to perform any resummation. This is the generic case in out of equilib-
rium statistical systems and for equilibrium ones involving fermions or gauge fields
(as, for instance, the electromagnetic field). Second, even in cases where five or six
loop series are known, resummations do not always lead to converged results. This is
the case, for instance, for frustrated spin systems in three dimensions [8–13]. Again,
contrary to the common belief, the problem is not just to compute accurately critical
exponents but to determine the qualitative behavior of the system at the transition:
either first or second order phase transition, belonging or not to a given universality
class, etc.
Of course, apart from these difficulties, more serious ones – not only related to
strong coupling behaviors – can exist and invalidate perturbation expansions. Many
examples exist in different physical contexts. The low temperature expansion of the
ferromagnetic XY (O(2)) model in two dimensions is identical at all orders to that
of a free theory. This expansion is invalidated at finite temperature by the existence
and liberation of vortices (Kosterlitz-Thouless transition) [3]. Quantum Chromody-
namics (QCD) is incapable, at any finite order of perturbation, of describing quark
confinement. The perturbation expansion of the β-function of the Kardar-Parisi-
Zhang (KPZ) equation describing the growth of interfaces is known at all orders (it
is given by the one-loop result) but is incapable of describing the scaling behaviors
in the rough phase above two dimensions [14,15]. The ε = 6 − d expansion of the
164
What can be learnt from the nonperturbative renormalization group?
random field O(N) model (RFO(N)M) is identical at all orders to the ε = 4 − d
expansion of the pure model (dimensional reduction) [16]. However, it has been rig-
orously proven that the RFIM in d = 3 and the pure Ising model in d = 1 display
different critical behaviors [17,18]. This implies that dimensional reduction should
break down, at least below a dimension between six and three.
In the two first examples, the problem is that the low energy excitations relevant
for the description of the physics at large scale are qualitatively different from those
at small scale: they are vortices for the XY model and bound states (actually confined
states) like hadrons for QCD. Since, on the one hand, these excitations dominate the
low energy physics and, on the other hand, they lead to non-analytic contributions
that are missed by perturbation expansions, the latter are incapable of describing
the large scale physics of these systems. The presence of non-analyticities is also
the reason for the breakdown of dimensional reduction in the RFO(N)M [19]. It is
not known, although suggested in the literature, whether these non-analyticities are
also related to the presence of a bound state [20,21], and neither it is for the KPZ
equation.
All these difficulties suggest that it is necessary to go beyond perturbation theory,
at least to be able to deal with strongly coupled systems without having to resort to
perturbation expansions at high orders and to resummations. Unfortunately, apart
from two dimensions or at large N , no systematic method for solving exactly a field
theory is known. One has to go back to the summation over the fluctuations in the
partition function and try to organize it differently. This is what Wilson did with
the momentum shell integration of rapid modes and the construction of effective
theories for slow modes [22]. His idea was to integrate out fluctuations scale by scale
and not order by order in a series expansion. Here, an important remark is of order.
If it were possible to perform exactly this integration between two different scales, it
would be possible to iterate this procedure until all fluctuations are integrated out.
This would amount to solving exactly the problem and, of course, this is impossible
in general. Actually, there are only very few examples where RG techniques have
enabled us to solve exactly a model that was not solved (in a simpler way) by another
method. In a sense, Wilson’s procedure is more subtle. Of course, the momentum
shell integration can be performed exactly in a formal way, that is by expressing the
RG flow of a quantity – e.g. an effective Hamiltonian – in terms of itself. This leads to
the formulation of an exact RG equation (that we derive in the following and which
is a functional differential equation) [22,23]. But this equation cannot be solved
in general. The important point in Wilson’s method is that the very idea behind
it suggests a natural way to approximate the integration over the rapid modes and
thus a natural truncation of the exact RG equation. Thus, RG techniques are almost
intrinsically linked with approximation methods: they are not very interesting when
exact solutions are known.
The aim of this article is to show how Wilson’s RG ideas can be concretely
implemented both at and out of equilibrium. In particular, we shall show how Wil-
son’s idea of a flow of effective Hamiltonians is conveniently replaced by that of a
flow of Gibbs free energies and we shall generalize it to non-equilibrium statistical
165
B.Delamotte, L.Canet
mechanics where the free energy no longer exists. One of the crucial parts of this
article will, of course, be devoted to the most developed truncations of this flow.
In particular, we shall show that they allow both to accurately determine universal
quantities, such as critical exponents even when the theories are strongly coupled,
and non-universal quantities, such as phase diagrams. This last point is a crucial
advantage of this method as compared with perturbation theory.
2. The non-perturbative renormalization group (NPRG)
formalism
2.1. The equilibrium case
Any implementation of Wilson’s momentum shell integration relies on a sepa-
ration between rapid and slow modes. This separation can be achieved through a
sharp cut-off in momentum space but this procedure leads to singularities when
approximations are performed. It is preferable to perform a smooth separation by
modifying the Boltzmann weights in such a way that the slow modes effectively
decouple from the model while the rapid ones are unaffected. This is conveniently
achieved by adding to the original partition function (for equilibrium systems) a
“momentum-dependent mass term” giving a large “mass” to slow modes and leavi-
ng unchanged the rapid ones. The slow modes are thus weakly correlated, they no
longer propagate and do not contribute to the long distance physics of the (modi-
fied) model. For a theory with one scalar field φ and Hamiltonian H [φ], we hence
define a scale dependent family of partition functions [24–29]:
Zk[B] =
∫
Dφ e−H[φ]−∆Hk[φ]+
∫
B.φ , (1)
where B is an external source, e.g. a magnetic field, k a momentum scale and ∆Hk[φ]
is the scale dependent mass term responsible for the decoupling of slow modes in
Zk:
∆Hk[φ] =
1
2
∫
ddx Rk(x− y)φ(x)φ(y) (2)
=
1
2
∫
ddq
(2π)d
Rk(q)φ(q)φ(−q). (3)
Rk is the cut-off function that should verify:
q2 � k2 : Rk(q
2) → 0 no mass for rapid modes, (4)
q2 � k2 : Rk(q
2) ∼ k2 large mass for slow modes, (5)
in order to achieve the decoupling. A typical and very simple function satisfying
these constraints is [30]
Rk(q) = (k2 − q2)θ(k2 − q2) (6)
which is a parabola with negative concavity on [0, k] and vanishes elsewhere. These
definitions yield that:
166
What can be learnt from the nonperturbative renormalization group?
1. when k = 0, Rk=0(q) = 0 identically and thus Zk=0 = Z, the original partition
function of the model under study. The latter is therefore recovered in the
limit k → 0, that is when all fluctuations have been integrated out;
2. when k = Λ ∼ inverse lattice spacing (or, more generally, the microscopic
scale), Rk(q) is very large for all modes so that all fluctuations are frozen.
Thus
logZk=Λ ∼ −H [MMF] − ∆Hk=Λ[MMF] +
∫
BMMF, (7)
where MMF is the configuration with the highest Boltzmann weight, that is
MMF is the mean field magnetization of the model with hamiltonian H+∆HΛ.
Then, if one defines the family of functionals Γk[M ] given by the Legendre trans-
form of the free energy Wk[B] = logZk[B] (up to the last term proportional to Rk)
[29]:
Γk[M ] + Wk[B] =
∫
BM − 1
2
∫
q
Rk(q
2)MqM−q (8)
with
M(x) =
δWk
δB(x)
, (9)
it follows that:
1. when k = 0, Γk=0 = Γ is the Gibbs free energy of the system since it be-
comes in this limit the standard Legendre transform of W (Rk=0 is identically
vanishing);
2. when k = Λ, Γk=Λ[M ] ∼ H [M ] from equations (7) and (8).
Thus, Γk[M ] smoothly interpolates between the microscopic Hamiltonian and the
Gibbs free energy when k is decreased from Λ to 0, that is, when more and more
flucuations are integrated out [29]. The interpretation of Γk and M is simple. When
B(x) is kept fixed, M as defined by equation (9) is k-dependent. It is the precursor
at scale k of the physical magnetization and corresponds to the magnetization of
a system composed of blocks of spins of size k−1, that is of a system where only
fluctuations within the blocks of size k−1 have been summed over. Γk[M ] is the
Gibbs free energy for the configuration with given magnetization M of this system.
Thus, contrary to Wilson’s effective Hamiltonian at scale k, which is the Hamiltonian
for the slow modes that have not yet been integrated out in Z, Γk[M ] is the free
energy for the rapid modes that have already been integrated out. It enables us, in
principle, to compute correlation functions of the rapid modes, contrary to Wilson’s
effective Hamiltonians. Γk[M ] is called the effective average action (Γk=0[M ] is called
the effective action in the context of quantum field theory; it is the generating
functional of the 1-Particle-Irreducible (1PI) correlation functions).
The exact evolution equation of Γk[M ] with k is obtained by differentiating equa-
tion (1) with respect to k and by eliminating Wk for Γk[M ] thanks to equation (8).
167
B.Delamotte, L.Canet
It writes [29]:
∂sΓk =
1
2
∫
q
Ṙk(q
2)(Γ
(2)
k (q,−q) +Rk(q
2))−1, (10)
where s = log(k/Λ), Ṙk = ∂sRk ,
Γ
(n)
k (q1, . . . , qn,M) =
δnΓk[M ]
δMq1
. . . δMqn
(11)
and (Γ
(2)
k (q,−q)+Rk(q
2))−1 is the inverse, in the operator sense, of Γ
(2)
k (q,−q,M)+
Rk(q
2). Equation (10) is exact and, as such, contains all perturbative and non-
perturbative physics. It is a functional (since Γ
(2)
k depends on M), partial differential
equation that requires truncations to be solved. The very idea of incomplete inte-
gration and effective theories is to preserve as much as possible the long distance
physics, approximating the short distance one. Thus, it is natural to expand Γk[M ]
in powers of derivatives of M . This is the so-called derivative expansion. In practice,
this amounts to proposing an ansatz for Γk[M ] which is polynomial in the deriva-
tives of the order parameter. In the case of a one component scalar field theory with
Z2 symmetry, this consists in considering for instance [25,29]:
Γk[M ] =
∫
ddx
{
Uk(ρ) +
1
2
Zk(ρ)(∇M)2 +O(∇4)
}
, (12)
where ρ = M2/2, Uk and Zk (that should not be confused with the partition function
Zk) are functions of M2 and k. For uniform magnetization, Γk[M ] reduces to Uk (up
to a volume factor) which is called the effective potential. The term proportional to
Zk(ρ) is the first correction to Uk taking into account the momentum dependence of
the correlation functions Γ
(n)
k (q1, . . . , qn). Zk(ρ) enables us to compute the anomalous
dimension of the field M . Of course, the ansatz (12) cannot be an exact solution of
Γk[M ] and thus Uk(ρ) and Zk(ρ) must be defined in such a way that their flows do
not get any contribution from the O(∇2n) terms with n > 1 that are supposed to
be neglected in the ansatz. It is convenient to define them as:
Uk(ρ) =
Γ(ρ)
V
, (13)
Zk(ρ) =
1
V
d
dq2
Γ
(2)
k (q, − q)(ρ)|
q2=0
, (14)
where V is the volume of the system and ρ is a uniform, that is x-independent, field
configuration. The insertion of these definitions in the RG flow equation (10) leads
to the RG equation for Uk(ρ) and Zk(ρ). These equations are partial differential
equations. It is interesting to notice that it is possible to further truncate Γk[M ]
while preserving qualitatively and semi-quantitatively the properties of the method.
This additional truncation consists in field expanding Zk(ρ) and/or the potential
Uk(ρ), that is [25,29]:
168
What can be learnt from the nonperturbative renormalization group?
1. Uk(ρ) is truncated by keeping only a polynomial in ρ of low degree. It is
convenient (and more accurate) in this case to expand Uk(ρ) around its running
minimum κk. The simplest non-trivial field truncation is: Uk(ρ) = λk/2(ρ −
κk)
2 +O((ρ− κk)
3).
2. Zk(ρ) is truncated by keeping only a field-independent coefficient: Zk(ρ) →
Zk ≡ Zk(ρ = κk). This is the simplest truncation that allows us to compute
a non-trivial anomalous dimension. Let us notice that it has for long been
believed that the anomalous dimension was essentially arbitrary within the
derivative expansion and that it was therefore useless to include a Zk(ρ) 6= 1.
Taking Zk(ρ) = 1 – and thus η = 0 – is called the Local Potential Approxi-
mation (LPA) and was first used (in a slightly different way) by Wegner and
Houghton back in the seventies [31].
If we combine truncations 1 and 2, the RG flow becomes a set of three simple
coupled differential equations for κk, λk and Zk [25]. A very important point with
this truncation when it is implemented on the O(N) models is that it allows us to
recover:
i) the one-loop result obtained perturbatively in d = 4 − ε,
ii) the N → ∞ result in any dimension and, for N > 3,
iii) the one-loop result obtained in d = 2 + ε from the non-linear sigma model
[12,29].
This is very different from the perturbative results that are very difficult to extrap-
olate beyond their domain of applicability. Let us also notice that for N = 2 and
d = 2, this truncation enables us to retrieve most of the qualitative as well as some
quantitative features of the Kosterlitz-Thouless transition [32].
There are two origins for the errors made with an ansatz such as equation (12)
supplemented or not by a field expansion of Uk(ρ) and/or Zk(ρ). First, the exact RG
equations for Uk and Zk involve the neglected terms of the derivative expansion of
order ∇4,∇6, . . .Moreover, if a field expansion of Uk(ρ) is performed as in truncation
1) above, the RG equations of κk and λk involve the coupling constants of the next
terms of the expansion, that is the coefficients of (ρ− κk)
n with n > 2. Second, the
choice of a cut-off function Rk that, in principle, has no effect since Rk(q
2) vanishes
identically in the limit k → 0, does matter once truncations are performed. Many
studies have been devoted to finding an optimal choice [33–37]. None of them gives
a complete solution to this problem.
Let us now work out the RG equations obtained with a standard truncation as
(12) for the effective potential and anomalous dimension. Let us first recall that
since at a second order phase transition the correlation length is infinite, the system
becomes scale invariant at the transition point (at sufficiently large distance com-
pared with the lattice spacing). Thus, in units of the (inverse) running block spin size
k, it becomes invariant under the RG transformations (for k � Λ). It is therefore
convenient to work with dimensionless quantities: in terms of them the transition
169
B.Delamotte, L.Canet
corresponds to a fixed point of the RG flow. Actually, to find out a fixed point, it is
also necessary to work with a renormalized magnetization. Indeed, at the transition
point, the magnitude of the running (spontaneous) magnetization M never stops
flowing towards 0 as k → 0, which a priori prevents us from finding a fixed point. It
is thus necessary to work with a (dimensionless) renormalized magnetization whose
magnitude tends to a finite value as k → 0. We thus define
ρ̃ = Zkk
2−dρ, (15)
uk(ρ̃) = k−dUk(ρ). (16)
With truncation 2) above and with the regulator of equation (6), the RG equations
for the potential uk(ρ̃) and for Zk become very simple:
∂suk(ρ̃) = −duk(ρ̃) + (d− 2 + ηk)ρ̃u
′
k(ρ̃)
+
4vd
d
(
1 − ηk
d+ 2
)
1
1 + u′k(ρ̃) + 2ρ̃u′′k(ρ̃)
, (17)
ηk = −∂s logZk =
16vd
d
λkκk
1 + 2λkκk
, (18)
where vd = 1/(2d+1πd/2Γ(d/2)), primes denote derivatives with respect to ρ̃, κk is
the minimum of uk(ρ̃) and λk = u′′k(κk). ηk is the running anomalous dimension
which goes to a fixed point value as k → 0 – the physical anomalous dimension – if
the potential itself reaches a fixed point solution.
This theoretical framework – or those deriving from more sophisticated trunca-
tions – have turned out to be very powerful. In what follows we show some results
that have been obtained for the Ising and O(N) models by ourselves and by other
groups. However, before this, let us show how to adapt this formalism for out of
equilibrium statistical systems.
2.2. The out of equilibrium case
For out of equilibrium systems, the very notion of free energy no longer exists in
general. However, it is still possible to define generating functionals of correlation
functions, analogous to the partition function Z. Moreover, even if it is no longer
possible to speak of Boltzmann weights and Hamiltonians, the generating functionals
Z are still often given by functional integrals of weights that, formally, enable us
to define “Hamiltonians”, called actions in this context. The Legendre transform
Γ of logZ is then analogous to the Gibbs free energy and is the generator of the
1PI correlation functions. There are two main means to build field theories in out of
equilibrium statistical mechanics: either from a Langevin formulation of the problem
or from a more microscopic approach – if it exists – à la Doi-Peliti [38,39], that is
from a master equation.
Even if approaches at and out of equilibrium are close together, some physical
differences are crucial, such as the irreversibility of the dynamics, the violation of the
fluctuation-dissipation theorem, etc. Technically, these differences have a translation
in the formalism:
170
What can be learnt from the nonperturbative renormalization group?
i) the field theories are written not only in terms of the “physical” field(s) but
also in terms of the response field(s),
ii) temporal fluctuations are taken into account by a single time derivative (con-
trary to the Laplacian for spatial fluctuations),
iii) the actions are not “hermitic”, that is, the physical and response fields do not
necessarily play symmetric roles [40].
However, all these differences do not prevent us from constructing an analog of the
effective average action method for out of equilibrium systems [41,42]. The non-
trivial point is to choose a Rk function. Should we take it space and time dependent
acting only on the physical field or on both the physical and response fields? Since
the free field theory corresponds to the following theory:
Z =
∫
DφDφ̄e−S0 (19)
with
S0 =
∫
ddxdt φ̄(x, t)
(
∂t −D∇2
)
φ(x, t) (20)
it is natural to define the Rk term by:
∆Sk =
∫
ddxddydtdt′ φ̄(x, t)Rk(x− y, t− t′)φ(y, t′). (21)
We have also chosen to take it time-independent so that the time integral in the RG
equation can be trivially performed [41]. It is not known whether this corresponds
to a “good” choice from the accuracy point of view. Once these choices have been
made, it is possible for a non-trivial model with action S to define:
Zk[J, J̄ ] =
∫
DφDφ̄ e−S[φ,φ̄]−∆Sk[φ,φ̄]+
∫
J.φ+
∫
J̄.φ̄. (22)
The effective average action is defined as in equation (8) by:
Γk[ψ, ψ̄] + Wk[J, J̄ ] =
∫
Jψ +
∫
J̄ ψ̄ −
∫
q
Rk(q
2)ψq,ωψ̄−q,−ω (23)
with
ψ(x, t) =
δWk
δJ(x, t)
and ψ̄(x, t) =
δWk
δJ̄(x, t)
. (24)
It is convenient to define both the 2×2 matrix Γ̂
(2)
k [ψ, ψ̄] of second functional deriva-
tives of Γk[ψ, ψ̄] with respect to ψ and ψ̄ and the 2×2 matrix R̂k, the off-diagonal
elements of which are Rk(q
2). Then, the out of equilibrium NPRG equation writes
[41]:
∂sΓk =
1
2
Tr
∫
q,ω
∂sR̂k
(
Γ̂
(2)
k (q, ω;−q,−ω) + R̂k(q
2)
)−1
. (25)
171
B.Delamotte, L.Canet
A typical ansatz for a reaction-diffusion problem is:
Γk[ψ, ψ̄] =
∫
ddxdt
{
Uk(ψ, ψ̄) + Zkψ̄
(
∂t −Dk∇2
)
ψ
}
. (26)
Within this formalism some problems of branching and annihilating random walks
(BARW) have been addressed [42] that we briefly sketch further on.
3. Some results obtained with the NPRG method
Among others, a crucial problem as for the NPRG method combined with the
derivative expansion, is to estimate the quantitative accuracy of the results obtained.
More precisely, it would be highly desirable to be able to estimate error bars. As
usual, this is a very difficult task when no analytic result is known on the behavior of
the series encountered in the method. As already mentioned, this question is closely
related to the choice of the cut-off function Rk.
Table 1. Critical exponents of the three dimensional Ising model. ∂0, ∂2 and ∂4
correspond to the order of the truncation of the derivative expansion (NPRG
method) [36]. For completeness, we have recalled in the last line, the results
obtained perturbatively [4].
order ν η
∂0 0.6506 0
∂2 0.6281 0.044
∂4 0.632 0.033
7-loops 0.6304(13) 0.0335(25)
We have studied the critical exponents of the Ising model in three dimensions as
a testing ground of the convergence of the derivative expansion at order ∇0 (LPA),
∇2 and ∇4 [36]. The rule of thumb adopted consists in evaluating the error bar
through the evolution of the values of the exponents with the order of the truncation.
In the different cases studied, we have found that the distance between the best
known results and ours decreases when increasing the order of the truncation of the
derivative expansion. Thus, it seems reasonable to believe that the error made at a
given order of the derivative expansion is given by the distance between the results of
the last two orders. More precisely, at each order of the truncation, we have studied
the dependence of the critical exponents on the choice of function Rk. Each time,
we have tried to find the best Rk by requiring the exponents to depend as weakly as
possible on the choice of Rk (since they verify this property in the exact theory). We
have therefore considered a one-parameter family of functions Rk and have computed
the critical exponents for all elements of this family. The function Rk that is selected
as optimal is that for which the exponents are stationary with respect to a change
of Rk (this is an implementation of the Principle of Minimal Sensitivity) [35]. At
172
What can be learnt from the nonperturbative renormalization group?
each order of the derivative expansion, this leads to a set of optimal exponents. At
order ∇4, these optimal exponents are quite close to the best known values, see
table 1. Thus, although only three orders of the derivative expansion are known, it
seems that, at least in the three dimensional Ising case, the derivative expansion
converges rapidly. It is important to notice that no resummation of any kind has
been necessary in order to obtain these results.
As a conclusion, one can see in table 1 that the results obtained from the NPRG
method are in a fairly good quantitative agreement with the best known results for
the three dimensional Ising model. This is, of course, very encouraging as for the
reliability of the method and the convergence of the derivative expansion. Moreover,
let us emphasize that the Ising model belongs to the very small class of models, the
perturbation expansion of which involves only one coupling constant and is known
to be Borel summable [4]. In many cases, things are much more complicated. A
remarkable example is given by the three dimensional frustrated spin systems, either
the triangular antiferromagnets or the helimagnets. Here, the results obtained by
means of the resummed perturbation series and those obtained from the NPRG are
qualitatively different [8–13]. The former predicts a second order phase transition
whereas the latter very weakly predicts first order transitions. The debate is still
open and it will be interesting to see which prediction is correct.
Let us now turn to out of equilibrium systems and more precisely to branching
and annihilating random walks. The problem is the following. On a d-dimensional
lattice, particles are diffusing (with rate D) and can create offsprings through the
reaction A → A + A, with rate σ. When they meet on the same site, they can be
annihilated through A+A → ∅ with rate λ. The problem is to determine whether the
system undergoes a continuous phase transition between an active and an inactive
(absorbing) phase and to which universality class it belongs. The field theory is
obtained from the master equation and the action reads [43]:
S =
∫
ddx dt
{
φ̄
(
∂t −D∇2
)
φ +
√
2σλ(φ̄φ2 − φ̄2φ) + λφ̄2φ2
}
. (27)
First of all, a mean field study predicts that the system is always in the active
phase for σ 6= 0. From a perturbative analysis performed at and below d = 2, Cardy
and Täuber have obtained the phase diagram for small λ/D and σ/D, showing
unambigously that fluctuations are capable of destabilizing the active phase so that
an absorbing phase exists at finite σ/D for d 6 2 [43]. As for the universality
class, this phase transition belongs to the Directed Percolation one. However, their
calculations also suggest that for all dimensions larger than two, the fluctuations are
not strong enough to destabilize the mean field result and that, therefore, the system
is always in the active phase. Using the NPRG method, we have re-examined the
physics of this system. We have been able to compute the critical exponents in all
dimensions and have shown that the phase diagram obtained perturbatively is wrong
[42]. More precisely, above dimension two, the system is indeed always in the active
phase at small σ/D and λ/D as predicted by the perturbative analysis. But there is
a threshold value of λ/D above which an absorbing phase can exist. By numerically
173
B.Delamotte, L.Canet
integrating the RG flow from dimensions 3 to 6 we have determined these threshold
values as well as the complete phase diagram [42]. In figure 1 there are displayed
both the NPRG results and those obtained from numerical simulations. Only one
free parameter has been used to match the numerical and analytical results. This
parameter takes into account the fact that both approaches differ at the microscopic
scale Λ since one is defined on the lattice and the other in the continuum. However,
it is known that both models should match after some RG iterations since, at a
sufficiently large scale, the lattice model becomes homogeneous and isotropic and
therefore can be described by a continuum theory. Thus, after a proper matching
of the ultraviolet scales Λ of both models – requiring one rescaling parameter –
the results can be compared quantitatively. The agreement is excellent and proves
that, also in this example, the NPRG method is reliable. Let us emphasize that the
phase diagram is a non-universal feature of the model. Computing it requires to
keep track all along the RG flow of both relevant and irrelevant couplings since they
all contribute to the non-universal behavior of the model. While this is in general a
formidable task in perturbation theory, that anyway is useless if these couplings are
large, it just consists in the NPRG approach in integrating the flow equations for
the potential with the bare potential of equation (27) as initial condition at scale Λ
[42,44,45].
Figure 1. Phase diagrams of BARW A
σ−→ 2A, 2A
λ−→ ∅ in dimensions 1 to 6.
Lines present NPRG results, rescaled as explained in the text. Symbols follow
from numerical simulations. For each dimension, the active phase lies on the left
of the transition line, the absorbing phase on the right.
4. Conclusion and some open problems
Let us now come to a conclusion. The NPRG method seems to be a promising
way to tackle the strong coupling regime of (at least some) field theories while
avoiding problematic resummation methods used in perturbative renormalization. It
also enables us to compute non-universal quantities such as phase diagrams because
174
What can be learnt from the nonperturbative renormalization group?
it can handle the functional nature of the effective potential with both its relevant
and irrelevant couplings, that are most of the time infinitely numerous [42,45]. For
the reaction-diffusion problem we have discussed in this article, the NPRG has
allowed us to find that a phase transition exists in all dimensions while perturbative
renormalization failed, probably even in a all-order analysis. This example (and it is
not unique, see for instance [12,42,46–49]) already shows that this method leads to
highly non-trivial results. Now, an interesting question is: can we classify problems
for which perturbative theory could fail whereas this method could still be relevant
and, reciprocally, can we list its weak points? Let us try.
We have already mentioned the problems encountered in the three dimensional
frustrated systems for which the NPRG and the six-loop perturbative computations
lead to different results [12]. It is interesting to understand why, since this problem
probably belongs to a whole class of problems where perturbative theory is prob-
lematic and where the NPRG method could be fruitful (see [12]). The problem is
the following. The Ginzburg-Landau-Wilson field theory (φ4-like) relevant for the
description of the critical physics of a system is determined by three properties of
the system:
i) the symmetry breaking scheme, G → H , between the high (G) and the low
(H) temperature phases,
ii) the number N of components of the order parameter,
iii) the space dimension d.
The number of coupling constants in the Hamiltonian depends on the symmetry
group G, and the lower this symmetry, the larger the number of invariant couplings
in the Hamiltonian. For all systems with a symmetry lower than O(N), at least
two coupling constants are present in the Hamiltonian. In these cases, the number
of fixed points is larger than one, contrary to the O(N) case where there is only
one. When N and/or d are varied, the fixed points move in the coupling constant
space and it is generically observed that, at fixed d, there exists a critical value Nc
of N where two of them collapse. Two possible situations can then occur: either
both fixed points disappear when N is further varied (they become complex) or
they exchange their stability. These two situations are, for instance, encountered in
frustrated systems and in ferromagnetic systems with cubic anisotropy. The difficulty
is that as d is varied, Nc changes and, of course, the problem is to compute Nc(d)
since it is crucial to determine if the fixed points exist in d = 3 for N = 1, 2, 3
and, if they do, which one is stable. This is a highly non-trivial task in perturbation
theory that requires, in the known cases, resummations of the series obtained at
least at four loops. Clearly, if the NPRG method leads to quantitatively reliable
results in these cases, this would avoid tedious perturbative calculations, that are
anyway useless if more than six loop results are necessary to obtain a converged
value of Nc(d = 3). These problems belong to the whole class of “strong coupling
problems” of field theory.
175
B.Delamotte, L.Canet
We have already mentioned another class of problems where NPRG could be use-
ful: those where perturbation theory fails at all orders to describe the critical physics.
An important example is given by the RFO(N)M where dimensional reduction has
been proven to be wrong [17,18]. Precisely on this example, the NPRG approach has
been used very recently [19]. The origin of the problem is that the effective potential
uk acquires non-analytic contributions at finite k that are missed by perturbation
theory. These non-analyticities have been successfully computed from the NPRG
method since the functional nature of the effective potential can be dealt with. It
will be very interesting to see if this method is capable of obtaining qualitatively
and quantitatively correct results for the Ising case in three dimensions. Anyhow,
the present results in this model already open the way to a renormalization group
which is both functional and non-perturbative.
A third instance where NPRG could turn out to be useful is the study of phase
transitions where topological defects play a role (especially in low dimensions). Let
us recall that almost nothing is known in the physics of phase transitions induced by
vortices in two dimensions apart from the Kosterlitz-Thouless transition for which
the decoupling between spin waves and vortices allows a perturbative study of the
transition. It could seem strange at first glance that, in situations where non-trivial
spatial configurations of the field are crucial, anything relevant can be learnt from
a derivative expansion that keeps only the lowest order(s) in the momentum de-
pendence of correlation functions. Let us however make two general remarks. First,
the Kosterlitz-Thouless transition has been studied by the NPRG method without
introducing the vortices by hand à la Villain. Almost all the physics is quantita-
tively recovered apart from the fact that the line of fixed points is replaced by a
line of quasi-fixed points where the β-function of the temperature is very small al-
though not strictly vanishing [48]. Second, spatially non-trivial field configurations
can play a great role in the summation over fluctuations, that is for the computa-
tion of the functional integral, while correlation functions, that are the results of
this summation, can be very smooth functions of the momenta. Thus, it is possible
that a derivative expansion succeeds in capturing the physics of topological defects.
This would be extremely interesting since in many cases other than the O(2) case,
no Villain trick is available to write down a field theory that can be studied pertur-
batively.
Let us now review some of the drawbacks of the NPRG method that would
clearly require much effort to be overcome. First, the derivative expansion forbids
us to compute the momentum dependence of correlation functions. It just focuses
on the flow of coupling constants with emphasis on those of the effective potential.
It is therefore insufficient to compute cross sections in particle physics or structure
factors. Second, it is not easy to point out the “small parameter” that underlies the
validity of the derivative expansion. We can postulate that the larger the anomalous
dimension, the worse its convergence. However, this important assumption has yet
not been supported by a systematic theoretical analysis. Third, the very way the
cut-off function Rk(q
2) is introduced can be problematic. For instance, it breaks
gauge symmetry and it is clearly one of the big challenges of the NPRG to allow
176
What can be learnt from the nonperturbative renormalization group?
gauge invariant computation [50]. Fourth, the choice of an ansatz for Γk relies on
the choice of its field content. In statistical mechanics, contrary to particle physics,
the microscopic degrees of freedom, at scale Λ, are known in general and the dif-
ficulty is to propose an ansatz in terms of the low energy degrees of freedom. As
already emphasized, they can qualitatively differ from those at high energy – e.g.
they are bound states – and it is non-trivial to know whether the RG flow of, say,
a potential will signal this change. The problem is that, although the physics can
still be described in terms of the microscopic degrees of freedom, it can then turn
out to be very complicated. For instance, while the physics is “local” in terms of
the bound state field, it can become non-local in terms of the microscopic degrees of
freedom. Thus, a derivative expansion formulated in terms of the bound state field
can be accurate whereas it is not in terms of the original fields [51]. The choice of
low energy degrees of freedom is therefore crucial for devicing an ansatz for Γk and
it is not clear so far if there exists a systematic way for, at least, detecting a bad
choice.
NPRG is still in its infancy. For sure, some of the above mentioned problems will
be solved in the future. Some others will remain out of reach. As usual, numerous
physical problems will challenge us and will force us to better understand the NPRG
method as well as its intrinsic limits.
Acknowledgements
We thank Yu.Holovatch who gave us the opportunity to contribute to the Fes-
chrift dedicated to the 60th birthday of R.Folk. We are grateful to D.Mouhanna,
M.Tissier, J.Vidal and N.Wschebor with whom parts of the results that have been
shown in this article have been obtained. The numerous discussions we have had with
them have been crucial for our understanding of the NPRG. The LPTL is unité mi-
xte du CNRS UMR 7600 and L.C. acknowledges financial support by the European
Community’s Human Potential Programme under contract HPRN–CT–2002–00307,
DYGLAGEMEM.
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178
What can be learnt from the nonperturbative renormalization group?
Чого можна навчитися із непертурбативної
ренормалізаційної групи?
Б.Делямот 1 , Л.Кане 2
1 Лабораторія теоретичної фізики рідин,
Університет П’єра і Марі Кюрі,
Париж VI, 75252 Париж Седекс 05, Франція
2 Школа фізики і астрономії,
Університет Манчестер,
М13 9PL, Великобританія
Отримано 2 грудня 2004 р.
Ми звертаємо увагу на певні обмеження пертурбативної ренормалі-
заційної групи, що використовується в рівноважній та нерівноважній
статистичній механіці. Ми стверджуємо, що формалізм неперт-
урбативної ренормалізаційної групи є добрим кандидатом для того,
щоб обійти деякі з них. Ми приводимо деякі недавно отримані в
літературі результати, що підтверджують наші думки. В кінці ми
перелічуємо деякі відкриті питання, для яких може бути корисний
цей формалізм, а також робимо огляд його недоліків.
Ключові слова: ренормалізаційна група, непертурбативна теорія
поля, динамічні критичні явища
PACS: 64.60.Ak, 11.10.Hi, 05.10.Cc
179
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