Renormalization group domains of the scalar Hamiltonian
Using the local potential approximation of the exact renormalization group (RG) equation, we show various domains of values of the parameters of the O(1) -symmetric scalar Hamiltonian. In three dimensions, in addition to the usual critical surface Sc (attraction domain of the Wilson-Fisher fixed...
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Bagnuls, C. Bervillier, C. 2017-06-13T12:22:06Z 2017-06-13T12:22:06Z 2000 Renormalization group domains of the scalar Hamiltonian / C. Bagnuls, C. Bervillier // Condensed Matter Physics. — 2000. — Т. 3, № 3(23). — С. 559-575. — Бібліогр.: 28 назв. — англ. 1607-324X DOI:10.5488/CMP.3.3.559 PACS: 05.10.Cc, 05.70.Jk, 11.10.Hi, 61.20.Qg https://nasplib.isofts.kiev.ua/handle/123456789/120996 Using the local potential approximation of the exact renormalization group (RG) equation, we show various domains of values of the parameters of the O(1) -symmetric scalar Hamiltonian. In three dimensions, in addition to the usual critical surface Sc (attraction domain of the Wilson-Fisher fixed point), we explicitly show the existence of a first-order phase transition domain Sf separated from Sc by the tricritical surface St (attraction domain of the Gaussian fixed point). Sf and Sc are two distinct domains of repulsion for the Gaussian fixed point, but Sf is not the basin of attraction of a fixed point. Sf is characterized by an endless renormalized trajectory lying entirely in the domain of negative values of the ϕ⁴ -coupling. This renormalized trajectory also exists in four dimensions making the Gaussian fixed point ultra-violet stable (and the ϕ⁴₄ renormalized field theory asymptotically free but with a wrong sign of the perfect action). We also show that a very retarded classical-to-Ising crossover may exist in three dimensions (in fact below four dimensions). This could be an explanation of the unexpected classical critical behaviour observed in some ionic systems. Використовуючи наближення локального потенціалу точного рівняння ренормалізаційної групи (РГ), ми показуємо різні області значень параметрів O(1) симетричного скалярного гамільтоніану. У трьох вимірах додатково до звичайної критичної поверхні Sc (область притягання фіксованої точки Вільсона-Фішера), ми явно показуємо існування області фазового переходу першого ряду Sf , відокремленої від Sc трикритичною поверхнею Sf (область притягання гаусової фіксованої точки). Sf і Sc є дві різні області відштовхування для гаусової фіксованої точки, а Sf не є в ділянці притягання фіксованої точки. Sf характеризується нескінченою ренормалізованою траєкторією, яка повністю лежить в області негативних значень констант взаємодії ϕ⁴ . Ця ренормалізована траєкторія також існує в чотирьох вимірах, роблячи гаусову фіксовану точку в ультрафіолетовій області стабільною (і ренормалізовану теорію поля ϕ⁴ асимптотично вільною, але з неправильним знаком ідеальної дії). Ми також показуємо, що дуже запізнений кросовер від класичної до ізінгівської поведінки може існувати у трьох вимірах (фактично нижче чотирьох вимірів). Це може бути поясненням для неочікуваної класичної критичної поведінки, яка спостерігається в деяких іонних системах. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Renormalization group domains of the scalar Hamiltonian Області ренормалізаційної групи скалярного гамільтоніану Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Renormalization group domains of the scalar Hamiltonian |
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Renormalization group domains of the scalar Hamiltonian Bagnuls, C. Bervillier, C. |
| title_short |
Renormalization group domains of the scalar Hamiltonian |
| title_full |
Renormalization group domains of the scalar Hamiltonian |
| title_fullStr |
Renormalization group domains of the scalar Hamiltonian |
| title_full_unstemmed |
Renormalization group domains of the scalar Hamiltonian |
| title_sort |
renormalization group domains of the scalar hamiltonian |
| author |
Bagnuls, C. Bervillier, C. |
| author_facet |
Bagnuls, C. Bervillier, C. |
| publishDate |
2000 |
| language |
English |
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Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Області ренормалізаційної групи скалярного гамільтоніану |
| description |
Using the local potential approximation of the exact renormalization group
(RG) equation, we show various domains of values of the parameters of
the O(1) -symmetric scalar Hamiltonian. In three dimensions, in addition to
the usual critical surface Sc (attraction domain of the Wilson-Fisher fixed
point), we explicitly show the existence of a first-order phase transition domain Sf separated from Sc by the tricritical surface St (attraction domain
of the Gaussian fixed point). Sf and Sc are two distinct domains of repulsion for the Gaussian fixed point, but Sf is not the basin of attraction
of a fixed point. Sf is characterized by an endless renormalized trajectory
lying entirely in the domain of negative values of the ϕ⁴ -coupling. This
renormalized trajectory also exists in four dimensions making the Gaussian fixed point ultra-violet stable (and the ϕ⁴₄ renormalized field theory
asymptotically free but with a wrong sign of the perfect action). We also
show that a very retarded classical-to-Ising crossover may exist in three
dimensions (in fact below four dimensions). This could be an explanation
of the unexpected classical critical behaviour observed in some ionic systems.
Використовуючи наближення локального потенціалу точного рівняння ренормалізаційної групи (РГ), ми показуємо різні області значень
параметрів O(1) симетричного скалярного гамільтоніану. У трьох вимірах додатково до звичайної критичної поверхні Sc (область притягання фіксованої точки Вільсона-Фішера), ми явно показуємо існування області фазового переходу першого ряду Sf , відокремленої від Sc трикритичною поверхнею Sf (область притягання гаусової фіксованої точки). Sf і Sc є дві різні області відштовхування для
гаусової фіксованої точки, а Sf не є в ділянці притягання фіксованої
точки. Sf характеризується нескінченою ренормалізованою траєкторією, яка повністю лежить в області негативних значень констант
взаємодії ϕ⁴ . Ця ренормалізована траєкторія також існує в чотирьох
вимірах, роблячи гаусову фіксовану точку в ультрафіолетовій області
стабільною (і ренормалізовану теорію поля ϕ⁴ асимптотично вільною, але з неправильним знаком ідеальної дії). Ми також показуємо,
що дуже запізнений кросовер від класичної до ізінгівської поведінки може існувати у трьох вимірах (фактично нижче чотирьох вимірів).
Це може бути поясненням для неочікуваної класичної критичної поведінки, яка спостерігається в деяких іонних системах.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120996 |
| citation_txt |
Renormalization group domains of the scalar Hamiltonian / C. Bagnuls, C. Bervillier // Condensed Matter Physics. — 2000. — Т. 3, № 3(23). — С. 559-575. — Бібліогр.: 28 назв. — англ. |
| work_keys_str_mv |
AT bagnulsc renormalizationgroupdomainsofthescalarhamiltonian AT bervillierc renormalizationgroupdomainsofthescalarhamiltonian AT bagnulsc oblastírenormalízacíinoígrupiskalârnogogamílʹtoníanu AT bervillierc oblastírenormalízacíinoígrupiskalârnogogamílʹtoníanu |
| first_indexed |
2025-11-24T04:21:34Z |
| last_indexed |
2025-11-24T04:21:34Z |
| _version_ |
1850841433794150400 |
| fulltext |
Condensed Matter Physics, 2000, Vol. 3, No. 3(23), pp. 559–575
Renormalization group domains of the
scalar Hamiltonian
C.Bagnuls 1 , C.Bervillier 2
1 Service de Physique de l’Etat Condensé,
CE Saclay, F91191 Gif-sur-Yvette Cedex, France
2 Service de Physique Théorique,
CE Saclay, F91191 Gif-sur-Yvette Cedex, France
Received March 9, 2000
Using the local potential approximation of the exact renormalization group
(RG) equation, we show various domains of values of the parameters of
the O(1) -symmetric scalar Hamiltonian. In three dimensions, in addition to
the usual critical surface Sc (attraction domain of the Wilson-Fisher fixed
point), we explicitly show the existence of a first-order phase transition do-
main Sf separated from Sc by the tricritical surface St (attraction domain
of the Gaussian fixed point). Sf and Sc are two distinct domains of re-
pulsion for the Gaussian fixed point, but Sf is not the basin of attraction
of a fixed point. Sf is characterized by an endless renormalized trajectory
lying entirely in the domain of negative values of the ϕ4 -coupling. This
renormalized trajectory also exists in four dimensions making the Gaus-
sian fixed point ultra-violet stable (and the ϕ4
4
renormalized field theory
asymptotically free but with a wrong sign of the perfect action). We also
show that a very retarded classical-to-Ising crossover may exist in three
dimensions (in fact below four dimensions). This could be an explanation
of the unexpected classical critical behaviour observed in some ionic sys-
tems.
Key words: renormalization group, critical, tricritical, first-order phase
transition, crossover
PACS: 05.10.Cc, 05.70.Jk, 11.10.Hi, 61.20.Qg
1. Introduction
The object of this paper1 is to carry on studying the local potential approxi-
mation of the exact renormalization group (RG) equation for the scalar theory [1].
In the previous publication [2] (to be considered as part I of the present work), we
had already considered this approximation with a view to qualitatively discuss the
connection between the standard perturbative renormalization of field theory (as it
1Based on a talk given at “RG 2000”, Taxco, Mexico, January 1999
c© C.Bagnuls, C.Bervillier 559
C.Bagnuls, C.Bervillier
can be found in most textbooks on field theory, see for example [3]) and the modern
view [4] in which the renormalized parameters of a field theory are introduced as
the “relevant” directions of a fixed point (FP) of a RG transform. Actually the local
potential approximation, which allows us to consider all the powers of the field ϕ
on the same footing, is an excellent textbook example of the way in which infinitely
many degrees of freedom are accounted for in (nonperturbative) RG theory. Almost
all the characteristics of the RG theory are involved in this approximation. The
only lacking features are related to phenomena highly correlated to the non-local
parts neglected in the approximation and, when the critical exponent η is small
(especially for d = 4 and d = 3), one expects the approximation to be qualitatively
correct regarding all aspects of the RG theory [1].
In the following we look at the domains of attraction or of repulsion of fixed
points in the O(1) scalar theory in three and four dimensions (d = 3 and d = 4).
At first sight, one could think that the issue considered is very simple since, with
regard to criticality, the O(1)-symmetric systems in three dimensions are known to
belong to the same class of universality (the Ising class). Now, because the Ising
class is associated to the domain of attraction of the unique (non-trivial) Wilson-
Fisher fixed point [5], then by adjusting one parameter (in order to reach the critical
temperature2) any O(1) scalar Hamiltonian should be driven to the Wilson-Fisher
fixed point under the action of renormalization. Consequently there would be only
two domains for the O(1) scalar theory: the critical subspace Sc (of codimension 1) in
the Wilson space (S) of infinite dimensions of the Hamiltonian parameters (in which
the RG transforms generate flows) and the complement to S of Sc (corresponding
to noncritical Hamiltonians).
In fact, this is not correct because there is another fixed point in S: the Gaussian
fixed point which, although trivial, controls tricritical behaviours in three dimen-
sions. Now each FP has its own basin of attraction in S [5]. The attraction domain
of the Gaussian FP is the tricritical subspace St of codimension 2 (with no intersec-
tion with Sc). In addition, we show that there is a second subspace of codimension
1 in S, called Sf , which is different from Sc, and thus which is not a domain of
attraction to the Wilson-Fisher fixed point. There is no FP to which a point of S f is
attracted. Sf is characterized by a negative sign of the ϕ4-Hamiltonian parameter u4
and is associated with systems undergoing a first-order phase transition [6]. We show
that an endless attractive RG trajectory is associated to this domain of first-order
transitions. It is a renormalized trajectory (denoted below by T ′′
1) that emanates
from the Gaussian fixed point. The frontier between Sf and Sc corresponds to the
tricritical subspace St which is the domain of attraction of the Gaussian fixed point
while Sf and Sc are two distinct domains of repulsion for the Gaussian fixed point.
Actually, the situation is in conformity with the usual view. Considering the
famous ϕ4-model [Landau-Ginzburg-Wilson (LGW) Hamiltonian] in which the as-
sociated coupling u4 is positive, the Hamiltonian at criticality is attracted exclusively
to the Wilson-Fisher fixed point, but if u4 is negative, a ϕ6-term is required for sta-
2We assume that the second relevant field, corresponding to the magnetic field for magnetic
systems, is set equal to zero.
560
Renormalization group domains
bility, but then one may get either a tricritical phase transition or a second- or a
first-order transition [7]. In the present study we do not truncate the Hamiltonian
which involves all the powers of the field ϕ.
We explicitly show that a system which would correspond to an initial point
lying very close to the frontier St in the critical side (in Sc) would display a retarded
classical-to-Ising crossover [8]. This result is interesting with regard to ionic systems
(for example) in which a classical behaviour has been observed while an Ising-like
critical behaviour was expected. The eventuality of a retarded crossover from the
classical to the Ising behaviour has previously been mentioned but without explain-
ing theoretically how this kind of crossover could develop [9]. In [8] a calculation
suggests that the RPM model for ionic systems would specifically correspond to a
scalar Hamiltonian with a negative sign for the ϕ4-Hamiltonian parameter (but the
order parameter chosen is not the bulk density [10]). This calculation has motivated
the present study.
We also indicate that the renormalized trajectory T
′′
1 still exists in four dimen-
sions. This makes the Gaussian fixed point ultraviolet stable and the scalar field
theory formally asymptotically free. However the associated “perfect” action [11]
would have the wrong sign to provide us with an acceptable (well defined) field
theory.
The paper is arranged as follows. In section 2 we briefly present the local potential
approximation of the exact RG equation to be studied. We introduce the strategy
we have chosen to solve the resulting nonlinear differential equation with a view
to show the trajectories of interest in the space S of infinite dimension. Because
the practical approach to the Gaussian fixed point is made difficult due to the
logarithmic slowness characteristic of a marginally irrelevant direction (for d = 3),
we found it useful to first test our numerical method with a close approach to the
Wilson-Fisher fixed point. We present the characteristic results of this approach and
various kinds of domains corresponding to u4 > 0 (a kind of a summary of [2])
In section 3 we describe various kinds of attraction or repulsion domains of the
Gaussian fixed point (for a negative value of the ϕ4-Hamiltonian parameter) corre-
sponding to tricritical, critical and first-order subspaces. Then we discuss the con-
sequences and especially explicitly show how a retarded crossover from the classical
to the Ising behaviour can be obtained.
We then shortly discuss the case d = 4 when u4 < 0.
In two appendices we report on some technical aspects of the numerical treatment
of the RG equation studied, in particular on the appearing of spurious nontrivial
tricritical fixed points (appendix A.1).
2. The RG equation studied
The local potential approximation was first considered by Nicoll et al. [12] from
the sharp cutoff version of the exact RG equation of Wegner and Houghton [13].
It was rederived by Tokar [14] by using approximate functional integrations and
rediscovered by Hasenfratz and Hasenfratz [15]. As in [2] we adopt the notation of
561
C.Bagnuls, C.Bervillier
the latter authors and consider the following nonlinear differential equation for the
simple function f(l, ϕ):
ḟ =
Kd
2
f ′′
1 + f ′
+
(
1−
d
2
)
ϕf ′ +
(
1 +
d
2
)
f (2.1)
in which a prime refers to a derivative with respect to the constant dimensionless
field ϕ (at constant l) and f(l, ϕ) = V ′(l, ϕ) is the derivative of the dimensionless
potential V (l, ϕ); ḟ stands for ∂f/∂l|ϕ in which l is the scale parameter defined
by Λ/Λ0 = e−l and corresponding to the reduction to Λ of an arbitrary initial
momentum scale of reference Λ0 (the initial sharp momentum cutoff). Finally, Kd
is the surface of the d-dimensional unit sphere divided by (2π)d.
A fixed point is a solution of the equation ḟ = 0. The study of the resulting
second order differential equation provides the following results:
• d > 4, no FP is found except the Gaussian fixed point.
• 3 6 d < 4, one nontrivial FP (the Wilson-Fisher fixed point [5]) is found
[15–17].
• A new nontrivial FP emanates from the origin (the Gaussian fixed point) below
each dimensional threshold dk = 2k/(k − 1), k = 2, 3, . . . ,∞ [18].
If one represents the function f(l, ϕ) as a sum of monomials of the form:
f(l, ϕ) =
∑
n
u2n (l)ϕ
2n−1
then, for d = 3, the Wilson-Fisher fixed point f ∗ is located in S at [2]: u∗
2 =
−0.461533 · · ·, u∗
4 = 3.27039 · · ·, u∗
6 = 14.4005 · · ·, u∗
8 = 32.31289 · · ·, etc.
Once the FP is known, one may study its vicinity which is characterized by
orthogonal directions corresponding to the infinite set of eigenvectors, solutions of
the differential equation (2.1) linearized at f ∗. The eigenvectors associated to positive
eigenvalues are said to be relevant; when the eigenvalues are negative they are said
to be irrelevant and marginal otherwise [19].
The relevant eigenvectors correspond to directions along which the RG trajec-
tories go away from the FP and the irrelevant eigenvectors correspond to directions
along which the trajectories go into the FP. A marginal eigenvector may be relevant
or irrelevant.
Our present FP f ∗ has only one relevant direction and infinitely many irrelevant
directions (no marginal direction, still see [1]). As already explained and shown in
[2], in order to approach f ∗ starting from an initial point in S, one must adjust one
parameter of the initial function f(0, ϕ). This amounts to fixing the temperature of
a system to its critical temperature.
Starting with a known initial function (at “time” l = 0) say:
f(0, ϕ) = u2 (0)ϕ+ u4 (0)ϕ
3,
562
Renormalization group domains
we adjust u2 (0) to the critical value uc
2 = −0.29958691 · · · corresponding to u4 (0) =
3 so that f(l, ϕ) [solution at time l of the differential equation (2.1)] approaches f ∗
when l → ∞. The approach to f ∗ is characterized by the least negative eigenvalue
λ2 = −1/ω1 (ω1 was noted ω in [2]). This means that, in the vicinity of f ∗ any
parameter un (l) evolves as follows (l → ∞):
un (l) ≃ u∗
n + an exp (−ω1l) .
l
0 2 4 6 8 10 12 14 16 18 20
ωeff
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7 ω1=0.5953
n=2
n=6
n=8
n=4
Figure 1. Evolutions for d = 3 of the first four Hamiltonian parameters u2(l),
u4(l), u6(l), u8(l) in a close approach to the Wilson-Fisher fixed point f ∗ along
T1 or T
′
1. The effective inverse eigenvalue ωeff (l) is given by equation (2.2) for
n = 2, 4, 6, 8. All these quantities reach the same universal value ω1 characteristic
of the least irrelevant eigendirection of f ∗. To get this close approach to f ∗ from
equation (2.1), the initial critical value uc
2 corresponding to u4(0) = 3, has been
determined with more than twenty digits.
Figure 1 illustrates this feature for the first four un (l)’s in the approach to f ∗. In
[2] the two associated attractive trajectories (locally tangent to the least irrelevant
eigenvector in the vicinity of f ∗) were noted T1 and T′
1.
One may also constrain the trajectory to approach f ∗ along the second irrelevant
direction (with the associated attractive trajectories noted T2 or T
′
2 in [2] and asso-
ciated with the second least negative eigenvalue λ3 = −1/ω2). In this case a second
parameter of the initial f must be adjusted, e.g., u4 (0) must be adjusted to uc
4 and
simultaneously u2 (0) to the corresponding uc
2, see [16,2]. Then, in the vicinity of f ∗,
any parameter un (l) will evolve as follows:
un (l) ≃ u∗
n + a′n exp (−ω2l) .
Looking for this kind of approach to f ∗, we have found that 6.66151663 < uc
4 <
6.66151669 and uc
2 = −0.58328898880579 · · · This has allowed us to estimate ω2
∼=
2.84. Although the shooting method is certainly not well adapted to determining
563
C.Bagnuls, C.Bervillier
l
0 2 4 6 8 10 12 14 16 18 20
ωeff
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
ω2=2.84
ω1
(T2)
(T'1)
(T1)
Figure 2. When a second condition is imposed on the initial Hamiltonian pa-
rameters, the approach to f∗ may be adjusted such as to asymptotically take the
second least irrelevant eigendirection. Here ωeff (l) is given by equation (2.2) for
n = 2 it clearly undergoes (full line) a flat inflection point at the value ω 2 = 2.84
corresponding to an approach to f ∗ along T2, the greater the critical parameter
uc4 is accurately determined the longer is the flat extremum. Because uc
4 is not
completely determined [within the available accuracy in solving equation (2.1)]
the trajectory leaves the direction of T2 to take one of the two directions of ap-
proach associated to the least irrelevant inverse eigenvalue ω1 (corresponding to
T1 or T
′
1 as indicated by dashed curves). Here, the trajectory corresponding to
the full line goes along T
′
1. Again a flat extremum of ωeff (l) indicates the ap-
proach along an eigenvector of f ∗ and requires an accurate determination of the
critical value uc
2. Because this determination is not complete, the trajectory ends
up going away from f ∗ as indicated by the sudden departure of ωeff (l) from ω1
for the large values of l.
the eigenvalues (see the huge number of digits required in determining u c
2 and uc
4),
our estimate is close to ω2
∼= 2.8384 found by Comellas and Travesset [20].
Because uc
4 cannot be perfectly determined, the trajectory leaves the trajectory
T2 before reaching f ∗ to take one of the two directions T1 or T
′
1 (corresponding to
ω1). Figure 2 illustrates this effect with the evolution, for n = 2, of the following
effective eigenvalue:
ω
(n)
eff (l) = −
d2un(l)/dl
2
dun(l)/dl
, (2.2)
the definition of which does not refer explicitly to f ∗. The evolution of ωeff (l) shows
a flat extremum (or a flat inflection point) at an RG eigenvalue of f ∗ each time the
RG flow runs along an eigendirection in the vicinity of f ∗.
Similar to uc
4, the value uc
2 cannot be perfectly determined. Consequently, the
trajectory ends up going away from the fixed point. This provides us with the op-
portunity of determining the only positive (the relevant) eigenvalue corresponding
to the critical exponent ν = λ1 = −1/ω0 [ωeff (l) shows then a flat extremum at ω0
when the flow still runs in close vicinity of f ∗]. Finally, far away from the fixed point,
564
Renormalization group domains
l
0 10 20 30 40 50 60 70
ωeff
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
ω2 (irrelevant)
ω1 (irrelevant)
ω0 =-1/ν
(relevant)
Trivial
A
p
p
ro
ac
h
A
w
ay
C
lo
se
F
ar
Figure 3. This figure is a continuation of figure 2. It shows the various plateaux
that ωeff (l) undergoes along a RG trajectory first adjusted to approach f ∗ along
the second irrelevant direction (plateau at ω2 = 2.84). Because it is not possible to
exactly determine the initial conditions, the trajectory always ends up going away
from the fixed point towards the trivial high temperature fixed point characterized
by the classical value 1
2 (for minus the inverse of ωeff (l), thus the final plateau
at −2). In-between, the RG flow has been influenced by the close vicinity of the
least irrelevant eigenvector (plateau at ω1) and that of the relevant eigenvector
(plateau at ω0 = − 1
ν
). The various regimes of the RG flows are indicated by the
vertical arrows on the left (direction of the flow with respect to the fixed point)
and on the right of the figure (distance to the fixed point).
the RG trajectory approaches the trivial high temperature fixed point characterized
by a classical eigenvalue (equal to 1
2
). The global picture summarizing the evolution
of ωeff (l) along the RG trajectory initialized in such a way as to approach f ∗ first
along T2, is drawn in figure 3.
The values we have determined by this shooting method are (for eigenvalues
other than the already mentioned ω2):
ω1
∼= 0.5953,
ν ∼= 0.68966,
which are close to the values found, for example, in [15,20]: ω1
∼= 0.5952 and ν ∼=
0.6895.
3. Trajectories for u4 < 0
In the preceding section, we have obtained a RG trajectory approaching the
Wilson-Fisher fixed point f ∗ along T2 by adjusting two parameters of the initial
565
C.Bagnuls, C.Bervillier
Hamiltonian (uc
4 and uc
2). This is exactly the procedure one must follow to determine
a tricritical RG trajectory approaching the Gaussian fixed point in three dimensions
(because of its two relevant directions). The only difficulty is to discover initial points
in S which are attracted to the Gaussian fixed point. To this end, we again use the
shooting method.
Based on the usual arguments regarding the LGW Hamiltonian as well as taking
into account the work done by Aharony on compressible ferromagnets [6], one ex-
pects to find the tricritical surface in the sector u4 < 0 (and with u2 > 0). Thus we
have tried to approach the Gaussian fixed point starting with initial function f(0)
of the form:
f(0, ϕ) = u2 (0)ϕ+ u4 (0)ϕ
3 + u6(0)ϕ
5 (3.1)
with (not large) negative values of u4 (0), for example u4 (0) = −1.
Because the Gaussian fixed point is twice unstable, we must adjust two param-
eters to approach it starting with (3.1). We do that by successive tries (shooting
method). For example, if we choose u4 (0) = −1 and u6(0) = 3 and determine a
value of u2 (0) such as to get a trajectory which does not go immediately towards
the trivial high temperature fixed point, the best we obtain is a trajectory which
approaches the Wilson-Fisher fixed point (thus the corresponding initial point be-
longs to the attraction domain of f ∗ although u4 (0) < 0 [6]). But if u6(0) = 2,
the adjustment of u2 (0) with a view to counterbalance the effect of the most rel-
evant direction of the Gaussian fixed point (which would drive the trajectory to-
ward the high temperature FP) yields a runaway RG flow towards larger and larger
negative values of u4 (l). From now on, the target is bracketed: the tricritical tra-
jectory corresponding to u4 (0) = −1 can be obtained with a value of u6(0) in the
range ]2, 3[ (we actually find a rather close approach to the Gaussian fixed point for
2.462280 > u6(0) > 2.4622788 and 6.4618440 · · · > u2(0) > 6.4618407 · · ·).
In order to understand the origin of the direction of runaway in the sector of
negative values of u4, it is worth studying the properties of the Gaussian fixed
point by linearization of the RG flow equation in the vicinity of the origin. If we
request the effective potential to be bounded by polynomials then the linearization
of equation (2.1) is identified with the differential equation of Hermite’s polynomials
of degree n = 2k − 1 for the set of discrete values of λ satisfying [15]:
2 + d− 2λk
d− 2
= 2k − 1 k = 1, 2, 3, . . . (3.2)
from which it follows that
• for d = 4: λk = 4− 2k, k = 1, 2, 3, . . ., there are two non-negative eigenvalues:
λ1 = 2 and λ2 = 0;
• for d = 3: λk = 3− k, k = 1, 2, 3, . . ., there are three non-negative eigenvalues:
λ1 = 2, λ2 = 1 et λ3 = 0.
If we denote by χk(ϕ) the eigenfunctions associated to the eigenvalue λk, it comes:
566
Renormalization group domains
-8 -6 -4 -2 0 2 4 6 8
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
C
B
A
u2
Simple fluid
u4
T1 (u4>0)
RT for u4<0
Tricritical RT
W-F FP
RPM-like ?
Gaussian FP
Figure 4. Domains of attraction and repulsion of the Gaussian fixed point. The
figure represents projections onto the plane {u2, u4} of various RG trajectories
running in the space S minus one dimension. The flows have been obtained by
solving equation (2.1). Black circles represent the Gaussian and the Wilson-Fisher
(W-F FP) fixed points. The arrows indicate the directions of the RG flows on
the trajectories. The ideal trajectory (dot line) which interpolates between these
two fixed points represents the usual renormalized trajectory T1 corresponding
to the so-called ϕ4
3 renormalized field theory in three dimensions (usual RT for
u4 > 0). White circles represent the projections onto the plane of initial critical
Hamiltonians. For u4(0) > 0, the effective Hamiltonians run toward the Ising
fixed point asymptotically along T1 (simple fluid). Instead, for u4(0) < 0 and
according to the values of Hamiltonian coefficients of higher order (u6, u8, etc.),
the RG trajectories either (A) meet an endless RT emerging from the Gaussian
FP (dashed curve) and lying entirely in the sector u4 < 0 or (B) meet the usual
RT T1 to reach the Ising fixed point. The frontier which separates these two
very different cases (A and B) corresponds to initial Hamiltonians lying on the
tricritical subspace St (white square C). This is a source of RG trajectories flowing
asymptotically toward the Gaussian FP along the tricritical RT. Notice that the
coincidence of the initial point B with the RG trajectory starting at point A is
not real (it is accidental, due to a projection onto a plane of trajectories lying in a
space of infinite dimension). The points A or B could correspond to the restricted
primitive model of ionic systems (see [8]).
567
C.Bagnuls, C.Bervillier
• χ+
1 = ϕ, χ+
2 = ϕ3 − 3
2
ϕ, χ+
3 = ϕ5 − 5ϕ3 + 15
4
ϕ, . . ., whatever the spatial
dimensionality d.
The superscript “+” is just a reminder of the fact that the eigenfunctions are
defined up to a global factor and thus the functions χ−
k (ϕ) = −χ+
k (ϕ) are also
eigenfunctions with the same eigenvalue λk.
3.1. Case d = 3
Similar to χ+
2 , the direction provided by χ−
2 in S is a direction of instability of
the Gaussian fixed point. Now χ+
2 is associated with the well known renormalized
trajectory T1 on which the usual (massless) ϕ4
3-field theory [16,2] is defined, for the
same reasons a renormalized trajectory T
′′
1 locally tangent to χ−
2 in the vicinity of
the origin of S exists with the same properties as T1 (see [2]). The difference is that
T
′′
1 lies entirely in the sector u4 < 0 and is endless (not ended by a fixed point).
This endless renormalized trajectory is associated with systems undergoing a
first-order phase transition. This is due to the absence of fixed point [21], in which
case the correlation length ξ cannot be made infinite although for some systems lying
close to T
′′
1 and attracted to it (i.e. at the transition temperature), ξ may be very
large (because T
′′
1 is endless), in which cases one may say that the transition is almost
of the second order [22]. Of course, a domain of the first-order phase transition in S
was expected out of the usual arguments [7,6]. We only better specify the conditions
of the first-order transition realization in S.
Figure 4 shows the attractive trajectory T
′′
1 together with the attractive tricritical
line approaching the Gaussian fixed point. The tricritical surface S t separates the
first-order surface Sf from the critical surface Sc. Figure 4 also shows that systems
lying close to the tricritical surface may still be attracted to the Wilson-Fisher fixed
point. In this case the effective exponents may undergo a very retarded crossover
to the asymptotic Ising values compared to usual systems corresponding to initial
points chosen in the sector u4 > 0 of S. Figure 5 illustrates how minus the inverse
of (2.2) provides us with different evolutions [calculated from (2.1)] of the effective
exponent νeff (τ) [with τ ∝ (T − Tc)/Tc] according to the initial point chosen in S.
It is worth explaining how we have defined νeff(τ).
We have seen at the end of section 2 that the quantity (2.2) undergoes a flat
extremum (or a flat inflection point) at an RG eigenvalue of f ∗ each time the RG
flow runs along an eigendirection in the vicinity of f ∗. Now it happens that this
extremum is less and less flat as one chooses larger and larger values of (u2(0)− uc
2)
(for the eigenvalue ν) but it still exists. This provides us with a way of expressing the
evolution of an effective exponent νeff when the RG-substitute to τ , namely (u2(0)−
uc
2)/u
c
2, is varied. Figure 6 shows such an evolution for some initial Hamiltonian
(with u4(0) = 4). Notice that for such a Hamiltonian, the extremum disappears
before νeff reaches the trivial value 1
2
(associated with the approach to the trivial
high temperature fixed point and to a regular – non critical – behaviour) while in
the case of a Hamiltonian initialized close to the tricritical surface, the classical-to-
Ising crossover is complete (see figure 4). This is because in the latter case the RG
568
Renormalization group domains
log10(τ)
-25 -20 -15 -10 -5 0 5
νeff
0.45
0.50
0.55
0.60
0.65
0.70
Figure 5. Evolutions of an effective exponent νeff(τ) [with τ ∝ (T −Tc)/Tc] along
three different families of RG trajectories (see text for additional details). The
full squares indicate the evolution of νeff(τ) for a family of trajectories initialized
in the sector u4 > 0 with u4(0) = 3 and for various values of u2(0) (the same
system at criticality corresponds to the white circle “Simple fluid” of figure 4).
When u2(0) → uc2 the effective exponent approaches the critical exponent value
ν ∼= 0.69 compatible with the present study. One observes that the crossover
towards the classical value 1
2 is not complete because νeff(τ) ceases to make sense
before τ becomes large. This is not the case of the evolution represented by the
full circles which correspond to trajectories initialized close to the Gaussian fixed
point. In this case the complete crossover reproduces the interpolation between
the Gaussian and the Wilson-Fisher fixed points and typically corresponds to
the usual answer given by field theory [28]. The third evolution (full triangles)
corresponds to a family of Hamiltonian initialized close to the tricritical surface
but still attracted to the Wilson-Fisher fixed point. One sees that the classical-
to-Ising crossover is complete but highly retarded compared to the two other
cases. This is because at criticality, the RG flow is first attracted to the Gaussian
fixed point (showing then an apparent classical value of ν) before interpolating
between the Gaussian and the Wilson-Fisher fixed point.
trajectory comes close to the Gaussian fixed point (and νeff(τ) has an extremum at
1
2
) before approaching f ∗. This reinforces the idea that the so-called classical-to-Ising
crossover actually exists only between the Gaussian and Wilson-Fisher fixed points
[23].
The same configuration displayed in figure 4 has been also obtained by Tetradis
and Litim [24] while studying analytical solutions of an exact RG equation in the
local potential approximation for the O(N)-symmetric scalar theory in the large N
limit. But they were not able to determine “the region in parameter space which
results in first-order transitions” [24].
569
C.Bagnuls, C.Bervillier
l
0 1 2 3 4 5
νeff
0.45
0.50
0.55
0.60
0.65
τ=10-1 τ=10-2
τ=10-0.5
τ=10+0.5
Figure 6. Illustration of the evolution of the extrema νeff(l) [minus the inverse of
equation (2.2)] for various values of τ = (u2(0) − uc2) /u
c
2 and for the family of RG
flows initialized at u4(0) = 3. The extremum (grey triangle) disappears at some
not very large value of τ (about 10−0.5) and does not reach the classical value 1
2 .
This induces the partial Ising-to-classical crossover drawn in figure 5 (squares).
3.2. Case d = 4
To decide whether the marginal operator (associated with the eigenvalue equal
to zero, i.e. λ2 in four dimensions, or λ3 in three dimensions) is relevant or irrelevant,
one must go beyond the linear approximation. The analysis is presented in [15] for
d = 4. If one considers a RG flow along χ+
2 such that g2(ϕ, l) = c(l)χ+
2 (ϕ), then
one obtains, for small c: c(l) = c(0) [1− Ac(0)l] with A > 0. Hence the marginal
parameter decreases as l grows. As it is well known, in four dimensions the marginal
parameter is irrelevant. However, if one considers the direction opposite to χ+
2 (i.e.
χ−
2 ) then the evolution corresponds to changing c → −c. This gives, for small values
of c: c(l) = c(0) [1 + A |c(0)| l] and the parameter becomes relevant. The parameter c
is the renormalized φ4 coupling constant uR and it is known that in four dimensions
the Gaussian fixed point is IR stable for uR > 0 but IR is unstable for uR < 0 [25].
We have verified that the trajectory T
′′
1 survives when d = 4 (contrary to T1,
see [2]). That trajectory T
′′
1 is a renormalized trajectory on which we could define a
continuum limit for the ϕ4
4-field theory and if the corresponding (perfect) action was
positive for all ϕ, one could say that the φ4
4-field theory with a negative coupling
is asymptotically free. Unfortunately, because the ϕ4-term is dominant for large ϕ
in the vicinity of the origin of S (due to the relevant direction provided by χ−
2 ),
the negative sign of the renormalized coupling prevents the path integral to be well
defined.
However, because the action to which one refers in the continuum limit (the
perfect action) is formal (because it involves an infinite number of parameters and
cannot be written down, see [2]) we wonder whether the wrong sign of the action is
actually a valid argument to reject the ϕ4
4-field theory with a negative renormalized
coupling. It is worth mentioning that the asymptotically free scalar field theory
570
Renormalization group domains
which has recently been considered on a lattice [26] could actually be the φ4
4-field
theory with a negative coupling to which we refer here.
4. Acknowledgements
We dedicate this article to Professor Yukhnovskii in grateful recognition of his
efficient and generous help in fostering the Ukrainian-French Symposium held in
Lviv in february 1993, with the hope that in the future the contacts between our
two communities will further develop.
A. The finite difference method used
For technical reasons, instead of studying equation (2.1), we consider the differ-
ential equation satisfied by g(ϕ) = f ′(ϕ) (i.e. the second derivative of the potential
with respect to the field):
ġ =
Kd
2
[
g′′
1 + g
−
(g′)2
(1 + g)2
]
+
(
1−
d
2
)
ϕg′ + 2g. (A.1)
Starting with a known initial function (at “time” l = 0), we follow its evolution
in S by approximating the differential equation (A.1) by finite differences and a two
dimensional grid with the uniform spacings dy = 0.01 and dl = 0.000390625. The
finite difference formulas for the derivatives g ′′ and g′ have been chosen with the
accuracy O(dy4):
g′(y) =
8
12dy
[g(y + dy)− g(y − dy)− g(y + 2dy) + g(y − 2dy)]
+ O(dy4), (A.2)
g′′(y) =
16
12dy2
[g(y + dy) + g(y − dy)− 30g(y)
− g(y + 2dy)− g(y − 2dy)] +O(dy4) (A.3)
The evolutionary function g(y, l) is known (calculated) at the discreet set of
points yi = i · dy with (i > 0) and a maximum value imax = 82. (This value is
large enough to study the approach to the Wilson-Fisher fixed point with great
accuracy but it is too small to precisely study the approach to the Gaussian fixed
point.) At each time lk = k · dl, the derivatives are estimated from g(yi, lk) =
g(yi, lk−1) + ġ(yi, lk−1) · dl by using equations (A.2), (A.3) which apply only for
1 < i < imax−1. For the marginal points i = 0, 1 we use the parity of g(y) [by
inserting g(−n · dy) = g(n · dy) for n = 1, 2 in equations (A.2), (A.3)]. For the two
other marginal points i = imax−1, imax of the grid, there is no fixed solution and we
shall alternately use the two following conditions [using the obvious abbreviation
g(i) instead of g(i · dy)]:
g′(i) = g′(i− 1),
g′′(i) = g′′(i− 1),
571
C.Bagnuls, C.Bervillier
g′(i) =
1
dy
[
25
12
g(i)− 4g(i− 1) + 3g(i− 2)−
4
3
g(i− 3)
+
1
4
g(i− 4) +O(dy4)
]
, (A.4)
g′′(i) =
1
dy2
[
915
244
g(i)−
77
6
g(i− 1) +
107
6
g(i− 2)− 13g(i− 3)
+
61
12
g(i− 4)−
5
6
g(i− 5)
]
+O(dy4). (A.5)
Condition 2 is more accurate than condition 1 but sometimes leads to strong
instabilities which do not appear when we first use condition 1 and then condition
2 after some finite “time” l0. The validity of the procedure is tested by, for example,
trying to approach a given fixed point (see the main part of the paper).
A.1. Spurious fixed points appearing in approaching the Gaussian fixed
point
In trying to determine the attractive tricritical trajectory (approaching the Gaus-
sian fixed point), we have encountered a spurious twice unstable fixed point lying at
some finite and non-negligible distance to the Gaussian fixed point. To understand
the origin of this undesirable numerical effect, it is necessary to shortly discuss the
solution of the fixed point equation ḟ = 0.
From (2.1) or (A.1), one sees that the fixed point equation is a second order
non-linear differential equation and a solution would be parametrized by two arbi-
trary constants. One of these two constants may easily be determined: since g∗(ϕ)
is expected to be an even function of ϕ [O(1) symmetry] then g∗′(0) = 0 may be
imposed. There remains one free parameter, thus a one-parameter family of (non-
trivial) fixed points are the solutions to the differential equation. But there is no
infinity of physically acceptable fixed points; all but a finite number of the solutions
in the family are singular at some ϕc [15,18,27]. Formally, by requiring the physical
fixed point to be defined for all ϕ, the acceptable fixed points are limited to the
Gaussian fixed point and (for d = 3) to the Wilson-Fisher fixed point.
However, in our study, because we numerically consider the function g(ϕ) in some
finite range of values of ϕ (see above: imax = 82), it appears that in approaching the
origin of S, infinitely many pseudo-fixed points exist which have there ϕc-singularity
located outside the finite range explicitly considered and there is at least one of
them which looks like a tricritical fixed point. When we enlarge the range of ϕ to
imax = 200, the previously observed nontrivial tricritical fixed point disappears to
the benefit of another one located closer to the origin. In conclusion, a larger and
larger number of grid-points must be considered as one tries to come closer and
closer to the Gaussian fixed point. This particularity together with the slowness of
the approach along a marginal direction makes it extremely difficult to come very
close to the Gaussian fixed point.
572
Renormalization group domains
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574
Renormalization group domains
Області ренормалізаційної групи скалярного
гамільтоніану
К.Банюльс 1 , К.Бервільє 2
1 Відділ фізики конденсованого стану, Саклє, Франція
2 Відділ теоретичної фізики, Саклє, Франція
Отримано 9 березня 2000 р.
Використовуючи наближення локального потенціалу точного рівнян-
ня ренормалізаційної групи (РГ), ми показуємо різні області значень
параметрів O(1) симетричного скалярного гамільтоніану. У трьох ви-
мірах додатково до звичайної критичної поверхні Sc (область при-
тягання фіксованої точки Вільсона-Фішера), ми явно показуємо іс-
нування області фазового переходу першого ряду Sf , відокремле-
ної від Sc трикритичною поверхнею Sf (область притягання гаусо-
вої фіксованої точки). Sf і Sc є дві різні області відштовхування для
гаусової фіксованої точки, а Sf не є в ділянці притягання фіксованої
точки. Sf характеризується нескінченою ренормалізованою траєк-
торією, яка повністю лежить в області негативних значень констант
взаємодії ϕ4 . Ця ренормалізована траєкторія також існує в чотирьох
вимірах, роблячи гаусову фіксовану точку в ультрафіолетовій області
стабільною (і ренормалізовану теорію поля ϕ4 асимптотично віль-
ною, але з неправильним знаком ідеальної дії). Ми також показуємо,
що дуже запізнений кросовер від класичної до ізінгівської поведін-
ки може існувати у трьох вимірах (фактично нижче чотирьох вимірів).
Це може бути поясненням для неочікуваної класичної критичної по-
ведінки, яка спостерігається в деяких іонних системах.
Ключові слова: ренормалізаційна група, критичний,
трикритичний, фазовий перехід першого роду, кросовер
PACS: 05.10.Cc, 05.70.Jk, 11.10.Hi, 61.20.Qg
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