We show that non-locality in the conservation of both the order parameter and
a noncritical density (model D dynamics) leads to new fixed points for critical
dynamics. Depending upon the parameters characterizing the non-locality in the
two fields, we find four regions: (i) model-A like where both the conservations
are irrelevant (ii) model B-like with the conservation in the order parameter
field relevant and the conservation in the coupling field irrelevant (iii)
model C like where the conservation in the order parameter field is irrelevant
but the conservation in the coupling field is relevant, and (iv) model D-like
where both the conservations are relevant. While the first three behaviours are
already known in dynamical critical phenomena, the last one is a novel
phenomena due entirely to the non-locality in the two fields.Comment: 4 pages revtex4; to appear in Journal of Physics A Letter

Bhattacharjee, Somendra M.

Sen, Parongama

English

arXiv

We show that non-locality in the conservation of both the order parameter and a non-critical density (model-D dynamics) leads to new fixed points for critical dynamics. Depending upon the parameters characterizing the non-locality in the two fields, we find four regions: (i) model-A-like, where both conservations are irrelevant; (ii) model-B-like, with the conservation in the order parameter field relevant and the conservation in the coupling field irrelevant; (iii) model-C-like, where the conservation in the order parameter field is irrelevant but the conservation in the coupling field is relevant; and (iv) model-D-like, where both conservations are relevant. While the first three behaviours are already known in dynamical critical phenomena, the last one is a novel phenomenon due entirely to the non-locality in the two fields

Publications of the IAS Fellows

INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL
J. Phys. A: Math. Gen. 35 (2002) L141–L146 PII: S0305-4470(02)31994-2
LETTER TO THE EDITOR
Is there a true model-D critical dynamics?
Parongama Sen1 and Somendra M Bhattacharjee2
1 Department of Physics, University of Calcutta, 92 APC Road, Kolkata-700009, India
2 Institute of Physics, Bhubaneswar-751005, India
E-mail: parongama@vsnl.net, paro@cubmb.ernet.in and somen@iopb.res.in
Received 17 December 2001
Published 8 March 2002
Online at stacks.iop.org/JPhysA/35/L141
Abstract
We show that non-locality in the conservation of both the order parameter and
a non-critical density (model-D dynamics) leads to new fixed points for critical
dynamics. Depending upon the parameters characterizing the non-locality in
the two fields, we find four regions: (i) model-A-like, where both conservations
are irrelevant; (ii) model-B-like, with the conservation in the order parameter
field relevant and the conservation in the coupling field irrelevant; (iii) model-
C-like, where the conservation in the order parameter field is irrelevant but the
conservation in the coupling field is relevant; and (iv) model-D-like, where
both conservations are relevant. While the first three behaviours are already
known in dynamical critical phenomena, the last one is a novel phenomenon
due entirely to the non-locality in the two fields.
PACS numbers: 64.60.Ht, 05.10.Cc, 45.30.+s
On the basis of long-time behaviour, the dynamics of different critical systems are classified [1]
into various dynamic universality classes. These classes are characterized by several factors,
such as the number of conserved fields and the non-vanishing Poisson brackets, in addition
to the usual quantities necessary to characterize equilibrium behaviour. Among the models
which have no propagating modes, model A is a system in which there are no conserved fields,
model B is a system with a conserved order parameter, while model C is the class of systems
which have a non-conserved order parameter field coupled to a conservative field. A fourth
class was also identified—the model-D system, in which both the order parameter field and
the coupled field are conserved. Model D, however, was shown to be equivalent to model B,
as the coupling to the field becomes irrelevant in the large-length long-timescale limit.
The dynamical behaviour in systems with one or more conserved fields, like the systems
of models B, C, and D in the above classification, is usually studied by considering local
conservation. However, conservation is by no means necessarily local and, in fact, non-
local conservation is a more general condition. A few recent studies on critical or near-critical
dynamics show that non-locality in the conserved field may change the dynamical class, besides
0305-4470/02/110141+06$30.00 © 2002 IOP Publishing Ltd Printed in the UK L141
L142 Letter to the Editor
leading to the obvious modifications in non-universal details. This kind of effect of non-locality
in the conservation has been noted in the study of critical dynamics of models B and C and in the
numerical study of the early-time effect of a model believed to belong to the class of model C,
and on the zero-temperature dynamics (phase-ordering kinetics or Ostwald ripening) [2–7].
Besides the theoretical interest in its own right, one of the important applications of non-local
dynamics is in speeding up numerical simulations, prompting implementation of various types
of non-local move [8]. Hence the need for a general classification of effects of non-locality in
dynamics.
In a sense, the model-D class is the most generalized class of dynamical critical phenomena
with dissipative dynamics. However, with local conservation, it is equivalent to the model-B
class showing no novel features. Motivated by the results for model C [4], we investigate
the problem of non-local conservation in model D in the most general way by keeping both
conservations—that of the order parameter field and that of the coupled field—non-local. One
of our main results is that a true model-D behaviour emerges in the non-local dynamics, unlike
in the local conservation case.
We introduce non-locality by using a non-local kernel [2–4] in the chemical potential,
keeping the continuity equation intact. Usingφ for the order parameter andm for the secondary
coupling parameter, the Hamiltonian of model D is given by
H =
∫
ddx [ 12 rφ2(x) + 12 (∇φ(x))2 + u˜φ4(x) + γφ2(x)m(x) + 12C−1m2(x)], (1)
where φ2(x) = ∑ni=1 φ2i (x), φ4(x) = [∑ni=1 φ2i (x)]2, r = 0 is the mean field critical point,
and C > 0 ensures non-criticality of m. There is a short-distance cut-off which in momentum
space is an ultraviolet cut-off. We shall set = 1 in the calculation. The above Hamiltonian,
equation (1), is the same as in model C [1, 4]. Making γ = 0 effectively reduces the above
model to model A.
The dynamics3 is given by the following equations:(
1
ρkρ
+
1
0
)
∂φk
∂t
= − δH
δφ−k
+ η(k, t) (2a)
(
1
λσ kσ
+
1
λ0
)
∂mk
∂t
= − δH
δm−k
+ ζ(k, t). (2b)
The noises η and ζ obey
〈η〉 = 0, 〈ζ 〉 = 0, (3a)
〈η(k, t)η(k′, t ′)〉 = 2
(
1
ρkρ
+
1
0
)
δ(k + k′)δ(t − t ′), (3b)
〈ζ(k, t)ζ(k′, t ′)〉 = 2
(
1
λσ kρ
+
1
λ0
)
δ(k + k′)δ(t − t ′). (3c)
Here, k = |k|, and we have introduced two different parameters ρ and σ to denote the non-
localities in the order parameter field and the coupling field respectively. The dissipative and
kinetic coefficients (0 and λ0) are labelled by the subscript 0 while the transport coefficients
(ρ and λσ ) are labelled by the corresponding powers of k. For the local case, both ρ
and σ assume a value equal to 2. Results are known for some limiting values of ρ and σ :
(a) (ρ = 0, γ = 0) corresponds to model A [1]; (b) (ρ = 2, γ = 0) and (ρ = 2, σ = 2) give
model-B-like behaviour [1]; (c) any value of ρ > 0 and γ = 0 give a non-local model B [2,5];
and (d) ρ = 0 with any σ > 0 gives non-local model C [4].
3 Additional torques for n = 3 (Heisenberg model) may lead to new dynamic universality classes [9]. We do not
consider such effects.
Letter to the Editor L143
A Hamiltonian of the type of equation (1) occurs near the critical wings of a tricritical
system or in critical systems with additional constraints. Model-D dynamics is relevant in such
cases—for example, in three- or four-component mixtures or in spin-1 or Potts models (see e.g.
[10]). One spin- 12 example—although a bit unrealistic—is an antiferromagnetic system with
conserved magnetization (as in model C [6,11]) as well as conserved staggered magnetization.
The Kawasaki dynamics can be used to conserve the magnetization, maintaining at the same
time that the exchanges between opposite spins have to be done in such a way that the order
parameter is unchanged.
A momentum-shell renormalization group approach is used to study the long-distance
long-time behaviour. In this approach, small-scale fluctuations in space (between e−δl and
 in momentum space) are integrated out, and the cut-off rescaled to . One then obtains an
effective Hamiltonian and effective equations for the dynamics valid for longer scales. The
universal features are obtained from the fixed points of the flow equations for the various
parameters of the problem. As in the usual RG scheme, any parameter that vanishes (grows) in
the long-scale limit is called an irrelevant (relevant) variable. A fixed point is then characterized
by the set of relevant parameters. In this scheme the four dynamics classes correspond to the
relevance and irrelevance of λσ and ρ as mentioned in the abstract.
The statics of this system is the same as for model C and the fixed-point values of the
parameters r , u, and γ 2C are as given in [4]. The dynamical equations, on the other hand, take
the forms
∂λ−1σ
∂l
=
(
−z + σ + α
ν
)
λ−1σ (4a)
∂λ−10
∂l
=
(
−z + α
ν
)
λ−10 + (nγ
2Kd)Q (4b)
∂−1ρ
∂l
= (−z + ρ + 2)−1ρ (4c)
∂−10
∂l
= (−z + 2)−10 + (4Cγ 2Kd)Q
1
1 + (Q/CP )
(4d)
where Q = −1ρ +−10 , P = λ−1σ + λ−10 , and Kd = 21−dπ−d/2(d/2). The parameters λσ , λ0,
ρ , 0, γ , and C appearing in the above equations are all l-dependent. To simplify notation,
this l-dependence has been suppressed.
From (4b) we get
0
λ0
= (1 + x)nγ
2Kd
z− α/ν (5)
where x = 0/ρ is the dimensionless ratio of the two coefficients for the primary order
parameter (note:  = 1). This quantity x satisfies
∂x
∂l
= (ρ − Y )x, where Y = 2α
νn
1
µ−1 + (x + 1)−1
. (6)
Here we have introduced the ratios of the transport and kinetic coefficients ofmwith the kinetic
coefficient of φ:
µσ = 0C
λσ
, µ0 = 0C
λ0
, and µ = µσ + µ0.
By using the fixed-point value of C and equation (5), µ0 can be rewritten as
µ0 = (x + 1)α/ν2(z− α/ν) .
L144 Letter to the Editor
The flow equation for µσ is
∂µσ
∂l
= µσ [σ − 2 + α/ν − Y ]. (7)
We consider the different solutions of the equations for x and µσ . Corresponding to the
relevance of the conserved quantities, one gets different regions in the ρ–σ plane with different
dynamic critical behaviours. It should be mentioned here that for n  4, the coupling γ scales
to zero and one effectively gets a non-local model B. This has already been considered and
shown to be model-A-like when globally conserved [2]. Our discussions are for the n < 4
region where the coupling survives.
There are three fixed points of x: x = 0, x = ∞, and a non-zero finite fixed-point value
for x from equation (6):
Case 1. x = 0 in the long-length-scale limit. This corresponds to non-local model C: the
conservation of the order parameter is irrelevant. The fixed points and their stabilities of non-
local model C, which were obtained in [4] from the solutions for µσ , now depend on the value
of ρ. In general, x = 0 will be a stable fixed point as long as ρ < Y . We focus on the stability
of the various regions for non-zero values of ρ.
(a) µσ has a finite fixed-point value and z = σ + α/ν. This is a stable solution in the region
(2/n)−1 > p > −1 (where p = (σ − 2)/(α/ν)) for ρ = 0. For non-zero ρ this remains
stable for ρ < p + 1.
(b) µσ = ∞: this is valid for p > (2/n) − 1. Here z = 2 + 2α/nν. This solution x = 0,
µσ = ∞ is stable as long as ρ < 2α/nν.
(c) For µσ = 0, the stability of x = 0 is valid only for p < −1 and ρ < O()2). Here
z = 2 + O()2) and the behaviour is model-A-like. Any non-zero value of ρ thus destroys
this model-A-like region. Therefore, a region where both the conservations are irrelevant
is restricted strictly to ρ = 0.
Case 2. A finite non-zero value of x in the long-length-scale limit. This occurs when ρ = Y .
In order to ensure that 0 reaches a finite fixed-point value, we must have z = 2 + ρ. In this
case, µσ again has the following fixed points:
(a) µσ = ∞ for σ − 2 + α/ν > ρ;
(b) µσ = 0 for σ − 2 + α/ν < ρ;
(c) µσ non-zero and finite for σ − 2 + α/ν = ρ.
Physically, a finite-valued fixed point of x means that the conservation in the order parameter
φ is relevant. If at the same time µσ attains a non-zero fixed-point value, the conservation of
the coupling field m is also relevant. This occurs if the fixed points (a) and (c) are stable.
For case (a), x = ((ρ − 2α)/νn)/(2α/νn), and for non-negative values of x, ρ > 2α/nν.
Hence between ρ = 2α/nν and ρ = σ − 2 + α/ν, a stable region is obtained where both
conservations are relevant.
For case (b), x + 1 = ρ(2z− α/ν)/(2α2/nν2). This has a finite-valued solution for x
only for ρ ∼ O()2). Hence for ρ > σ − 2 + α/ν, x →∞ is the only solution for µ = 0. It is
a conventional model-B fixed point.
For case (c), z = σ +α/ν and, in principle, along the entire line σ−2+α/ν = ρ a solution
exists where both the conservations are relevant. However, there is a restriction on the value
of σ arising from the condition of existence of a non-zero finite value of µσ . We find that
µσ + µ0 = (σ − 2 + α/ν)(x + 1)2 − σ − α/ν + (2α/nν)(x + 1) . (8)
Letter to the Editor L145
IIIII
V
/
IV
ρ
 = p + 1
IV
ρ
ρ
−1 0 1 p
= 2
II
−
−
S
Figure 1. The different regions in the non-local model D with n = 1 are shown in the rescaled
ρ¯–p plane where ρ¯ = ρ/(α/ν) and p = (σ − 2)/(α/ν). Regions I and II are model-C-like with
z = σ + α/ν and z = 2 + 2α/ν respectively. Region III′ is model-A-like (as in model C of [4])
with z = 2 + O()2). Region IV is the model-D region, with z = ρ + 2. Region V is model-B-like,
with z = ρ + 2. The line p + 1 = ρ is also model-D-like up to point S (shown by the dashed line).
There are no discontinuities of z along any boundary except at the boundary between III′ and V.
Hence with σ = 2+pα/ν, we get the condition (1+p) < 2(x+1)/n. Since x is also non-zero,
this implies that this solution with both x and µσ finite is valid for p + 1 = ρ/(α/ν) with at
least p < (2/n)− 1.
In the rest of the ρ–σ plane, the only solution is x = ∞ and 1/λσ = 0, for which only
the conservation of the order parameter is relevant. Here z = ρ + 2.
All the above possibilities for different dynamic behaviours are summarized in figure 1
in the ρ–σ plane (for n = 1). The possibility of seeing a region where both conservation
conditions are relevant raises new issues—such as the early-time effect and boundary or surface
effects (see e.g. [6, 12])—in the new regime. These remain to be studied.
To summarize, model D can be viewed as the most general dynamic model with purely
dissipative dynamics and we do indeed obtain all four types of behaviour as soon as non-locality
in the conservations (as in equation (2)) is introduced. Since our general model subsumes the
previously studied non-local model C and local model D, it is natural to expect A-like, B-like,
and C-like regimes. The model-A-like region, however, turns out to be unstable for any finite
value of ρ. The model-C-like regions are stable for small non-zero values of ρ but ultimately
disappear for larger values of ρ. The most non-trivial result, for general values of ρ and σ ,
is the appearance (region IV in figure 1) of a region with a new dynamical behaviour where
both conservations are relevant. This we call a true model-D-like region—a consequence of
non-local conservation laws.
PS acknowledges financial support from DST project SP/S2/M-11/99.
References
[1] Hohenberg P C and Halperin B I 1977 Rev. Mod. Phys. 49 435
[2] Bray A J 1994 Adv. Phys. 43 357
[3] Bray A J 1989 Phys. Rev. B 41 6724
Bray A J 1989 Phys. Rev. Lett. 62 2841
[4] Sen P 1999 J. Phys. A: Math. Gen. 32 1623
[5] Sire C and Majumdar S N 1995 Phys. Rev. E 52 244
[6] Zheng B and Lou H J 2001 Phys. Rev. E 63 066130
[7] Conti M, Meerson B, Peleg A and Sasorov P V 2001 Preprint cond-mat/0109157
L146 Letter to the Editor
[8] Tamayo P and Klein W 1989 Phys. Rev. Lett. 63 2757
Tamayo P and Klein W 1991 Phys. Rev. Lett. 66 2049
Mosley L L, Gibbs P W and Jan N 1992 J. Stat. Phys. 67 813
Rutenberg A D 1996 Phys. Rev. E 54 972
Moukarzel C and Parga N 1989 J. Phys. A: Math. Gen. 22 943
Stauffer D 1997 Int. J. Mod. Phys. C 8 1263
[9] Das J and Rao M 1998 Phys. Rev. E 57 5069
Das J and Rao M 2000 Phys. Rev. E 62 1601
[10] Lawrie I D and Sarbach S 1984 Phase Transitions and Critical Phenomena vol 9, ed C Domb and J Lebowitz
(London: Academic)
[11] Sen P, Dasgupta S and Stauffer D 1998 Eur. Phys. J. B 1 107
[12] Mukherji S 1999 Eur. Phys. J. B 8 423
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Oerding K and Jenssen H K 1993 J. Phys. A: Math. Gen. 26 3369
Sen P and Dasgupta S 2002 J. Phys. A: Math. Gen. at press (cond-mat/0202235)


Is there a true model-D critical dynamics?

Indian Academy of Sciences

INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL
J. Phys. A: Math. Gen. 35 (2002) L141–L146 PII: S0305-4470(02)31994-2
LETTER TO THE EDITOR
Is there a true model-D critical dynamics?
Parongama Sen1 and Somendra M Bhattacharjee2
1 Department of Physics, University of Calcutta, 92 APC Road, Kolkata-700009, India
2 Institute of Physics, Bhubaneswar-751005, India
E-mail: parongama@vsnl.net, paro@cubmb.ernet.in and somen@iopb.res.in
Received 17 December 2001
Published 8 March 2002
Online at stacks.iop.org/JPhysA/35/L141
Abstract
We show that non-locality in the conservation of both the order parameter and
a non-critical density (model-D dynamics) leads to new fixed points for critical
dynamics. Depending upon the parameters characterizing the non-locality in
the two fields, we find four regions: (i) model-A-like, where both conservations
are irrelevant; (ii) model-B-like, with the conservation in the order parameter
field relevant and the conservation in the coupling field irrelevant; (iii) model-
C-like, where the conservation in the order parameter field is irrelevant but the
conservation in the coupling field is relevant; and (iv) model-D-like, where
both conservations are relevant. While the first three behaviours are already
known in dynamical critical phenomena, the last one is a novel phenomenon
due entirely to the non-locality in the two fields.
PACS numbers: 64.60.Ht, 05.10.Cc, 45.30.+s
On the basis of long-time behaviour, the dynamics of different critical systems are classified [1]
into various dynamic universality classes. These classes are characterized by several factors,
such as the number of conserved fields and the non-vanishing Poisson brackets, in addition
to the usual quantities necessary to characterize equilibrium behaviour. Among the models
which have no propagating modes, model A is a system in which there are no conserved fields,
model B is a system with a conserved order parameter, while model C is the class of systems
which have a non-conserved order parameter field coupled to a conservative field. A fourth
class was also identified—the model-D system, in which both the order parameter field and
the coupled field are conserved. Model D, however, was shown to be equivalent to model B,
as the coupling to the field becomes irrelevant in the large-length long-timescale limit.
The dynamical behaviour in systems with one or more conserved fields, like the systems
of models B, C, and D in the above classification, is usually studied by considering local
conservation. However, conservation is by no means necessarily local and, in fact, non-
local conservation is a more general condition. A few recent studies on critical or near-critical
dynamics show that non-locality in the conserved field may change the dynamical class, besides
0305-4470/02/110141+06$30.00 © 2002 IOP Publishing Ltd Printed in the UK L141
L142 Letter to the Editor
leading to the obvious modifications in non-universal details. This kind of effect of non-locality
in the conservation has been noted in the study of critical dynamics of models B and C and in the
numerical study of the early-time effect of a model believed to belong to the class of model C,
and on the zero-temperature dynamics (phase-ordering kinetics or Ostwald ripening) [2–7].
Besides the theoretical interest in its own right, one of the important applications of non-local
dynamics is in speeding up numerical simulations, prompting implementation of various types
of non-local move [8]. Hence the need for a general classification of effects of non-locality in
dynamics.
In a sense, the model-D class is the most generalized class of dynamical critical phenomena
with dissipative dynamics. However, with local conservation, it is equivalent to the model-B
class showing no novel features. Motivated by the results for model C [4], we investigate
the problem of non-local conservation in model D in the most general way by keeping both
conservations—that of the order parameter field and that of the coupled field—non-local. One
of our main results is that a true model-D behaviour emerges in the non-local dynamics, unlike
in the local conservation case.
We introduce non-locality by using a non-local kernel [2–4] in the chemical potential,
keeping the continuity equation intact. Usingφ for the order parameter andm for the secondary
coupling parameter, the Hamiltonian of model D is given by
H =
∫
ddx [ 12 rφ
2(x) + 12 (∇φ(x))2 + ũφ4(x) + γφ2(x)m(x) + 12C−1m2(x)], (1)
where φ2(x) = ∑ni=1 φ2i (x), φ4(x) = [∑ni=1 φ2i (x)]2, r = 0 is the mean field critical point,
and C > 0 ensures non-criticality ofm. There is a short-distance cut-off which in momentum
space is an ultraviolet cut-off. We shall set = 1 in the calculation. The above Hamiltonian,
equation (1), is the same as in model C [1, 4]. Making γ = 0 effectively reduces the above
model to model A.
The dynamics3 is given by the following equations:(
1
ρkρ
+
1
0
)
∂φk
∂t
= − δH
δφ−k
+ η(k, t) (2a)
(
1
λσ kσ
+
1
λ0
)
∂mk
∂t
= − δH
δm−k
+ ζ(k, t). (2b)
The noises η and ζ obey
〈η〉 = 0, 〈ζ 〉 = 0, (3a)
〈η(k, t)η(k′, t ′)〉 = 2
(
1
ρkρ
+
1
0
)
δ(k + k′)δ(t − t ′), (3b)
〈ζ(k, t)ζ(k′, t ′)〉 = 2
(
1
λσ kρ
+
1
λ0
)
δ(k + k′)δ(t − t ′). (3c)
Here, k = |k|, and we have introduced two different parameters ρ and σ to denote the non-
localities in the order parameter field and the coupling field respectively. The dissipative and
kinetic coefficients (0 and λ0) are labelled by the subscript 0 while the transport coefficients
(ρ and λσ ) are labelled by the corresponding powers of k. For the local case, both ρ
and σ assume a value equal to 2. Results are known for some limiting values of ρ and σ :
(a) (ρ = 0, γ = 0) corresponds to model A [1]; (b) (ρ = 2, γ = 0) and (ρ = 2, σ = 2) give
model-B-like behaviour [1]; (c) any value of ρ > 0 and γ = 0 give a non-local model B [2,5];
and (d) ρ = 0 with any σ > 0 gives non-local model C [4].
3 Additional torques for n = 3 (Heisenberg model) may lead to new dynamic universality classes [9]. We do not
consider such effects.
Letter to the Editor L143
A Hamiltonian of the type of equation (1) occurs near the critical wings of a tricritical
system or in critical systems with additional constraints. Model-D dynamics is relevant in such
cases—for example, in three- or four-component mixtures or in spin-1 or Potts models (see e.g.
[10]). One spin- 12 example—although a bit unrealistic—is an antiferromagnetic system with
conserved magnetization (as in model C [6,11]) as well as conserved staggered magnetization.
The Kawasaki dynamics can be used to conserve the magnetization, maintaining at the same
time that the exchanges between opposite spins have to be done in such a way that the order
parameter is unchanged.
A momentum-shell renormalization group approach is used to study the long-distance
long-time behaviour. In this approach, small-scale fluctuations in space (between e−δl and
 in momentum space) are integrated out, and the cut-off rescaled to . One then obtains an
effective Hamiltonian and effective equations for the dynamics valid for longer scales. The
universal features are obtained from the fixed points of the flow equations for the various
parameters of the problem. As in the usual RG scheme, any parameter that vanishes (grows) in
the long-scale limit is called an irrelevant (relevant) variable. A fixed point is then characterized
by the set of relevant parameters. In this scheme the four dynamics classes correspond to the
relevance and irrelevance of λσ and ρ as mentioned in the abstract.
The statics of this system is the same as for model C and the fixed-point values of the
parameters r , u, and γ 2C are as given in [4]. The dynamical equations, on the other hand, take
the forms
∂λ−1σ
∂l
=
(
−z + σ + α
ν
)
λ−1σ (4a)
∂λ−10
∂l
=
(
−z + α
ν
)
λ−10 + (nγ
2Kd)Q (4b)
∂−1ρ
∂l
= (−z + ρ + 2)−1ρ (4c)
∂−10
∂l
= (−z + 2)−10 + (4Cγ 2Kd)Q
1
1 + (Q/CP )
(4d)
whereQ = −1ρ +−10 , P = λ−1σ + λ−10 , andKd = 21−dπ−d/2(d/2). The parameters λσ , λ0,
ρ , 0, γ , and C appearing in the above equations are all l-dependent. To simplify notation,
this l-dependence has been suppressed.
From (4b) we get
0
λ0
= (1 + x)nγ
2Kd
z− α/ν (5)
where x = 0/ρ is the dimensionless ratio of the two coefficients for the primary order
parameter (note:  = 1). This quantity x satisfies
∂x
∂l
= (ρ − Y )x, where Y = 2α
νn
1
µ−1 + (x + 1)−1
. (6)
Here we have introduced the ratios of the transport and kinetic coefficients ofmwith the kinetic
coefficient of φ:
µσ = 0C
λσ
, µ0 = 0C
λ0
, and µ = µσ + µ0.
By using the fixed-point value of C and equation (5), µ0 can be rewritten as
µ0 = (x + 1)α/ν
2(z− α/ν) .
L144 Letter to the Editor
The flow equation for µσ is
∂µσ
∂l
= µσ [σ − 2 + α/ν − Y ]. (7)
We consider the different solutions of the equations for x and µσ . Corresponding to the
relevance of the conserved quantities, one gets different regions in the ρ–σ plane with different
dynamic critical behaviours. It should be mentioned here that for n  4, the coupling γ scales
to zero and one effectively gets a non-local model B. This has already been considered and
shown to be model-A-like when globally conserved [2]. Our discussions are for the n < 4
region where the coupling survives.
There are three fixed points of x: x = 0, x = ∞, and a non-zero finite fixed-point value
for x from equation (6):
Case 1. x = 0 in the long-length-scale limit. This corresponds to non-local model C: the
conservation of the order parameter is irrelevant. The fixed points and their stabilities of non-
local model C, which were obtained in [4] from the solutions for µσ , now depend on the value
of ρ. In general, x = 0 will be a stable fixed point as long as ρ < Y . We focus on the stability
of the various regions for non-zero values of ρ.
(a) µσ has a finite fixed-point value and z = σ + α/ν. This is a stable solution in the region
(2/n)−1 > p > −1 (where p = (σ − 2)/(α/ν)) for ρ = 0. For non-zero ρ this remains
stable for ρ < p + 1.
(b) µσ = ∞: this is valid for p > (2/n) − 1. Here z = 2 + 2α/nν. This solution x = 0,
µσ = ∞ is stable as long as ρ < 2α/nν.
(c) For µσ = 0, the stability of x = 0 is valid only for p < −1 and ρ < O(ε2). Here
z = 2 + O(ε2) and the behaviour is model-A-like. Any non-zero value of ρ thus destroys
this model-A-like region. Therefore, a region where both the conservations are irrelevant
is restricted strictly to ρ = 0.
Case 2. A finite non-zero value of x in the long-length-scale limit. This occurs when ρ = Y .
In order to ensure that 0 reaches a finite fixed-point value, we must have z = 2 + ρ. In this
case, µσ again has the following fixed points:
(a) µσ = ∞ for σ − 2 + α/ν > ρ;
(b) µσ = 0 for σ − 2 + α/ν < ρ;
(c) µσ non-zero and finite for σ − 2 + α/ν = ρ.
Physically, a finite-valued fixed point of x means that the conservation in the order parameter
φ is relevant. If at the same time µσ attains a non-zero fixed-point value, the conservation of
the coupling field m is also relevant. This occurs if the fixed points (a) and (c) are stable.
For case (a), x = ((ρ − 2α)/νn)/(2α/νn), and for non-negative values of x, ρ > 2α/nν.
Hence between ρ = 2α/nν and ρ = σ − 2 + α/ν, a stable region is obtained where both
conservations are relevant.
For case (b), x + 1 = ρ(2z− α/ν)/(2α2/nν2). This has a finite-valued solution for x
only for ρ ∼ O(ε2). Hence for ρ > σ − 2 + α/ν, x → ∞ is the only solution for µ = 0. It is
a conventional model-B fixed point.
For case (c), z = σ +α/ν and, in principle, along the entire line σ−2+α/ν = ρ a solution
exists where both the conservations are relevant. However, there is a restriction on the value
of σ arising from the condition of existence of a non-zero finite value of µσ . We find that
µσ + µ0 = (σ − 2 + α/ν)(x + 1)
2 − σ − α/ν + (2α/nν)(x + 1) . (8)
Letter to the Editor L145
II
III
V
/
IV
ρ  = p + 1
IV
ρ
ρ
−1 0 1 p
= 2
II
−−
S
Figure 1. The different regions in the non-local model D with n = 1 are shown in the rescaled
ρ̄–p plane where ρ̄ = ρ/(α/ν) and p = (σ − 2)/(α/ν). Regions I and II are model-C-like with
z = σ + α/ν and z = 2 + 2α/ν respectively. Region III′ is model-A-like (as in model C of [4])
with z = 2 + O(ε2). Region IV is the model-D region, with z = ρ + 2. Region V is model-B-like,
with z = ρ + 2. The line p + 1 = ρ is also model-D-like up to point S (shown by the dashed line).
There are no discontinuities of z along any boundary except at the boundary between III′ and V.
Hence with σ = 2+pα/ν, we get the condition (1+p) < 2(x+1)/n. Since x is also non-zero,
this implies that this solution with both x and µσ finite is valid for p + 1 = ρ/(α/ν) with at
least p < (2/n)− 1.
In the rest of the ρ–σ plane, the only solution is x = ∞ and 1/λσ = 0, for which only
the conservation of the order parameter is relevant. Here z = ρ + 2.
All the above possibilities for different dynamic behaviours are summarized in figure 1
in the ρ–σ plane (for n = 1). The possibility of seeing a region where both conservation
conditions are relevant raises new issues—such as the early-time effect and boundary or surface
effects (see e.g. [6, 12])—in the new regime. These remain to be studied.
To summarize, model D can be viewed as the most general dynamic model with purely
dissipative dynamics and we do indeed obtain all four types of behaviour as soon as non-locality
in the conservations (as in equation (2)) is introduced. Since our general model subsumes the
previously studied non-local model C and local model D, it is natural to expect A-like, B-like,
and C-like regimes. The model-A-like region, however, turns out to be unstable for any finite
value of ρ. The model-C-like regions are stable for small non-zero values of ρ but ultimately
disappear for larger values of ρ. The most non-trivial result, for general values of ρ and σ ,
is the appearance (region IV in figure 1) of a region with a new dynamical behaviour where
both conservations are relevant. This we call a true model-D-like region—a consequence of
non-local conservation laws.
PS acknowledges financial support from DST project SP/S2/M-11/99.
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Is there a true Model-D critical dynamics?

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