Page 1

arXiv:0812.0675v1 [cond-mat.stat-mech] 3 Dec 2008

Field theory of bicritical and tetracritical points. III. Relaxational

dynamics including conservation of magnetization (Model C)

R. Folk,1, ∗Yu. Holovatch,2,1, †and G. Moser3, ‡

1Institute for Theoretical Physics, Johannes Kepler University Linz, Altenbergerstrasse 69, A-4040, Linz, Austria

2Institute for Condensed Matter Physics, National Academy of

Sciences of Ukraine, 1 Svientsitskii Str., UA–79011 Lviv, Ukraine

3Department for Material Research and Physics,

Paris Lodron University Salzburg, Hellbrunnerstrasse 34, A-5020 Salzburg, Austria

(Dated: December 3, 2008)

We calculate the relaxational dynamical critical behavior of systems of O(n?)⊕O(n⊥) symmetry

including conservation of magnetization by renormalization group (RG) theory within the minimal

subtraction scheme in two loop order. Within the stability region of the Heisenberg fixed point

and the biconical fixed point strong dynamical scaling holds with the asymptotic dynamical critical

exponent z = 2φ/ν − 1 where φ is the crossover exponent and ν the exponent of the correlation

length. The critical dynamics at n?= 1 and n⊥ = 2 is governed by a small dynamical transient

exponent leading to nonuniversal nonasymptotic dynamical behavior. This may be seen e.g. in the

temperature dependence of the magnetic transport coefficients.

PACS numbers: 05.50.+q, 64.60.Ht

I.INTRODUCTION

In two preceding papers1,2we have considered the crit-

ical statics and relaxational dynamics of O(n?) ⊕ O(n⊥)

physical systems near the multicritical point where the

two phase transition lines of the system corresponding

to O(n?)-symmetry and O(n⊥)-symmetry meet.

space of the order parameter (OP) dimensions n?and

n⊥ decomposes in regions where the multicritical be-

havior is described by different fixed points (FPs) - the

O(n = n?+ n⊥)- isotropic FP, the biconical FP, the de-

coupling FP - and a region where no stable FP is found

(run away region) (see Fig 1 in Ref.1). In the resummed

two loop order field theoretic treatment it was found that

for integer values of n?and n⊥the biconical FP is stable

only for a system with n?= 1, n⊥= 2, and its symmetric

counterpart. For specific initial conditions of the nonuni-

versal parameters of the system also the O(n = n?+n⊥)-

isotropic FP (Heisenberg FP) might be reached. Such a

system is physically represented by an antiferromagnet

in an external magnetic field. The two cases mentioned

above correspond to tetracritical and bicritical multicrit-

ical points correspondingly. If no FP is reached the mul-

ticritical point might be of first order, i.e. a triple point.

The

The dynamics of the antiferromagnet in a magnetic

field is quite complicated and the equations of motion

have been formulated for the slow densities by Dohm

and Janssen3. These Eqs. contain reversible and irre-

versible coupling terms between the OPs (the compo-

nents of the staggered magnetization parallel and perpen-

dicular to the magnetic field) and one conserved density

(the parallel component of the magnetization). In fact

there is a second conserved density (CD) - the energy

density - which in general should be taken into account

but it will not be included here since in two loop order

the specific heat exponent at the biconical FP turned out

to be negative4for the case n?= 1 and n⊥= 2. Con-

cerning the new static results1a simplified dynamical

model3has been reconsidered2consisting of two relax-

ational equations for the two OP. The timescale ratio v

between the two relaxation rates Γ?and Γ⊥introduces a

very small dynamical transient since the dynamical FP

lies very near to the stability boundary separating strong

and weak dynamical scaling. The strong dynamical scal-

ing FP is governed by v⋆finite and different from zero

whereas the weak dynamical scaling FP by v⋆= 0or∞

correspondingly.

A further step to the complete model is to include the

diffusive dynamics of the slow CD leading to a model C

like extension. In this extended model a new timescale

ratio appears defined by the ratio of one of the OP-

relaxation rate to the kinetic coefficient λ of the con-

served density m. This model has been studied in one

loop order in Refs3,5,6taking into account only a part

of dynamical two loop order terms and one loop statics.

Here we present a complete two loop order calculations.

The inclusion of further densities beside the OP makes

it necessary to extend the static functional of the usual

φ4-theory, although the OP alone would be sufficient to

describe the static critical behavior. Such an extended

static functional for isotropic systems (O(n) symmetry)

with short range interaction has the form7,8

H(C)=

?

ddx

?

1

2

˚˜ r?φ0·?φ0+1

2

n

?

2˚ γm0?φ0·?φ0−˚hm0

i=1

∇i?φ0· ∇i?φ0+1

2m2

0

+

˚˜ u

4!

??φ0·?φ0

?2

+1

?

. (1)

Here the order parameter?φ0≡?φ0(x) is assumed to be a

n-component real vector, the symbol · denotes the scalar

product. The secondary density m0≡ m0(x) is consid-

ered as a scalar quantity and˚h is the conjugated field

Page 2

2

to m0. It is chosen to have a vanishing average value

?m0? = 0. Within statics, the above functional is equiv-

alent to the Ginzburg-Landau-Wilson(GLW)-functional

HGLW=

?

ddx

?

1

2˚ r?φ0·?φ0+1

2

n

?

i=1

∇i?φ0· ∇i?φ0

+˚ u

4!

??φ0·?φ0

?2?

,(2)

where ˚ r is proportional to the temperature distance to

the critical point and ˚ u is the fourth order coupling in

which perturbation expansion is usually performed. The

GLW-functional (2) is obtained by integrating out the

CD, which appears only in Gaussian order in (1), in the

corresponding partition function. The parameters˚˜ r,˚˜ u

and ˚ γ in (1) and, ˚ r and ˚ u in (2) are related by

˚ r =˚˜ r +˚ γ˚h ,˚ u =˚˜ u − 3˚ γ2. (3)

The extended static functional appears in the driving

force of the equations of motion for the OP?φ0and the

CD m0. The ratio of the kinetic coefficient˚Γ in the re-

laxation equation for the OP and the kinetic coefficient

˚λ in the diffusive equation for the CD defines the dynam-

ical parameter w whose FP value governs the dynamical

scaling of the model.

It is worthwhile to summarize some results for model C

at a usual critical point, where the CD can be identified

with an energy like density and the value of specific heat

exponent α (in any case) governs the relevance of the

asymmetric static coupling ˚ γ between the OP and the

CD. Namely, this coupling is irrelevant in the RG sense

- it vanishes at the FP - if the specific heat exponent is

negative, i.e. the specific heat of the system does not

diverge at the critical point. If the specific heat diverges

then there remain two possibilities for the dynamical FP:

either the FP value of the time scale ratio w between the

timescale of the OP and the CD is different from zero

and finite, or its FP value is zero or infinite. In the first

case strong dynamical scaling with one timescale for the

OP and the CD is realized with one dynamical scaling

exponent z = 2+α/ν (ν the exponent of the correlation

length). In the second case weak dynamical scaling is

present and the time scale of the OP is different from the

timescale of the CD, both represented by a corresponding

dynamical critical exponent.

dynamical scaling FP is tiny (see e.g Fig. 1 in Ref.8). One

should note that at the usual critical point an asymmetric

coupling to the OP as given in Eq. (1) is always ’energy-

like’ independent of its physical origin. That means the

divergence of the CD susceptibility is always described

by the specific heat exponent α.

In the case of a multicritical point treated here the sit-

uation is more complicated since there are two OPs and

a CD might couple to both of these OPs. As has been

shown in paper I after a proper rotation (Eq. (64)) in

the OP space temperature and magnetic like field direc-

This region of the weak

tions can be identified. In consequence at the multicriti-

cal point one has to discriminate the case of energy and

magnetization conservation.

The paper is organized as follows. In section II we ex-

tend model C of the O(n) symmetrical critical system to

the case of a O(n?) ⊕ O(n⊥) symmetrical multicritical

point as considered in papers I and II. The renormaliza-

tion is performed in section III and the field theoretic

functions are calculated in section IV. Then we discuss

the possible FP and their stability in section V. Effec-

tive dynamical critical behavior is considered in section

VI followed in section VII by a short summary of the

results and an outlook on further work to be done.

II.MODEL C FOR MULTICRITICAL POINTS

A. Static functional

In order to describe the multicritical behavior the n-

dimensional space of the order parameter components is

split into two subspaces with dimensions n⊥and n?with

the property n⊥+n?= n. The order parameter separates

into

?φ0=

??φ⊥0

?φ?0

?

, (4)

where?φ⊥0is the n⊥-dimensional order parameter of the

n⊥-subspace, and?φ?0is the n?-dimensional order param-

eter of the n?-subspace. Introducing this separation into

the GLW-functional (2) one obtains

HBi=

?

ddx

?

1

2˚ r⊥?φ⊥0·?φ⊥0+1

2

n⊥

?

n?

?

?2

i=1

∇i?φ⊥0· ∇i?φ⊥0

+1

2˚ r??φ?0·?φ?0+1

2

i=1

+˚ u?

∇i?φ?0· ∇i?φ?0

+˚ u⊥

4!

+2˚ u×

??φ⊥0·?φ⊥0

??φ⊥0·?φ⊥0

multicritical

The properties of this functional con-

cerning the renormalization, regions of stable FPs, and

corresponding type of multicritical behavior has been ex-

tensively discussed in paper I (see9for earlier references).

The separation (4) has now to be performed in (1). The

resulting functional is

4!

??φ?0·?φ?0

???φ?0·?φ?0

Ginzburg-Landau-

?2

4!

??

. (5)

which

Wilson model.

representsa

H(C)

Bi=

?

ddx

?

1

2

˚˜ r⊥?φ⊥0·?φ⊥0+1

2

n⊥

?

i=1

∇i?φ⊥0· ∇i?φ⊥0

+1

2

˚˜ r??φ?0·?φ?0+1

2

n?

?

i=1

∇i?φ?0· ∇i?φ?0+1

2m2

0

Page 3

3

+

˚˜ u⊥

4!

??φ⊥0·?φ⊥0

+2˚˜ u×

4!

?2

+

˚˜ u?

4!

??φ?0·?φ?0

???φ?0·?φ?0

?2

?

?

??φ⊥0·?φ⊥0

2˚ γ?m0?φ?0·?φ?0−˚hm0

+1

2˚

γ⊥m0?φ⊥0·?φ⊥0+1

. (6)

Integrating the contributions of the secondary density in

the corresponding partition function, (6) reduces to the

static functional (5). Relations analogous to (3) between

the parameters of the two static functionals arise. They

read

˚ r⊥=˚˜ r⊥+˚ γ⊥˚h ,

˚ r?=˚˜ r?+˚ γ?˚h ,

˚ u⊥=˚˜ u⊥− 3˚ γ2

˚ u?=˚˜ u?− 3˚ γ2

˚ u×=˚˜ u×− 3˚ γ⊥˚ γ?.

⊥,(7)

?,(8)

(9)

Because the partition function calculated from (6) is re-

ducible to a partition function based on (5) by integra-

tion, the correlation functions, or vertex functions respec-

tively, of the secondary density m0are exactly related to

correlation functions of the order parameter. This leads

to several relations which are important for the renor-

malization. In particular, the average value of m0 and

the two-point correlation function are defined as

?m0?≡

1

N(C)

Bi

1

N(C)

Bi

?

?

D(φ⊥0,φ?0,m0) m0e−H(C)

Bi ,(10)

?m0m0?≡

D(φ⊥0,φ?0,m0) m0m0e−H(C)

Bi ,

(11)

with N(C)

tion constant and D(φ⊥0,φ?0,m0) as a suitable integral

measure. Performing the integration over m0in (10) and

using Eqs.(7)-(9), the average value of m0reads

Bi=?D(φ⊥0,φ?0,m0) e−H(C)

Bi as the normaliza-

?m0? =˚h −˚ γ⊥

?1

2

?φ2

⊥0

?

−˚ γ?

?1

2

?φ2

?0

?

(12)

where?φ2denotes quadratic insertions of the order pa-

rameter. Their average values on the right hand side of

(12),

?1

2

?φ2

αi0

?

=

1

NBi

?

D(φ⊥0,φ?0)1

2

?φ2

αi0e−HBi, (13)

are now calculated with the static functional (5) and

NBi=?D(φ⊥0,φ?0) e−HBi. In order to obtain ?m0? = 0

˚h =˚ γ⊥

2

the conjugated external field is chosen to

?1

?φ2

⊥0

?

+˚ γ?

?1

2

?φ2

?0

?

(14)

Quite analogous by integrating m0in (11) one obtains the

following relation for the two-point correlation function

of the secondary density

?m0m0?c= 1 −˚? γ

T·˚Γ

(0,2)·˚? γ .(15)

In (15) where we have introduced the column matrix

˚? γ ≡

?˚ γ⊥

˚ γ?

?

.(16)

The superscriptTindicates a transposed vector or ma-

trix, while the subscript c on the average at the left

hand side of (15) denotes the cummulant ?A B?c ≡

?A B? − ?A??B?. The matrix

˚Γ

(0,2)

=

?˚Γ(0,2)

˚Γ(0,2)

?

;⊥⊥

˚Γ(0,2)

;⊥?

˚Γ(0,2)

;??;?⊥

?1

?1

?

= −

2?φ2

2?φ2

⊥0

1

2?φ2

1

2?φ2

⊥0?c ?1

?0?c ?1

2?φ2

2?φ2

⊥0

1

2?φ2

1

2?φ2

?0?c

?0?c

⊥0

?0

?

(17)

of two-point vertex functions is related to correlations of

φ2-insertions. The vertex functions generally have been

introduced in paper I (section III Renormalization), and

especially the matrix (17) (renormalized counterpart) in

Eq.(83) therein. A third important relation can be ob-

tained by differentiating the average value (10) by˚h at

fixed parameter △˚˜ rα,˚˜ uα,˚ γα. In △˚˜ rα =˚˜ rα−˚˜ rαic the

shift of the critical temperature has been taken into ac-

count (for more details see Appendix A in paper I). As a

result one obtains

∂

∂˚h

?m0(x)?

???△˚˜ rα,˚˜ uα,˚ γα=

?

dx′?m0(x) m0(x′)?c. (18)

Due to relation (14) the external field is function of

△˚ rα. The h-derivative in (18) can be rewritten as △˚ rα-

derivatives. Finally one obtains

?

dx′?m0(x) m0(x′)?c=˚? γ

T·

∂

∂ △˚? r

?m0(x)?

???△˚˜ rα,˚˜ uα,˚ γα

(19)

where we have defined

∂

∂ △˚? r

≡

?

∂

∂△˚ r⊥

∂

∂△˚ r?

?

.(20)

All static vertex functions, for the order parameter as

well as for the secondary density, may be calculated with

(6) in perturbation expansion as functions of the cor-

relation lengths {ξ} ≡ {ξ⊥,ξ?}, the set of quartic cou-

plings {˚˜ u} ≡ {˚˜ u⊥,˚˜ u?,˚˜ u×}, the set of asymmetric cou-

plings {˚ γ} ≡ {˚ γ⊥,˚ γ?}, and the wave vector modulus k.

The parameters in the order parameter vertex functions

˚ ¯Γ

α1···αN;i1···iL(for the notation see Appendix A in pa-

per I) via relations (7) - (9) combine the corresponding

parameters of the multicritical GLW-model (5). Thus

all order parameter vertex functions calculated with (6)

have the property

(N,L)

˚ ¯Γ

(N,L)

α1···αN;i1···iL

˚Γ(N,L)

?{ξ},k,{˚˜ u},{˚ γ}?=

α1···αN;i1···iL

?{ξ},k,{˚ u}?.

(21)

Page 4

4

meaning that they are identical to corresponding func-

tions of the multicritical GLW-model (5). For this rea-

son no distinction between the correlation lengthes en-

tering the left and right hand side of (21) is necessary.

The correlation lengths are defined from the two-point

order parameter vertex functions at the left side with

(6), and on the right side with (5) (see Eqs.(A7) and

(A8) in paper I). Vertex functions of the secondary den-

sity can be expressed as functions of {˚ u} instead of

{˚˜ u} by using (7)-(9). Especially the two-point function

˚ ¯Γmm= ?m0m0?−1

lowing, can be written as

c, which will be of interest in the fol-

˚ ¯Γmm

?{ξ},k,{˚˜ u},{˚ γ}?=˚Γmm

?{ξ},k,{˚ u},{˚ γ}?

(22)

B.Dynamical model

The dynamical equations of model A in paper II have

now to be extended by appending a diffusion equation

for the secondary density. One obtains

∂?φ⊥0

∂t

∂?φ?0

∂t

= −˚Γ⊥δH(C)

Bi

δ?φ⊥0

+?θφ⊥,(23)

= −˚Γ?δH(C)

Bi

δ?φ?0

+?θφ?, (24)

∂m0

∂t

=˚λ∇2δH(C)

Bi

δm0

+ θm.(25)

In addition to the two kinetic coefficients˚Γ⊥and˚Γ?of

the order parameter in the corresponding subspaces, a

kinetic coefficient˚λ of diffusive type for the conserved

secondary density is now present. The stochastic forces

?θφ⊥,?θφ?and θmfulfill Einstein relations

?θα

φ⊥(x,t) θβ

?θi

φ⊥(x′,t′)?=2˚Γ⊥δ(x − x′)δ(t − t′)δαβ,(26)

φ?(x,t) θj

?θm(x,t) θm(x′,t′)?=−2˚λ∇2δ(x − x′)δ(t − t′) ,(28)

φ?(x′,t′)?=2˚Γ?δ(x − x′)δ(t − t′)δij, (27)

with indices α,β = 1,...,n⊥and i,j = 1,...,n?corre-

sponding to the two subspaces. The dynamical two-point

vertex function of the secondary density has a general

structure quite analogous to the corresponding functions

of the order parameter (see Eqs.(6) and (7) in paper II).

One can write

˚Γm ˜ m

?{ξ},k,ω?= −iω˚Ωm ˜ m

where˚Γmm

?{ξ},k?

genuine dynamical function8. ˜ m is the auxiliary density

corresponding to m. For shortness we have dropped the

couplings and kinetic coefficients in the argument lists of

(29).

?{ξ},k,ω?+˚Γmm

?{ξ},k?˚λ

(29)

is the static two-point function dis-

cussed in the previous subsection and˚Ωm ˜ m

?{ξ},k,ω?is a

III. RENORMALIZATION

A.Renormalization of the static parameters

As a consequence of the discussion at the end of subsec-

tion IIA all vertex functions will be expanded in powers

of the quartic couplings {˚ u} of the multicritical GLW-

model and the asymmetric couplings {˚ γ}. The renor-

malization scheme introduced in section III in paper I

remains valid and will be used in the following. The cor-

responding definitions and relations can be found therein

and will not be repeated here. In particular, we imple-

ment the minimal subtraction RG scheme10,11directly

at d = 3 to the two loop order. In the current extended

model additional renormalizations for the secondary den-

sity m0and the asymmetric couplings {˚ γ} have to be con-

sidered. The renormalized counterparts of the secondary

density and the asymmetric couplings are introduced as

m0 = Zmm

˚? γ = κ−ε/2Z−1

(30)

mZ−1

φ· Zγ·? γA−1/2

d

(31)

where κ is the usual reference wave vector modulus and

ε = 4 − d. The geometrical factor Adand the diagonal

matrix Zφ has been defined in Eq.(8) and Eq.(17) of

paper I. With (30) and (31) at hand, the renormalization

for the CD-CD two-point vertex Γmm function readily

follows

Γmm= Z2

m˚Γmm,(32)

The additional Z-factor Zm and the matrix Zγ are re-

lated to the known renormalization factors of the multi-

critical GLW-model as a consequence of the reducibility

of the extended model to the multicritical GLW-model.

From the condition that (19) is also valid for the renor-

malized counterparts of the appearing quantities the re-

lation

Zγ= Z2

mZr= Z2

mZφ· ZT

φ2

(33)

follows. For the second equality relation (18) of paper

I has been used. The Z-factors of the asymmetric cou-

plings are determined by the renormalizations of the sec-

ondary density and the φ2-insertions in the multicritical

GLW-model.

Relation (15) establishes a connection between the cor-

relation functions of the CD and the φ2-insertions in the

multicritical GLW-model. This relation should be invari-

ant under renormalization. Thus the renormalization of

the secondary density is related by

Z−2

m= 1 +? γT· A({u}) ·? γ(34)

to the additive renormalization A({u}) of the correla-

tion function of the φ2-insertions (17) in the multicritical

GLW-model introduced in Eq.(15) in paper I.

Page 5

5

B. Renormalization of the dynamical parameters

The general form of the renormalization of the auxil-

iary densities˜φ⊥0 ,˜φ?0, and the kinetic coefficients˚Γ⊥

and˚Γ?has been presented within model A in subsection

III A in paper II. It remains valid and will be used in the

following. Of course new contributions occur to the dy-

namical renormalization factors especially of the kinetic

coefficients

˚Γ⊥= ZΓ⊥Γ⊥,

˚Γ?= ZΓ?Γ?.(35)

due to the asymmetric coupling ? γ.

Within model C additional renormalizations only are

necessary for the auxiliary density ˜ m0 and the kinetic

coefficient˚λ. Thus we introduce

˜ m0= Z˜ m˜ m ,

˚λ = Zλλ.(36)

In the case of conserved densities the dynamical function

˚Ωm ˜ m

?{ξ},k,ω?in (29) does not contain new dimensional

sity ˜ m0 needs no independent renormalization. The Z-

factor Z˜ mis determined by the relation

singularities. Therefore the corresponding auxiliary den-

Z˜ m= Z−1

m. (37)

Due to the absence of mode coupling terms the renor-

malization of the kinetic coefficient λ is completely de-

termined by the static renormalization and Zλis

Zλ= Z2

m. (38)

IV.

ζ - AND β - FUNCTIONS

As already mentioned in the preceding section the

renormalization of the GLW-functional remains valid.

This validates also all ζ- and β-functions introduced in

section IV in paper I. We do not repeat them here, al-

though they will be used in the following.

A.Static functions

Apart from the three β-functions βu⊥, βu×and βu?,

and the two ζ-matrices Bφ2 and ζφ2 appearing in the

multicritical GLW-model (see section IV in paper I),

an additional ζ-function ζm and a column matrix of β-

functions for the asymmetric coupling (16) have to be

introduced. The relations between the renormalization

factors discussed in subsection IIIA give rise to corre-

sponding relations between the ζ- and β-functions. It

follows immediately from (34)

ζm({u},{γ}) ≡dlnZ−1

m

dlnκ

=1

2? γT· Bφ2({u}) ·? γ ,(39)

where Bφ2({u}) has been defined in Eq.(30) in paper I.

The κ-derivatives, also in the following definitions, always

are taken at fixed unrenormalized parameters. Inserting

the two loop expression of Bφ2({u}) (see Eq.(31) in paper

I) we obtain

ζm({u},{γ}) =n⊥

4

γ2

⊥+n?

4

γ2

?.(40)

The column matrix of the β-functions for the asymmetric

coupling ? γ is defined as

?βγ({u},{γ}) ≡ κd? γ

dκ

(41)

Inserting Eq.(31) into the above definition one obtains

together with relation (33) the expression

?βγ({u},{γ}) =

??

−ε

2+ ζm

?

1 + ζT

φ2({u})

?

·? γ .(42)

There 1 denotes the two dimensional unit matrix. The

matrix ζφ2({u}) has been introduced in paper I (see

Eq.(22)). The ζ-function ζmis exactly known from (39).

Thus finally we arrive at

?βγ({u},{γ}) =

??

−ε

2+12? γT· Bφ2({u}) ·? γ

?

1

+ζT

φ2({u})

?

·? γ . (43)

The above expression is valid in all orders of perturbation

expansion. Bφ2({u}) and ζφ2({u}) are calculated in loop

expansion within the multicritical GLW-model.

two loop expressions have been given in Eq.(31) and (23)-

(26) in paper I.

Their

B. Dynamical functions

Using relation (38) the ζ-function ζλcorresponding to

the kinetic coefficient λ is simply given by

ζλ({u},{γ}) ≡dlnZ−1

λ

dlnκ

= 2ζm({u},{γ})(44)

The dynamical ζ-functions of the kinetic coefficients of

the order parameter are defined by

ζ(C)

Γα({u},{γ},{w}) ≡dlnZ−1

Γα

dlnκ

,α = ?,⊥ .(45)

In the model C dynamics, they get non-trivial contribu-

tions from the asymmetric couplings γ⊥ and γ?. They

read now in two loop order

ζ(C)

Γ⊥

?{u},{γ},{w})?=¯ζ(C⊥)?u⊥,γ⊥,w⊥

w⊥γ⊥γ?

1 + w⊥

1 + w⊥

?

−n?

4

?

2

3u×+w⊥γ⊥γ?

??

1 + ln

2v

1 + v

−

?

1 +2

v

?

ln2(1 + v)

2 + v

?

+ ζ(A)

Γ⊥

?u⊥,u×,v?,

(46)

Page 6

6

ζ(C)

Γ?

?{u},{γ},{w})?=¯ζ(C?)?u?,γ?,w?

w?γ?γ⊥

1 + w?

1 + w?

?

−n⊥

4

?

2

3u×+w?γ?γ⊥

??

1 + ln

2

1 + v

−(1 + 2v)ln2(1 + v)

1 + 2v

?

+ ζ(A)

Γ?

?u?,u×,v?

(47)

where we have defined the timescale ratios

w⊥=Γ⊥

λ

,w?=Γ?

λ

(48)

The ratio v is equally defined to paper II as the ratio

v ≡

Γ?

Γ⊥

=w?

w⊥

,(49)

and is therefore a function of w⊥ and w?. In (46) and

(47) several ζ-functions of known subsystems already has

been introduced. ζ(A)

Γα

?uα,u×,v?with α = ? or ⊥ are the

explicitly in paper II (see Eqs (14) and (15) therein),

ζ-functions of the full multicritical model A presented

ζ(A)

Γ⊥

=

n⊥+ 2

36

n?

36u2

n?+ 2

36

n⊥

36

u2

⊥

?

3ln4

3−1

2

+ ln(1 + v)2

?

(50)

+

×

?2

u2

vln2(1 + v)

2 + v

?

?

v(2 + v)−1

2

?

,

ζ(A)

Γ?

=

?

3ln4

3−1

2

?

+ ln(1 + v)2

1 + 2v

(51)

+

u2

×

2v ln2(1 + v)

1 + 2v

−1

2

?

,

where¯ζ(Cα)?u,γ,w?are the genuine ζ-functions of model

fourth order coupling terms (pure model A terms). They

have been given explicitly in12for a n-component sys-

tem in two loop order. These contributions of model C

in the n-component subspaces without the corresponding

model A terms are8

C within the nα-component subspaces without pure

¯ζ(Cα)?u,γ,w?=

?

−

wγ2

1 + w

?

1

−1

2

?

nα+ 2

3

u1 − 3ln4

3

?

+

wγ2

1 + w

?

nα

2

w

1 + w−3(nα+ 2)

ln(1 + w)2

1 + 2w

2

ln4

3

−(1 + 2w)

1 + w

???

.

(52)

The β-functions corresponding to the timescale ratios

(48) and (49) can be expressed in terms of the corre-

sponding ζ-functions of the kinetic coefficients:

βv ≡ κdv

dκ= v?ζ(C)

Γ?− ζ(C)

Γ⊥

?, (53)

βw?≡ κdw?

dκ

= w?

?ζ(C)

?ζ(C)

Γ?− ζλ

?,

?,

(54)

βw⊥≡ κdw⊥

dκ

= w⊥

Γ⊥− ζλ

(55)

with the κ-derivatives taken at fixed unrenormalized pa-

rameters.

Note that these equations are not independent but one

of the three equations can be eliminated by the relation

v(l) = w?(l)/w⊥(l), which of course holds also for the

initial conditions.

V.FIXED POINTS AND THEIR STABILITY

A.Static fixed points

The FPs of the couplings ua and their stability have

been studied in section V of paper I. Their values and

the corresponding transient exponents have been listed

in Table I there. Let us recall that depending on the

values of n?and n⊥one of the following FPs is stable and

governs multicritical behavior: the isotropic Heisenberg

FP H(n?+ n⊥) with u⋆

D with u⋆

FP with u⋆

FPs of paper I one can now determine the FP values for

γ⊥and γ?from the Eq.

?= u⋆

⊥= u⋆

×= 0, and the biconical

⊥?= 0, and u⋆

×, the decoupling FP

??= 0, u⋆

??= 0, u⋆

⊥?= 0 and u⋆

×?= 0. For each of the

?βγ({u⋆},{γ⋆}) =

??

−ε

2+12? γ⋆T· Bφ2({u⋆}) ·? γ⋆

?

1

+ζT

φ2({u⋆})

?

·? γ⋆= 0.(56)

This splits each FP of paper I into a set of FPs in the

combined {u}-{γ}-space, which are equivalent in statics

but different in dynamics. Given the formula (43) for

?βγ, one can see, that Eq. (56) includes two equations

which have to be solved for given FP values of the quar-

tic couplings {u⋆}. The static FPs of paper I can be

roughly separated into three classes: i) the Gaussian FP

G, with u⋆

a= 0 for all couplings; ii) the decoupling FPs

H(n⊥),H(n?),D where u⋆

berg and biconical FPs H(n⊥+ n?),B where all u⋆

different from zero. Henceforth we list the FPs values

of the corresponding asymmetric couplings γ⋆

Tab. I, that summarizes our analysis given below. Note,

that γ⋆

?= 0 is of course always a solution of

equation (56), independent which values {u⋆} have. We

do not list this trivial solution explicitly in Table I, al-

though this may be the stable FP for definite values of

n⊥and n?in some cases.

×= 0; iii) the isotropic Heisen-

aare

?and γ⋆

⊥in

⊥= 0, γ⋆

Page 7

7

1.Gaussian fixed point G

At this FP one has ζφ2 = 0 and the two equations in

(56) reduce to the condition

n⊥

2

γ⋆2

⊥+n?

2γ⋆2

?= ε ,(57)

which is valid in all orders of perturbation expansion.

The above equation defines a line of FPs.

2. Heisenberg and decoupling fixed points H(n⊥),H(n?),D

At these FPs, where the cross coupling u× vanishes,

the matrix ζφ2 has the form

ζφ2(u×= 0) =

?

ζ(n⊥)

φ2

(u⊥)0

0ζ(n?)

φ2 (u?)

?

.(58)

The function ζ(nα)

the nα-component isotropic system. Eq.(56) reduces to

φ2

(uα) is the well known ζ-function of

?

−ε

2+12? γ⋆T· Bφ2({u⋆}) ·? γ⋆+ ζ(n⊥)

φ2

(u⋆

⊥)

?

γ⊥= 0,

(59)

?

−ε

2+12? γ⋆T· Bφ2({u⋆}) ·? γ⋆+ ζ(n?)

φ2 (u⋆

?)

?

γ?= 0,

(60)

where the matrix Bφ2 is of the form

Bφ2(u×= 0) =

?

B(n⊥)

φ2

(u⊥)0

0B(n?)

φ2 (u?)

?

. (61)

The non trivial FP values for γ⊥and γ?resulting from

Eqs.(59) and (60) are listed in Table I. They are valid in

all orders of perturbation expansion.

We want to remark that in Table I only FP which

exist for arbitrary order parameter component numbers

are given. For special n-values a line of FPs exist where

both asymmetric couplings γα are different from zero.

In the case n⊥ = n? = n/2 one has u⋆

and the ζ-functions in (60) and (60) are equal leading to

γ⋆2

φ2

⊥= u⋆

?= ¯ u⋆

⊥+ γ⋆2

?=?ε − 2ζ(n/2)

(¯ u⋆)?/B(n/2)

φ2

(¯ u⋆).

3. Isotropic Heisenberg and biconical FPs H(n⊥+ n?),B

At the isotropic Heisenberg FP H(n⊥+ n?) and the

biconical FP B, where all couplings uaare different from

zero, it is more convenient to transform the matrix ζφ2

into its diagonal form with the transformation

?ζ+

0

0 ζ−

?

= P−1· ζT

φ2 · P

(62)

introduced in section VI.B in paper I. Inserting (62) into

Eq.(56) leads to the transformed β-function

?βγ= P ·?βγ±

(63)

with

?βγ±=

??

−ε

2+12? γT· Bφ2 ·? γ

?

1 +

?ζ+

0

0 ζ−

??

·? γ±.

(64)

Here, the transformed asymmetric coupling column ma-

trix is defined as

? γ±≡

?γ+

γ−

?

= P−1·? γ .(65)

Note, that the scalar quantity

? γT· Bφ2 ·? γ = ? γT

±· B(±)

φ2 ·? γ±,(66)

where B(±)

tion. Therefore in (64) it is written in the untransformed

form. At the FP Eq.(63) reduces to the condition

φ2 = PT·Bφ2·P, is invariant under transforma-

?βγ±({u⋆},{γ⋆}) = 0(67)

because the determinant of matrix P does not vanish.

Subsequently, Eq. (64) leads to two FP equations

?

?

− ε +? γ⋆T· B⋆

φ2 ·? γ⋆+ 2ζ⋆

+

?

?

γ⋆

+= 0(68)

− ε +? γ⋆T· B⋆

φ2 ·? γ⋆+ 2ζ⋆

−

γ⋆

−= 0(69)

In the above equations we have introduced the short hand

notations B⋆

transformed asymmetric couplings γ⋆

ent from zero, the above two equations lead to the con-

dition ζ⋆

−. This condition is not valid if all quartic

couplings u⋆

aare different from zero. Thus at least one

of the two transformed asymmetric couplings, γ+or γ−,

has to be zero at the FP. The transformation matrix P

has been presented in Eq.(64) in paper I. Expressed in

terms of the ζ-functions it reads

φ2 ≡ Bφ2({u⋆}) and ζ⋆

±≡ ζ±({u⋆}). If both

+and γ⋆

−are differ-

+= ζ⋆

P =

?P11 P12

P21 P22

?

=

1

?ζφ2

?

21

ζ−−?ζφ2

?

11

?ζφ2

?

12

ζ+−?ζφ2

?

22

1

(70)

where [ζφ2]ijare the elements of the matrix ζφ2 (for the

two loop expressions see Eqs.(23)-(26) in paper I).

Page 8

8

FPγ⋆

⊥/γ⋆

?

γ⋆2

⊥

γ⋆2

?

G

line of FPs (57)

0

line of FPs (57)

0

line of FPs (57)

2

n?ε

H(n⊥)

∞

ε−2ζ(n⊥)

φ2

B(n⊥)

φ2

2

n⊥ε

(u⋆

⊥)

(u⋆

⊥)

0

∞

0

H(n?)

00

ε−2ζ

(n?)

φ2

(n?)

φ2

(u⋆

?)

B

(u⋆

?)

∞

ε−2ζ(n⊥)

φ2

B(n⊥)

φ2

(u⋆

⊥)

(u⋆

⊥)

0

D

00

ε−2ζ

(n?)

φ2

(n?)

φ2

(u⋆

?)

B

(u⋆

?)

ζ⋆

+−?ζ⋆

φ2?

12

22

?ζ⋆

?ζ⋆

φ2

?

2(ε−2ζ⋆

? ?ζ⋆

2(ε−2ζ⋆

?

+)

n⊥+n?

φ2?

12

ζ⋆

+−?ζ⋆

−−?ζ⋆

φ2?

22

?2

2(ε−2ζ⋆

+)

n⊥

?

ζ⋆

+−?ζ⋆

2(ε−2ζ⋆

? ?ζ⋆

φ2?

12

22

?ζ⋆

φ2?

?2

+n?

H(n⊥+ n?), B

φ2?

21

ζ⋆

−−?ζ⋆

φ2?

11

−)

n⊥+n?

ζ⋆

φ2

?

11

?ζ⋆

φ2?

21

?2

−)

n⊥

φ2

?

21

ζ⋆

−−?ζ⋆

φ2?

11

?2

+n?

TABLE I: Fixed points of the asymmetric couplings γ⊥ and γ? of the extended O(n?) ⊕ O(n⊥) model. The values of the

isotropic Heisenberg FP H(n⊥+ n?) and the Biconical FP B are valid in two loop order because Eq.(75) has been used. For

all other FPs the expressions are valid in all orders of perturbation expansion.

Let us now consider the two cases where one of the

asymmetric couplings is nonzero.

Case a: γ⋆

−= 0: Taking into account Eq. (65)

the condition for a vanishing γ⋆

0. Given the matrix elements (70) it can be rewritten as

+?= 0, γ⋆

−reads −P21γ⋆

⊥+P11γ⋆

?=

γ⋆

γ⋆

⊥

?

=

?ζ⋆

φ2?

12

φ2?

ζ⋆

+−?ζ⋆

22

(71)

At finite γ⋆

results in the condition

+the bracket in Eq.(68) has to vanish, which

2ζ⋆

m= ? γ⋆T· B⋆

φ2 ·? γ⋆= ε − 2ζ⋆

+=α

ν.

(72)

The last equality uses the definition of the asymptotic

exponents derived in paper I (Eq. (90) there).

Case b: γ⋆

−?= 0: In this case Eq. (65) leads im-

mediately to the condition P22γ⋆

(70) gives

+= 0, γ⋆

⊥−P12γ⋆

?= 0. Inserting

γ⋆

γ⋆

⊥

?

=

?ζ⋆

φ2?

21

φ2?

ζ⋆

−−?ζ⋆

11

. (73)

At finite γ⋆

results in the condition

−the bracket in Eq.(69) has to vanish, which

2ζ⋆

m= ? γ⋆T· B⋆

φ2 ·? γ⋆= ε − 2ζ⋆

−= 2φ

ν− d .(74)

Again, the last equality uses the definition of the asymp-

totic exponents derived in paper I (Eq. (82) there). The

above equations (71), (72) and (73), (74) respectively, de-

termine the FP values of the two asymmetric couplings

in the corresponding cases. The relations are valid in all

orders of perturbation expansion.

Since in two loop order B⋆

dent of the couplings {u} (see Eq.(31) in paper I) the left

hand sides of Eqs (72) and (74) read

φ2 is diagonal and indepen-

? γ⋆T· B⋆

φ2 ·? γ⋆= γ⋆2

⊥

n⊥

2

+ γ⋆2

?

n?

2.(75)

In consequence the asymmetric static couplings are zero

when the exponent expressions on the right hand side of

(72) and (74) are zero. This is the case if the specific

heat like CD and/or the magnetic like CD susceptibility

do not diverge.

Using (75) together with (71)-(74) leads to the FP val-

ues of the asymmetric couplings γ⋆2

Case a: γ⋆

−= 0:

?and γ⋆2

⊥

+?= 0, γ⋆

γ⋆2

⊥=

2(ε − 2ζ⋆

+)

n⊥+ n?

?

?ζ⋆

φ2

?

12

ζ⋆

+−?ζ⋆

φ2

?

22

?2,(76)

Page 9

9

?

?

FIG. 1: Exponents α/ν and 2φ/ν −3 appearing in Eqs. (72)

and (74) at n?= 1 as function of n⊥. Note that via relation

(105) 2φ/ν −3 = z−2 the strong dynamical scaling exponent

z results.

γ⋆2

?=

2(ε − 2ζ⋆

+)

?2

n⊥

?

ζ⋆

+−?ζ

⋆

φ2

?

?

22

?ζ

⋆

φ2

12

+ n?

.(77)

Case b: γ⋆

+= 0, γ⋆

−?= 0:

γ⋆2

⊥=

2(ε − 2ζ⋆

−)

n⊥+ n?

?

ζ⋆

−−?ζ

⋆

φ2

?

?

11

?ζ⋆

φ2

21

?2, (78)

γ⋆2

?=

2(ε − 2ζ⋆

−)

n⊥

?

?ζ

⋆

φ2

?

21

ζ⋆

−−?ζ⋆

φ2

?

11

?2

+ n?

.(79)

Note that the ratios in (71) and (73) might be negative

leading to a negative product γ⋆

The explicit values of the above FPs depend on wether

the isotropic Heisenberg or the biconical FP is inserted

into the ζ-functions. Eqs.(76)-(79) are valid up to two

loop order. In three loop order it is known from the

isotropic GLW-model that the function Bφ2 gets u2-

contributions13. In the multicritical GLW-model the ma-

trix Bφ2 may also be non diagonal, and then Eq.(75) does

not hold in this simple form.

In the case of the isotropic Heisenberg FP u⋆

u⋆

ratios of the elements of the ζφ2-matrix reduce to

?γ⋆

⊥.

⊥=

?= u⋆

×= u⋆Eqs.(76)-(79) simplify considerably. The

?ζ⋆

φ2?

12

φ2?

ζ⋆

+−?ζ⋆

22

= 1 =

γ⋆

γ⋆

⊥

?

(80)

?

?

FIG. 2: Fixed point values of the asymmetric static couplings

γ?and γ⊥for Case b (γ⋆

(n⊥< 1.61) and the biconical FP (1.61 < n⊥< 2.18).

⊥= 0, γ⋆

??= 0) at the Heisenberg FP

for Case a and

?ζ⋆

φ2?

21

φ2?

ζ⋆

−−?ζ⋆

11

= −n?

n⊥

=γ⋆

⊥

γ⋆

?

, (81)

for Case b. For the second equalities (71) and (73) have

been used. Together with the relations ε−2ζ⋆

ε−2ζ⋆

−= φ/ν−d, which introduce the critical exponents

and follow from Eqs. (80), (81) and (82) in paper I, the

values for the isotropic Heisenberg FP are:

Case a: γ⋆

−= 0:

+= α/ν and

+?= 0, γ⋆

γ⋆2

⊥= γ⋆2

?=

2

n⊥+ n?

α

ν.

(82)

Case b: γ⋆

+= 0, γ⋆

−?= 0:

γ⋆2

⊥=

2

n⊥+ n?

n?

n⊥

?

2φ

ν− d

?

,(83)

γ⋆2

?=

2

n⊥+ n?

n⊥

n?

?

2φ

ν− d

?

.(84)

Note that due to the sign in Eq.(81) the relation

γ⋆

?γ⋆

⊥= −

2

n⊥+ n?

?

2φ

ν− d

?

(85)

holds in this case. For n?= 1 and n⊥ = 2 our results

agree with those of Ref.6.

Page 10

10

4. Resummation procedure

As in our former papers1,2of this series, in order to

get numerical estimates we proceed within fixed dimen-

sion RG technique, i.e. we evaluate RG expansions in

couplings {u?,u⊥,u×} at fixed d = 3. Furthermore, as

far as the expansions are known to have zero radius of

convergence we use resummation technique14to get reli-

able numerical estimates. The results given below were

obtained within such a technique applied to the two-loop

RG expansions. One of the ways to judge about typical

numerical accuracy of our data, is to give an estimate

for some cases where the expansions (and, subsequently,

their numerical estimates) are known within much higher

order of loops. As far as the static exponents α and ν

explicitly enter many of formulas considered above, let

us take them as an example. Namely, let us estimate

relations

(2φ/ν − d)|d=3 ≡ 2/ν−− 3, ,

α/ν|d=3 ≡ 2/ν+− 3,

(86)

(87)

that enter the formulas for the couplings γ⊥, γ?. The

exponents ν+ and ν− have been defined in Eqs. (80)

and (81) of paper I. Fig.

2φ/ν − 3 and of α/ν on the order parameter component

numbers n⊥ at fixed n?= 1. Recall that of main in-

terest for us will be the physical case n?= 1, n⊥ = 2

indicated by the arrow. The region of n⊥shown in the

figure covers also the region of stability of the Heisenberg

O(n)-symmetrical FP, with n = n?+n⊥. In particular it

starts at n⊥= 1 near the marginal field dimension ncat

which the exponent α changes its sign. For the O(n) vec-

tor model an estimate based on the fixed d = 3 six loop

RG expansions reads15: nc= 1.945 ± 0.002. We get for

the correlation length critical exponent in the Heisenberg

O(2) FP: ν = 0.684, via hyperscaling relation this leads

to α = −0.053. Our estimate correctly reproduces the

absence of a divergency in the specific heat of O(2) model

(α is negative), however the value of nc≃ 1.6 we get is

rather underestimated. Note however, that the fixed d

approach we exploit in two loop approximation is essen-

tially better than the corresponding ε expansion. Indeed,

in two-loop ε-expansion one gets: nc= 4−4ε, which does

not lead to reasonable estimates16. The two loop esti-

mate of the massive field theory at d = 3, mc≃ 2.0117,

is more close to the most accurate value of Ref.15, how-

ever it gives a wrong sign for the exponent α. In any

case the negative value of α for n⊥= 2 agrees with other

calculations as reported in paper I.

It turns out (see below) that Case b is the stable FP

for the asymmetric couplings. In order to evaluate nu-

merically the values of couplings γ2

substitute the resummed fixed point values of the static

couplings {u?,u⊥,u×} into formulas (78), (79) and resum

the resulting expression. In principle, one can use differ-

ent ways for such an evaluation. Indeed, as we proceeded

before, one can present these formulas in the form of ex-

pansion in renormalized couplings (keeping the two-loop

1 shows the dependence of

?, γ2

⊥, we therefore

terms) and resum the resulting second-order polynomial.

Alternatively, based on the observation that numerator

and denominator of Eqs. (78), (79) contain combinations

of critical exponents, one can resum the numerator and

the denominator separately. We will exploit both ways

which naturally will lead to slightly different numerical

estimates. This difference may also serve to get an idea

about typical numerical accuracy of the results. Sepa-

rately, we will evaluate the ratios γ⊥/γ?. Again, it will

be done by resummation of the series for this ratio, Eq.

(73), as well as by using resummed values for γ2

particular, for n?= 1, n⊥ = 2 we get: (γ⋆

(γ?⋆)2= 0.286 (when denominator and numerator are re-

summed separately), and (γ⋆

(when an entire expression is resummed). Resulting dif-

ferences of the order of several percents bring about a

typical numerical accuracy of the estimates. In Fig. 2

we plot the FP values of the asymmetric couplings and

their ratio obtained within resummation of the entire ex-

pressions. These values will be used below to calculate

the critical dynamics.

?, γ2

⊥.In

⊥)2= 0.034,

⊥)2= 0.031, (γ?⋆)2= 0.293

B. Static transient exponents

The stability of the fixed points is determined by the

sign of the corresponding transient exponents. Latter can

be found from the eigenvalues of the matrix

∂βγα

∂γβ

(88)

with α,β =⊥,?. Inserting (43) into (88) the correspond-

ing eigenvalues read

λ(±)=1

2

?

− ε + n⊥γ2

⊥+ n?γ2

?+?ζφ2?

11+?ζφ2?

22

±

??n⊥γ2

⊥− n?γ2

2

?

+?ζφ2?

11−?ζφ2?

22

?2

+?n⊥γ⊥γ?+ 2?ζφ2?

The above eigenvalues are valid in two loop order because

Eq.(75) already has been used. The transient exponents

12

??n?γ⊥γ?+ 2?ζφ2?

21

?

?1/2?

.(89)

ω(±)

γ

≡ λ(±)?

{u} = {u⋆},{γ} = {γ⋆}

?

(90)

are calculated by inserting the fixed point values of the

static couplings into the eigenvalues.

point values (82) - (84) and Eq. (89) we obtain for the

isotropic Heisenberg FP the transient exponents

Case a: γ⋆

−= 0:

With the fixed

+?= 0, γ⋆

ω(+)=α

ν,

ω(−)= −¯ W⋆.(91)

Page 11

11

Case b: γ⋆

+= 0, γ⋆

−?= 0:

ω(+)= 2φ

ν− d ,ω(−)=¯ W⋆. (92)

¯ W⋆is the root

¯ W ≡

???ζφ2?

11−?ζφ2?

22

?2

+ 4?ζφ2?

12

?ζφ2?

21

(93)

taken at the fixed point values of the couplings. It is

always positive and reads at the isotropic Heisenberg FP

in two loop order

¯ W⋆=n⊥+ n?

6

u⋆

?

1 −u⋆

3

?

. (94)

Thus one concludes that Case b is the stable FP even if

α would be positive19. For the biconical FP the stability

of Case b can be verified explicitly by the flow of the

couplings.

C. Dynamical fixed points

Calculations of the dynamical FPs values are done by

solving the FP equations for the dynamical β-functions.

Since only two of the equations (53)-(55) are independent

the third equation serves as consistency check of the so-

lution found. It is useful to choose for this purpose Eqs

(54) and (55) for the time scale ratios w?and w⊥. Then

one has to solve

βw?(w?,w⊥,w?/w⊥) = 0,

βw⊥(w?,w⊥,w?/w⊥) = 0.

(95)

To find the dynamical FP values coordinates, the re-

summed FP values of static couplings u∗

are inserted into these equations18.

The dynamical FPs depend on which static FP is con-

sidered. There might be several dynamical FPs for one

static FP, which could be either strong dynamical or weak

dynamical scaling FPs. Since also unstable static FPs

might be reached in the asymptotics if one starts with

static initial conditions in the attraction region of this

FP (a subspace in the space the static couplings see e.g.

Fig. 3 in paper I) at least both the Heisenberg FP and

the biconical FP have to be taken into consideration. It

turns out that in both cases apart from the trivial unsta-

ble FP where all timescale ratios are zero (see below) two

dynamical FPs are found: (i) an unstable weak dynami-

cal scaling FP corresponding to model A and (ii) a stable

new strong dynamical scaling FP. In the physical inter-

esting case n?= 1 and n⊥= 2 these cases correspond to

dynamical behavior at a multicritical point of bicritical

and tetracritical type respectively. The different types of

weak and strong dynamical FPs are shown in Tab. II.

?,u∗

⊥,u∗

×,γ∗

⊥,γ∗

?

?

?

??

?

?

?

FIG. 3:

and w?for the static stable FPs: the isotropic Heisenberg FP

(n⊥< 1.61) and the biconical FP (1.61 < n⊥< 2.18). Strong

dynamical scaling is valid up to the stability borderline to the

decoupling FP. In the biconical region the values of vC and

1/wC⊥are finite but not to distinguish from zero on this scale.

The notation of the dynamical FPs correspond to the notation

in Tab II. The dashed curve shows the unstable model A FP

(see text).

Fixed point values of the timescale ratios v, 1/w⊥

1.Strong dynamical scaling fixed point

In order to find the strong dynamical scaling FPs it is

not necessary to discriminate between the static Heisen-

berg FP or the biconical FP although the dynamical

equations to be solved simplify in the first case a little

bit. Thus we use the results for the FP values derived in

paper I for the quartic couplings {u} and the FP values

for the asymmetric couplings {γ} of Case b (see Eqs.

(73), (78) and (79)). At the strong scaling dynamical

FP all timescale ratios have to be nonzero and finite.

Moreover due to the definitions of the timescale ratios it

follows that w⋆

⊥. This dynamical strong scaling

FP value is found by setting the differences of two of the

three dynamical ζ-functions, (44)-(47) to zero leading to

three equations. Since there are only two independent

time ratios the third equation can be used to check the

results.

The FP values (FP with subscript C in Tab. II) have

been plotted in Fig. 3 for different n⊥ at n?= 1 and

the numerical values for n⊥ = 2 are collected in Tab.

III. This shows that the FP value of the timescale ratio

for v is different from the FP value found in the pure

relaxational model A. These were at the Heisenberg FP

v⋆

in approaching the stability borderline to the decoupling

fixed point (see Fig. 1 in paper II).

A numerical problem arises in finding the FP values of

?= v⋆w⋆

A= 1 and at the biconical FP v⋆= vB

Awith vB

A→ ∞

Page 12

12

FPscaling typevw?

w⊥= w?/vzφ?

zφ⊥

zm

HCw

HA

HC

weak

weak

strong

0

v(H)

A

vH

C

wH

0

wH

?Cw

∞

2φ

2 + cη

2φ

ν− 1

ν− 1infinite fast

2 + cη

2φ

ν− 1

2φ

2φ

2φ

ν− 1

ν− 1

ν− 1

0

wH

?

⊥

BCw

BA

BC

weak

weak

strong

0

v(B)

A

v(B)

C

wB

0

wB

?Cw

∞

2φ

zB

2φ

ν− 1infinite fast

zB

2φ

ν− 1

2φ

2φ

2φ

ν− 1

ν− 1

ν− 1

0

wB

?

⊥

ν− 1

TABLE II: Types of dynamical FPs for the static Heisenberg and biconical FP. Not included is the trivial unstable fixed point

with all times scale ratios equal to zero. The value of c reads c = 6ln(4/3) − 1. Note that for the weak scaling FP the result

is only valid in two loop order, whereas the relation of the dynamical critical exponent in the strong scaling FP holds in all

orders.

FPu⋆

?

u⋆

⊥

u⋆

×

γ⋆2

?

0.29378

γ⋆2

⊥

0.03170

v⋆

6.09592 · 10−43

7.29393·10−5

7.30771·10−5

w⋆

1.24285 · 1042

1.55665·104

1.55372·104

⊥

w⋆

?

BC, a

HC, b

HC, a

1.287451.12769 0.301290.75763

1.13541

1.13541

1.001561.001561.001560.725540.18139

TABLE III: FP values of couplings and timescale ratios for n?= 1, n⊥ = 2. a FP values of the timescale ratios found via

approximation using Eqs. (100), (101), as described in the text with the values for FP B: A = 0.09770, B = 0.00101 and the

FP H: A = 0.31534, B = 0.03311. b numerical solution for the FP values of the timescale ratios;

v and 1/w⊥when they reach very small values. It cannot

be numerically decided wether the FP values are zero or

finite. In order to clarify the existence or nonexistence of

a weak scaling FP one has to look for an analytic expres-

sion for the small FP values. However it is numerically

easy to find the FP value of w?which is nonzero and finite

in the whole region up to the stability borderline between

the biconical and decoupling FP. In order to solve this

problem the dependence of the ζ-functions are studied

within this region. One observes that there are logarith-

mic terms which would diverge in the limit v → 0 under

the condition w⊥= w?/v. Thus one obtains two equa-

tions for the FP value of v and w?. In the equation for

the FP of w?one might safely perform the limit v → 0

and w⊥→ ∞. This leads to

0 = ζ(C)

Γ?

?{u⋆},{γ⋆},v = 0,w?,w⊥→ ∞?− 2ζ⋆

Using limiting functions

m. (96)

ζ(A)

Γ?=n?+ 2

36

u⋆2

?

?

3ln4

3−1

2

?

−n⊥

72

u⋆2

×

(97)

and

ζ(C?)

Γ?

?u⋆

1 − 3ln4

?,γ⋆

?,w|

?=

+

w?γ⋆2

1 + w?

?

?

1 −1

2

?

n?+ 2

3

u⋆

?

?

3

?

w?γ⋆2

1 + w?

?

?

n?

2

−

w?

1 + w|

−

(98)

3(n?+ 2)

2

ln4

3−(1 + 2w?)

1 + w?

ln(1 + w?)2

1 + 2w?

???

then Eq. (96) reads

0 =

w?γ⋆2

1 + w?

?

?

1 −1

2

?

n?+ 2

3

u⋆

?

?

1 − 3ln4

3

?

+

w?γ⋆2

1 + w?

?

?

n?

2

−

w?

1 + w?

−3(n?+ 2)

2

ln4

3

−1 + 2w?

1 + w?

ln(1 + w?)2

1 + 2w?

???

(99)

−n⊥

4

w?γ⋆

1 + w?

?γ⋆

⊥

?

?

2

3u×+

w?γ⋆

1 + w?

?γ⋆

⊥

?

+n?+ 2

36

u⋆2

?

?

3ln4

3−1

2

−n⊥

72

u⋆2

×− 2ζ⋆

m

This equation is solved numerically to give the value of

w⋆

In order to find v⋆one collects the logarithmic diverging

terms in the equation for v⋆,

?which then is inserted into the second equation for v⋆.

ζ(C)

Γ?(v,w?,w?/v) − ζ(C)

Γ⊥(v,w?,w?/v) = 0. (100)

In the remaining terms the limit v → 0 can safely per-

formed. Then the solution reads

lnv⋆= −A

B

(101)

Page 13

13

1

0

3

4100

200w_perp

300

400

2

2.0

0

1

1.5

w_par

2

v

1.0

3

0.5

4

0.0

5

n_perp=1.2

1

0

3

100

2

4200w_perp

300

400

2.0

0

1

1.5

w_par

2

v

1.0

3

0.5

4

0.0

5

n_perp=1.7

FIG. 4: Dynamical flow at n?= 1 and different n⊥values for

different dynamical initial conditions numbered 1 to 4. The

static couplings are chosen to be fixed at their stable FP values

(the isotropic Heisenberg FP for n⊥ = 1.2, the biconical FP

for n⊥ = 1.7). The dynamical FP values are v⋆= 0.399,

0.004, w⋆

respectively. Also shown is the surface v = w?/w⊥ to which

the flow is restricted.

?= 1.661, 1.300 and w⋆

⊥= 4.159, 351.06 at HC, BC

with

A = 2ζ⋆

m− γ⋆2

⊥

?

1 −1

2

?

n⊥+ 2

3

u⋆

⊥

?

1 − 3ln4

3

?

1

0

3

100

2

4200w_perp

300

400

2.0

0

1

1.5

w_par

2

v

1.0

3

0.5

4

0.0

5

n_perp=2.0

FIG. 5:

dynamical initial conditions numbered 1 to 4. The static cou-

plings are chosen to be fixed at their biconical FP values. The

static and dynamical FP values of BC are given in Tab. III.

The dynamical FP lies outside the region shown. Also shown

is the surface v = w?/w⊥ to which the flow is restricted.

Dynamical flow at n?= 1 and n⊥ = 2 for different

+γ⋆2

⊥

?

n⊥

2

− 1 −3(n⊥+ 2)

2

ln4

3− 2ln

w⋆

?

2

???

+n?

4γ⋆

⊥γ⋆

?

?

3ln4

2

3u⋆

×+ γ⋆

⊥γ⋆

?

?

ln2 −n⊥+ 2

36

u⋆2

⊥

?

3−1

2

?

−n?

72u⋆2

×

?

1 − 2ln2

?

(102)

and

B = γ⋆4

⊥+n?

36u⋆2

×+n?

4γ⋆

⊥γ⋆

?

?

2

3u⋆

×+ γ⋆

⊥γ⋆

?

?

.(103)

It can be shown that A and B are positive. Approach-

ing the stability borderline to the decoupling FP A stays

finite and B goes to zero since u⋆

In consequence v⋆goes to zero and w⋆

in and only in this limit. The analytic solution found

within this region joins smoothly to the numerical solu-

tion found for larger values of the timescale ratios. Thus

it is proven that in the whole region where the Heisen-

berg FP or the biconical FP is stable dynamical strong

scaling holds.

Considering the FP values for the timescale ratios for

the Heisenberg FP in the region of n⊥> 1.7 (where it is

reached only for static initial conditions in a subspace of

the fourth order couplings) one also finds a small value

for v⋆, a very large value for w⋆

value for w⋆

?. However contrary to the biconical FP now

A and B stay finite at the stability borderline between

×and γ⋆

⊥goes to infinity

⊥go to zero.

⊥and a nonzero finite

Page 14

14

the biconical FP and the decoupling FP at n⊥ ∼ 2.18.

Indeed the values calculated for the Heisenberg FP from

Eqs (102) and (103) are A = 0.34866, B = 0.02558

whereas for the biconical FP one obtains A = 0.05268,

B = 4.27958· 10−9.

The asymptotic dynamical exponents are obtained

from the values of the ζ-functions at the FP:

zφ?= 2+ζ⋆

Γ?

zφ⊥= 2+ζ⋆

Γ⊥

z⋆

m= 2+ζλ. (104)

At the strong dynamical scaling FP all dynamical ζ-

functions are equal to twice the static ζ-function ζm.

Therefore the CD induces the value of the dynamical

critical exponent z

z = 2 + 2ζ⋆

m= 2φ

ν− 1(105)

according to Eq. (39) and Eq. (74). The values for the

static exponents depend on which static FP is stable. For

n?= 1 the n⊥-dependence of z − 2 is shown in Fig. 1.

2.Weak dynamical scaling fixed point

Weak dynamical scaling FPs are solutions of the dy-

namical FP equations where one or more of the FP values

of the timescale ratios are zero or infinite. Such a weak

dynamical scaling FP has already been found in model

A and it became stable at the stability borderline to the

decoupling FP.

Indeed Eqs (95) allow solutions where both timescale

ratios w?and w⊥are zero. In such a case one has to rely

on the third equation for the ratio v = w?/w⊥ to find

the limiting FP value. However in the limit w?→ 0 and

w⊥ → 0 Eq. (53) for v reduces to the FP equation of

model A (FP with subscript A in Tab. II). Thus one

recovers the model A FPs in this case.

There is no solution w⋆

finite due to the lnv term in (50). For a similar reason

no FP with w⋆

is possible. However a FP with w⋆

v⋆= 0 and w⊥= ∞ is possible (FP with subscript Cwin

Tab. II). The values of w⋆

?are obtained from Eq. (99),

but now these values are not an approximation but the

exact CwFP values for any n?and n⊥.

The dynamical critical exponents may be different in

the case of weak dynamical scaling. For the weak model

C FP (subscript Cw) w⋆

ζ⋆

ν− 1.

timescale for the OP φ?. Inserting w⋆

leads due to logarithmic diverging terms to an infinite

value of the corresponding dynamical exponent zφ⊥. This

indicates that the density φ⊥ is much faster than the

other densities. It is especially much faster than the other

OP φ?.

In the case where both FP values of the timescale ratios

w?and w⊥are zero and v is finite and nonzero, both OPs

?= v⋆= 0 and w⊥nonzero and

?nonzero and finite, w⋆

⊥= 0 and v⋆= ∞

?nonzero and finite,

?is finite and nonzero therefore

λand zφ?= 2φ

Γ?= ζ⋆

Thus the CD sets the

⊥= ∞ into ζΓ⊥

have the same timescale with the dynamical exponent (of

model A) z = 2 + cη different from the exponent of the

CD zm= 2φ

ν− 1.

D.Dynamical transient exponents

The dynamical transient exponents can be calculated

from the matrix of the derivatives of the β-function with

respect to the timescale ratios v, w?and w⊥. Since only

two timescale ratios are independent only two are con-

sidered in the stability matrix. The eigenvalues of the

2 × 2-matrix have to be positive for the overall stable

FP otherwise the FP is unstable. In the following the

timescale ratios w?and w⊥are chosen as independent.

The model A type FP with v⋆= vAnonzero and finite

is unstable since the two eigenvalues,

ωw?= ζ⋆

Γ?− 2ζ⋆

m

andωw⊥= ζ⋆

Γ⊥− 2ζ⋆

m

(106)

are negative. In fact they are equal because ζ⋆

Their values are calculated with 2ζ⋆

inserting the model A FP value v⋆= vA(see Fig 3) and

w⋆

Γ?−2φ/ν+1 < 0 and ωw?= −0.126

for n?= 1 and n?= 2.

Similarly the instability of the FP with v⋆= w⋆

w⋆

⊥= 0 can be shown. However some care has to be taken

due to the vanishing timescale ratio v. The eigenvalues

are again given by Eq. (106) but now they are different.

Whereas ωw?is negative ωw⊥goes to ∞ due to the lnv

term in the model A ζ function (50).

The transient exponent for the strong scaling FP are

the eigenvalues of the matrix of derivatives of the β-

functions according to the timescale ratios at the FP

Γ?= ζ⋆

Γ⊥.

m= 2φ/ν − 1 and

?= w⋆

⊥= 0. Then ζ⋆

?=

∂βw?

∂w?

∂βw⊥

∂w?

∂βw?

∂w⊥

∂βw⊥

∂w⊥

⋆

=

w?

?∂ζΓ?

?∂ζΓ⊥

∂w?

?

?

w?

?∂ζΓ?

?∂ζΓ⊥

∂w⊥

?

?

w⊥

∂w?

w⊥

∂w⊥

(107)

⋆

.

Use has been made from the independence of ζλon the

timescale ratios. The nondiagonal elements depend on

the timescale ratios by which they are derived only via

v and therefore are proportional to 1/w⊥. In the region

where w⊥is very large the two eigenvalues are then given

by the diagonal elements.

ωw?= w⋆

?

?∂ζΓ?

∂w?

?⋆

,ωw⊥= w⋆

⊥

?∂ζΓ⊥

∂w⊥

?⋆

.(108)

Near the stability borderline to the decoupling FP the

second eigenvalue goes to zero according to

ωw⊥= B + O(1/w⊥)(109)

with B from Eq. (103), being exactly zero at the bor-

derline. The value of the slow transient at n?= 1 and

n⊥= 2 is given by ωw⊥= 0.001 Thus as shown in the

next section in this case nonasymptotic effects are present

in the physical accessible region.

Page 15

15

As already mentioned for the Heisenberg FP HC B

does not reach zero at the borderline but its value is

one order smaller than the static transient exponents.

At n⊥ = 2 and n⊥ = 2.18 the value of the dynamical

transient exponent is given by ωw⊥= 0.033 and ωw⊥=

0.026 respectively.

VI.DYNAMICAL FLOWS AND EFFECTIVE

EXPONENTS

The flow of the timescale ratios is described by the RG

equations

l∂v

∂l

= βv({u},{γ},{w}),

l∂w⊥

∂l

l∂w?

∂l

= βw⊥({u},{γ},{w}),(110)

= βw?({u},{γ},{w}),

with the β-functions Eqs. (53)-(54). Note, that the dy-

namical β-functions depend also on the RG equations of

the static quartic couplings (Eqs (33)-(36) of paper I)

and the RG equations of the asymmetric couplings

l∂? γ

∂l=?βγ({u},{γ}) (111)

with the β-function Eq. (43).

In order to simplify the picture it is assumed that the

static couplings have already reached the FP by which

they are attracted from their initial conditions. Then the

RG flows are displayed in the three-dimensional space of

the time-scale ratios w?,w⊥,v in Figs. 4 and 5 for the

physical interesting case at n?= 1, n⊥= 2. The static

biconical FP is stable for this case in general. However

for initial conditions on the surface separating the stable

biconical FP from the Heisenberg FP the flow is attracted

to the Heisenberg FP. Therefore one may also fix the

static parameters to this FP.

To give an overview of the different patterns of the

flows three different value of n⊥are chosen for fixed n?=

1: (i) n⊥= 1.2, where the static Heisenberg FP is stable

(Fig. 4), (ii) n⊥= 1.7 (Fig. 4) and (iii) n⊥= 2 (Fig. 5)

where the biconical FP is stable.

In all cases the FP values of the timescales are nonzero

and finite but the values of v⋆becomes very small and

w⋆

⊥very large. The asymptotic approach to the FP in

cases (ii) and (iii) occurs in the direction of the w⊥-axis

almost at v ∼ 0 and w|∼ w⋆

?.

A.Effective exponents

We define the effective exponents by:

z?eff(l) = 2 + ζΓ?({u(l)},{γ(l)},{w(l)}),

z⊥eff(l) = 2 + ζΓ⊥({u(l)},{γ(l)},{w(l)}),

zmeff(l) = 2 + ζλ({u(l)}).(112)

?

?

?

?

?

?

?

?

?

?

FIG. 6: Effective dynamical exponents z?, z⊥, and zm calcu-

lated along the different RG flows of Fig. 5 (indicated by the

numbers). The insert shows that even for flow parameters

as small as lnl = −2000 the effective exponent z⊥ has not

reached its asymptotic value 2.18.

These exponents appear e.g. in the critical temperature

and/or wave vector dependence of the transport coeffi-

cients describing the relaxation of the alternating magne-

tization or the diffusion of the magnetization in the direc-

tion of the external magnetic field. They are in principle

experimentally accessible. An interesting feature is the

independence of the effective dynamical scaling exponent

zmof the CD from the dynamical timescales. Therefore

its nonasymptotic value is only due to nonasymptotic ef-

fects within statics. This allows to trace back nonasymp-

totic effects in dynamical quantities to the slow transients

in statics or those appearing in dynamics.

In order to calculate the numerical values of the ef-

fective exponents, we substitute into Eq. (112) the re-

summed coordinates of the static FP {u∗

the values of timescale ratios {w?(ℓ),w⊥(ℓ),v(ℓ)} along

the RG flow. Choosing the FP values of the static cou-

plings fixes the asymptotic values of the effective dy-

namical exponents since they are expressed by the static

asymptotic exponents for the strong dynamical scaling

FP.

The results shown in Fig. 6 correspond to the case

n?= 1, n⊥= 2. We evaluate the timescale ratios along

the previously obtained flows 1-4 in Fig. 5. The effective

exponents for the different initial conditions are shown

by numbered solid lines, As one can observe from this

figure, the exponents calculated along several flows do

not coincide for the values of the flow parameter shown.

However one sees the merging of the different values for

z?to their asymptotic value z⋆

static value corresponding to the CD (the constant line

?,u∗

⊥,u∗

×} and

⊥= z⋆

m= 2.18 given by the