Critical thermodynamics of two-dimensional N-vector cubic model in the five-loop approximation
The critical behavior of the two-dimensional N-vector cubic model is studied within the field-theoretical renormalization-group (RG) approach. The β functions and critical exponents are calculated in the five-loop approximation, RG series obtained are resummed using Pade-Borel-Leroy and ´ confor...
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | Critical thermodynamics of two-dimensional N-vector cubic model in the five-loop approximation / P. Calabrese, E.V. Orlov, D.V. Pakhnin, A.I. Sokolov // Condensed Matter Physics. — 2005. — Т. 8, № 1(41). — С. 193–211. — Бібліогр.: 31 назв. — англ. |
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irk-123456789-1194832017-06-08T03:04:21Z Critical thermodynamics of two-dimensional N-vector cubic model in the five-loop approximation Calabrese, P. Orlov, E.V. Pakhnin, D.V. Sokolov, A.I. The critical behavior of the two-dimensional N-vector cubic model is studied within the field-theoretical renormalization-group (RG) approach. The β functions and critical exponents are calculated in the five-loop approximation, RG series obtained are resummed using Pade-Borel-Leroy and ´ conformal mapping techniques. It is found that for N = 2 the continuous line of fixed points is well reproduced by the resummed RG series and an account for the five-loop terms makes the lines of zeros of both β functions closer to each other. For N > 3 the five-loop contributions are shown to shift the cubic fixed point, given by the four-loop approximation, towards the Ising fixed point. This confirms the idea that the existence of the cubic fixed point in two dimensions under N >2 is an artifact of the perturbative analysis. In the case N = 0 the results obtained are compatible with the conclusion that the impure critical behavior is controlled by the Ising fixed point. В рамках теоретико-польового підходу ренормалізаційної групи (РГ) вивчається критична поведінка двовимірної N-векторної кубічної моделі. β функції і критичні показники обчислюються в п’ятипетлевому наближенні, отримані РГ ряди пересумовуються з використанням техніки Паде-Бореля-Лєруа і конформного перетворення. Знайдено, що для N = 2 неперервна лінія нерухомих точок добре відтворюється пересумованими РГ рядами і врахування п’ятипетлевих членів робить лінії нулів обох β функцій ближчими один до одного. Показано, що для N > 3 п’яти-петлеві внески зсувають кубічну нерухому точку, отриману в чотири-петлевому наближенні, до нерухомої точки Ізинґа. Це підтверджує ідею, що існування кубічної нерухомої точки в двох вимірах під N > 2 є результатом пертурбативного аналізу. У випадку N = 0 отримані результати є сумісні з висновком, що критична поведінка, пов’язана з домішками, контролюється нерухомою точкою Ізинґа. 2005 Article Critical thermodynamics of two-dimensional N-vector cubic model in the five-loop approximation / P. Calabrese, E.V. Orlov, D.V. Pakhnin, A.I. Sokolov // Condensed Matter Physics. — 2005. — Т. 8, № 1(41). — С. 193–211. — Бібліогр.: 31 назв. — англ. 1607-324X PACS: 75.10.Hk, 05.70.Jk, 64.60.Fr, 11.10.Kk DOI:10.5488/CMP.8.1.193 http://dspace.nbuv.gov.ua/handle/123456789/119483 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
The critical behavior of the two-dimensional N-vector cubic model is studied
within the field-theoretical renormalization-group (RG) approach. The
β functions and critical exponents are calculated in the five-loop approximation,
RG series obtained are resummed using Pade-Borel-Leroy and ´
conformal mapping techniques. It is found that for N = 2 the continuous
line of fixed points is well reproduced by the resummed RG series and an
account for the five-loop terms makes the lines of zeros of both β functions
closer to each other. For N > 3 the five-loop contributions are shown to
shift the cubic fixed point, given by the four-loop approximation, towards
the Ising fixed point. This confirms the idea that the existence of the cubic
fixed point in two dimensions under N >2 is an artifact of the perturbative
analysis. In the case N = 0 the results obtained are compatible with the
conclusion that the impure critical behavior is controlled by the Ising fixed
point. |
| format |
Article |
| author |
Calabrese, P. Orlov, E.V. Pakhnin, D.V. Sokolov, A.I. |
| spellingShingle |
Calabrese, P. Orlov, E.V. Pakhnin, D.V. Sokolov, A.I. Critical thermodynamics of two-dimensional N-vector cubic model in the five-loop approximation Condensed Matter Physics |
| author_facet |
Calabrese, P. Orlov, E.V. Pakhnin, D.V. Sokolov, A.I. |
| author_sort |
Calabrese, P. |
| title |
Critical thermodynamics of two-dimensional N-vector cubic model in the five-loop approximation |
| title_short |
Critical thermodynamics of two-dimensional N-vector cubic model in the five-loop approximation |
| title_full |
Critical thermodynamics of two-dimensional N-vector cubic model in the five-loop approximation |
| title_fullStr |
Critical thermodynamics of two-dimensional N-vector cubic model in the five-loop approximation |
| title_full_unstemmed |
Critical thermodynamics of two-dimensional N-vector cubic model in the five-loop approximation |
| title_sort |
critical thermodynamics of two-dimensional n-vector cubic model in the five-loop approximation |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119483 |
| citation_txt |
Critical thermodynamics of two-dimensional N-vector cubic model in the five-loop approximation / P. Calabrese, E.V. Orlov, D.V. Pakhnin, A.I. Sokolov // Condensed Matter Physics. — 2005. — Т. 8, № 1(41). — С. 193–211. — Бібліогр.: 31 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT calabresep criticalthermodynamicsoftwodimensionalnvectorcubicmodelinthefiveloopapproximation AT orlovev criticalthermodynamicsoftwodimensionalnvectorcubicmodelinthefiveloopapproximation AT pakhnindv criticalthermodynamicsoftwodimensionalnvectorcubicmodelinthefiveloopapproximation AT sokolovai criticalthermodynamicsoftwodimensionalnvectorcubicmodelinthefiveloopapproximation |
| first_indexed |
2025-07-08T15:57:13Z |
| last_indexed |
2025-07-08T15:57:13Z |
| _version_ |
1837094912872415232 |
| fulltext |
Condensed Matter Physics, 2005, Vol. 8, No. 1(41), pp. 193–211
Critical thermodynamics of
two-dimensional N -vector cubic model
in the five-loop approximation
P.Calabrese∗1 , E.V.Orlov 2 , D.V.Pakhnin 2 , A.I.Sokolov† 2
1 Scuola Normale Superiore and INFN,
Piazza dei Cavalieri 7, I–56126 Pisa, Italy
2 Department of Physical Electronics,
Saint Petersburg Electrotechnical University,
Professor Popov Street 5, St. Petersburg 197376, Russia
Received November 12, 2004
The critical behavior of the two-dimensional N -vector cubic model is stud-
ied within the field-theoretical renormalization-group (RG) approach. The
β functions and critical exponents are calculated in the five-loop approx-
imation, RG series obtained are resummed using Padé-Borel-Leroy and
conformal mapping techniques. It is found that for N = 2 the continuous
line of fixed points is well reproduced by the resummed RG series and an
account for the five-loop terms makes the lines of zeros of both β functions
closer to each other. For N > 3 the five-loop contributions are shown to
shift the cubic fixed point, given by the four-loop approximation, towards
the Ising fixed point. This confirms the idea that the existence of the cubic
fixed point in two dimensions under N >2 is an artifact of the perturbative
analysis. In the case N = 0 the results obtained are compatible with the
conclusion that the impure critical behavior is controlled by the Ising fixed
point.
Key words: renormalization group expansions, 2D cubic model
PACS: 75.10.Hk, 05.70.Jk, 64.60.Fr, 11.10.Kk
1. Introduction
The two-dimensional (2D) model with N -vector order parameter and cubic ani-
sotropy is known to have a rich phase diagram; it contains, under different values of
N and of the anisotropy parameter, the Ising-like and Kosterlitz-Thouless critical
points, lines of the first-order phase transitions, and the line of the second-order
transitions with continuously varying critical exponents (see, e. g. [1–3] for review).
∗E-mail: calabres@df.unipi.it
†E-mail: ais2002@mail.ru
c© P.Calabrese, E.V.Orlov, D.V.Pakhnin, A.I.Sokolov 193
P.Calabrese et al.
This model is related to many other familiar models in various particular cases,
while for N → 0 it describes the critical behavior of 2D weakly disordered Ising
systems. Moreover, exact solutions are known for the 2D cubic model in the several
limits such as an Ising decoupled limit, the limit of extremely strong anisotropy
for N > 2 [2,4] and the replica limit N → 0 [2,5,6]. The mappings, in particular
regions of the phase diagram, with the N -color Ashkin-Teller models, discrete cubic
models, and planar model with fourth order anisotropy give further information
about the critical behavior. All these issues are reviewed in [3] and we will not
repeat here. These features make the 2D N -vector cubic model a convenient and,
perhaps, unique testbed for evaluation of the analytical and numerical power of
perturbative methods widely used nowadays in the theory of critical phenomena.
The field-theoretical renormalization-group (RG) approach in physical dimensions
is among of them.
Recently, the critical behavior of the 2D N -vector cubic model was explored using
the renormalization-group technique in the space of fixed dimensionality [3]. The
four-loop expansions for the β-functions and critical exponents were calculated and
analyzed using the Borel transformation combined with the conformal mapping and
Padé-approximant techniques as a tool for resummation of the divergent RG series.
The most part of predictions obtained within the renormalization group approach
turned out to be in accord with the known exact results. At the same time, some
findings were quite new. In particular, for N > 2 the resummed four-loop RG
expansions for β-functions were found to yield a cubic fixed point with (almost)
marginal stability; this point does not correspond to any of the critical asymptotes
revealed by exact methods ever applied. Although the stability properties of the
cubic fixed point look very similar to those of its Ising counterpart, these points
were found to lie too far from each other (for moderate N) to consider the distance
between them as a splitting caused by the limited accuracy of the RG approximation
employed.
It is worth noting that this situation is quite different from what we have in three
dimensions. Indeed, for the 3D cubic model the structure of the RG flow diagram is
known today with a rather high accuracy. Recent five-loop [7–9] and six-loop [10,11]
RG calculations certainly confirmed that for N > 2 the cubic fixed point does not
merge with any other fixed point and governs the specific anisotropic mode of critical
behavior, distinguishable from the Ising and Heisenberg modes (see, e. g. [12]).
It is very desirable, therefore, to clear up to what extent the location of the cubic
fixed point in two dimensions is sensitive to the order of the RG calculations and,
more generally, whether this point really exists at the flow diagram or its appearance
is the approximation artifact caused by the finiteness of the perturbative series and
by an ignorance of the confluent singularities significant in two dimensions [13–16].
Of prime interest is also the situation with the line of fixed points that should run,
under N = 2, from the Ising fixed point to the Heisenberg one. Within the four-loop
approximation, the zeros of β-functions for the O(N)-symmetric and anisotropic
coupling constants form two lines that for N = 2 are practically parallel to each
other and separated by the distance that is smaller than the error bar appropriate
194
Critical thermodynamics of two-dimensional N -vector cubic model
to the working approximation [3]. Will the higher-order contributions keep these
two lines parallel? Will an account for the higher-order terms further diminish the
distance between these lines or their splitting should be attributed, at least partially,
to the effect of the singular terms just mentioned?
To answer the above questions, it is necessary to analyze the critical behavior of
the 2D cubic model in the higher perturbative orders. Recently, the renormalization-
group expansions for the 2D O(N)-symmetric model were obtained within the five-
loop approximation [15]. In the course of this study, all the integrals corresponding to
the five-loop four-leg and two-leg Feynman graphs have been evaluated. This makes
it possible to investigate the critical thermodynamics of anisotropic 2D models with
several couplings in the five-loop approximation. In this paper, such an investigation
will be carried out for the 2D N -vector model with cubic anisotropy.
2. Renormalization group expansions
In order to study the effect of cubic anisotropies one usually considers the φ4
theory [17,18]:
H =
∫
ddx
{
1
2
N
∑
i=1
[
(∂µφi)
2 + rφ2
i
]
+
1
4!
N
∑
i,j=1
(u0 + v0δij) φ2
i φ
2
j
}
, (1)
in which the added cubic term breaks explicitly the O(N) invariance leaving a resid-
ual discrete cubic symmetry given by the reflections and permutations of the field
components. In two dimensions the effect of anisotropy is particularly important:
systems possessing continuous symmetry do not exhibit conventional long-range or-
der at finite temperature, while models with discrete symmetry do undergo phase
transitions into conventionally ordered phase.
In general, the model (1) has four fixed points: the trivial Gaussian one, the
Ising one in which the N components of the field decouple, the O(N)-symmetric
and the cubic fixed points. The Gaussian fixed point is always unstable, and so
is the Ising fixed point for d > 2 [17]. Indeed, in the latter case, it is natural
to interpret equation (1) as the Hamiltonian of N Ising-like systems coupled by
the O(N)-symmetric term. But this interaction is the sum of the products of the
energy operators of the different Ising systems. Therefore, at the Ising fixed point,
the crossover exponent associated with the O(N)-symmetric quartic term should be
given by the specific-heat critical exponent αI of the Ising model, independently of
N . Since αI is positive for all d > 2 the Ising fixed point is unstable. Obviously,
in two dimensions this argument only told us that the crossover exponent at this
fixed point vanishes. Higher order corrections to RG equation may lead either to
a marginally stable fixed point or to a line of fixed points. It was argued that for
N > 3 the first possibility is realized, while for N = 2 the second one holds (see
[1–3] and references therein).
The stability properties of the O(N)-symmetric and of the cubic fixed points
depend on N . For sufficiently small values of N , N < Nc, the O(N)-symmetric
195
P.Calabrese et al.
fixed point is stable and the cubic one is unstable. For N > Nc, the opposite is
true: the renormalization-group flow is driven towards the cubic fixed point, which
now describes the generic critical behavior of the system. At N = Nc, the two fixed
points should coincide for d > 2. At d = 2, it is expected that Nc = 2 and a line of
fixed points connecting the Ising and the O(2)-symmetric fixed points exists [1,3].
The fixed-dimension field-theoretical approach is known to be a powerful tool in
studying the critical properties of three-dimensional systems belonging to the O(N)
and more complicated universality classes (see, e.g., [18–20]). In this approach one
performs an expansion in powers of appropriately defined zero-momentum quartic
couplings and renormalizes the theory by a set of zero-momentum conditions for the
(one-particle irreducible) two-point and four-point correlation functions:
Γ
(2)
ab (p) = δabZ
−1
φ
[
m2 + p2 + O(p4)
]
, (2)
Γ
(4)
abcd(0) = Z−2
φ m2
[u
3
(δabδcd + δacδbd + δadδbc) + v δabδacδad
]
. (3)
They relate the inverse correlation length (mass) m and the zero-momentum quartic
couplings u and v to the corresponding Hamiltonian parameters r, u0, and v0:
u0 = m2uZuZ
−2
φ , v0 = m2vZvZ
−2
φ . (4)
In addition, one introduces the function Zt defined by the relation
Γ
(1,2)
ab (0) = δabZ
−1
t , (5)
where Γ(1,2) is the (one-particle irreducible) two-point function with an insertion
of φ2/2.
From the pertubative expansions of the correlation functions Γ(2), Γ(4), and Γ(1,2)
and the above relations, one derives the functions Zφ(u, v), Zu(u, v), Zv(u, v), and
Zt(u, v) as double expansions in u and v.
The fixed points of the theory are given by the common zeros of the β-functions
βu(u, v) = m
∂u
∂m
∣
∣
∣
∣
u0,v0
, βv(u, v) = m
∂v
∂m
∣
∣
∣
∣
u0,v0
. (6)
The stability properties of the fixed points are controlled by the eigenvalues ωi of
the matrix
Ω =
∂βu(u, v)
∂u
∂βu(u, v)
∂v
∂βv(u, v)
∂u
∂βv(u, v)
∂v
, (7)
computed at the given fixed point: a fixed point is stable if both eigenvalues are
positive. The eigenvalues ωi are related to the leading scaling corrections, which
vanish as ξ−ωi ∼ |t|∆i where ∆i = νωi.
One also introduces the functions
ηφ,t(u, v) =
∂ ln Zφ,t
∂ ln m
∣
∣
∣
∣
u0,v0
= βu
∂ ln Zφ,t
∂u
+ βv
∂ ln Zφ,t
∂v
, (8)
196
Critical thermodynamics of two-dimensional N -vector cubic model
so that the critical exponents are obtained from
η = ηφ(u
∗, v∗), (9)
ν = [2 − ηφ(u
∗, v∗) + ηt(u
∗, v∗)]−1 , (10)
γ = ν(2 − η), (11)
where (u∗, v∗) is the position of the stable fixed point.
Here, we present the perturbative expansions of the RG functions (6) and (8) up
to five loops. The results are written in terms of the rescaled couplings
u ≡
8π
3
RN u, v ≡
8π
3
v, (12)
where RN = 9/(8 + N).
Table 1. The coefficients b
(u)
ij , cf. equation (13).
i, j (N + 8)ib
(u)
ij
2,0 −47.6751 − 10.335 N
1,1 −8.39029
0,2 −0.21608
3,0 524.377 + 149.152 N + 5.00028 N2
2,1 144.813 + 7.27755 N
1,2 10.0109 + 0.0583278 N
0,3 0.231566
4,0 −7591.11 − 2611.15 N − 179.698 N2 − 0.088843 N3
3,1 −2872.08 − 291.254 N + 0.126813 N2
2,2 −330.599 − 5.97086 N
1,3 −16.0559 − 0.0578955 N
0,4 −0.311695
5,0 133972. + 53218.6 N + 5253.56 N2 + 80.3097 N3 − 0.0040796 N4
4,1 64819.7 + 9554.31 N + 164.916 N2 + 0.145241 N3
3,2 10584.2 + 439.25 N + 1.29693 N2
2,3 818.21 + 8.3695 N
1,4 32.7458 + 0.0796603 N
0,5 0.555161
The resulting series are
β̄u = −u + u2 +
2
3
uv + u
∑
i+j>2
b
(u)
ij uivj, (13)
β̄v = −v + v2 +
12
8 + N
uv + v
∑
i+j>2
b
(v)
ij uivj , (14)
197
P.Calabrese et al.
ηφ =
∑
i+j>2
e
(φ)
ij uivj , (15)
ηt = −
2(2 + N)
(8 + N)
u −
2
3
v +
∑
i+j>2
e
(t)
ij uivj , (16)
where
β̄u =
3
16π
R−1
N βu, β̄v =
3
16π
βv . (17)
The coefficients b
(u)
ij , b
(v)
ij , e
(φ)
ij , and e
(t)
ij are reported in the tables 1, 2, 3, and 4. Note
that due to the rescaling (17), the matrix element of Ω are two times the derivative
of β̄ with respect to u and v.
Table 2. The coefficients b
(v)
ij , cf. equation (14).
i, j (N + 8)ib
(v)
ij
2,0 −92.6834 − 5.83417 N
1,1 −17.392
0,2 −0.716174
3,0 1228.63 + 118.503 N − 1.83156 N2
2,1 358.882 + 2.84758 N
1,2 31.4235
0,3 0.930766
4,0 −20723.1 − 2692. N − 25.4854 N2 − 0.824655 N3
3,1 −8273.27 − 233.78 N + 0.574757 N2
2,2 −1134.8 − 1.91402 N
1,3 −68.4022
0,4 −1.58239
5,0 414915. + 67526.8 N + 1868.92 N2 − 13.7618 N3 − 0.4602 N4
4,1 211041. + 10633.2 N − 22.0443 N2 + 0.044688 N3
3,2 39732.3 + 365.816 N − 0.399537 N2
2,3 3666.92 − 0.93257 N
1,4 171.066
0,5 3.26042
We have verified the exactness of our series by the following relations:
(i) β̄u(u, 0), ηφ(u, 0) and ηt(u, 0) reproduce the corresponding functions of the
O(N)-symmetric model [15,21].
(ii) β̄v(0, v), ηφ(0, v) and ηt(0, v) reproduce the corresponding functions of the
Ising-like (N = 1) φ4 theory.
198
Critical thermodynamics of two-dimensional N -vector cubic model
Table 3. The coefficients e
(φ)
ij , cf. equation (15).
i, j (N + 8)ie
(φ)
ij
2,0 1.83417 + 0.917086 N
1,1 0.611391
0,2 0.0339661
3,0 −0.873744 − 0.54609 N − 0.054609 N2
2,1 −0.436872 − 0.054609 N
1,2 −0.054609
0,3 −0.00202255
4,0 41.5352 + 29.2512 N + 4.05641 N2 − 0.0926845 N3
3,1 27.6901 + 5.65571 N − 0.123579 N2
2,2 5.40424 + 0.1328 N
1,3 0.410151
0,4 0.0113931
5,0 −426.896 − 325.329 N − 57.7615 N2 − 1.0524 N3 − 0.07092 N4
4,1 −355.747 − 93.2339 N − 1.5176 N2 − 0.1182 N3
3,2 −94.4586 − 5.65221 N − 0.0262601 N2
2,3 −11.0421 − 0.084259 N
1,4 −0.61813
0,5 −0.0137362
(iii) The following relation holds close to Heisenberg [22,23]
∂ηφ,t
∂v
∣
∣
∣
∣
(u,0)
=
N + 8
3(N + 2)
∂ηφ,t
∂u
∣
∣
∣
∣
(u,0)
, (18)
∂β̄v
∂v
∣
∣
∣
∣
(u,0)
+
3(N + 2)
N + 8
∂β̄u
∂v
∣
∣
∣
∣
(u,0)
=
∂β̄u
∂u
∣
∣
∣
∣
(u,0)
, (19)
and Ising fixed points [23]
RN
(
∂β̄v
∂v
∣
∣
∣
∣
(0,v)
−
∂β̄u
∂u
∣
∣
∣
∣
(0,v)
)
=
∂β̄v
∂u
∣
∣
∣
∣
(0,v)
, (20)
RN
∂ηφ
∂v
∣
∣
∣
∣
(0,v)
=
∂ηφ
∂u
∣
∣
∣
∣
(0,v)
. (21)
No analog of such relations exists close to the Ising fixed point for ηt.
(iv) The following relations hold for N = 1:
β̄u(u, x− u) + β̄v(u, x− u) = β̄v(0, x),
199
P.Calabrese et al.
Table 4. The coefficients e
(t)
ij , cf. equation (16).
i, j (N + 8)ie
(φ)
ij
2,0 13.5025 + 6.751258N
1,1 4.50084
0,2 0.250047
3,0 −96.7105 − 65.1686 N − 8.40668 N2
2,1 −48.3553 − 8.40668 N
1,2 −6.19023 − 0.116656 N
0,3 −0.233588
4,0 1135.04 + 844.5 N + 139.656 N2 + 0.583377 N3
3,1 756.697 + 184.652 N + 0.777836 N2
2,2 149.468 + 7.55346 N
1,3 11.5154 + 0.115791 N
0,4 0.323089
5,0 −16885.3 − 13691.4 N − 2885.83 N2 − 130.427 N3 + 0.14672 N4
4,1 −14071.1 − 4373.98 N − 217.868 N2 + 0.244533 N3
3,2 −3777.55 − 367.372 N − 2.33704 N2
2,3 −449.218 − 11.589 N
1,4 −25.4411 − 0.159321 N
0,5 −0.568897
ηφ(u, x − u) = ηφ(0, x),
ηt(u, x − u) = ηt(0, x). (22)
(v) For N = 2, one can easily obtain the identities [24,3]
β̄u(u +
5
3
v,−v) +
5
3
β̄v(u +
5
3
v,−v) = β̄u(u, v),
β̄v(u +
5
3
v,−v) = −β̄v(u, v),
ηφ(u +
5
3
v,−v) = ηφ(u, v),
ηt(u +
5
3
v,−v) = ηt(u, v). (23)
These relations are also exactly satisfied by our five-loop series. Note that,
since the Ising fixed point is (0, g∗
I ), and g∗
I is known with high precision [25]
g∗
I = 1.7543637(25), (24)
the above symmetry gives us the location of the cubic fixed point: ((5/3)g∗
I ,−g∗
I ).
(vi) In the large-N limit the critical exponents of the cubic fixed point are related
to those of the Ising model: η = ηI and ν = νI. One can easily see that,
200
Critical thermodynamics of two-dimensional N -vector cubic model
for N → ∞, ηφ(u, v) = ηI(v), where ηI(v) is the perturbative series that
determines the exponent η of the Ising model. Therefore, the first relation is
trivially true. On the other hand, the second relation ν = νI is not identically
satisfied by the series, and is verified only at the critical point [10].
(vii) The series reproduces the previous four-loop results [3].
The obtained RG series are asymptotic and some resummation procedure is
needed to extract accurate numerical values for the physical quantities. Exploiting
the property of Borel summability of φ4 theories in two and three dimensions, we
resum the divergent asymptotic series by a Borel transformation combined with a
method for the analytic extension of the Borel transform. This last procedure can
be obtained by a Padé extension or by a conformal mapping [26] which maps the
domain of analyticity of the Borel transform onto a circle (see [19,26] for details).
The conformal mapping method takes advantage of the knowledge of the large
order behavior of the perturbative series F (u, z) =
∑
k fk(z)uk [3,10]
fk(z) ∼ k! (−a(z))k kb
[
1 + O(k−1)
]
with a(z) = −1/ub(z), (25)
where ub(z) is the singularity of the Borel transform closest to the origin at fixed
z = v̄/ū, given by [3]
1
ub(z)
= −a (RN + z) for 0 < z, (26)
1
ub(z)
= −a
(
RN +
1
N
z
)
for 0 > z > −
2NRN
N + 1
,
where a = 0.238659217 . . .
It should be noted that these results do not apply to the case N = 0. Indeed, in
this case, additional singularities in the Borel transform are expected [27].
For each perturbative series R(ū, v̄), we obtain the following approximants
E(R)p(α, b; u, v) =
p
∑
k=0
Bk(α, b; v/u) ×
∫ ∞
0
dt tbe−t y(ut; v/u)k
[1 − y(ut; v/u)]α
, (27)
where
y(x; z) =
√
1 − x/ub(z) − 1
√
1 − x/ub(z) + 1
, (28)
and the coefficients Bk are determined by the condition that the expansion of
E(R)p(α, b; u, v) in powers of u and v gives R(u, v) to order p.
Within the second resummation procedure, the Borel-Leroy transform is analyt-
ically extended by means of a generalized Padé approximant technique, using the
resolvent series trick (see, e. g. [9]). Explicitly, once introduced the resolvent series
of the perturbative one R(ū, v̄)
P̃ (R)(u, v, b, λ) =
∑
n
λn
n
∑
k=0
P̃k,n
(n + b)!
un−kvk , (29)
201
P.Calabrese et al.
which is a series in powers of λ with coefficients being uniform polynomials in u, v.
The analytical continuation of the Borel transform is the Padè approximant [N/M ]
in λ at λ = 1. Obviously, the approximant for each perturbative series depends on
the chosen Padé approximant and on the parameter b.
An important issue in the fixed dimension approach to critical phenomena (and
in general of all the field theoretical methods) concerns the analytic properties of
the β-functions. As shown in [16] for the O(N) model, the presence of confluent
singularities in the zero of the perturbative β function causes a slow convergence of
the results given by the resummation of the perturbative series to the correct fixed
point value. The O(N) two-dimensional field-theory estimates of physical quantities
[15,26] are less accurate than the three-dimensional ones due to the stronger non-
analyticities at the fixed point [13,14,16], to say nothing about the stronger growth
of the series coefficients themselves. In [16] it is shown that the nonanalytic terms
may cause large imprecisions in the estimate of the exponent related to the leading
correction to the scaling ω. At the same time, the result for the fixed point location
turns out to be a rather good approximation of the accurate one.
3. Analysis of five loop series
3.1. Stability of O(N) and Ising fixed points
First of all, we analyze the stability properties of the O(N)-symmetric fixed point.
Since ∂uβv|(u,0) = 0, the stability of the fixed point with respect to an anisotropic
cubic perturbation is given by
ω2 = 2
∂β̄v
∂v
(u∗, 0), (30)
where u∗ is the fixed-point value of the O(N) vector model. In table 5 we report
the results for ω2 for several values of N . The O(N) fixed point results unstable for
N > 3. For N = 2 our result is compatible with ω2 = 0, which is essential in the
context of a continuous line of the fixed points expected.
Table 5. Half of the exponent ω2 at the O(N) fixed point. CM is the value
obtained using conformal mapping technique and PB the one using Padé-Borel.
N CM 4-loop PB 4-loop CM5-loop PB 5-loop
2 0.03(3) 0.06(4) 0.025(40) 0.00(5)
3 −0.08(3) −0.07(3) −0.10(6) −0.10(5)
4 −0.18(4) −0.17(5) −0.20(3) −0.22(4)
8 −0.45(5) −0.44(6) −0.48(4) −0.50(5)
Then we focus our attention on the stability properties of the Ising fixed point.
Also in this case the stability is given by
ω2 = 2
∂β̄u
∂u
(0, v∗), (31)
202
Critical thermodynamics of two-dimensional N -vector cubic model
where v∗ is the fixed-point value of the Ising model (24). As expected, the series
ω2(v) is independent of N
ωI
2(v)
2
= −1 +
2
3
v − 0.2161v2 + 0.23157v3 − 0.31169v4 + 0.555161v5. (32)
The fixed point value of this exponent is ωI
2/2 = −0.09(8), using the conformal
mapping method, and −0.08(10) using the Padé-Borel analysis. We note that the
[4/1] approximant with b = 1 leads to ωI
2/2 = −0.031. These values are compatible
with the exact known result αI = 0.
3.2. Analysis for N > 3
We first analyze the position of the cubic fixed point for N > 3, previously found
in the four-loop approximation [3]. It was claimed in [3] that the quite peculiar
features of this fixed point (like the marginal instability) make its existence quite
doubtful and that it might be an artifact of the relatively short series available at
that time. Now we are in position to analyze longer series and to further confirm or
to reject this statement.
Table 6. Coordinates of the cubic fixed point in the 4-loop and 5-loop approxi-
mations.
N 4-loop 5-loop Padé [4/1] b=1
3 [0.83(12),1.12(9)] [0.54(6),1.35(4)] [0.050,1.757]
4 [0.54(10),1.43(8)] [0.32(5),1.58(4)] [0.031,1.774]
8 [0.24(8),1.72(10)] [0.14(4),1.74(4)] [0.015,1.788]
The results obtained with the conformal mapping methods are reported in table 6
together with the old four loop results, in order to make the comparison visible at
first sight. This table shows that the position of the cubic fixed point drastically
moves close to the Ising fixed point with increasing the order of perturbation theory
from four to five loops. Both at four and at five loops, the quoted errors are less than
the difference between the two estimates, leading to the conclusion that the reported
uncertainty is actually an underestimate of the correct one. We remind to the reader
that this error comes from the so-called stability criterion, i. e. it is obtained looking
at those approximants that minimize the difference between the estimates at the
two highest available orders. These considerable discrepancies between the four-
and five-loop results lead to serious doubts in the existence of the cubic fixed point
in two dimensions.
In order to better understand these strange results, we report the values of the
coordinate u∗ obtained for the cubic fixed point using several Padé approximants for
N = 0, 3, 4, 8; the estimates are presented in figure 1 as functions of b. Let us consider
first the case N = 3 as a typical example. If one limits himself with only three
lower-order approximants [2/1], [3/1], and [2/2], he easily finds that they minimize
203
P.Calabrese et al.
0 1 2 3
b
-0.4
-0.2
0
0.2
0.4
0.6
0.8
u*
[2/1]
[3/1]
[4/1]
[2/2]
[3/2]
N=4
0 0.5 1 1.5 2 2.5 3
b
-0.4
-0.2
0
0.2
0.4
0.6
0.8
u*
N=8
0 0.5 1 1.5 2 2.5 3
b
-0.4
-0.2
0
0.2
0.4
0.6
0.8
u*
RIM
0 1 2 3
b
-0.4
-0.2
0
0.2
0.4
0.6
0.8
u*
N=3
Figure 1. Coordinate u∗ of the cubic fixed point within Padé-Borel method for
some values of N.
their differences under b ∼ 0, leading in such a way to the estimate u∗ ∼ 0.7.
Taking into account two non-defective Padé approximants [4/1] and [3/2] (note the
oscillating behavior of the [3/2] approximant for b < 1, signaling the presence of
close singularities) existing at the five-loop level shifts the zone of stability to b ∼ 2,
thus leading to the estimate u∗ ∼ 0.5. Moreover, the approximant [4/1] with b = 1,
that is usually considered as one of the best approximants, results in the estimate
u∗ = 0.050 very close to zero. Because of the alternative character of RG expansions,
it looks very likely that the unknown six-loop contribution (and the higher-order
ones) will locate the stability region somewhere near b ∼ 1, leading finally to the
coalescence of the Ising and cubic fixed point. According to this scenario, the cubic
fixed point, found at finite order in perturbation theory, is probably only an artifact
due to the finiteness of the perturbative series.
The same scenario is also possible for other values of N . From figure 1 we see that
the region of maximum stability always shifts from b ∼ 0 to b ∼ 2 with increasing
the order of approximation from four to five loops, moving the coordinate of the
fixed point from u∗ ∼ 0.5 for N = 4 (u∗ ∼ 0.2 for N = 8) to u∗ ∼ 0.3 (u∗ ∼ 0.1). Let
us stress again that the value given by the approximant [4/1] with b = 1 is always
very close to zero, as is seen from table 6. Note that, with increasing N, the distance
of the cubic fixed point from the Ising one reduces fastly.
In the limit N → ∞ the series simplify as at the four-loop level (see [3]). We only
mention that with increasing the length of the RG series the coordinate u∗ of the
cubic fixed point shifts from u∗ ∼ 0.08 to u∗ ∼ 0.03 that again is much closer to zero.
204
Critical thermodynamics of two-dimensional N -vector cubic model
Finally, we briefly discuss the fate of the random fixed point governing the critical
behavior of the weakly-disordered Ising model which is described by the Hamiltonian
(1) in the replica limit N → 0. We do not use here advanced resummation procedures
developed [28] to avoid Borel non-summability at fixed u/v [27], but limit ourselves
by a simple Padé analysis, since it is sufficient for our aims. We find, for the majority
of the considered approximants, a fixed point with negative u∗. A possible estimate,
according to stability criteria is u∗ = −0.1(1), but if we concentrate on some certain
approximants we obtain u∗ = −0.090 for the [3/1] and u∗ = −0.030 for the [4/1]
(both with b = 1). In particular, the last value is very close to zero, i. e. to the
value predicted by the asymptotically exact solution that has been obtained in the
framework of the fermionic representation [2,5,6]. It is worth noting that the five-
loop results for N = 0 seem to be less scattered than analogous four-loop estimates
obtained by means of Chisholm-Borel resummation technique [29], and they look
more precise than their five-loop counterparts for finite N .
3.3. Analysis for N = 2
The four-loop analysis of [3] for N = 2 turned out to be compatible with the
presence of a line of the fixed points joining the O(2)-symmetric and the decoupled
Ising fixed points. The lines of zeros of the two β functions were found to be practi-
cally parallel and the quoted error was bigger than the distance between them. This
line of fixed points with continuously varying critical exponents is in agreement with
what is expected from the correspondence, at the critical point, between the cubic
model and the Ashkin-Teller and the planar model with fourth-order anisotropy
[1,3]. We are now in a position to verify this statement at the five-loop level.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
u
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
v
[3/1]
[4/1]
[2/2]
[3/2]
[3/1]
[4/1]
[2/2]
[3/2]
Figure 2. Zeros of βu (continuous lines) and βv (points) for several Padé approx-
imants (all with b = 1).
First, we analyze the series with the conformal mapping method. Again we find
that zeros of two β functions form two parallel lines, while the apparent uncertainties
seem to be smaller than their separation. Of course, this fact may simply indicate
205
P.Calabrese et al.
that the model has no fixed point at all. Let us, however, look more attentively at
this result and, in particular, at the accuracy of the quoted error. In fact, as we
have already seen for N > 2, the error coming from stability criteria is likely an
underestimate of the correct one. To better understand the situation, we use the
Padé-Borel method. In figure 2 we report the curves of zeros of the two β functions
given by several Padé approximants under b = 1, the value that for N > 2 always
leads to good results and that is the best for the fixed point values of O(N) and Ising
model [15]. The four approximants for βu are always well-defined. They are hardly
distinguishable close to the O(2) fixed point and their separation slowly increases
moving toward the v-axis. The coordinate of the Ising fixed point v ∼ 1.8 is obtained
using the [4/1] approximant, since approximants of [L-1/1] type proved to give rather
precise estimates for a fixed point location both in two and three [30,31] dimensions.
The situation is a bit worse for βv function. In fact, the working approximants are
well defined close to the Ising fixed point, but approaching the u axis they become
defective. The approximant [3/2] starts oscillating around u ∼ 0.8, while [4/1] is
bad in the range u ∈ [1, 1.5] and [3/1] for u > 1.3. Also the values of zeros of the
approximant [4/1] for u > 1.5 are not reliable enough since they may suffer from
the effect of close singularities.
Despite these shortcomings, we can obtain a rich information from figure 2.
Indeed, the line of zeros of βu given by the approximant [4/1] practically coincides
with those of the βv from the Ising fixed point up to u ∼ 0.8. For greater u, various
approximants for βv result in the lines of zeros that diverge leaving, however, the
line [4/1] of βu zeros between them. Keeping in mind a finite length of the RG series
and the effect of non-analytic terms missed by the perturbation theory, we retain
this fact as a strong evidence in favor of the continuous line of fixed points. The
best estimate for this line is believed to be that given by the approximant [4/1]
for βu. Thus it will be used in what follows to calculate continuous varying critical
exponents.
0 5 10 15 20 25
-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
[2/1]
[3/1]
[4/1]
[2/2]
Figure 3. Smallest eigenvalues of the Ω matrix along the line of fixed points
for several Padé approximant. The various curves corresponding to the same
approximant correspond to different choices of b from 0 to 1.
206
Critical thermodynamics of two-dimensional N -vector cubic model
First, we check that the smallest eigenvalues of the Ω matrix are compatible
with zero, which is a necessary condition for getting a line of fixed points. Some
Padé-Borel results are shown in figure 3. The figure seems somewhat complicated.
However, one can realize at first glance that the smallest eigenvalue is always very
close to zero.
When evaluating the critical exponents, one should keep in mind that the limit
z → 0 is not simply accessible perturbatively since it corresponds to the 2D XY
model which is known to behave in a quite specific manner. In particular, it does
not undergo an ordinary transition into the ordered phase at any finite temperature
and its critical behavior is essentially controlled by the vortex excitations. Such ex-
citations lead to an exponentially diverging correlation length at finite temperature.
This behavior cannot be accounted for within the λφ4 model equation (1) dealt with
in this paper. Since a new physics emerges after arriving at the point z = 0, it is
natural to assume that the observables as functions of the fixed point location may
be non-analytic near this point. Hence, what we really can explore trusting upon
our five-loop expansions is a domain corresponding to finite (and not too small)
values of z. Oppositely, the limit z → ∞ (or 1/z → 0) looks not at all dangerous
in the above sense since it corresponds to the critical behavior close to that of the
Ising model, which was shown not to be considerably effected by the non-analytic
terms even in two dimensions [15]. Note that the above presented results concerning
the line of the fixed point confirm this idea. Indeed, as seen from figure 2, at the
“Ising side”, i. e. for 0 < u ∼ 0.8, the zeros of both β functions form smooth curves
running very close to each other. However, the closer the “XY side” is the stronger
the estimates for β function zeros are scattered, probably indicating the increasing
impact of non-analytic contributions.
The expected value of η is 1/4 independent of the location of a fixed point
within the line. A clever way to check the constantness of η along the line of fixed
points is probably to write the RG function in terms of u and x = u + v, i.e.
ηx(x, u) = η(u, x− u). Then one resums the difference ∆(x, u) = ηx(x, u)− ηx(x, 0).
Along the line for all the five-loop approximants we always find |∆(x, u)| < 8×10−3.
This leads us to conclude that the two-dimensional LGW approach is capable of
keeping the constantness of η within an error of 3%. Note that the previous quoted
problems about non-analyticities close to the O(2) side do not significantly affect
the estimates of η
For the exponent y = η − ηt = 2 − 1/ν it was conjectured in [3] that it should
behave like
y =
2
1 + x
, where, x =
2
π
arctan
v∗
u∗
. (33)
A direct reliable quantitative estimate of this exponent is impossible due to the
effect of nonanalyticities, in particular close to the O(2) fixed point for the reasons
explained above. In fact, we know from [15] that the resummation of y at the Ising
fixed point y provides 1.04, very close to the exact value 1. Instead, at the O(2) fixed
point one has y ∼ 1.25, which is quite far from a diverging ν, i.e. y = 2. A direct
resummation of this exponent is reported in figure 4, and as expected it seems to
reproduce the correct critical behavior only close to the Ising fixed point.
207
P.Calabrese et al.
0 0.2 0.4 0.6 0.8 1
x
1
1.2
1.4
1.6
1.8
2
y
CM 5 loop
Pade [4/1]
Pade [2/1] constrained
Pade [3/1] constrained
Pade [2/2] constrained
First approximation
Second approximation
Figure 4. The exponent y as function of x from free and constrained resummation.
The solid line is the conjecture equation (34).
In [3] it was proposed to use the method of constraining the exponents to assume
the exactly known value along the axes in order to get better quantitative results.
We apply here a quite different method of constrained analysis with respect to [3].
Without going into technical details we have preferred to constrain the series of y
expressed in terms of u and x = u+v (as previously for the exponent η). The results
obtained by Padé-Borel method are again reported in figure 4. They are practically
indistinguishable from the unconstrained ones up to x ∼ 0.6, and then they start
oscillating, signaling the presence of singularities and of bad quantitative estimates.
The conformal mapping results are practically equivalent. The numerical data thus
obtained are well fitted by the conjectured curve:
y =
4
4 − (1 − x)(2 − x)
, (34)
but we are not able to estimate the goodness of our resummation and, as a result, to
verify the new conjecture equation (34). Perhaps, the exact behavior of the exponent
ν along the line of fixed points requires a new method of analysis of the perturbative
series and, in any case, it deserves additional studies on the subject like Monte Carlo
simulation or high temperature expansion.
4. Conclusions
To summarize, the critical behavior of the two-dimensional cubic model with the
N -vector order parameter has been studied. The five-loop contributions to the β
functions and critical exponents have been calculated and the five-loop RG series
have been resummed by means of Padé-Borel-Leroy procedure and using the con-
formal mapping technique. For N = 2 we have found that the continuous line of
fixed points connecting the Heisenberg and the Ising ones is well reproduced by the
resummed five-loop RG series. Moreover, the five-loop terms make the lines of zeros
208
Critical thermodynamics of two-dimensional N -vector cubic model
of β functions for u and v closer to each other, thus improving the results of the
lower-order approximation. For N > 2, the five-loop contributions have been shown
to shift the cubic fixed point, given by the four-loop approximation, towards the Isi-
ng fixed point. This may be considered as an argument in favor of the idea that the
existence of cubic fixed point in two dimensions for N > 3 is an artifact of the per-
turbative analysis. The model with N = 0 describing the critical thermodynamics
of 2D weakly-disordered Ising systems has been also studied. The results obtained
have been found to be compatible with the conclusion that in two dimensions the
impure critical behavior is governed by the Ising fixed point.
Acknowledgements
We are grateful to Pietro Parruccini and Ettore Vicari for discussions. The au-
thors acknowledge the financial support of EPSRC under Grant No. GR/R83712/01
(P.C.), the Russian Foundation for Basic Research under Grant No. 04–02–16189
(A.I.S., E.V.O., D.V.P.), and the Ministry of Education of Russian Federation un-
der Grant No. E02–3.2–266 (A.I.S., E.V.O., D.V.P.). A.I.S. has much benefitted
from the warm hospitality of Scuola Normale Superiore and Dipartimento di Fisica
dell’Universitá di Pisa, where the major part of this research was done.
References
1. José J.V., Kadanoff L.P., Kirkpatrick S., Nelson D.R., Phys. Rev. B, 1977, 16, No. 3,
1217–1241.
2. Shalaev B.N., Phys. Reports, 1994, 237, No. 3, 129–188.
3. Calabrese P., Celi A., Phys. Rev. B, 2002, 66, No. 18, 184410.
4. Shalaev B.N., Fiz. Tverd. Tela, 1989, 31, No. 1, 93–101 (in Russian);
Sov. Phys. Solid State, 1989, 31, No. 1, 51.
5. Dotsenko Vl.S., Dotsenko Vik.S., Adv. Phys., 1983, 32, 129.
6. Shalaev B.N., Fiz. Tverd. Tela, 1984, 26, No. 10, 3002–3005 (in Russian);
Sov. Phys. Solid State, 1984, 26, No. 10, 1811–1813.
7. Kleinert H., Schulte-Frohlinde V., Phys. Lett. B, 1995, 342, No. 1–4, 284–296;
Kleinert H., Thoms S., Phys. Rev. D, 1995, 52, No. 10, 5926–5943;
Kleinert H., Thoms S., Schulte-Frohlinde V., Phys. Rev. B, 1997, 56, No. 22, 14428–
14434.
8. Shalaev B.N., Antonenko S.A., Sokolov A.I., Phys. Lett. A, 1997, 230, No. 1–2, 105–
110.
9. Pakhnin D.V., Sokolov A.I., Phys. Rev. B, 2000, 61, No. 22, 15130–15135.
10. Carmona J.M., Pelissetto A., Vicari E., Phys. Rev. B, 2000, 61, No. 22, 15136–15151.
11. Folk R., Holovatch Yu., Yavors’kii T., Phys. Rev. B, 2000, 62, No. 18, 12195–12200;
12. Pakhnin D.V., Sokolov A.I., Phys. Rev. B, 2001, 64, No. 9, 094407.
13. Nickel B.G., Phase Transitions, ed. by Lévy M., Le Guillou J.C., Zinn-Justin J.
Plenum, New York-London, 1982;
Nickel B.G., Physica A, 1991, 177, No. 1, 189–196.
209
P.Calabrese et al.
14. Pelissetto A., Vicari E., Nucl. Phys. B, 1998, 519, No. 3, 626–660;
Nucl. Phys. B (Proc. Suppl.), 1999, S73, 775–777.
15. Orlov E.V., Sokolov A.I., Fiz. Tverd. Tela, 2000, 42, No. 11, 2087–2093 (in Russian);
Phys. Solid State, 2000, 42, No. 11, 2151–2158.
16. Calabrese P., Caselle M., Celi A., Pelissetto A., Vicari E., J. Phys. A, 2000, 33, No. 46,
8155–8170.
17. Aharony A., Phase Transitions and Critical Phenomena, ed. by Domb C., Lebowitz J,
vol. 6, p. 357. Academic Press, New York, 1976.
18. Pelissetto A., Vicari E., Phys. Reports, 2002, 368, No. 6, 549–727.
19. Zinn-Justin J. Quantum Field Theory and Critical Phenomena, fourth edition. Claren-
don Press, Oxford, 2002.
20. Calabrese P., Pelissetto A., Rossi P., Vicari E., Int. J. Mod. Phys. B, 2003, 17, No. 31–
32, 5829–5838.
21. Unfortunately, the five-loop contribution to the critical exponent γ (ηt) reported in
[15] contains numerical errors. Here, the correct values of corresponding coefficients
are presented (table 4).
22. Calabrese P., Pelissetto A., Vicari E., Phys. Rev. B, 2003, 67, No. 2, 024418;
Acta Phys. Slov., 2002, 52, 311–316.
23. Calabrese P., Parruccini P., Pelissetto A., Vicari E., in preparation.
24. Korzhenevskii A.L., Zh. Eksp. Teor. Fiz., 1976, 71, No. 4, 1434–1442 (in Russian);
Sov. Phys. JETP, 1976, 44, No. 4, 751–757.
25. Caselle M., Hasenbusch M., Pelissetto A., Vicari E., J. Phys. A, 2000, 33, No. 46,
8171–8180;
J. Phys. A, 2001, 34, No. 14, 2923–2948.
26. Le Guillou J. C., Zinn-Justin J., Phys. Rev. Lett., 1977, 39, No. 2, 95–98;
Phys. Rev. B, 1980, 21, No. 9, 3976–3998.
27. Bray A.J., McCarthy T., Moore M.A., Reger J.D., Young A.P., Phys. Rev. B, 1987,
36, No. 4, 2212–2219;
McKane A.J., Phys. Rev. B, 1994, 49, No. 17, 12003–12009;
Alvarez G., Martin-Mayor V., Ruiz-Lorenzo J.J., J. Phys. A, 2000, 33, No. 5, 841–850.
28. Pelissetto A., Vicari E., Phys. Rev. B, 2000, 62, No. 10, 6393–6409.
29. Mayer I.O., Sokolov A.I., Shalaev B.N., Ferroelectrics, 1989, 95, No. 1, 93–96.
30. Antonenko S.A., Sokolov A.I., Phys. Rev. E, 1995, 51, No. 3, 1894–1898.
31. Sokolov A.I., Fiz. Tverd. Tela, 1998, 40, No. 7, 1284–1290 (in Russian);
Phys. Solid State, 1998, 40, No. 7, 1169–1174.
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Critical thermodynamics of two-dimensional N -vector cubic model
Критична термодинаміка двовимірної N -векторної
кубічної моделі в п’яти-петлевому наближенні
П.Калабрезе 1 , Е.В.Орлов 2 , Д.В.Пахнін 2 , А.І.Соколов 2
1 Вища нормальна школа та INFN,
Пл. Кавалерів 7, I–56126 Піза, Італія
2 Факультет фізичної електроніки,
С.-Петербурзький Електротехнічний Університет,
вул. професора Попова 5, С.-Петербург 197376, Росія
Отримано 12 листопада 2004 р.
В рамках теоретико-польового підходу ренормалізаційної групи (РГ)
вивчається критична поведінка двовимірної N -векторної кубічної
моделі. β функції і критичні показники обчислюються в п’яти-
петлевому наближенні, отримані РГ ряди пересумовуються з вико-
ристанням техніки Паде-Бореля-Лєруа і конформного перетворення.
Знайдено, що для N = 2 неперервна лінія нерухомих точок добре
відтворюється пересумованими РГ рядами і врахування п’яти-
петлевих членів робить лінії нулів обох β функцій ближчими один
до одного. Показано, що для N > 3 п’яти-петлеві внески зсувають
кубічну нерухому точку, отриману в чотири-петлевому наближенні,
до нерухомої точки Ізинґа. Це підтверджує ідею, що існування
кубічної нерухомої точки в двох вимірах під N > 2 є результатом
пертурбативного аналізу. У випадку N = 0 отримані результати є
сумісні з висновком, що критична поведінка, пов’язана з домішками,
контролюється нерухомою точкою Ізинґа.
Ключові слова: розклади ренормалізаційної групи, 2D кубічна
модель
PACS: 75.10.Hk, 05.70.Jk, 64.60.Fr, 11.10.Kk
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