The strong-weak coupling symmetry in 2D Φ⁴ field models
It is found that the exact beta-function β(g) of the continuous 2D gΦ⁴ model possesses two types of dual symmetries, these being the KramersWannier (KW) duality symmetry and the strong-weak (SW) coupling symmetry f(g), or S-duality. All these transformations are explicitly constructed. The S-dua...
Збережено в:
| Дата: | 2005 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут фізики конденсованих систем НАН України
2005
|
| Назва видання: | Condensed Matter Physics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/119486 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | The strong-weak coupling symmetry in 2D Φ⁴ field models / B.N. Shalaev // Condensed Matter Physics. — 2005. — Т. 8, № 1(41). — С. 113–122. — Бібліогр.: 17 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-119486 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1194862017-06-08T03:06:10Z The strong-weak coupling symmetry in 2D Φ⁴ field models Shalaev, B.N. It is found that the exact beta-function β(g) of the continuous 2D gΦ⁴ model possesses two types of dual symmetries, these being the KramersWannier (KW) duality symmetry and the strong-weak (SW) coupling symmetry f(g), or S-duality. All these transformations are explicitly constructed. The S-duality transformation f(g) is shown to connect domains of weak and strong couplings, i.e. above and below g*. Basically it means that there is a tempting possibility to compute multiloop Feynman diagrams for the β-function using high-temperature lattice expansions. The regular scheme developed is found to be strongly unstable. Approximate values of the renormalized coupling constant g* found from duality symmetry equations are in an agreement with available numerical results. Знайдено, що точні бета-функції β(g) неперервної 2D gΦ⁴ моделі володіють двома типами дуальної симетрії, а саме дуальною симетрією Крамерса-Ванньє (КВ) і симетрією сильно-слабкого (СС) зв’язку f(g), або S-дуальністю. Усі ці перетворення конструюються явно. Показано, що S-дуальне перетворення f(g) зв’язує домени слабкого і сильного зв’язку, тобто вище і нижче g*. По суті це означає, що є приваблива можливість обчислювати багатопетлеві діаграми Фейнмана для β-функцій, використовуючи високотемпературні ґраткові розклади. Знайдено, що розвинута регулярна схема є дуже нестійкою. Знайдені з рівнянь дуальної симетрії наближені значення g* узгоджуються з доступними чисельними результатами. 2005 Article The strong-weak coupling symmetry in 2D Φ⁴ field models / B.N. Shalaev // Condensed Matter Physics. — 2005. — Т. 8, № 1(41). — С. 113–122. — Бібліогр.: 17 назв. — англ. 1607-324X PACS: 11.25.Hf, 74.20.-z,05.10.Cc DOI:10.5488/CMP.8.1.113 http://dspace.nbuv.gov.ua/handle/123456789/119486 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 |
It is found that the exact beta-function β(g) of the continuous 2D gΦ⁴
model possesses two types of dual symmetries, these being the KramersWannier
(KW) duality symmetry and the strong-weak (SW) coupling symmetry
f(g), or S-duality. All these transformations are explicitly constructed.
The S-duality transformation f(g) is shown to connect domains of
weak and strong couplings, i.e. above and below g*. Basically it means
that there is a tempting possibility to compute multiloop Feynman diagrams
for the β-function using high-temperature lattice expansions. The regular
scheme developed is found to be strongly unstable.
Approximate values of the renormalized coupling constant g* found from
duality symmetry equations are in an agreement with available numerical
results. |
| format |
Article |
| author |
Shalaev, B.N. |
| spellingShingle |
Shalaev, B.N. The strong-weak coupling symmetry in 2D Φ⁴ field models Condensed Matter Physics |
| author_facet |
Shalaev, B.N. |
| author_sort |
Shalaev, B.N. |
| title |
The strong-weak coupling symmetry in 2D Φ⁴ field models |
| title_short |
The strong-weak coupling symmetry in 2D Φ⁴ field models |
| title_full |
The strong-weak coupling symmetry in 2D Φ⁴ field models |
| title_fullStr |
The strong-weak coupling symmetry in 2D Φ⁴ field models |
| title_full_unstemmed |
The strong-weak coupling symmetry in 2D Φ⁴ field models |
| title_sort |
strong-weak coupling symmetry in 2d φ⁴ field models |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119486 |
| citation_txt |
The strong-weak coupling symmetry in 2D Φ⁴ field models / B.N. Shalaev // Condensed Matter Physics. — 2005. — Т. 8, № 1(41). — С. 113–122. — Бібліогр.: 17 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT shalaevbn thestrongweakcouplingsymmetryin2dph4fieldmodels AT shalaevbn strongweakcouplingsymmetryin2dph4fieldmodels |
| first_indexed |
2025-07-08T15:57:29Z |
| last_indexed |
2025-07-08T15:57:29Z |
| _version_ |
1837094931326304256 |
| fulltext |
Condensed Matter Physics, 2005, Vol. 8, No. 1(41), pp. 113–122
The strong-weak coupling symmetry in
2D Φ
4 field models
B.N.Shalaev
A.F.Ioffe Physical & Technical Institute,
Russian Academy of Sciences,
Polytechnicheskaya str. 26,
194021 St.Petersburg, Russia
Received November 9, 2004
It is found that the exact beta-function β(g) of the continuous 2D gΦ4
model possesses two types of dual symmetries, these being the Kramers-
Wannier (KW) duality symmetry and the strong-weak (SW) coupling sym-
metry f(g), or S-duality. All these transformations are explicitly construc-
ted. The S-duality transformation f(g) is shown to connect domains of
weak and strong couplings, i.e. above and below g∗. Basically it means
that there is a tempting possibility to compute multiloop Feynman diagrams
for the β-function using high-temperature lattice expansions. The regular
scheme developed is found to be strongly unstable.
Approximate values of the renormalized coupling constant g∗ found from
duality symmetry equations are in an agreement with available numerical
results.
Key words: Kramers-Wannier duality, S-duality, renormalization group,
beta-function
PACS: 11.25.Hf, 74.20.-z,05.10.Cc
1. Introduction
The 2D Ising model and some other lattice spin models are known to possess
the remarkable Kramers-Wannier(KW) duality symmetry, playing an important role
both in statistical mechanics, quantum field theory [1,2] and in superstring models
[3]. The self-duality of the isotropic 2D Ising model means that there exists an exact
mapping between the high-T and low-T expansions of the partition function [2].
In the transfer-matrix language this implies that the transfer-matrix of the model
under discussion is covariant under the duality transformation. If we assume that the
critical point is unique, the KW self-duality would yield the exact Curie temperature
of the model. This holds for a large set of lattice spin models including systems with
quenched disorder (for a review see [4]). Recently, the Kramers-Wannier duality
c© B.N.Shalaev 113
B.N.Shalaev
symmetry was extended to the continuous 2D gΦ4 model [5] in the strong-coupling
regime, i.e. for g > g∗.
This beta-function β(g) is to date known only in the five-loop approximation
within the framework of the conventional perturbation theory at the fixed dimension
d = 2 [6].
The strong coupling expansion for the calculation of the beta-function of the
2D scalar gΦ4 theory as an alternative approach to the convemtional perturbation
theory (described in the excellent textbook, known among experts in the field as
Bible [7]) and was developed in [8].
It is well known from quantum field theory and statistical mechanics that any
strong coupling expansions are closely connected with the high-temperature (HT)
series expansions for lattice models. From the field-theoretical point of view the
HT series are nothing but strong coupling expansions for field models, the lattices
being considered as a technical device to define cut-off field theories (see [8–11] and
references therein).
Calculations of beta-functions are of great interest in statistical mechanics and
in quantum field theory. The beta-function contains the essential information on the
renormalized coupling constant g∗, this being important in constructing the equation
of state of the 2D Ising model, [7].
In this paper we study other duality symmetries of the beta-function β(g) for
the 2D gΦ4 theory regarded as a non-integrable continuum limit of the exactly
solvable 2D Ising model. The main purpose is to construct exlicitly the strong-weak
(SW) coupling duality transformation f(g) connecting domains of weak and strong
couplings, i.e. above and below g∗. The last transformation allows one to compute
the yet unknown multiloop orders (6, 7, . . .) of the β-functions based on the lattice
expansions [11].
The paper is organized as follows. In section 2 we set up basic notations and define
both the correlation length and beta-function β(g). In section 3 the duality symmetry
transformation g̃ = d(g) is derived. Then it is proved that β(d(g)) = d′(g)β(g).
An approximate expression for d(g) is also found. Section 4 contains an explicit
derivation of the strong-weak coupling transformation whilst in section 5 in order
to illustrate our approach the sixth-order term of β(g) is approximately computed.
Section 4 contains disussion and some concluding remarks.
2. The correlation length and the Coupling constant
We begin by considering the classical Hamiltonian of the 2D Ising model (in
the absence of an external magnetic field), defined on a square lattice with periodic
boundary conditions; as usual:
H = −J
∑
〈i,j〉
σiσj , (1)
where 〈i, j〉 indicates that the summation is over all nearest-neighboring sites; σi =
±1 are spin variables and J is a spin coupling. The standard definition of the
114
The strong-weak coupling symmetry in 2D Φ4 field model
spin-pair correlation function reads:
G(R) = 〈σRσ0〉, (2)
where 〈· · · 〉 stands for the conventional Gibbs average.
The statistical mechanics definition of the correlation length is given by [9]
ξ2 =
d lnG(p)
dp2
|p=0 . (3)
The quantity ξ2 is known to be conveniently expressed in terms of the spherical
moments of the spin correlation function itself, namely
µl =
∑
R
(
R
a
)l
G(R) (4)
with a being some lattice spacing. It is easy to see that
ξ2 =
µ2
2dµ0
, (5)
where d is the spatial dimension (in our case d = 2).
In order to extend the KW duality symmetry to the continuous field theory
we have a need for a “lattice” model definition of the coupling constant g, equiv-
alent to the conventional one exploited in the RG approach. The renormalization
coupling constant g of the gΦ4 theory is closely related to the fourth derivative
of the “Helmholtz free energy”, namely ∂4F (T, m)/∂m4, with respect to the order
parameter m = 〈Φ〉. It may be defined as follows (see [7])
g(T, h) = −
(∂2χ/∂h2)
χ2ξd
+ 3
(∂χ/∂h)2
χ3ξd
, (6)
where χ is the homogeneous magnetic susceptibility
χ =
∫
d2xG(x). (7)
It is in fact easy to show that g(T, h) in equation (6) is merely the standard four-
spin correlation function taken at zero external momenta. The renormalized coupling
constant of the critical theory is defined by the double limit
g∗ = lim
h→0
lim
T→Tc
g(T, h) (8)
and it is well known that these limits do not commute with each other. As a result,
g∗ is a path-dependent quantity in the thermodynamic (T, h) plane [8].
Here we are mainly concerned with the coupling constant on the isochore line
g(T > T∗, h = 0) in the disordered phase and with its critical value
g∗
+ = lim
T→T
+
c
g(T, h = 0) = −
∂2χ/∂h2
χ2ξd
|h=0 . (9)
The “lattice” coupling constant g∗
+ defined in equation (9) is a given function of
the temperature Tc.
115
B.N.Shalaev
3. The Kramers-Wannier symmetry
The standard KW duality tranformation is known to be as follows [1–3]
sinh(2K̃) =
1
sinh(2K)
. (10)
We shall see that it will be more convenient to deal with a new variable s =
exp(2K) tanh(K), where K = J/T .
It follows from the definition that s transforms as s̃ = 1/s; this implies that
the correlation length of the 2D Ising model given by ξ2 = s/(1 − s)2 is a self-dual
quantity [5]. Now, on the one hand, we have the formal relation
ξ
ds(g)
dξ
=
ds(g)
dg
β(g), (11)
where s(g) is defined as the inverse function of g(s), i.e. g(s(g)) = g and the beta-
function is given, as usual, by
ξ
dg
dξ
= β(g). (12)
On the other hand, it is known from [5]
ξ
ds
dξ
=
2s(1 − s)
(1 + s)
. (13)
From equations (11)–(13), a useful representation of the beta-function in terms of
the s(g) function thus follows
β(g) =
2s(g)(1 − s(g))
(1 + s(g)) (ds(g)/dg)
. (14)
It follows from the representation (14) that the beta-function vanishes at the
point g = g∗ where s(g∗) = 1. If one assumes that the fixed point is not singular
(although this is not neccesary and not obvious),then from this equation it would
follow that ω = β ′(g)|g=gc
= 1 in agreement with the classical paper [12].
The correct solution of this non-trivial problem was found in remarkable papers
belonging to the Italian group of researchers [13,15,16]. The main result is that
ω = β ′(g)|g=gc
= 1.75.
Let us define the dual coupling constant g̃ and the duality transformation func-
tion d(g) as
s(g̃) =
1
s(g)
, g̃ ≡ d(g) = s−1
(
1
s(g)
)
, (15)
where s−1(x) stands for the inverse function of x = s(g). It is easy to check that a
further application of the duality map d(g) gives back the original coupling constant,
i.e. d(d(g)) = g, as it should be. Notice also that the definition of the duality
116
The strong-weak coupling symmetry in 2D Φ4 field model
transformation given by equation (15) has a form similiar to the standard KW
duality equation, equation (10).
Consider now the symmetry properties of β(g). We shall see that the KW duality
symmetry property, equation (10), results in the beta-function being covariant under
the operation g → d(g):
β(d(g)) = d′(g)β(g). (16)
To prove it let us evaluate β(d(g)). Then equation (14) yields
β (d(g)) =
2s(g̃)(1 − s(g̃))
(1 + s(g̃)) (ds(g̃)/dg̃)
. (17)
Bearing in mind equation (15) one is led to
β(d(g)) =
2s(g) − 2
s(g)(1 + s(g)) (ds(g̃)/dg̃)
. (18)
The derivative in the r.h.s. of equation (18) should be rewritten in terms of s(g) and
d(g). It may be easily done by applying equation (15):
ds(g̃)
dg̃
=
d
dg̃
1
s(g)
= −
s′(g)
s2(g)
1
d′(g)
. (19)
Substituting the r.h.s. of equation (19) into equation (18) one obtains the desired
symmetry relation, equation (16).
Therefore, the self-duality of the model allows us to determine the fixed point
value in another way, namely from the duality equation d(g∗) = g∗.
Making use of a rough approximation for s(g), one gets [5]
s(g) '
2
g
+
24
g2
'
2
g
1
1 − 12/g
=
2
g − 12
. (20)
Combining this Padé-approximant with the definition of d(g), equation (15), one is
led to
d(g) = 4
3g − 35
g − 12
. (21)
The fixed point of this function, d(g∗) = g∗, is easily seen to be g∗
+ = 14. On the
one hand, that is a rough approximation, on the other hand best numerical and
analytical estimates obtained by making use of lattice and conformal field theory
yield g∗
+ = 14.697323(20) see [13,15,16]). and references therein).
4. The weak-strong coupling dual symmetry
The beta-function of the model under discussion possesses the important alge-
braic property (14) (KW duality) which permits to develop the weak-strong-duality
transformation f(g) connecting both the weak-coupling and strong coupling regimes.
117
B.N.Shalaev
Nowadays both the five-loop approximation results ([6]) and the strong coupling
expansion for the beta-function [9] are known rather well. These are given by
β1(g) = 2g − 2g2 + 1.432347241g3 − 1.861532885g4
+ 3.164776688g5 − 6.520837458g6 + O(g7), (22)
β2(g) = −2g +
12
π
−
9
π2g
+
27
π3g2
+
81
8π4g3
−
3645
16π5g4
−
15309
32π6g5
+
2187
64π7g6
+ O(g−7). (23)
Here indices 1, 2 stand for the weak and strong coupling regimes, respectively.
The main goal of this section is to determine a dual transformation f(g) such as
f [f(g)] = g relating beta-functions β1(g) and β2(g).
From equation (14) one can easily find the functions S1(g), S2(g) and their inverse
functions G1(s) = S−1
1 (g), G2(s) = S−1
2 (g) corresponding to two regimes. Simple but
cumbersome calculations lead to
G1(s) = s + s2 + 0.3580868104s3 − 0.1166327797s4
− 0.1968226859s5 − 0.1299831557s6 + O(s7),
S1(g) = g − g2 + 1.6419131896g3 − 3.09293317g4
+ 6.361881481g5 − 13.78545095g6 + O(g7),
s ∈ [0, 1], g ∈ [0, g∗], (24)
G2(s) =
3
4πs
+
9
2π
−
9s
4π
+
18s2
π
−
108s3
π
+
618s4
π
−
3474s5
π
+
19494s6
π
+ O(s7),
S2(g) =
2 ∗ 3
8πg
+
24 ∗ 32
(8πg)2
+
264 ∗ 33
(8πg)3
+
2976 ∗ 34
(8πg)4
+
35136 ∗ 35
(8πg)5
+
423680 ∗ 36
(8πg)6
+
5149824 ∗ 37
(8πg)7
+
63275520 ∗ 38
(8πg)8
+ O(g−9),
s ∈ [0, 1], g ∈ [g∗,∞). (25)
118
The strong-weak coupling symmetry in 2D Φ4 field model
Being equipped with these formulas one may easily construct two branches of the
same duality transformation function f12(g) and f21(g) defined in different domains
of g. The functions are
1
f21(g)
≡
1
G2(S1(g))
=
4πg
3
−
28πg2
3
+ 220.5059303g3
− 1766.8145g4 + 14816.94007g5 − 127842.5955g6 ,
g ∈ [0, g∗], f21(g) ∈ [g∗,∞], (26)
f12(g) ≡ G1(S2(g)) =
3
4πg
+
63
16π2g2
+
0.61714739472
g3
+
0.9560453953
g4
+
1.502156783
g5
+
2.368311503
g6
+ O(g7),
g ∈ [g∗,∞), f12(g) ∈ [0, g∗]. (27)
Functions found above look like inversion, but they are not so simple. A nontrivial
example of the 2D model disordered Dirac fermions was discovered in [14]. It was
shown that the beta-function of the (nonintegrable) model under consideration also
exhibits the strong-weak coupling duality such as g∗ → 1/g [14].
It is worth noting that the transformation found is dual indeed
f12(f21(g)) = f21(f12(g)) ≡ g. (28)
Moreover, by definition weak-strong coupling expansions of β(g) are related to
each other in the following way:
β2(g) =
β1(f12(g))
f ′
12(g)
, (29)
β1(g) =
β2(f21(g))
f ′
21(g)
. (30)
It is rather amusing that equation (27) looks like a geometric series. Making use
of the Pade method we arrive at
f12(g) ≈
0.2387324146g2 − 0.0745907136g + 0.0850867165
g3 − 1.983571753g2 + 1.086109562g − 0.6919672492
g ∈ [g∗,∞), f12(g) ∈ [0, g∗]. (31)
The weak-strong duality equation and strong-coupling expansion yield the fol-
lowing numerical values (the “exact” estimates one may find in the previous section)
f12(g) − g = 0, g∗ = 14.38,
β2(g
∗) = 0, g∗ = 14.63. (32)
119
B.N.Shalaev
5. Higher-order terms of the beta-function
Finally, let us consider how one can compute the β(g) in the multiloop approxi-
mation via the strong-coupling expansion and the S-duality function. This is in order
to find that one should exploit equation (29), equation (23) and the approximate
expression for f12(g) given by equation (31).
After some tedious but routine calculations we arrive at some polynomial of 7th
degree for β1(g).
β1(g) = 2g − 2g2 + 1.432347241g3 − 1.861532885g4
+ 3.164776688g5 − 6.520837458g6 − 331.454743g7 . (33)
It is easily seen that the first 6 terms except for the 7th one are the exact
perturbation expansion for β1(g) [6]. It would be tempting but wrong to regard
equation (33) as a β(g)-function in the 7th loop approximation. In fact, the function
in equation (31) is approximate, so that we have to estimate an accuracy of our
calculations.
Suppose a difference between the “exact” duality function f exact
12 (g) and the ap-
proximate one given by equation (31) reads
f exact
12 (g) =
0.2387324146g2 − 0.0745907136g + 0.0850867165
g3 − 1.983571753g2 + 1.086109562g − 0.6919672492
+ b/g7 (34)
with b being a fitting parameter. The straightforward calculation shows that a “new”
7th loop contribution computed by making use of the equation (34) depends on the
fitting parameter b and differs vastly from the previous one, it being
β1(g) = 2g − 2g2 + 1.432347241g3 − 1.861532885g4
+ 3.164776688g5 − 6.520837458g6
+ (−331.454743 + 271519.803807b)g7 . (35)
Therefore, we see that the approach proposed above provides a regular algorithm
for computing higher-order corrections to the β(g)-function based on the lattice high-
order expansions. In other words, one obtaines a tempting possibility to compute
(approximately) multiloop Feynman diagrams based on the equation (30) and of
high-temperature expansions [9]. Unfortunately, it is very difficult to bring into
action this method because of its strong instability, [17].
6. Discussion and concluding remarks
We have shown that the β-function of the 2D gΦ4 theory does have the two
types of dual symmetries, these being the Kramers-Wannier symmetry and the weak-
strong coupling symmetry (S-duality).
Our proof of the KW symmetry is based on the properties of g(s), s(g) defined
only for 0 6 s 6 1; g∗ 6 g < ∞ and therefore does not cover the weak-coupling
120
The strong-weak coupling symmetry in 2D Φ4 field model
region, 0 6 g 6 g∗. It means that the KW symmetry holds only in the strong-
coupling region.
We established the existence of the dual function f(g) or S-duality connecting
two domains of both weak coupling and strong coupling. Given both perturbative
RG calculations and lattice high-temperature expansions, this function f(g) can be
approximately computed. We also explicitly computed high-order terms for β(g). A
close analysis of the scheme developed shows that this is strongly unstable.
7. Acknowledgements
The author is most grateful to A.PeIissetto and E.Vicari for sending him ex-
tremely interesting comments. The discussions with A.I.Sokolov were also of great
benefit to the author. This work was supported by the Russian Foundation for Basic
Research (Grant No. 05–02–17807).
References
1. Kramers H.A., Wannier G.H., Phys.Rev., 1941, 60, 252.
2. Savit R., Rev.Mod.Phys., 1980, 52, 453;
Kogut J.B., Rev.Mod.Phys., 1979, 51, 659.
3. Gukov S.G., Phys. Usp., 1998, 41, 627.
4. Shalaev B.N., Phys. Rep., 1994, 237, 129.
5. Jug G., Shalaev B.N., J. Phys. A, 1999, 32, 7249.
6. Sokolov A.I., Orlov E.V., Fiz. Tverd. Tela, 2000, 42, 2087.
7. Zinn-Justin J. Quantum Field Theory and Critical Phenomena (3rd ed.). Clarendon,
Oxford, 1999.
8. Baker (Jr.) G.A., Nickel B.G., Meiron D.I., Phys. Rev. B, 1978, 17, 1365.
9. Tarko H.B., Fisher M.E., Phys. Rev. B, 1975, 11, 1217.
10. Zinn S.-Y., Lai S.-N., Fisher M.E., Phys.Rev. E, 1996, 54, 1176.
11. Butera P., Comi M., Phys. Rev. B, 1996, 54, 15828.
12. Barouch E., McCoy B.M., Wu T.T., Phys. Rev. Lett., 1973, 31, 1409.
13. Calabrese P., Caselle M., Celi A., Pelissetto A., Vicari E., J. Phys. A, 2000, 33, 8155,
(hep-th/0005254).
14. LeClair A., Phys. Rev. Lett., 2000, 84, 1292.
15. Caselle M., Hasenbuch M., Pelissetto A., Vicari E., J. Phys. A, 2000, 33, 8171,
(hep-th/000309).
16. Caselle M., Hasenbuch M., Pelissetto A., Vicari E., J. Phys. A, 2001, 34, 2923,
(cond-mat/0011305).
17. Shalaev B.N., TMP, 2002, 132, 621.
121
B.N.Shalaev
Симетрія сильно-слабкого зв’язку в 2D Φ
4 польових
моделях
Б.Н.Шалаєв
Фізико-технічний інститут ім. А.Ф.Йоффе,
Російська Академія Наук,
вул. Політехнічна 26,
194021 С.-Петербург, Росія
Отримано 9 листопада 2004 р.
Знайдено, що точні бета-функції β(g) неперервної 2D gΦ4 моделі
володіють двома типами дуальної симетрії, а саме дуальною симе-
трією Крамерса-Ванньє (КВ) і симетрією сильно-слабкого (СС)
зв’язку f(g), або S-дуальністю. Усі ці перетворення конструю-
ються явно. Показано, що S-дуальне перетворення f(g) зв’яз-
ує домени слабкого і сильного зв’язку, тобто вище і нижче g∗.
По суті це означає, що є приваблива можливість обчислювати
багатопетлеві діаграми Фейнмана для β-функцій, використовую-
чи високотемпературні ґраткові розклади. Знайдено, що розвинута
регулярна схема є дуже нестійкою.
Знайдені з рівнянь дуальної симетрії наближені значення g∗ узго-
джуються з доступними чисельними результатами.
Ключові слова: Крамерса-Ванньє дуальність, S-дуальність,
ренормалізаційна група, бета-функція
PACS: 11.25.Hf, 74.20.-z,05.10.Cc
122
|