Interacting N-vector order parameters with O(N) symmetry
We consider the critical behavior of the most general system of two Nvector order parameters that is O(N) invariant. We show that it may have a multicritical transition with enlarged symmetry controlled by the chiral O(2) ⊗ O(N) fixed point. For N = 2, 3, 4, if the system is also invariant under...
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Інститут фізики конденсованих систем НАН України
2005
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| Цитувати: | Interacting N-vector order parameters with O(N) symmetry / A. Pelissetto, E. Vicari // Condensed Matter Physics. — 2005. — Т. 8, № 1(41). — С. 87–101. — Бібліогр.: 28 назв. — англ. |
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irk-123456789-1193862017-06-07T03:03:23Z Interacting N-vector order parameters with O(N) symmetry Pelissetto, A. Vicari, E. We consider the critical behavior of the most general system of two Nvector order parameters that is O(N) invariant. We show that it may have a multicritical transition with enlarged symmetry controlled by the chiral O(2) ⊗ O(N) fixed point. For N = 2, 3, 4, if the system is also invariant under the exchange of the two order parameters and under independent parity transformations, one may observe a critical transition controlled by a fixed point belonging to the mn model. Also in this case there is a symmetry enlargement at the transition, the symmetry being [SO(N) ⊕ SO(N)] ⊗ C₂, where C₂ is the symmetry group of the square. Ми розглядаємо критичну поведінку найбільш загальної системи двох N-векторних параметрів порядку, яка є O(N) інваріантною. Ми показуємо, що вона може мати мультикритичний перехід з розширеною симетрією контрольованою чіральною O(2) ⊗ O(N) нерухомою точкою. Для N = 2, 3, 4, якщо система є також інварінтною відносно обміну двох параметрів порядку і відносно незалежного перетворення парності, можна спостерегти критичний перехід контрольований нерухомою точкою, що належить mn моделі. Також у цьому випадку відбувається розширення симетрії при переході, симетрія стає [SO(N) ⊕ SO(N)] ⊗ C₂, де C₂ є групою симетрії на квадраті. 2005 Article Interacting N-vector order parameters with O(N) symmetry / A. Pelissetto, E. Vicari // Condensed Matter Physics. — 2005. — Т. 8, № 1(41). — С. 87–101. — Бібліогр.: 28 назв. — англ. 1607-324X DOI:10.5488/CMP.8.1.87 PACS: 05.70.Jk, 64.60.Fr, 75.10.Hk http://dspace.nbuv.gov.ua/handle/123456789/119386 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
| description |
We consider the critical behavior of the most general system of two Nvector
order parameters that is O(N) invariant. We show that it may have
a multicritical transition with enlarged symmetry controlled by the chiral
O(2) ⊗ O(N) fixed point. For N = 2, 3, 4, if the system is also invariant
under the exchange of the two order parameters and under independent
parity transformations, one may observe a critical transition
controlled by a fixed point belonging to the mn model. Also in this case
there is a symmetry enlargement at the transition, the symmetry being
[SO(N) ⊕ SO(N)] ⊗ C₂, where C₂ is the symmetry group of the square. |
| format |
Article |
| author |
Pelissetto, A. Vicari, E. |
| spellingShingle |
Pelissetto, A. Vicari, E. Interacting N-vector order parameters with O(N) symmetry Condensed Matter Physics |
| author_facet |
Pelissetto, A. Vicari, E. |
| author_sort |
Pelissetto, A. |
| title |
Interacting N-vector order parameters with O(N) symmetry |
| title_short |
Interacting N-vector order parameters with O(N) symmetry |
| title_full |
Interacting N-vector order parameters with O(N) symmetry |
| title_fullStr |
Interacting N-vector order parameters with O(N) symmetry |
| title_full_unstemmed |
Interacting N-vector order parameters with O(N) symmetry |
| title_sort |
interacting n-vector order parameters with o(n) symmetry |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119386 |
| citation_txt |
Interacting N-vector order parameters with O(N) symmetry / A. Pelissetto, E. Vicari // Condensed Matter Physics. — 2005. — Т. 8, № 1(41). — С. 87–101. — Бібліогр.: 28 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT pelissettoa interactingnvectororderparameterswithonsymmetry AT vicarie interactingnvectororderparameterswithonsymmetry |
| first_indexed |
2025-07-08T15:46:57Z |
| last_indexed |
2025-07-08T15:46:57Z |
| _version_ |
1837094282043850752 |
| fulltext |
Condensed Matter Physics, 2005, Vol. 8, No. 1(41), pp. 87–101
Interacting N -vector order parameters
with O(N) symmetry
A.Pelissetto 1 , E.Vicari 2
1 Dip. Fisica dell’Università di Roma “La Sapienza” and INFN,
P.le Moro 2, I–00185 Roma, Italy
2 Dip. Fisica dell’Università di Pisa and INFN,
V. Buonarroti 2, I–56127 Pisa, Italy
Received September 21, 2004
We consider the critical behavior of the most general system of two N -
vector order parameters that is O(N) invariant. We show that it may have
a multicritical transition with enlarged symmetry controlled by the chiral
O(2) ⊗ O(N) fixed point. For N = 2, 3, 4, if the system is also in-
variant under the exchange of the two order parameters and under in-
dependent parity transformations, one may observe a critical transition
controlled by a fixed point belonging to the mn model. Also in this case
there is a symmetry enlargement at the transition, the symmetry being
[SO(N)⊕SO(N)]⊗C2, where C2 is the symmetry group of the square.
Key words: N -vector model, O(N) symmetry, multicritical transitions
PACS: 05.70.Jk, 64.60.Fr, 75.10.Hk
1. Introduction
The critical behavior of a system with a single N -vector order parameter is well
known [1,2]. In this paper we investigate the critical behavior of a system with two
N -vector parameters that is invariant under O(N) transformations and independent
parity transformations.
If the two N -vector order parameters are identical, i.e. the model is symmetric
under their exchange, the most general Landau-Ginzburg-Wilson (LGW) Φ4 Hamil-
tonian is given by
Hcr =
∫
d3x
[
1
2
∑
µ
(∂µφ · ∂µφ+ ∂µψ · ∂µψ) +
r
2
(
φ2 + ψ2
)
+
u0
4!
(
φ4 + ψ4
)
+
w0
4!
φ2ψ2 +
z0
4!
(φ · ψ)2
]
, (1)
c© A.Pelissetto, E.Vicari 87
A.Pelissetto, E.Vicari
where ψa and φa are two N -dimensional vectors. This Hamiltonian is well defined
for u0 > 0, 2u0 +w0 > 0, and 2u0 +w0 + z0 > 0. Hamiltonian (1) is invariant under
the transformations
(Z2)exch ⊗ (Z2)par ⊗ O(N). (2)
The first Z2 group is related to the exchange transformations φ ↔ ψ, while the
second group is related to the parity transformations φ → −φ, ψ → ψ, or, equiva-
lently, φ→ φ, ψ → −ψ (note that the transformation φ→ −φ, ψ → −ψ is already
accounted for by the O(N) group).
If the two order parameters are not identical and therefore the symmetry is only
(Z2)par ⊗ O(N), (3)
the corresponding LGW Φ4 Hamiltonian is
Hmcr =
∫
d3x
[
1
2
∑
µ
(∂µφ · ∂µφ+ ∂µψ · ∂µψ) +
r1
2
φ2 +
r2
2
ψ2
+
u0
4!
φ4 +
v0
4!
ψ4 +
w0
4!
φ2ψ2 +
z0
4!
(φ · ψ)2
]
, (4)
that is well defined for u0 > 0, v0 > 0, w0 + 2
√
u0v0 > 0, and w0 + z0 + 2
√
u0v0 > 0.
Hamiltonian (4) has two different mass terms and thus it gives rise to a variety
of critical and multicritical behaviors. It generalizes the multicritical Hamiltonian
considered in [3] that has w0 = 0 and is symmetric under the larger symmetry group
O(N) ⊕ O(N).
For N = 2 there is a transformation of the fields and couplings that leaves
Hamiltonians (1) or (4) invariant. If we transform the fields as φ′
a =
∑
b εabφb,
ψ′
a = ψa and the couplings as
u′0 = u0, v′0 = v0, w′
0 = w0 + z0, z′0 = −z0, (5)
we reobtain Hamiltonians (1) and (4) expressed in terms of the primed fields and
couplings [4]. This implies that, for any FP with z > 0, there exist an equivalent
one with the same stability properties and z < 0.
Finally, if we do not require the invariance of the model under independent parity
transformations, i.e., the model is only O(N) symmetric, we must add an additional
quadratic term φ · ψ and two additional quartic terms, (φ · ψ)φ2 and (φ · ψ)ψ2. In
this case, the general analysis becomes more complex since we have to deal with a
multicritical theory with three quadratic terms.
In this paper we investigate whether theories (1) and (4) have stable fixed points
(FPs) in three dimensions. We do not determine the renormalization-group (RG)
flow in the full theory, but rather we show that stable FPs can be identified by an
analysis of the submodels whose RG flow is already known. We consider the stable
FPs of the submodels and determine their stability properties with respect to the
perturbations that are present in the complete theory. In this way we are able to
88
Interacting N -vector order parameters with O(N) symmetry
identify three stable FPs. For any N , there is an O(2) ⊗O(N) symmetric FP. This
FP may be the relevant one for models with z0 > 0, which may therefore show
a symmetry enlargement at the (multi)critical transition. For N = 2 there is an
equivalent O(2) ⊗ O(2)/Z2 symmetric FP with z < 0, a consequence of symmetry
(5), which may be the relevant one for models with z0 < 0. For N = 2, 3, 4 we
find that Hcr – but not the multicritical theory Hmcr – has another stable FP that
belongs to the so-called mn model [5] with n = 2 and m = N . Also in this case
there is a symmetry enlargement at the transition: the FP is symmetric under the
group [SO(N) ⊕ SO(N)] ⊗ C2 where C2 is the symmetry group of the square. It is
interesting to note that the chiral O(2)⊗O(N) FP is also stable if we do not require
the model to be invariant under independent parity transformations. Indeed, the
additional terms (φ ·ψ)φ2 and (φ ·ψ)ψ2 are irrelevant perturbations at the chiral FP.
In the analysis we mainly use the minimal-subtraction (MS) scheme without
ε expansion (henceforth indicated as 3d-MS scheme) in which no ε expansion is
performed and ε is set to the physical value ε = 1 [6]. In order to generate the relevant
perturbative series we use a symbolic manipulation program that generates the
diagrams and computes symmetry and group factors. For the Feynman integrals we
use the results reported in [7]. In this way we obtained five-loop 3d-MS expansions.
2. Mean-field analysis
The mean-field analysis of the critical behavior of Hamiltonian Hcr is quite
straightforward. If r > 0 the system is paramagnetic, with φ = ψ = 0. For r < 0
there are three possible low-temperature phases:
(a) For w0 > 2u0 and z0 > 2u0 − w0, we have φ 6= 0 and ψ = 0 (or vice versa).
The corresponding symmetry-breaking pattern is (Z2)exch ⊗ (Z2)par ⊗O(N) →
(Z2)par ⊗O(N − 1).
(b) For z0 < 0 and −2u0 < w0 + z0 < 2u0, we have φ = ψ 6= 0. The corresponding
symmetry-breaking pattern is (Z2)exch⊗(Z2)par⊗O(N) → (Z2)exch⊗O(N−1).
(c) For z0 > 0 and −2u0 < w0 < 2u0, we have |φ| = |ψ| 6= 0, φ · ψ = 0.
The corresponding symmetry-breaking pattern is (Z2)exch ⊗ (Z2)par ⊗O(N) →
(Z2)exch ⊗ O(N − 1).
The analysis of the mean-field behavior of Hmcr is presented for N = 4 in [8] and
it is easily extended to the present case. There are three possible phase diagrams:
(a1) For z0 < 0 and −2
√
u0v0 < w0 + z0 < 2
√
u0v0, the multicritical point is
tetracritical, see figure 1. Phase 1 is paramagnetic with φ = ψ = 0, in phase
2 φ 6= 0 and ψ = 0, while in phase 3 the opposite holds, φ = 0 and ψ 6= 0; in
phase 4 φ 6= 0, ψ 6= 0 with φ‖ψ. All transitions are of second order. Transitions
1–2 and 1–3 are associated with the symmetry breaking Z2 ⊗ O(N) → Z2 ⊗
O(N−1). In the presence of fluctuations these transitions belong to the O(N)
89
A.Pelissetto, E.Vicari
O(N)
O(N)
Is
Is
(a1)
1
3
2
4
O(N)
O(N)
O(N-1)
O(N-1)
(a2)
1
3
2
4
O(N)
O(N)(b)
1
3
2
Figure 1. Possible multicritical phase diagrams. Thin lines indicate second-order
transitions, while the thick line in case (b) corresponds to a first-order transition.
“Is” indicates an Ising transition.
universality class. Transitions 2–4 and 3–4 are associated with the symmetry-
breaking pattern Z2 ⊗O(N − 1) → O(N − 1). In the presence of fluctuations
they should belong to the Ising universality class.
(a2) For z0 > 0 and −2
√
u0v0 < w0 < 2
√
u0v0 the multicritical point is tetracritical,
see figure 1. Phases 1, 2, and 3 as well as transitions 1–2 and 1–3 are identical
to those discussed in case (a1). In phase 4 φ 6= 0, ψ 6= 0 with φ · ψ = 0. All
transitions are second-order ones. Transitions 2–4 and 3–4 are associated with
the symmetry-breaking pattern Z2 ⊗O(N − 1) → Z2 ⊗O(N − 2) and, in the
presence of fluctuations, they should belong to the O(N−1) universality class.
(b) For w0 > 2
√
u0v0 and w0 + z0 > 2
√
u0v0 the multicritical point is bicritical,
see figure 1. Phases 1, 2, and 3 as well as transitions 1–2 and 1–3 are identical
to those discussed in case (a1). The transition between phases 2 and 3 is of
first order.
90
Interacting N -vector order parameters with O(N) symmetry
3. Analysis of some particular cases
3.1. Particular models and fixed points
The three-dimensional properties of the RG flow are determined by its FPs.
Some of them can be identified by considering particular cases in which some of the
quartic parameters vanish. For Hcr we can easily recognize two submodels:
(a) The O(2) ⊗ O(N) model with Hamiltonian [9]
Hch =
∫
ddx
1
2
∑
ai
[
∑
µ
(∂µΦai)
2 + rΦ2
ai
]
+
g1,0
4!
(
∑
ai
Φ2
ai)
2 +
g2,0
4!
∑
i,j
(
∑
a
ΦaiΦaj
)2
−
(
∑
ai
Φ2
ai
)2
, (6)
where Φai is an N × 2 matrix, i.e., a = 1, . . . , N and i = 1, 2. Hamiltonian (1)
reduces to (6) for 2u0 − w0 − z0 = 0, if we set Φa1 = φa, Φa2 = ψa, u0 = g1,0,
w0 = 2(g1,0 − g2,0), and z0 = 2g2,0. The properties of O(2)⊗O(N) models are
reviewed in [2,10–12]. In three dimensions perturbative calculations within the
three-dimensional massive zero-momentum (MZM) scheme [13,14] and within
the 3d-MS scheme [12] indicate the presence of a stable FP with attraction
domain in the region g2,0 > 0 for all values of N (only for N = 6 the evidence is
less clear: a FP is identified in the 3d-MS scheme but not in the MZM scheme).
For N = 2, these conclusions have been recently confirmed by a Monte Carlo
calculation [12]. On the other hand, near four dimensions, a stable FP is found
only for large values of N , i.e., N > Nc = 21.80−23.43ε+7.09ε2+O(ε3) [9,15–
17]. A stable FP with attraction domain in the region g2,0 < 0 exists for N = 2
(it belongs to the XY universality class) [9], for N = 3 ([18]), and N = 4 ([4]).
Note that nonperturbative approximate RG calculations have so far found no
evidence of stable FPs for N = 2 and 3 [11,19,20]. In the following we call the
FP with g2 > 0 chiral FP (we indicate it with g∗1,ch, g
∗
2,ch > 0), while the FP
with g2 < 0 is named collinear FP and indicated with g∗1,cl, g
∗
2,cl < 0.
(b) The so-called mn model with Hamiltonian [5]
Hmn =
∫
ddx
{
1
2
∑
ai
[
∑
µ
(∂µΦai)
2 + rΦ2
ai
]
+
g1,0
4!
(
∑
ai
Φ2
ai)
2 +
g2,0
4!
∑
abi
Φ2
aiΦ
2
bi
}
, (7)
where Φai is an m×n matrix, i.e., a = 1, . . . , m and i = 1, . . . , n. Hamiltonian
(1) reduces to (6) for n = 2, m = N , and z0 = 0, if we set Φa1 = φa, Φa2 = ψa,
u0 = g1,0 + g2,0 and w0 = 2g1,0. A stable FP is the O(m) FP with g1 = 0 and
91
A.Pelissetto, E.Vicari
g2 = g∗m, where g∗m is the FP value of the renormalized coupling in the O(m)
model. In App. A we show that the model has a second stable FP with g2 < 0
for n = 2 and m = 2, 3, and 4. We name this FP the mn FP and we label the
corresponding coordinates by g∗1,mn and g∗2,mn.
The presence of these two submodels that have one parameter less than the original
model imply that the quartic parameter space splits into four regions such that the
RG flow does not cross the two planes z = 0 and 2u − w − z = 0. Note that, for
N = 2, because of symmetry (5), we should only consider the region z > 0.
The results for models (a) and (b) allow us to identify four possible FPs that are
candidates for being stable FPs of the full theory:
(1) u = g∗1,ch, w = 2(g∗1,ch − g∗2,ch), z = 2g∗2,ch; this FP may be the stable FP of the
trajectories that start in the region z0 > 0;
(2) u = g∗1,cl, w = 2(g∗1,cl − g∗2,cl), z = 2g∗2,cl; this FP may be the stable FP of the
trajectories that start in the region z0 < 0;
(3) u = g∗N , w = 0, z = 0; this FP may be the stable FP of the trajectories that
start in the region 2u0 − w0 − z0 > 0;
(4) u = g∗1,mn+ g∗2,mn, w = 2g∗1,mn, z = 0 for N = 2, 3, 4; this FP may be the stable
FP of the trajectories that start in the region 2u0 − w0 − z0 < 0.
Note that, because of symmetry (5), for N = 2 there is also a chiral (resp. collinear)
FP with z < 0 (resp. z > 0).
The analysis of the particular cases of Hamiltonian (4) is very similar. There are
two relevant submodels:
(a) The forementioned O(2) ⊗ O(N) model for u0 = v0 and w0 + z0 = 2u0. The
identication is u0 = v0 = g1,0, w0 = 2g1,0 − 2g2,0, and z0 = 2g2,0.
(b) The O(N) ⊕O(N) model [3]:
Hmcr,2 =
∫
d3x
[
1
2
∑
µ
(∂µφ · ∂µφ+ ∂µψ · ∂µψ) +
r1
2
φ2 +
r2
2
ψ2
+
f1,0
4!
φ4 +
f2,0
4!
ψ4 +
f3,0
4!
φ2ψ2
]
. (8)
Hamiltonian Hmcr reduces to this model for z0 = 0, with the obvious identifica-
tion of the parameters. Hamiltonian (8) describes the multicritical behavior of
a model with two N -vector order parameters that is symmetric under indepen-
dent O(N) transformations of the two order parameters, i.e. that is invariant
under the symmetry group O(N) ⊕ O(N) [3]. In the case we are interested
in, i.e. for N > 2, the stable FP is the decoupled FP [21,22], i.e., f3 = 0,
f1 = f2 = g∗N (see also App. A).
92
Interacting N -vector order parameters with O(N) symmetry
Note that in this case the O(2) ⊗ O(N) model has two parameters less than the
original one and thus its presence does not imply any separation of the RG flow.
Instead, the second model implies that the quartic parameter space splits into two
regions such that the RG flow does not cross the plane z = 0. The analysis of the
possible FPs is identical to that presented above, since the FPs we have identified
are exactly those we have already described.
Table 1. Estimates of the RG dimensions y2,2, y4,2, and y4,4 of the operators
V (2,2), V (4,2), and V (4,4) at the chiral (ch) FP and at the collinear (cl) FP. They
have been obtained from a conformal-mapping analysis of the corresponding 3d-
MS 5-loop perturbative expansions. The results for y2,2 are taken from [4]. The
results at the collinear FP for N = 2 have been computed by using the mapping
with the XY model and the results of [21].
y2,2 y4,2 y4,4
ch,2 1.34(15) −1.6(1.1) −0.9(4)
ch,3 1.21(9) −1.4(8) −1.0(3)
ch,4 1.17(8) −1.1(5) −1.0(3)
ch,6 1.13(9) −0.9(4) −0.9(2)
ch,8 1.13(8) −0.9(5) −0.9(2)
ch,16 1.08(2) −1.0(3) −0.94(11)
ch,∞ 1 −1 −1
cl,2 1.9620(8) 0 0.532(12)
cl,3 2.05(15) 0.9(3)
cl,4 2.05(15) 0.9(7)
3.2. Stability of the O(2) ⊗ O( N ) fixed points
In this section we study the stability properties of the two FPs that appear in
the O(2) ⊗ O(N) model. For this purpose we need to classify the perturbations of
the O(2) ⊗ O(N) model that do not break the O(N) invariance. The multicritical
Hamiltonian (4) can be rewritten as
Hmcr = Hch +
1
2
r2,2V
(2,2) +
1
4!
f4,4V
(4,4) +
1
4!
f4,2V
(4,2), (9)
where Hch is theO(2)⊗O(N) Hamiltonian (6) with r = (r1+r2)/2, g1,0 = (2u0+2v0+
w0 + z0)/6, g2,0 = (2u0 +2v0−2w0 + z0)/6, and r2,2 = (r1 − r2)/2, f4,2 = (u0−v0)/2,
and f4,4 = (u0 + v0 − w0 − z0)/6. Here, V (2,2), V (4,4), and V (4,2), are respectively a
quadratic term that transforms as a spin-2 operator under the O(2) group and two
quartic terms that transform as a spin-4 and as a spin-2 operator respectively. Their
explicit expressions are:
V (2,2) ≡ φ2 − ψ2,
93
A.Pelissetto, E.Vicari
V (4,2) ≡ (φ2 + ψ2)V (2,2),
V (4,4) ≡ (φ2)2 + (ψ2)2 − 2φ2ψ2 − 4(φ · ψ)2. (10)
A detailed description of all possible perturbations of the O(2) ⊗ O(N) FP that
leave the O(N) group invariant can be found in App. B of [4]. Note that for Hcr we
have r2,2 = f4,2 = 0, so that one must only consider the spin-4 quartic perturbation.
Let us first discuss the chiral FP (a) that has g∗2,ch > 0. In order to estimate the
RG dimensions y4,2 and y4,4 of the above-reported perturbations, we computed the
corresponding five-loop MS series and we analyzed them within the 3d-MS scheme.
The perturbative series, that are not reported here but are available on request,
were analyzed using the conformal-mapping method and the Padé-Borel method,
following closely [23], to which we refer for details. The error on the conformal-
method results takes into account the spread of the results as the parameters α and
b are varied (cf. [23] for definitions) and the error due to the uncertainty of the FP
location (we use the estimates reported in [4,12–14]). The results of the analyses
using the conformal-mapping method are reported in table 1. Completely consistent
results are obtained by using Padé-Borel approximants. As it can be seen, y4,2 and
y4,4 are always negative, indicating that the chiral FP is stable for any N . This FP is
therefore expected to be the relevant FP whenever the RG flow starts in the region
z0 > 0. For N = 2, symmetry (5) implies that a chiral FP [the equivalent one that
is obtained by using (5)] may also be reached from the region z0 < 0.
It is of interest to compute also the RG dimension y2,2 of the quadratic pertur-
bation. For the multicritical Hamiltonian it is related to the crossover exponent φ:
φ = νy2,2, where ν is the correlation-length exponent at the chiral FP (see [12–14]
for numerical estimates). The exponent y2,2 has already been computed for several
values of N in [4]: indeed, y2,2 = y4, where y4 is the RG dimension of the operator
O(4) defined in App. C of [4]. Numerical estimates, taken from [4], are reported in
table 1.
Now, let us consider the collinear FP (b) that has g∗2,cl(N) < 0 for 2 6 N 6 4. For
N = 2 the RG dimensions at the collinear FP can be related to the RG dimensions
of operators in the XY model. Indeed, the O(2)⊗O(2) collinear FP is equivalent to
an XY FP. The mapping is as follows. One defines two fields ai and bi, i = 1, 2, and
considers [9]
φ11 = (a1 − b2)/
√
2,
φ22 = (a1 + b2)/
√
2,
φ12 = (b1 − a2)/
√
2,
φ21 = (b1 + a2)/
√
2. (11)
At the collinear FP, fields a and b represent two independent XY fields. Using
this mapping it is easy to show that V (4,2) ∼ O(3,1)
i (a)bj , O(3,1)
i (b)aj , and V (4,4) ∼
T11(a)T11(b), T12(a)T12(b), where
O(3,1)
i (a) ≡ aia
2, Tij(a) ≡ aiaj −
1
2
δija
2. (12)
94
Interacting N -vector order parameters with O(N) symmetry
Thus, if y3,1 and y2 are the RG dimensions of O(3,1)
i and Tij in the XY model, we
have
y4,2 = yh + y3,1 − 3, y4,4 = 2y2 − 3. (13)
By using the equations of motion, one can relate O(3,1)
i to ai [24] and obtain y3,1 =
3−yh, so that y4,2 = 0 exactly (this holds in three dimensions; in generic dimension d,
y4,2 = 3−d). For y2 we can use the result reported in [21], obtaining y4,4 = 0.532(12).
The analysis of the perturbative series gives results that are fully consistent: y4,2 =
0.0(1), y4,4 = 0.57(4).
In order to determine y4,2 and y4,4 forN = 3 and 4 we analyzed the corresponding
5-loop 3d-MS expansions. The results for y4,4 are reported in table 1. They indicate
that y4,4 is positive, which implies that the spin-4 quartic perturbation is relevant
and therefore the collinear FP is unstable. We do not quote any result for y4,2. The
perturbative analysis does not allow us to obtain any reliable result: the estimates
vary significantly with the parameters b and α and with the perturbative order.
It is interesting to observe that, based on the group-theoretical analysis reported
in App. B of [4], any quartic perturbation of the chiral FP that leaves the O(N)
symmetry invariant is a combination of spin-2 and spin-4 operators. Thus, the re-
sults presented here indicate that the chiral O(2) ⊗ O(N) FP is stable under any
perturbation that preserves the O(N) symmetry. In particular, it is also stable un-
der a perturbation of the form (φ · ψ)(aφ2 + bψ2) that may arise if the model is not
invariant under independent parity transformations. Indeed, such a term is nothing
but a particular combination of spin-2 and spin-4 perturbations. In the notations of
App. B of [4] (note that M and N of [4] should be replaced by N and 2 respectively)
we have
(φ · ψ)(aφ2 + bψ2) =
1
2
(a + b)O(4,2,1)
12 +
1
3
(a− b)O(4,4)
1112 . (14)
Moreover, the additional quadratic term φ · ψ is nothing but a component of the
spin-2 quadratic term that breaks the O(2) group, so that the associated crossover
exponent is again φ = νy2,2, y2,2 being reported in table 1. This is a general result
that follows from the analysis of [4]: any quadratic perturbation of the O(2)⊗O(N)
FP that does not break the O(N) invariance is a combination of the components of
the spin-2 quadratic operator. Thus, any perturbation is always associated with the
same crossover exponent φ = νy2,2.
3.3. Stability of the decoupled O(N) ⊕ O(N) fixed point
We now consider FP (3) discussed in section 3.1. In order to check its stability,
we must determine the RG dimensions at the FP of the perturbations
PE ≡ φ2ψ2, PT ≡
∑
ij
Tφ,ijTψ,ij , (15)
where Tφ,ij = φiφj − 1
2
δijφ
2. Simple RG arguments show that the RG dimensions
are given by
yE =
2
νN
− 3 =
αN
νN
, yT = 2y2 − 3, (16)
95
A.Pelissetto, E.Vicari
where αN and νN are the critical exponents of the 3-dimensional O(N) universali-
ty class (see [2] for a comprehensive review of results), while y2 is the exponent
associated with the quadratic spin-2 perturbation in the O(N) model [21,25,26].
Since αN < 0 for N > 2 we have yE < 0, i.e. the perturbation PE is always irrel-
evant. As for yT we can use the results reported in [21,25,26]. The spin-2 exponent is
equal to y2 = 1.766(6), 1.790(3), 1.813(6) for N = 2, 3, 4 and increases towards 2 as
N → ∞. Correspondingly yT = 0.532(12), 0.580(6), 0.626(12), increasing towards 1
as N → ∞. It follows that PT is always relevant. Thus, the decoupled FP is always
irrelevant.
3.4. Stability of the mn fixed point
Here, we wish to consider the stability of the mn FP. For this purpose we must
consider the two perturbations
P1 ≡ (φ2)2 − (ψ2)2, P2 ≡ (φ · ψ)2 − 1
N
φ2ψ2. (17)
Note that P1 is not symmetric under interchange of φ and ψ and is therefore not
of interest for Hcr. The corresponding RG dimensions y1 and y2 are computed in
App. A: y1 = 0.4(3), 0.2(2), 0.2(2) for N = 2, 3, 4; y2 = −0.9(5), −1.0(8), −0.8(5)
for the same values of N . They indicate that P1 is relevant and P2 is irrelevant at the
mn FP. Therefore, the mn FP is a stable FP for Hcr (only P2 should be considered
in this case) and an unstable one for Hmcr.
4. Conclusions
In this paper we have investigated the critical behavior of systems described by
Hamiltonians (1) and (4). We find that Hcr has three possible stable FPs: for any N ,
except possibly N = 6 (for such a value of N the evidence of this FP is less robust
[12]), there is the O(2) ⊗ O(N) chiral FP that is relevant for systems with z0 > 0;
for N = 2 there is a stable chiral FP with z < 0 [equivalent to the previous one by
symmetry (5)], that is relevant for systems with z0 < 0; for N = 2, 3, 4, there is the
mn FP that is relevant for systems with 2u0−w0−z0 < 0. In the multicritical theory
(4) only the chiral FPs are stable. Thus, systems with z0 > 0 (or, for N = 2, with
z0 6= 0) may show a multicritical continuous transition with the larger O(2)⊗O(N)
symmetry.
It is interesting to note that the most general O(N)-invariant LGW Hamiltonian
for two N -vector parameters includes other couplings beside those present in (1)
and (4). One should consider
Hcr,ext = Hcr +
r2
2
φ · ψ +
a0
4!
(φ · ψ)(φ2 + ψ2), (18)
Hmcr,ext = Hmcr +
r3
2
φ · ψ +
1
4!
(φ · ψ)(a1φ
2 + a2ψ
2), (19)
96
Interacting N -vector order parameters with O(N) symmetry
depending whether one wants to preserve the symmetry under the exchange of the
two fields. As we discussed in section 3.2, the chiral FP is a stable FP also for these
two extended models.
Hamiltonians (4) and (18) have two mass parameters and thus symmetry en-
largement can be observed only at the multicritical point, where the singular part
of the free energy has the form
Fsing = µ2−α
t f(µgµ
−φ
t ), (20)
where µt and µg are two linear scaling fields (linear combinations of the temperature
and of another relevant parameter), α and φ are the specific-heat and the crossover
exponents at the O(2)⊗O(N) model. Note that the same expression, with the same
α and φ, applies to both models, apart from nonuniversal normalization constants.
In Hmcr,ext there are three quadratic parameters and thus the chiral multicritical
point can be observed only if three relevant parameters are properly tuned. The
singular part of the free energy becomes
Fsing = µ2−α
t f(µg1µ
−φ
t , µg2µ
−φ
t ), (21)
where µg1 and µg2 are two linear scaling fields associated with the same crossover
exponent φ.
A. The mn model: new fixed points
In this Appendix we consider the mn model defined by Hamiltonian (7), focusing
on the case n = 2 that is of interest for the present paper. Within the ε expansion one
finds four FPs, the stable one being the O(m) FP with g1 = 0 and g2 = g∗m, where g∗m
is the FP value of the renormalized zero-momentum coupling in the O(m) model, see
[2,5,27] and references therein. For m = 2 (and n = 2) the mn model is equivalent
[9] to the O(2)⊗O(2) model defined by Hamiltonian (6). For this model, the results
of [12,13] indicate the presence of a new FP that is not predicted by the ε-expansion
analysis. Because of the mapping, this implies the presence of a new FP in the mn
model with g2 < 0. In the MZM scheme the results of [13] imply the presence of a
stable FP at g1 = 4.4(2) and g2 = −4.5(2), where the renormalized couplings g1 and
g2 are normalized so that g1 = 3g1,0/(16πR2mm), g2 = 3g2,0/(16πRmm) at tree level
(m is the renormalized zero-momentum mass), where Rk = 9/(8 + k). In the 3d-MS
scheme, by using the results of [12], we obtain g1 = 2.25(13) and g2 = −2.31(21),
where gi = gi,0µ
−ε/Ad with Ad = 2d−1πd/2Γ(d/2). It is thus of interest to check
whether additional FPs are also present for other values of m > 2. As we shall show
below we find an additional FP for m = 3 and m = 4. For m > 5 no new FP is
found.
In order to check for the presence of additional FPs we considered the six-loop
MZM expansions of [28] and we generated 5-loop 3d-MS expansions. For the analysis
we used the conformal-mapping method: the position of the Borel singularity in the
MZM scheme is reported in [28], while in the 3d-MS we used its trivial generaliza-
tion. The two β functions were resummed by using several different approximants
97
A.Pelissetto, E.Vicari
Table 2. Results for the mn model for n = 2 in two different schemes. The index
in column “scheme”, 4l, 5l, 6l, refers to the number of loops. We report the
coordinate of the FP g1, g2, the percentage of approximants that find the zero
(pFP), and the percentage of approximants that indicate that the FP is stable
(pst). In the column “info” we report the number of approximants that give real
(first number) and complex eigenvalues (second number).
m scheme g1 g2 pFP pst info
2 3d-MS5l 2.3(2) −2.3(2) 24/24 19/24 6/24, 18/24
3d-MS4l 2.4(2) −2.5(3) 13/24 2/13 13/13, 0/24
MZM6l 4.60(8) −4.51(11) 24/24 24/24 0/24, 24/24
MZM5l 4.7(3) −4.6(4) 24/24 20/24 4/24, 20/24
3 3d-MS5l 2.5(2) −2.5(2) 23/24 22/23 18/23, 5/23
3d-MS4l 2.5(3) −2.6(5) 13/24 2/13 13/13, 0/13
MZM6l 5.6(3) −5.2(3) 23/24 23/23 13/23, 9/23
MZM5l 5.2(2) −4.8(2) 24/24 24/24 1/24, 23/24
4 3d-MS5l 2.9(4) −2.9(3) 19/24 18/19 19/19, 0/19
3d-MS4l 3.0(3) −3.0(4) 8/24 0/8 8/8, 0/8
MZM6l 6.6(6) −6.0(6) 15/24 15/15 13/15, 2/15
MZM5l 5.9(3) −5.2(3) 24/24 24/24 12/24, 12/24
depending on two parameters, b and α (see [23] for definitions). For simplicity, each
time we resummed the two β functions by using the same b and α and then deter-
mined their common zeroes. In principle, it would have been more natural to consider
different values of b and α for the two β functions and all possible combinations.
However, as we already tested in the O(2)⊗O(N) model, the two choices give fully
equivalent results. In the analysis we used α = −1, 0, 1, 2 and b = 4, 6, . . . , 14, which
appeared to be an optimal choice. We report the results in table 2. For comparison,
we also performed the analysis for m = 2, obtaining results completely consistent
with those reported above. In the table we also give the percentage of cases in which
a FP was found (pFP) and in which this FP was stable (pst). Finally, we also indicate
the number of cases in which the stability eigenvalues were real or complex.
For m = 2 and m = 3 the presence of a new FP is unambiguous. Essentially all
considered approximants at five and six loops in both schemes give a stable FP. For
m = 4, the percentages are smaller, although the overall results are still in favor of
a new stable FP. For m > 5 there is essentially no evidence. For m = 2 the stability
eigenvalues are complex, in agreement with the results for the O(2) ⊗ O(2) model
[12–14]. For m = 3 and m = 4 the numerical results favor real eigenvalues instead.
It is interesting to note that the 3d-MS FPs lie at the boundary of the region in
which the expansions are Borel summable, g1 + g2 > 0. This is not the case for the
MZM ones (Borel summability requires R2mg1+Rmg2 > 0). Thus, the 3d-MS results
should be more reliable in these models.
The mn model for n = 2 is invariant under the group [SO(m) ⊕ SO(m)] ⊗
98
Interacting N -vector order parameters with O(N) symmetry
C2 where C2 is the symmetry group of the square. We now consider two quartic
operators that break such a symmetry:
P1 ≡ (Φ1 · Φ1)
2 − (Φ2 · Φ2)
2, (22)
P2 ≡ (Φ1 · Φ2)
2 − 1
m
(Φ1 · Φ1)(Φ2 · Φ2), (23)
where the scalar products are taken in the O(m) space. The first operator is the
only quartic one that preserves the continuous symmetry and breaks C2 → Z2 ⊕Z2,
while the second preserves C2 but breaks SO(m) ⊕ SO(m) → SO(m). Note that
in general P1 mixes with the lower-dimensional operator Φ1 · Φ1 − Φ2 · Φ2. Such a
mixing should be taken into account in the MZM scheme, but does not occur in the
massless MS scheme. The operators P1 and P2 are relevant for the stability of the
FPs of the mn theory in larger models with smaller symmetry group.
We computed the anomalous dimensions of P1 and P2 at the new FPs of table 2
by analyzing the corresponding 5-loop 3d-MS series. The exponent y1 was obtained
from the analysis of the inverse series 1/y1; the direct analysis of the series of y1 was
very unstable. For y2 we used instead the corresponding series. The results were not
very stable and should be taken with caution. They are:
m = 2 : y1 = 0.4(3), y2 = −0.9(5);
m = 3 : y1 = 0.2(2), y2 = −1.0(8);
m = 4 : y1 = 0.2(2), y2 = −0.8(5).
It is interesting to note that the mn model is a submodel of the multicritical
Hamiltonian (8) for r1 = r2 and f1,0 = f3,0 if we set Φa1 = φa, Φa2 = ψa, f1,0 =
f3,0 = g1,0 +g2,0, and f2,0 = 2g1,0. This implies that the new FPs may be relevant for
the multicritical behavior of Hmcr,2. To investigate this possibility we must compute
the anomalous dimension of the operator that breaks [SO(m) ⊕ SO(m)] ⊗ C2 →
O(m)⊕O(m), i.e., the operator P1. As it can be seen, y1 > 0 in all cases, indicating
that the mn FP is unstable in the full theory. Thus, the decoupled FP appears to
be the only stable FP of the multicritical model (8) [21,22].
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100
Interacting N -vector order parameters with O(N) symmetry
Взаємодіючі N -векторні параметри порядку з O(N)
симетрією
А.Пеліссетто 1 , Е.Вікарі 2
1 Фізичний факультет, Університет “La Sapienza” Рим та INFN,
пл. ле Моро 2, I–00185 Рим, Італія
2 Фізичний факультет, Університет Піза та INFN,
вул. Буонарроті 2, I–56127 Піза, Італія
Отримано 21 вересня 2004 р.
Ми розглядаємо критичну поведінку найбільш загальної системи
двох N -векторних параметрів порядку, яка є O(N) інваріантною.
Ми показуємо, що вона може мати мультикритичний перехід з
розширеною симетрією контрольованою чіральною O(2) ⊗ O(N)
нерухомою точкою. Для N = 2, 3, 4, якщо система є також
інварінтною відносно обміну двох параметрів порядку і відносно
незалежного перетворення парності, можна спостерегти критичний
перехід контрольований нерухомою точкою, що належить mn
моделі. Також у цьому випадку відбувається розширення симетрії
при переході, симетрія стає [SO(N) ⊕ SO(N)] ⊗ C2, де C2 є гру-
пою симетрії на квадраті.
Ключові слова: N -векторна модель, O(N) симетрія,
мультикритичні переходи
PACS: 05.70.Jk, 64.60.Fr, 75.10.Hk
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