Revisiting (logarithmic) scaling relations using renormalization group
We explicitly compute the critical exponents associated with logarithmic corrections (the so-called hatted exponents) starting from the renormalization group equations and the mean field behavior for a wide class of models at the upper critical behavior (for short and long range φ n -theories) an...
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irk-123456789-1565472019-06-19T01:26:05Z Revisiting (logarithmic) scaling relations using renormalization group Ruiz-Lorenzo, J.J. We explicitly compute the critical exponents associated with logarithmic corrections (the so-called hatted exponents) starting from the renormalization group equations and the mean field behavior for a wide class of models at the upper critical behavior (for short and long range φ n -theories) and below it. This allows us to check the scaling relations among these critical exponents obtained by analysing the complex singularities (Lee-Yang and Fisher zeroes) of these models. Moreover, we have obtained an explicit method to compute the ϙˆ exponent [defined by ξ ∼ L(logL) ϙˆ ] and, finally, we have found a new derivation of the scaling law associated with it. Ми явно обчислюємо критичнi показники, пов’язанi з логарифмiчними поправками, виходячи з рiвнянь ренормгрупи i середньопольової поведiнки для широкого класу моделей як при вищiй критичнiй вимiрностi (для коротко- i далекосяжних φ n -теорiй), так i нижче вiд неї. Це дозволяє нам перевiрити спiввiдношення скейлiнгу, що пов’язують критичнi показники, аналiзуючи комплекснi сингулярностi (нулi Лi-Янга i Фiшера) цих моделей. Окрiм того, ми запропонували явний метод для обчислення показника ϙˆ [означеного як ξ ∼ L(logL) ϙˆ ] i, накiнець, ми отримали нове виведення закона скейлiнгу, пов’язаного з цим показником. 2017 Article Revisiting (logarithmic) scaling relations using renormalization group / J.J. Ruiz-Lorenzo // Condensed Matter Physics. — 2017. — Т. 20, № 1. — С. 13601: 1–10. — Бібліогр.: 25 назв. — англ. 1607-324X PACS: 64.60-j,05.50+q,05.70.Jk,75.10.Hk DOI:10.5488/CMP.20.13601 arXiv:1702.05072 http://dspace.nbuv.gov.ua/handle/123456789/156547 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We explicitly compute the critical exponents associated with logarithmic corrections (the so-called hatted exponents) starting from the renormalization group equations and the mean field behavior for a wide class of models
at the upper critical behavior (for short and long range φ
n
-theories) and below it. This allows us to check the
scaling relations among these critical exponents obtained by analysing the complex singularities (Lee-Yang and
Fisher zeroes) of these models. Moreover, we have obtained an explicit method to compute the ϙˆ exponent
[defined by ξ ∼ L(logL)
ϙˆ
] and, finally, we have found a new derivation of the scaling law associated with it. |
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Article |
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Ruiz-Lorenzo, J.J. |
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Ruiz-Lorenzo, J.J. Revisiting (logarithmic) scaling relations using renormalization group Condensed Matter Physics |
| author_facet |
Ruiz-Lorenzo, J.J. |
| author_sort |
Ruiz-Lorenzo, J.J. |
| title |
Revisiting (logarithmic) scaling relations using renormalization group |
| title_short |
Revisiting (logarithmic) scaling relations using renormalization group |
| title_full |
Revisiting (logarithmic) scaling relations using renormalization group |
| title_fullStr |
Revisiting (logarithmic) scaling relations using renormalization group |
| title_full_unstemmed |
Revisiting (logarithmic) scaling relations using renormalization group |
| title_sort |
revisiting (logarithmic) scaling relations using renormalization group |
| publisher |
Інститут фізики конденсованих систем НАН України |
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2017 |
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http://dspace.nbuv.gov.ua/handle/123456789/156547 |
| citation_txt |
Revisiting (logarithmic) scaling relations using renormalization group / J.J. Ruiz-Lorenzo // Condensed Matter Physics. — 2017. — Т. 20, № 1. — С. 13601: 1–10. — Бібліогр.: 25 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT ruizlorenzojj revisitinglogarithmicscalingrelationsusingrenormalizationgroup |
| first_indexed |
2025-07-14T08:53:03Z |
| last_indexed |
2025-07-14T08:53:03Z |
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1837611807766740992 |
| fulltext |
Condensed Matter Physics, 2017, Vol. 20, No 1, 13601: 1–10
DOI: 10.5488/CMP.20.13601
http://www.icmp.lviv.ua/journal
Revisiting (logarithmic) scaling relations using
renormalization group
J.J. Ruiz-Lorenzo1,2,3
1 Departamento de Física, Universidad de Extremadura, 06071 Badajoz, Spain
2 Instituto de Computación Científica Avanzada (ICCAEx), Universidad de Extremadura, 06071 Badajoz, Spain
3 Instituto de Biocomputación y Física de los Sistemas Complejos (BIFI), Zaragoza, Spain
Received January 11, 2017, in final form January 23, 2017
We explicitly compute the critical exponents associated with logarithmic corrections (the so-called hatted expo-
nents) starting from the renormalization group equations and themean field behavior for a wide class of models
at the upper critical behavior (for short and long range φn -theories) and below it. This allows us to check the
scaling relations among these critical exponents obtained by analysing the complex singularities (Lee-Yang and
Fisher zeroes) of these models. Moreover, we have obtained an explicit method to compute the ϙ̂ exponent
[defined by ξ∼ L(logL)ϙ̂] and, finally, we have found a new derivation of the scaling law associated with it.
Key words: renormalization group, scaling, logarithms, mean field
PACS: 64.60-j,05.50+q,05.70.Jk,75.10.Hk
1. Introduction
One of the main achievements of Wilson’s [1] renormalization group (RG) was the definition of uni-
versality class by means of a finite number of critical exponents. These critical exponents determine the
divergences of some observables at the critical point [2–6].
In particular circumstances, logarithmic corrections arisemultiplicatively in these critical laws. These
logarithms are of a paramount importance in some materials (for example, dipolar magnets in three di-
mensions, which is the upper critical dimension of the system [7]) and can be accessed experimentally [8].
Moreover, their effects are very important in the non-perturbative definition of quantum field theories
in four dimensions (the so-called triviality problem) [9].
In [10–12], the scaling relations of the exponents which characterize the logarithmic corrections were
derived using the Lee-Yang [13] and Fisher zeroes [14] techniques in a model-independent manner. In
this paper, we will explicitly compute, using RG and field theory, the value of these exponents and then
check the (scaling) relations among them. We have done this for a wide class of models [φn
models at
their upper critical dimensions with short (SR) and long range (LR) interactions] and can also be applied
to the models in low dimensions (as the four-state Potts model in two dimensions).
In the presence of logarithmic corrections, the scaling laws for the observables near the critical point
must be modified as [2–6]
1
ξ ∼ |t |−ν∣∣ log |t |∣∣ν̂ , (1.1)
C ∼ |t |−α∣∣ log |t |∣∣α̂ , (1.2)
m ∼ |t |β∣∣ log |t |∣∣β̂ for t < 0, (1.3)
χ ∼ |t |−γ∣∣ log |t |∣∣γ̂ , (1.4)
1
We use in the definition of the critical exponents the standard notation, see, for example, [2–6, 10, 11].
© J.J. Ruiz-Lorenzo, 2017 13601-1
https://doi.org/10.5488/CMP.20.13601
http://www.icmp.lviv.ua/journal
J.J. Ruiz-Lorenzo
m ∼ h1/δ| logh|δ̂ for t = 0, (1.5)
rLY ∼ |t |∆∣∣ log |t |∣∣∆̂ , (1.6)
G(r ) ∼ (logr )η̂
r d−2+η for t = 0, (1.7)
which define the so-called hatted exponents (d being the dimension). The standard critical exponents
(e.g., α, β, γ, etc.) satisfy the classic scaling laws (see, for example, [2–6]). In addition, in [10–12], it was
shown that the hatted exponents satisfy the following scaling relations:
2
∆̂ = β̂− γ̂ , (1.8)
β̂(δ−1) = δδ̂− γ̂ , (1.9)
η̂ = γ̂− ν̂(2−η) . (1.10)
Finally, see [10–12], at the infinite volume critical point, the correlation length of the system defined on a
finite box of size L behaves as3
ξ∼ L(logL)ϙ̂ (1.11)
and the associated related scaling relation is
α̂= d ϙ̂−d ν̂ . (1.12)
When α = 0 and the impact angle of the Fisher zeros satisfies φ , π/4, the previous relation should be
modified as [11]
α̂= 1+d ϙ̂−d ν̂ . (1.13)
Finally, an additional scaling relation can be written [12]
2β̂− γ̂= d ϙ̂−d ν̂ . (1.14)
In this paper we will mainly analyze generic φn
theories, with Hamiltonian (for simplicity we write
the scalar version for the short range model):
4
H =
∫
dd x
[
1
2
(∂µφ)2 + 1
2
r0φ
2 + 1
n
gnφ
n
]
. (1.15)
2. Some mean field results
We will use RG to analyze the critical behavior of the models, and after a finite number of RG step
we will finish in the parameter region in which we can apply mean field results. In this section we will
briefly review the basic facts of the scaling in this mean field region [4, 5].
We start with the free energy per spin for a φn
-theory:
f (m) = r0
2
m2 + gn
n
mn . (2.1)
Minimizing f (m), for r0 < 0, we obtain magnetization as:
m =
( |r0|
gn
)1/(n−2)
∼ 1
g pm
n
, (2.2)
2
Recall δ= (d +2−η)/(d −2+η).
3
In this paper we avoid the mean field region by working at and below the upper critical region.
4
Using power counting, we can compute when the coupling gn is marginal, obtaining the so-called upper critical dimension, that
for short range models is
du = 2n
n −2
and for long range models (with propagator 1/qσ)
du = nσ
n −2
.
For σ= 2, we recover the short range result.
13601-2
Revisiting (logarithmic) scaling relations
where pm = 1/(n −2) and
fmin∝
r n/(n−2)
0
g 2/(n−2)
n
. (2.3)
The susceptibility is
χ∝|r0| , (2.4)
and the specific heat
C ∝ r 2/(n−2)
0
g 2/(n−2)
n
∼ 1
g pc
n
, (2.5)
where pc = 2/(n −2). Finally, we can add a magnetic field [which induces a term −hm in equation (2.1)]
and compute the minimum of the free energy just at the critical point, r0 = 0 (which is relevant in the
computation of the critical isotherm)
fmin(r0 = 0,h) ∝ hn/(n−1)
g 1/(n−1)
n
, (2.6)
and the magnetization at criticality is
m(r0 = 0,h) ∝
(
h
gn
)ph
, (2.7)
where ph = 1/(n − 1).5 Hence, since n > 2, gn is an irrelevant dangerous variable for magnetization,
critical isotherm and specific heat, yet, χ is free of this problem.
3. Revisiting logarithmic corrections
The starting point is the behavior of the singular part of the free energy density (that we denote simply
as f and denoting gn by g ) under a RG transformation
f (t0,h0, g0) = 1
bd
f [t (b), g (b),h(b)] , (3.1)
where b is the RG scaling factor and t (b), h(b) and g (b) (the running couplings) denote the evolution
of different couplings under a RG transformation, which are obtained solving the following differential
equations (we write them for the LR model)
dt
dlogb
= t [σ+γ(g )] , (3.2)
dlogh
dlogb
= d
2
+1− γ
2
, (3.3)
dg
dlogb
= βW(g ) , (3.4)
which define the functions βW,γ and γ.6 For further use we define two functions F (b) and ζ(b) and we
assume the following asymptotic behavior [g0 ≡ g (1)]
F (g ) ≡ exp
g (b)∫
g0
dg
γ(g )
βW(g )
∼ ba(logb)p , (3.5)
ζ(g ) ≡ exp
−1
2
g (b)∫
g0
dg
γ(g )
βW(g )
∼ bc (logb)x . (3.6)
5
The introduction of pm , ph and pc will be useful at the upper critical dimension to collect the extra logs yielded by the g
renormalizing to zero in a logarithmic way. Below the upper critical dimension, gn is not a dangerous irrelevant variable: in this
situation, we will use pm = pc = ph = 0, i.e., there will be no extra logs from the gn (b) in the mean field region.
6
We can compute the thermal and magnetic critical exponents by means of η = γ(g∗) and 1/ν = σ+γ(g∗), where g∗
satisfies
βw(g∗) = 0 [3, 15].
13601-3
J.J. Ruiz-Lorenzo
The solutions are (we also write the asymptotic behavior as b →∞) as follows:
t (b) = t0bσ exp
g (b)∫
g0
dg
γ(g )
βW(g )
∼ t0bσ+a(logb)p , (3.7)
h(b) = h0b
d
2 +1 exp
−1
2
g (b)∫
g0
dg
γ(g )
βW(g )
∼ h0b
d
2 +1+c (logb)x , (3.8)
logb = exp
g (b)∫
g0
dg
1
βW(g )
. (3.9)
In the asymptotic regime (and for the models under consideration in this paper where βW∝ g s
), the last
equation can be written as
g (b) ∼ (logb)−r , (3.10)
and this defines the r exponent (1/r = s −1). In particular, the useful relation t (b∗) = 1 can be written as
b∗ ∼ t−1/(σ+a)
0 (log t0)−p/(σ+a) . (3.11)
Therefore, we can identify ν= 1/(σ+a) and ν̂=−p/(σ+a) =−pν. From the form of h(b), one can obtain
c = −η/2. By computing suitable derivatives of the free energy per spin [see equation (3.1)] and using
the renormalized couplings given by equations (3.7)–(3.9), and in the case of the upper critical dimension
using the expression of the intensive free energy in the mean field regime [equations (2.2), (2.4)–(2.6)],
we can obtain the following relations for the exponents which control the logarithmic corrections (see
the appendix for more details)
α̂=−d ν̂+ r pc , (3.12)
γ̂= ν̂(2−η)+2x , (3.13)
β̂=−ν̂
(
d
2
−1+ η
2
)
+x + r pm , (3.14)
δ̂= 2xd
d +2−η + r ph , (3.15)
∆̂=−ν̂
(
d
2
+1− η
2
)
−x + r pm , (3.16)
η̂= 2x . (3.17)
These equation must be read at the upper critical dimension with η= 0 (SR) or η= 2−σ (LR) and d =
du , otherwise, below du , all the p ’s from the mean field are zero (pm = pc = ph = 0). With these explicit
expressions for the hatted exponents, it is easy to re-derive the scaling relations given by equations (1.8)–
(1.10), (1.14).
In models with α = 0 and impact angle of the Fisher zeroes φ , π/4, a circumstance equivalent to
A−/A+ = 1 (being A± the critical amplitudes of the specific heat) [16], the scaling of the free energy is
modified as
7
f (t0,h0, g0) = 1
bd
f
(
t (b), g (b),h(b)
)+ 1
bd
(logb) fl
(
t (b), g (b),h(b)
)
, (3.18)
where the functions f and fl satisfy additional constraints to generate the right logarithmic correc-
tions (for more details see [16] and references therein). This decomposition of the free energy can be
7
As described in [16] the appearance of this extra log term in the free energy can be explained either as a resonance between the
thermal and the identity operators or as an interplay between the singular and regular parts of the free energy.
13601-4
Revisiting (logarithmic) scaling relations
also understood in terms of a Lee-Yang and Fisher zeros analysis, see [11, 12]. For instance, in the two-
dimensional pure Ising model, only the “energy”-sector develops logarithmic corrections, and these cor-
rections (for the free energy, energy and specific heat) are provided by the term proportional to fl . How-
ever, the scaling of the “magnetic”-sector is given by the standard term, proportional to f . In the two
dimensional diluted Ising model, the magnetic sector also shows logarithmic corrections, provided by the
(standard) term proportional to f , whereas the corrections for the energy-sector are given by the term
proportional to fl . Hence, only the relation of α̂ (which is computed with the fl -term) should be modified
α̂=−d ν̂+ r pc +1. (3.19)
We have checked that these equations provide correct hatted exponents in O(N )-φ4
models
8
in the
short range and long range interactions, tensor (short range) φ3
(which includes percolation, m-compo-
nent spin glasses and Lee-Yang singularities, and can also be related with lattice animals), all of them
at their upper critical dimension and in the four-state Potts models, pure Ising model and diluted Ising
model in two dimensions [12, 15, 17–20]. The logarithmic scaling relations for all these models were
thoroughly checked in [12].
9
Finally, using this theoretical framework we have been able to compute ∆̂ for the four-state two-
dimensional Potts model, ∆̂, β̂, η̂ and δ̂ for SR tensor φ3
-theories and ∆̂ for the LR O(N ) φ4
-theories.
Finally, ϙ̂ has been computed for the LR O(N ) φ4
-theories. The numerical values for all these exponents
were derived in references [10–12] using the logarithmic scaling relations (1.8)–(1.10), (1.12), (1.14). See
[12] for the values of these hatted exponents.
4. A re-derivation of α̂= d ϙ̂−d ν̂
We start with the dependence of a singular part of the intensive free energy on L
fsing∝ L−d . (4.1)
This is the key point of the derivation. Below the upper critical dimension, one has L ∼ ξ and one can
write fsing ∝ ξ−d
, but due to the logarithmic corrections which appear at the upper critical dimension
this is no longer true.
We can also write the singular part of the free energy, using the scaling of the specific heat [see equa-
tion (1.2)], as
fsing∝ L−d ∝ t 2−α(log t )α̂ . (4.2)
Using equations (1.1) and (1.11) one can write
L−d ∼ ξ−d (logξ)d ϙ̂ ∼ tνd (log t )d ϙ̂−d ν̂ ∼ t 2−α(log t )α̂ . (4.3)
Identifying the exponents of log t of the last two expressions we obtain the scaling relation given by
equation (1.12).
When α = 0 and φ , π/4 [11], the free energy scales as f ∝ L−d logL [see equation (3.18) and the
discussion of section 3]. This extra-log, using the previous arguments, provides the following scaling law:
α̂= 1+d(ϙ̂− ν̂) , (4.4)
obtaining equation (1.13).
8
Where N is the number of components of the field.
9
In [12] other exponents were defined (e.g., ε̂, ν̂c and α̂c ). It is straightforward to compute them using the theoretical framework
of this paper.
13601-5
J.J. Ruiz-Lorenzo
5. Computation of the ϙ̂-exponent
We will compute the exponent ϙ̂ for a generic φn
theory at its upper critical dimension for both short
and long range models. The starting point is the expression of χ in terms of the free energy10
χ∼ b2ζ2 ∂
2 f
(
t (b), g (b),h
)
∂h2
h=0
. (5.1)
This can be written as [using t (b∗) = 1 and b∗ ∼ ξ]
χ∼ ζ(ξ)2ξ2 ∝ ξ2+2c (logξ)2x . (5.2)
In a φn
theory we can rescale the field via φ′ = g 1/nφ [21], and the free energy per spin verifies
f (t0, g0,h0) = L−dG
(
t (L)
g 2/n
,
h(L)
g 1/n
)
. (5.3)
Differentiating twice equation (5.3) with respect to the magnetic field (h0), we obtain
χ∝ L−d
[
∂h(L)
∂h0
]2 ∂2
∂h(L)2 G
(
t (L)
g (L)2/n
,
h(L)
g (L)1/n
)
h0=t0=0
∼ L2ζ(L)2 1
g (L)2/n
. (5.4)
Comparing with equation (5.2), we finally obtain
ξ∼ L
g (L)2/[n(2+2c)]
, (5.5)
and assuming the asymptotic behavior of g (L) given in equation (3.10) we finally get
ϙ̂= 2r
n(2+2c)
= 2r
n(2−η)
. (5.6)
For the short range φ4
theory (σ= 2, n = 4, η=−2c = 0 and r = 1) we obtain ϙ̂= 1/4. For the short range
φ3
theory (σ= 2, n = 3, η=−2c = 0 and r = 1/2) we get ϙ̂= 1/6. In addition, for the long range φ4
model
[n = 4, η=−2c = (2−σ) and r = 1], ϙ̂= 1/(2σ).
Another way to obtain ϙ̂ is to use the scaling relation provided by equation (1.12) and equation (3.12)
ϙ̂= α̂
d
+ ν̂= r pc
d
(5.7)
or for α= 0 and φ,π/4, equations (1.13), (3.19)
ϙ̂= α̂
d
+ ν̂− 1
d
= r pc
d
, (5.8)
obtaining the same final result irrespectively of the value of α and the impact angle φ.
So, ϙ̂= 0 below du since pc = 0 therein; at the upper critical dimension (SRmodels) d = du = 2n/(n−2),
then ϙ̂ = r /n as computed before. For LR models, du = nσ/(n −2) and then we recover the result given
by equation (5.6).
In [15, 22], the ϙ̂ exponent was computed using a misidentification of the correlation length for a
lattice of size L = 1 [see equations (4.1), (4.2) and (3.11), (3.12) of [22] and [15], respectively], providing,
however, with the correct value of ϙ̂ in generalφ4
theories (and, in particular, for the four dimensional di-
lutedmodel, see reference [22], where the right value of ϙ̂= 1/8was obtained [22]) but not inφ3
ones [15].
In this section we have developed a new general method which avoids the previous misidentification of
ξ. In particular, we have obtained the correct value of ϙ̂ = 1/6 for the general class of φ3
theories, see
above.
10
Since, in this section, we work with the susceptibility, we take into account only the term proportional to f in equation (3.18)
independently of the value of α and φ. See discussion of section 3.
13601-6
Revisiting (logarithmic) scaling relations
In [23] it was conjectured that there is a relationship between ϙ̂ and 1/du which is frequently an
equality but not always so. Indeed, It was already known [24] that ϙ̂= 1/8 for the four dimensional Ising
model which is described by a φ4
theory which has du = 4. In this paper we have provided the general
relation between ϙ̂ and du .
To finish this section, we present two examples in which ϙ̂ , 1/du to understand the reasons behind
the modification of this behavior. The first one is based on the study of φ2k
-theories with k > 2 and the
second one is the two parameter φ4
-theory which describes the four dimensional diluted Ising model.
5.1. φ2k
-theories with k > 2 and short range interactions
The upper critical dimension for these models is du = 2k/(k −1). One can compute the RG equations
at du obtaining [5]
dg2k
dlogb
∝ g 2
2k (5.9)
and so g2k ∝ 1/logL (r = 1), that using equation (5.6) provides ϙ̂ = 1/(2k) which is different to ϙ̂ = (k −
1)/(2k) (only works for k = 2).11
5.2. Diluted Ising model
One can obtain an effective field theoretical version of the diluted Ising model by using the replica
trick, with effective Hamiltonian given by [22, 25]
Heff[φi ] =
∫
dd x
[
1
2
n∑
i=1
(
∂µφi
)2 + r
2
n∑
i=1
φ2
i +
u
4!
(
n∑
i=1
φ2
i
)2
+ v
4!
n∑
i=1
φ4
i
]
, (5.10)
where v is related with the original Ising coupling and u is a function of the disorder strength. In the
replica trick it is mandatory to take the limit of the number of replicas, n, to zero (i = 1, . . . ,n). The RG
equations are, in d = 4 and n = 0,
dr
dlogb
= 2r +4(2u +3v)(1− r ) , (5.11)
dv
dlogb
= −12v(4u +3v) , (5.12)
du
dlogb
= −8u(4u +3v) . (5.13)
In the standard φ4
theory one gets β∝ g 2
. Hence, g ∝ 1/logL and ϙ̂= 1/4. However, the RG flow of the
diluted model asymptotically finishes on the line 4u +3v =O(u2), so we need to include the next (cubic)
terms in the perturbative expression and the RG β-functions are no longer quadratic in the couplings.
Finally, one finds that u(b)2 ∼ v(b)2 ∼ 1/logb: hence, ϙ̂= 1/8 as derived in [22, 24].
6. Conclusions
By explicitly computing the hatted critical exponents for a wide family of models we have been able
to check the scaling relations among them using the RG framework and the behavior in the mean field
regime. Some of these hatted exponents (for some of the models) have been previously derived by using
the logarithm scaling relations.
In addition, we have generalized a conjecture regarding a relationship between ϙ̂ and du and de-
rived it.
Finally, we have found a new method to derive the scaling relation associated with ϙ̂ and we have
briefly discussed the logarithmic corrections to the free energy when the Fisher zeros have an impact
angle other than π/4 and α= 0.
11
In addition, working at du for short range models, η= 0 and so c = 0.
13601-7
J.J. Ruiz-Lorenzo
Acknowledgements
I dedicate this paper to Y. Holovatch to celebrate his 60th birthday.
I acknowledge interesting discussions with R. Kenna, B. Berche andM. Dudka. This workwas partially
supported by Ministerio de Economía y Competitividad (Spain) through Grants No. FIS2013-42840-P and
FIS2016-76359-P (partially funded by FEDER) and by Junta de Extremadura (Spain) through Grant No.
GRU10158 (partially funded by FEDER).
A. Appendix
In this appendix we give additional details of the computation of the hatted exponents, see section 3.
By differentiating once the free energy (3.1) with respect to the magnetic field, then renormalizing to
t (b∗) = 1, and finally evaluating the magnetization using the mean field behavior (2.2), we obtain
m ∝ (b∗)−d (b∗)d/2+1 exp
−1
2
g (b∗)∫
g0
dg
γ(g )
βW(g )
1
g (b∗)pm
∼ (b∗)−d/2+1+c (logb∗)x+pm r . (A.1)
The susceptibility is obtained by differentiating twice the free energy with respect to the magnetic field
[notice that there is no dependence on g in the mean field region (2.4)]:
χ∝ (b∗)−d (b∗)d+2 exp
− g (b∗)∫
g0
dg
γ(g )
βW(g )
∼ (b∗)2+2c (logb∗)2x . (A.2)
To obtain the specific heat, we differentiate twice the free energy with respect to the temperature, renor-
malize to t (b∗) = 1, and evaluate the specific heat using the mean field behavior (1.2), obtaining
C ∝ (b∗)−d (b∗)2σ exp
2
g (b∗)∫
g0
dg
γ(g )
βW(g )
1
g (b∗)pc
∼ (b∗)−d+2σ+2a(logb∗)2p+r pc . (A.3)
The correlation length is obtained from t (b∗) = 1
ξ∝ b∗ ∼ t−1/(σ+a)
0 (log t0)−p/(σ+a) . (A.4)
By putting the previous relation between b∗
and t0 in equations (A.1)–(A.3) and matching the l.h.s. log-
arithms [given by equations (1.2)–(1.4)] with the r.h.s. ones [given by equations (A.1)–(A.3)] we obtain
equations (3.12)–(3.14).
To compute the Lee-Yang edge, the starting point is the renormalized potential [5]
V
(
t (b), g (b),h(b)
)= t (b)
2
m2 + g (b)
n
mn −h(b)m . (A.5)
From the constraints ∂V /∂m = 0 and ∂2V /∂m2 = 0 and working in the broken phase with t (b∗) =−1, it
is possible to show that
h(b∗) ∼ m(b∗) ∼ 1/g (b∗)pm , (A.6)
that can be written as
h(b∗) = h0(b∗)
d
2 +1 exp
−1
2
g (b∗)∫
g0
dg
γ(g )
βW(g )
∼ h0(b∗)
d
2 +1+c (logb∗)x ∼ (logb∗)r pm , (A.7)
which allows us to compute h0 as a function of b∗
, and knowing b∗(t0) (A.4), we can easily obtain h0(t0).
The comparison of the logarithm of h0(t0) with that of equation (1.6) provides us with relation (3.16).
13601-8
Revisiting (logarithmic) scaling relations
Relation (3.17) can be obtained taking the Fourier transform of equation (1.7) at b ∼ 1/q (q being the
momentum), and comparing this with the renormalized propagator in momentum space (see [6]).
Finally, for the critical isotherm, we start with the free energy computed at the critical point
f (0,h0, g0), differentiate once with respect to h0 to compute the critical magnetization, then renormalize
to h(b∗) = 1 and use the mean field behavior of the critical magnetization (2.7), obtaining
m ∝ (b∗)−d h(b∗)
h0
[
h(b∗)
g (b∗)
]ph
∼ (b∗)−d 1
h0
g (b∗)−ph ∼ (b∗)−d 1
h0
(logb∗)r ph , (A.8)
where h(b∗) = 1
b∗ ∼ h−2/(d+2+2c)
0 (logh0)−2x/(d+2+2c) . (A.9)
By matching the l.h.s. and r.h.s. logarithms of equations (1.5) and (A.8), respectively, we obtain the rela-
tion (3.15).
Remember that 2c =−η, ν= 1/(σ+a) and ν̂=−pν.
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J.J. Ruiz-Lorenzo
Перегляд (логарифмiчних) спiввiдношень скейлiнгу з
використанням ренормгрупи
Х.Х. Руiс-Лоренсо1,2,3
1 Фiзичний факультет, Унiверситет Екстремадури, 06071 Бадахос, Iспанiя
2 Iнститут передових наукових обчислень (ICCAEx), Унiверситет Екстремадури, 06071 м. Бадахос, Iспанiя
3 Iнститут бiообчислень i фiзики складних систем (BIFI), м. Сарагоса, Iспанiя
Ми явно обчислюємо критичнi показники, пов’язанi з логарифмiчними поправками, виходячи з рiвнянь
ренормгрупи i середньопольової поведiнки для широкого класу моделей як при вищiй критичнiй вимiр-
ностi (для коротко- i далекосяжних φn -теорiй), так i нижче вiд неї. Це дозволяє нам перевiрити спiввiдно-
шення скейлiнгу, що пов’язують критичнi показники, аналiзуючи комплекснi сингулярностi (нулi Лi-Янга
i Фiшера) цих моделей. Окрiм того, ми запропонували явний метод для обчислення показника ϙ̂ [озна-
ченого як ξ ∼ L(logL)ϙ̂] i, накiнець, ми отримали нове виведення закона скейлiнгу, пов’язаного з цим
показником.
Ключовi слова: ренормгрупа, скейлiнг, логарифми, середнє поле
13601-10
Introduction
Some mean field results
Revisiting logarithmic corrections
A re-derivation of = d -d
Computation of the -exponent
2 k-theories with k>2 and short range interactions
Diluted Ising model
Conclusions
Appendix
|