On the universality class of a magnetic liquid in an external field
Time scale ratios play an important role in critical dynamics. Their fixed point values govern the dynamical scaling behavior of the relevant fields and their dynamic correlation functions as well as the hydrodynamic transport coefficients. As an example for a system with several time scale ratios w...
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| Cite this: | On the universality class of a magnetic liquid in an external field / R. Folk, G. Moser // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23601: 1-14. — Бібліогр.: 35 назв. — англ. |
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| citation_txt | On the universality class of a magnetic liquid in an external field / R. Folk, G. Moser // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23601: 1-14. — Бібліогр.: 35 назв. — англ. |
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| description | Time scale ratios play an important role in critical dynamics. Their fixed point values govern the dynamical scaling behavior of the relevant fields and their dynamic correlation functions as well as the hydrodynamic transport coefficients. As an example for a system with several time scale ratios we consider a magnetic liquid near the gas-liquid transition.
Зазначено, що коефіцієнти часових масштабів відіграють важливу роль у критичній динаміці. Значення їх фіксованих точок керують динамічною скейлінговою поведінкою полів, динамічними кореляційними функціями, а також гідродинамічними коефіцієнтами переносу. Як приклад системи з декількома коефіцієнтами часових масштабів розглянуто магнітну рідину поблизу переходу газ - рідина.
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| first_indexed | 2025-12-07T16:45:22Z |
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Condensed Matter Physics 2010, Vol. 13, No 2, 23601: 1–14
http://www.icmp.lviv.ua/journal
On the universality class of a magnetic liquid in an
external field
R. Folk1, G. Moser2
1 Institute for Theoretical Physics, Johannes Kepler University, Altenberger Str. 69, A–4040 Linz, Austria
2 Department for Material Research and Physics, Paris Lodron University Salzburg, Hellbrunnerstrasse 34,
A–5020 Salzburg, Austria
Received February 9, 2010, in final form February 25, 2010
Time scale ratios play an important role in critical dynamics. Their fixed point values govern the dynamical
scaling behavior of the relevant fields and their dynamic correlation functions as well as the hydrodynamic
transport coefficients. As an example for a system with several time scale ratios we consider a magnetic liquid
near the gas-liquid transition.
Key words: phase transition, critical dynamics
PACS: 64.60.Ht, 05.70.Jk, 75.50.Mm, 64.70.Fx
1. Introduction
Collective modes in simple fluids, binary and ternary mixtures and more complex systems like
magnetic or ionic fluids have been of interest for I. Mryglod over a long time in his scientific carrier
(see his publication list). It is a pleasure for one of us (R.F.) having the chance to participate in this
field with him over several years. Many aspects of such systems have been studied with different
methods to get a more or less complete picture of the model and to understand the behavior of
real systems [1–7]. The phase diagram of a magnetic liquid may show different topologies. The
topology do change if such a liquid is exposed to an external magnetic field and as a result a line
of gas-liquid critical points might exist [8–10].
In the following the critical dynamics of a magnetic fluid in an external magnetic field is
discussed. After some general remarks concerning time scales and their ratios appearing in critical
dynamics, the steps for defining the equation of motion near the gas-liquid critical point are
indicated. As a conclusion one finds that the critical dynamics of the magnetic fluid can be mapped
on the critical dynamics of a mixture. Taking this result into consideration the consequences for
the critical dynamics and the behavior of the transport coefficients are discussed.
2. General considerations on dynamics
The critical dynamics is described by the dynamics of the slow density which are the order
parameter density and the densities of conserved variables. The order parameter might be conserved
or non-conserved. In any case its dynamics is slow due to a critical slowdown when the transition is
approached. Each slow density has its own time scale set by the relaxation or diffusion coefficients
appearing in the dynamic equations and driving the system to its thermodynamic equilibrium.
Time scale ratios are then defined by the ratio of these kinetic coefficients. The question arises
which densities have to be taken into account in order to define the universality class of the critical
dynamical behavior. E.g., the densities describing the sound mode are conserved but the coupling
of the sound mode to the dynamics of the order parameter equation has no effect on the genuine
dynamical critical behavior of the order parameter. This is argued by considering the sound mode
as faster than the diffusive modes defining the dynamical universality class of the order parameter.
c© R. Folk, G. Moser 23601-1
http://www.icmp.lviv.ua/journal
R. Folk, G. Moser
Dynamic scaling or more precisely strong dynamic scaling is realized when all time scales present
in the dynamic model describing critical dynamics are related by finite nonzero time scale ratios.
In consequence the critical dynamics is described by one time scale. This time scale may be defined
by the characteristic frequency of the order parameter ωOP(k) = AOPkzOP at the transition tem-
perature Tc. This is given by a power law in the wave vector modulus characterized by the dynamic
critical exponent zOP. The amplitude A is proportional to the relaxation or diffusion coefficient of
the order parameter. The dynamical critical exponent zOP is calculated in field theory from the
so-called ζ-function at the dynamical fixed point. The renormalization procedure, which removes
singularities (for a general introduction see [11]), leads to a renormalized kinetic coefficient Γ in the
equation of motion for the order parameter. From these renormalization factors, the ζ-functions
are obtained and their values at the stable dynamic fixed point (characterized by a star) give the
values of dynamical exponents. For the order parameter this reads
zOP = 2 + ζ?
Γ . (1)
One approach to describe the slow dynamics is by using Langevin equations for the dynamical
variables. The simplest situation may arise when the order parameter is non-conserved and there
exists an additional conserved density. Such a situation may arise for an anisotropic (uniaxial)
antiferromagnet in an external field where the order parameter is given by the staggered mag-
netization and the conserved density by the magnetization both in the direction of the external
field [12, 13], which is an example for the so-called model C [14, 15]. In such a case an additional
kinetic coefficient λ appears in the equation for the conserved density.Thus a second characteristic
frequency ωCD(k) = ACDkzCD might be defined. In addition a second dynamical critical exponent
might be defined by the corresponding ζ-function for the conserved density
zCD = 2 + ζ?
λ . (2)
The fixed point value of the time scale ratio w of the two kinetic coefficients
w =
Γ
λ
(3)
is found according to its definition from the stationary solution of the renormalization equation
`
dw
d`
= w(`)
(
ζΓ(w(`), . . . ) − ζλ(w(`), . . . )
)
. (4)
The dots indicate other possible renormalizing parameters of the dynamical equations. If the fixed
point value of w is different from zero and finite, the right hand side of (4) has to be zero. This
immediately leads to the relation
ζ?
Γ = ζ?
λ (5)
and consequently to only one dynamic critical exponent z = zOP = zCD, strong scaling is valid. If
however the two exponents are different and the two kinetic coefficients scale differently, two time
scales are present, the situation being called weak dynamic scaling. Such a situation arises for the
anisotropic antiferromagnet at higher values for the external field. The dynamics belongs the to
the universality class of model E [16] or if non-asymptotic effects are taken into account to model
F [16].
A different situation arises in cases where the order parameter is already a conserved density.
Then the time scale ratio to another conserved density is an irrelevant parameter since its naive
dimension is e.g. −2, whereas if the order parameter is non-conserved, the naive dimension of the
time scales is zero. The first case arises at the gas-liquid critical point in a fluid where the order
parameter, the entropy density (or a linear combination with the mass density), is conserved and
dynamically couples with the transverse momentum current ~jt. These two densities describe the
critical modes of heat diffusion and the shear mode. Then two time scales are present from the
beginning and two exponents describing the divergence of the thermal conductivity and the weak
divergence of the shear viscosity have to be calculated. The time scale ratio has to be set to zero in
23601-2
Time scale ratios
the minimal subtraction scheme used to calculate the dynamical exponents [17]. The second case
arises in the above mentioned model C.
As already mentioned above a similar situation arises for the densities describing the sound
mode when the minimal subtraction scheme is used [18, 19]. There the unrenormalized coupling
constant between the density and the longitudinal part of the momentum current has to go to
infinity. This procedure ensures that even the non-asymptotic critical dynamics of the order pa-
rameter is not effected by the sound mode and, on the other hand, the critical effect in the sound
mode can be calculated.
In what follows, the critical dynamics of a magnetic fluid in an external magnetic field is consi-
dered and the steps leading to the dynamical model describing its critical dynamics are presented.
It turns out [20] that the critical dynamics belongs to the universality class of a pure fluid [21]
(model H [16]), whereas the non-asymptotic behavior belongs to the case of a mixture [19, 22].
3. Hydrodynamics of Heisenberg fluids in an external magnetic field
Magnetic fluids are well known systems combining translational and magnetic degrees of free-
dom. A subclass of such systems are those magnetic liquids where on the microscopic level the
magnetic moments (spins) interact via a short ranged Heisenberg interaction [23–25]. In order
to obtain the hydrodynamic equations for a Heisenberg fluid in a magnetic field ~Hex we have to
consider the corresponding slow extensive densities per volume near the gas liquid critical point.
These are first of all those densities already describing the critical dynamics of a simple liquid,
namely the entropy density s(x), the mass density ρ(x) and the momentum density ~j(x), which is
a vector in the coordinate space with components ji(x) (i = x, y, z). Additionally, one has to con-
sider the components of a classical spin density mµ(x), where µ = x, y, z denotes the components
in spin space, describing the magnetic degrees of freedom in the direction of the external field.
Thus, independent of the nature of the spin, either Ising, XY or Heisenberg, only one component is
involved in the dynamics. An intensive conjugate field corresponds to each density. These fields are
the temperature T (x), the chemical potential µ(x), the velocity ~v(x) and the magnetic field ~H(x).
The introduced densities define four hydrodynamical modes of the system: the thermal diffusion,
the shear mode, the sound mode and the spin diffusion.
The reversible Euler equations for the densities result from generalized Poisson bracket relations
and a corresponding “Hamiltonian” H. Denoting the extensive densities per volume generally with
αi(x), the reversible equations have the form
∂αi(x)
∂t
= {αi(x),H}. (6)
The “Hamiltonian” in the present macroscopic case is effectively a static functional
H =
∫
d3xe
(
{αi(x)}
)
. The whole ensemble {αi(x)} consists of the extensive densities αi(x). The
total energy density
e = u + ekin −
∑
µ
Hµ
exm
µ (7)
of the system is composed of the internal energy u, where
du = Tds + µdρ +
∑
µ
Hµdmµ , (8)
the kinetic energy ekin = 1
2ρ~v 2 and the energy of the spin density in the external field −∑µ Hµ
exm
µ.
The magnetic field Hµ =
(
∂u
∂mµ
)
s,ρ,~j
represents the internal magnetic field produced by the spin
density.
Following [26], the generalized Poisson brackets {αi, αj} between the conserved densities are
determined by the group generators of operations in coordinate and spin space. The momentum
23601-3
R. Folk, G. Moser
density, as the generator of translations in coordinate space, determines the Poisson brackets,
{s(x),~j(x′)} = s(x′)~∇′δ(x − x′) , (9)
{ρ(x),~j(x′)} = ρ(x′)~∇′δ(x − x′) , (10)
{ji(x), jk(x′)} = ji(x
′)∇′
kδ(x − x′) − jk(x)∇iδ(x − x′) , (11)
where the prime on ~∇′ indicates that it acts on x′. In quantum mechanics, the spin operator is the
generator of rotations in spin space, acting only on the spin components rather than on quantities
in coordinate space. The components of the spin density are introduced as mµ(x) =
∑
i Sµ
i δ(x−xi).
In a coarse-grained theory the Sµ
i represent the mean µ− component of the spin in a small volume
located at xi. The Poisson bracket generated by the spin rotation is
{mµ(x), mν(x′)} = −
∑
λ
εµνλmλ(x)δ(x − x′) . (12)
The spin components Sµ
i are not affected by transformations in coordinate space such as translation
or rotation. However, the spin density mµ(x) is affected by such transformations due to the scalar δ-
function. It generates a Poisson bracket between the magnetization m and the momentum density,
which has the same structure as for the entropy density
{mµ(x),~j(x′)} = mµ(x′)~∇′δ(x − x′) . (13)
Then the following Euler equations are obtained
∂s
∂t
+ ~∇ · (s~v) = 0 , (14)
∂ρ
∂t
+ ~∇ · (ρ~v) = 0 , (15)
∂ji
∂t
+ ∇iP +
∑
l
∇lvlji = 0 , (16)
∂mµ
∂t
+ εµνλ(Hν − Hν
ex)m
λ +
∑
l
∇lvlm
µ = 0 . (17)
So far, equations (14)–(17) do not contain dissipative processes. We have made use of the fact that
the external field is homogeneous, otherwise an additional term proportional to the scalar product
between the magnetisation and the gradient of the field appears. This would considerably change
the resulting critical model and is not discussed here. In fact, it would lead to anisotropies in the
fluid system [27].
The total energy H has to remain constant in time, i.e. ∂H/∂t =
∫
d3x (∂e/∂t) = 0, which
can be immediately verified by using the continuity equations from above. Using the relation
~j(x) = ρ(x)~v(x) and introducing the tensor [P ]ki ≡ Pki = Pδki, one immediately obtains the
equation for the velocity ~v from the conservation equations for momentum density (16), and finally
the hydrodynamic equation for the kinetic energy
∂ekin
∂t
= −~∇ · [ekin~v + P · ~v] +
∑
i,k
Pki∇kvi . (18)
The first term on the right hand side in (18) represents the divergence of the kinetic energy current
ekin~v + P · ~v. In addition, an energy source term appears in (18) because the kinetic energy is not
a conserved density. The total energy density e is conserved and therefore obeys the continuity
equation,
∂e
∂t
= −~∇ · ~Je , (19)
where ~Je is the total energy current. Without dissipative effects, the total energy current is the
sum of the convective flow e~v and the flow of mechanical work P · ~v.
23601-4
Time scale ratios
In order to treat dissipation we have to append the dissipative currents [28] to equations (14),
(16), (17) and (19). The total energy current contains two dissipative processes. The first of these
accounts for the finite viscosity of the fluid. Viscosity causes additional mechanical work when
different fluid layers shift against each other, and is taken into account using an extension of the
pressure tensor,
Pki = Pδki + Πki , (20)
where Πki is a symmetric tensor. The second dissipative process takes into account the energy
transport by heat conduction, which may be included by adding a heat current ~Jq to the total
energy current. Thus we get
~Je = e~v + P · ~v + ~Jq (21)
for the total energy current with dissipation. The contribution of the Heisenberg interaction to the
energy of a spin system depends only on the orientation of the spins. Therefore the spin density
does not contribute to the energy transport and in (21) no magnetic current is present.
Since in the following we will consider a fluid in an external magnetic field ~Hex = Hex~ez, with
~ez being the unit vector in the z-direction in the spin space. Thus only the z component mz(x) of
the spin density is conserved. From (17) it follows that the z-component now fulfills the equation
∂mz
∂t
+
∑
l
∇lvlm
z = −~∇ ~Jz
m , (22)
where ~Jz
m is the spin current vector.
Inserting equation (20) into the equation for the momentum current equations (16)
∂ji
∂t
+ ∇iP +
∑
l
∇lvlji = −
∑
l
∇lΠ
(s)
li −∇iΠ (23)
the dissipative processes are included. In the above equation, the symmetric tensor [Π]ik = Πik =
Π
(s)
ik + Πδik introduced in (20) has been separated into the traceless contribution [Π(s)]ik = Π
(s)
ik
and the trace Π = 1
3
∑
l Πll.
The dissipative current in the entropy equation is found from the dissipative currents already
introduced in the energy, the magnetic density and the momentum current. From (8) we also
determine the equation for the entropy density conservation
T
∂s
∂t
=
∂u
∂t
−
∑
µ
Hµ ∂mµ
∂t
. (24)
From (7) we obtain the hydrodynamic equation for the internal energy u
∂u
∂t
=
∂e
∂t
− ∂ekin
∂t
+
∑
µ
Hµ
ex
∂mµ
∂t
. (25)
Inserting (19), (18) and (22) we obtain for the internal energy the equation,
∂u
∂t
= −~∇ · [u~v + ~Jq] −
∑
i,k
Pki∇kvi −
∑
µ,ν,λ
εµνλHµmνHλ
ex . (26)
The final equation for the entropy density is found by inserting (26) into (24)
∂s
∂t
+ ~∇ · (s~v) = −~∇ ·
(
~Jq − ~Jz
mHz
T
)
+ qs , (27)
with qs the entropy source term,
qs = −( ~Jq − ~Jz
mHz) ·
~∇T
T 2
− ~Jz
m ·
~∇Hz
T
−Π
(s) :
∇v
(s)
T
− Π
~∇ · ~v
T
. (28)
23601-5
R. Folk, G. Moser
Some notations of tensor algebra have been introduced in the above relation, namely ab denotes
the dyadic product [ab]ij = aibj of two vectors ~a and ~b. The symbol colon ‘:’ expresses the total
contraction of two tensors of the second rank, A : B =
∑
i,j AijBij .
From (28) we can see that the entropy source has the structure [29],
qs = −
∑
i
JiXi , (29)
with the currents Ji and corresponding thermodynamic forces Xi. In the present case the ther-
modynamic forces are ~∇ · ~v/T , ~∇T/T 2, ~∇Hz/T and ∇v
(s)/T The thermodynamic forces have
different tensor characters. The first one is a scalar, the second and the third are polar vectors,
the fourth is a polar tensor of the second rank. The same is true for the currents. The currents
Ji = Ji({Xj}) are functions of the thermodynamic forces. For small gradients we can assume a
linear dependence Ji =
∑
j LijXj , where Lij are the Onsager coefficients. Generally, all currents
may depend on all thermodynamic forces that define a set of Onsager coefficients, which, of course,
have different tensor character depending on the tensor character of the connected current and
force. But the interaction symmetry and the tensor character of thermodynamic forces restrict
the form of thermodynamic forces in a current and reduces the number of non-vanishing Onsager
coefficients. For an isotropic system, the form of thermodynamic forces follows from invariance
under coordinate inversion and arbitrary rotation, so that only tensor characters of the same type
can have cross coefficients. In (28) two polar vectors ~∇T/T 2 as well as ~∇Hz/T are present. Thus,
the resulting currents are,
~Jq − ~Jz
mHz = −γ~∇T − Tβ~∇Hz , (30)
~Jz
m = −β~∇T − α~∇Hz , (31)
Π = −2η̄∇v
(s) , (32)
Π = −ζ ~∇ · ~v . (33)
In equations (30) and (31) we have already used the symmetry properties of the Onsager coef-
ficients. Linearizing the hydrodynamic equations about the mean values s̄, m̄z, ρ̄ and ~̄j = 0 we
obtain,
∂s
∂t
+ s̄~∇ · ~v =
γ
T
∇2T + β∇2Hz , (34)
∂mz
∂t
+ m̄z ~∇ · ~v = β∇2T + α∇2Hz , (35)
∂ρ
∂t
+ ρ̄~∇ · ~v = 0 , (36)
∂~j
∂t
+ ~∇P = η̄∇2~v +
(
ζ +
4
3
η̄
)
~∇(~∇ · ~v) . (37)
A statistical treatment of generalized hydrodynamic equations can be found in [30].
The Onsager coefficients α, β and γ introduced in (30) and (31) can be related to the thermal
conductivity κT or the thermal diffusion coefficient DT respectively, the thermomagnetic diffusion
ratio kT and the spin diffusion coefficient Dm. The coefficients η̄ and ζ represent the shear and
bulk viscosity. Assuming a vanishing spin current ~Jz
m = 0 in (31), the gradients of the temperature
and magnetic field are related by ~∇Hz = −(β/α)~∇T . Inserting this relation into (30), we obtain
for the heat current
~Jq = −
(
γ − Tβ2
α
)
~∇T , (38)
from which we identify the thermal conductivity κT at vanishing spin current to be,
κT = γ − Tβ2
α
. (39)
23601-6
Time scale ratios
At constant spin density per mass µm = mz/ρ and constant pressure P , the temperature gradient
is proportional to the gradient of the entropy density per unit mass σ = s/ρ,
~∇T =
(
∂T
∂σ
)
µm,P
~∇σ . (40)
The equation for σ is easily derived from equations (34) and (36),
∂σ
∂t
=
κT
ρT
(
∂T
∂σ
)
µm,P
∇2σ , (41)
and this describes heat conduction in the magnetic fluid. From this equation we identify the thermal
diffusion coefficient to be
DT =
κT
ρT
(
∂T
∂σ
)
µm,P
. (42)
Generally, in the case of local thermodynamic equilibrium the gradient of the internal magnetic
field at constant pressure can be written as
~∇Hz =
(
∂Hz
∂T
)
mz,P
~∇T +
(
∂Hz
∂mz
)
T,P
~∇mz . (43)
Inserting this into the equation for the spin current (31), we obtain,
~Jz
m = −α
(
∂Hz
∂mz
)
T,P
{
~∇mz +
(
∂mz
∂Hz
)
T,P
[
(
∂Hz
∂T
)
mz,P
+
β
α
]
~∇T
}
. (44)
Considering a magnetic fluid at rest (~v = ~0), at constant temperature and at constant pressure
from equation (35) the spin diffusion equation for the magnetization density per mass is given by
∂µm
∂t
=
α
ρ
(
∂Hz
∂µm
)
T,P
∇2µm . (45)
Thus, the spin diffusion coefficient reads
Dm =
α
ρ
(
∂Hz
∂µm
)
T,P
. (46)
At vanishing spin current and at constant pressure, a temperature gradient causes a spin density
gradient. From (44) we immediately obtain,
~∇µm = −
(
∂µm
∂Hz
)
T,P
[
(
∂Hz
∂T
)
µm,P
+
β
α
]
~∇T , (47)
which leads to the identification of the thermomagnetic diffusion ratio,
kT
T
=
(
∂µm
∂Hz
)
T,P
[
(
∂Hz
∂T
)
µm,P
+
β
α
]
. (48)
This transport coefficient corresponds to the thermodiffusion ratio in binary mixtures.
4. Static functional of the Heisenberg fluid
In order to find the static functional for the magnetic fluid near the gas-liquid phase transition
we expand the general functional in the fluctuations of the densities αi(x) per volume. The entropy
density per volume represents the order parameter implying that the fluctuation terms of this
23601-7
R. Folk, G. Moser
density have to be taken into account up to the fourth order. The remaining terms in the static
functional are the quadratic terms of all the other densities per volume mentioned and a quadratic
gradient term for the order parameter. Because the order parameter is a scalar quantity, there is
also a third order term allowed, consistent to the symmetry properties. In order to obtain a suitable
static functional, several steps are necessary to be performed:
i) The densities per volume αi(x) (except the mass density itself) will be replaced by densi-
ties per mass αi(x)/ρ(x). The correlation functions will then be related to thermodynamic
derivatives at fixed pressure instead of fixed chemical potential.
ii) In order to decouple the order parameter from the secondary densities a shift in the densities
per mass has to be performed. This shift eliminates the quadratic terms between the order
parameter and secondary densities and the third order term of the order parameter leading
to the usual φ4-model.
iii) For a discussion of the slow heat and magnetic diffusion modes only, it is not necessary to keep
the fast sound mode in the considerations. In this case it is convenient to integrate out the
mass density both in the static and in the dynamic functional. Consequently the assignment
of the background parameters in the quadratic part of the static functional changes from
thermodynamic derivatives at a fixed chemical potential to corresponding quantities at a
fixed pressure.
In the following we will consider the slowest modes only. In this case from the momentum
density, separated into the longitudinal component ~jl defined by ~∇ ∧ ~jl = ~0, and the transverse
component ~jt, defined by ~∇ · ~jt = 0, only the transverse part has to be considered. This finally
leads to the static functional in an external magnetic field
H =
∫
d3x
{
1
2
τ̊φ0
2+
1
2
(~∇φ0)
2 +
˚̃u
4!
φ0
4 +
1
2
aqq0
2 +
1
2
aj
~jt · ~jt +
1
2
γ̊qq0φ0
2 − h̊qq0
}
. (49)
The fluctuations are defined as
φ0(x) =
√
NA
[
4 σ(x) − 〈σ(x)〉
]
, (50)
q0(x) =
√
NA
[
4µm(x) −
(
∂µm
∂σ
)
H,P
(4σ(x) − 〈σ(x)〉)
]
. (51)
NA denotes the Avogadro number. The nonzero correlations are those of the order parameter and
the density q0.
Introducing the magnetic susceptibility
χσ,P =
(
∂µm
∂Hz
)
σ,P
(52)
the static correlation function and the corresponding vertex function Γ̊qq are
〈q0 q0〉c =
1
Γ̊qq
=
RT
ρ
χσ,P . (53)
Thus, the vertex function contains the magnetic susceptibility, which behaves like (∂µm/∂Hz)σ,P ∼
t−α in the asymptotic critical region. The coefficient aq corresponds to constant background values
of thermodynamic derivatives. In the present case it is given by
aq =
ρ
RTχ
(0)
σ,P
=
1
Γ̊
(0)
qq
, (54)
where Γ̊
(0)
qq represents the zeroth order of the vertex function in (53).
23601-8
Time scale ratios
Apart from the Gaussian terms of the transverse spin components, which do not contribute to
the critical behavior, the static functional (49) is the same as those of binary liquid mixtures at the
plait point [19]. All thermodynamic relations can be taken over from this system if one replaces the
concentration c and the chemical potential ∆ of the liquid mixture by the spin density µm and the
magnetic field Hz of the magnetic liquid. Therefore the static critical behavior of the quantities in
the magnetic liquid is the same as the static critical behavior of the corresponding quantities in
the liquid mixture at the plait point.
5. Dynamical model for Heisenberg fluids near the gas-liquid critical point
Using Zwanzig’s projection operator method, one can derive dynamic equations that describe
the time evolution of slow densities (see [31, 32]). Considering a set {αk(x, t)} of slow densities,
one obtains,
∂αi(x, t)
∂t
= Vi({αk(x, t)}) −
∑
j
Λij
δH({αk(x, t})
δαj(x, t)
+ θi(x, t) (55)
with Λij = −Lij∇2 for conserved densities and Λij = Lij for non-conserved densities. The term,
Vi({αk(x, t)}) =
∫
dx′
∑
j
[
Qij(x, t; x′, t′)
δH({αk(x, t})
δαj(x, t)
− δQij(x, t; x′, t′)
δαj(x, t)
]
(56)
constitutes the reversible part of equations. The functions Qij are determined by,
Qij(x, t; x′, t′) = kBT{αi(x, t), αj(x
′, t′)} , (57)
where the brackets {., .} denote generalized Poisson brackets and kB the Boltzmann constant.
H represents a static functional. The coefficients Lij are related to the Onsager coefficients of
hydrodynamic theory. The functions θi(x, t) include contributions of fast variables that will be
considered as stochastic forces. Assuming that these stochastic forces θi(x, t) are determined by a
Markovian process, the coefficients Λij fulfill the relations
〈θi(x, t) θj(x
′, t′)〉 = Λijδ(x − x′)δ(t − t′). (58)
In the case of Heisenberg fluids, the densities {αk(x, t)} are represented by hydrodynamic den-
sities s(x, t), ρ(x, t), mz(x, t) and ~j(x, t) considered in section 3. The generalized Poisson brackets
(9)–(13) determine the reversible part (56), which is independent of the external magnetic field. The
coefficients Lij are related to Onsager coefficients introduced in section 3. They may be identified
by comparing the general structure (55) of dynamic equations with the linearized hydrodynamic
equations (34)–(37). In the present case one has
∂s
∂t
= kBγ∇2 δH
δs
+ kBTβ∇2 δH
δm
+ Vs({αk}) + θs , (59)
∂mz
∂t
= kBTβ∇2 δH
δs
+ kBTα∇2 δH
δmz
+ Vmz({αk}) + θm , (60)
∂ρ
∂t
= Vρ({αk}) , (61)
∂~j
∂t
= kBT (ζ(0) +
4
3
η̄(0))∇2 δH
δ~jl
+ kBT η̄(0)∇2 δH
δ~jt
+ ~Vj({αk}) + ~θj . (62)
23601-9
R. Folk, G. Moser
The reversible contributions determined by generalized Poisson brackets are
Vs({αk}) = −kBT ~∇·
(
s
δH
δ~j
)
, (63)
Vmz ({αk}) = −kBT
∑
i
∇im
z δH
δji
, (64)
Vρ({αk}) = −kBT ~∇·
(
ρ
δH
δ~j
)
, (65)
~Vj({αk}) = −kBT
(
s~∇δH
δs
+ ρ~∇δH
δρ
+mz ~∇ δH
δmz
)
−kBT
∑
i
(
ji
~∇δH
δji
+ ∇i
~j
δH
δji
)
. (66)
The Onsager coefficients α, β, γ, ζ(0) and η̄(0) have been introduced in (30)–(33). The superscript
(0) indicates uncritical background values. The equation of mass conservation (61) does not contain
dissipative terms (Onsager coefficients) or stochastic forces.
With the set of dynamic equations, a dynamic functional can be established using the method of
[33]. Within this functional, the critical behavior of thermal modes as well as the sound mode may
be determined. The mass density and longitudinal momentum density, which are necessary only if
one wants to consider the critical behavior of the sound, can be integrated out of the dynamical
functional. The remaining dynamic functional corresponds to a reduced set of dynamic equations
including the entropy density, the magnetization density and the transverse momentum density
only, and a reduced static functional which is identical to (49) presented in the previous section.
Introducing the order parameter (50) and the secondary density (51), the set of reduced dynamic
equations reads
∂φ0
∂t
= Γ̊∇2 δH
δφ0
+ L̊∇2 δH
δq0
− g̊
δH
δ~jt
· ~∇φ0 + θφ , (67)
∂q0
∂t
= L̊∇2 δH
δφ0
+ µ̊∇2 δH
δq0
− g̊
δH
δ~jt
· ~∇q0 + θq , (68)
∂~jt
∂t
= λ̊t∇2 δH
δ~jt
+ g̊
[
δH
δφ0
~∇φ0 +
δH
δq0
~∇q0
]
− g̊T
∑
i
(
ji
~∇δH
δji
+ ∇i
~j
δH
δji
)
+ ~θt . (69)
The corresponding static functional H is given by (49). The kinetic coefficients appearing in (67)
and (68) are related to the Onsager coefficients introduced in (30) and (31) by,
Γ̊ =
Rγ
ρ2
, L̊ =
RT
ρ2
(
β +
(
∂µm
∂σ
)
Hz ,P
γ
T
)
, (70)
µ̊ =
RT
ρ2
(
α − 2
(
∂µm
∂σ
)
Hz ,P
β +
(
∂µm
∂σ
)2
Hz ,P
γ
T
)
. (71)
The non-linear mode coupling is defined as
g̊ =
RT√
NA
. (72)
From power counting arguments we obtain the cut-off (Λ) dimensions of kinetic coefficients which
are Γ̊ ∼ Λ0, L̊ ∼ Λ1, µ̊ ∼ λ̊t ∼ Λ2, and g̊ ∼ Λ1+ε/2. Within the perturbation expansion it is
convenient to introduce time scale ratios
ẘ =
Γ̊
µ̊
∼ Λ−2 , ẘ3 =
L̊
√
Γ̊µ̊
∼ Λ0 (73)
23601-10
Time scale ratios
and a mode coupling parameter
f̊2
t =
g̊2
Γ̊λ̊t
∼ Λε (74)
which has a finite fixed-point value. The time scale ratio ẘ is irrelevant as can be seen from its
cut-off dimension and, therefore, has to be set to zero in expressions within perturbation expansion
in order to obtain a theory renormalizable with minimal subtraction.
The dynamic equations (67)–(69) have the same form as in the case of binary liquid mixtures
at the plait point [19]. Having considered the dynamical critical phenomena in the Heisenberg
liquid, one can now proceed along the lines developed in [32, 33]. From the dynamical functional
one calculates the dynamical correlations by a perturbation expansion in the mode couplings and
the static couplings.
6. Applying the field theoretic approach
In order to proceed in the calculation of physical measurable quantities, e.g., the temperature
dependence of a transport coefficient of the magnetic fluid, a standard procedure of field theory is
followed. These steps are not repeated here. They are:
(i) A perturbational calculation (loop expansion) of the physical quantities.
(ii) The introduction of renormalized counterparts of unrenormalized static and dynamic model
parameters presented so far leading to renormalization factors which contain the dimensional
poles.
(iii) The calculation of the so-called ζ-functions following from the renormalization factors, which
determine the temperature behavior of the model parameters from the background to the
asymptotic region (usually called the flow of parameters).
The transport coefficients in hydrodynamic limit (small frequencies and small modulus of the wave
vector) are calculated from the vertex functions in this limit. In the asymptotic limit the result
leads to power laws in the relative temperature distance from the gas-liquid critical point. The
exponents are given by the ζ-functions taken at the fixed point of the flow equations for kinetic
coefficients or their ratios, and the static and dynamic couplings. The renormalized counterparts
Γ, L and µ of the kinetic coefficients fulfill the flow equations
`
dΓ(`)
d`
= Γ(`)ζΓ , `
dL(`)
d`
= L(`)(−1 + ζL) , `
dµ(`)
d`
= µ(`)(−2 + ζµ). (75)
The flow parameter ` is related to the reduced temperature t by the matching condition,
ξ−2(t)
(ξ−1
0 `)2
= 1 , (76)
with the correlation length ξ(t). The renormalized counterpart of the time scale ratio ẘ3 introduced
in (73) is given by
w3(`) =
L(`)
√
Γ(`)µ(`)
. (77)
The ζ-function ζΓ, introduced in (75), can be separated into a static contribution and a purely
dynamic part ζ
(d)
Γ [34]. The remaining two ζ-functions ζL and ζµ do not contain dynamic contri-
butions. They fulfill the exact relations
ζΓ = ζφ + ζ
(d)
Γ , ζL =
1
2
ζφ , ζµ = 0. (78)
The static ζ-function, ζφ(u), depends only on the fourth order coupling of the φ4-model u(`). It
is obtained from renormalization of the k2 terms of the static two point order parameter vertex
23601-11
R. Folk, G. Moser
function. Thus, a kinetic coefficient Γ(d) may be introduced which is obtained from Γ by separating
the static contributions. From (75), (77) and (78) it follows that the dynamic flow is determined
by the equations
`
dΓ(d)(`)
d`
= Γ(d)(`)ζ
(d)
Γ , `
dw3(`)
d`
= −1
2
w3(`)ζ
(d)
Γ , (79)
with proper initial conditions. The functions ζφ and ζ
(d)
Γ are calculated from the static and dynamic
renormalization factors within the perturbation expansion. The renormalized counterpart of the
mode coupling parameter f̊t introduced in (74) is
f2
t (`) =
g2(`)
Γ(`)λt(`)
. (80)
It is determined by the flow equation
`
dft(`)
d`
= −1
2
ft(`)(ε + ζφ + ζ
(d)
Γ + ζλt
) . (81)
The function ζλt
is obtained from renormalization of the k2 terms of transverse momentum current
density two point function. The kinetic coefficient λt(`) is determined by the flow equation,
`
dλt(`)
d`
= λt(`)(−2 + ζλt
). (82)
In order to find the FP values of the time scale ratio w3 one might use quite general arguments
valid in all loop orders. Since the non-classical dynamical critical exponent of the OP is set by
the FP value of ζ
(d)
Γ one expects its value nonzero and positive. Therefore it follows that the
flow equation (79) drives w3 to zero. Then at the FP all dynamical ζ-functions reach their pure
fluid form concerning the other parameters and thus from the FP equation for the mode coupling
equation (81) one obtains the pure fluid value. Thus the asymptotic properties are the same as in
pure fluids and the non-asymptotic properties are described by the same equations as for mixtures
where concentration is replaced by magnetization in the direction of external field.
Let us note that this is not a general feature of mixtures. E.g., in mixtures of 4He-3He at the
superfluid transition, a similar time scale ratio w3 appears involving only the kinetic coefficients of
the conserved densities, entropy and concentration rather than the OP. In this case the FP value
is nonzero [21].
7. Critical behavior of the hydrodynamic transport coefficients
In the previous section we have shown that the dynamic model for the Heisenberg fluid in an
external homogeneous magnetic field reduces to the model known from binary liquid mixtures.
The relations of hydrodynamic transport coefficients to the model parameters and their explicit
calculation within renormalization group theory for a liquid mixture have been treated in [19].
Therefore, we shall not repeat these steps in the present paper, and only summarize the results we
obtained.
From calculations analogous to those for binary liquid mixtures at the plait point [19], we
obtain, at zero frequency, the thermal conductivity,
κT(`)
ρT
=
1
Dm(`)
( ρ
RT
)2
(
∂Hz
∂µm
)
T,P
µ̊Γ(d)(`)
(
1 − w2
3(`) + G(w3(`), ft(`), u(`))
)
, (83)
where G(w3, ft, u) contains contributions of perturbation expansion of the amplitude function for
the dynamic order parameter two point function. In one loop order, one has G(w3, ft, u) = −f2
t /16.
The spin diffusion coefficient Dm appearing in (83) is given by,
Dm(`) =
ρ
RT
(
∂Hz
∂µm
)
T,P
[
µ̊ + 2aL̊ + a2Γ(d)(`)(1 + G(w3(`), ft(`), u(`)))
]
. (84)
23601-12
Time scale ratios
The parameter a in the above equation is determined by thermodynamic derivative,
a =
(
∂µm
∂σ
)
Hz ,P
, (85)
which only weakly depends on the temperature and, therefore, can be considered as a constant
in the critical region. The thermomagnetic diffusion ratio related to the cross effect between heat
conduction and spin diffusion is,
kT(`)
T
= − ρ
RT
1
Dm(`)
[
L̊ +
µ̊
a
]
−
(
∂µm
∂T
)
σ,P
. (86)
From the expressions (83), (84) and (86) for transport coefficients, the following power laws in the
limit ` → 0 together with the matching condition (76) are found
κT(t) ∼ const. , Dm(t) ∼ tγ−xλν , kT(t) ∼ t−γ+xλν , (87)
where the exponents γ, xλ and ν can be taken from the renormalization theory for pure fluids (e.g.
[21]). The value of the dynamic exponent in two-loop order reads xλ = 0.9103 [35].
The shear viscosity can be calculated as in pure fluids and liquid mixtures. At zero frequency
the resulting expression for the shear viscosity is
η̄(`) =
1
RT
(κ`)2λt(`)
(
1 + Et(ft(`), w3(`))
)
. (88)
The amplitude function Et(ft(`), w3(`)) contains contributions of perturbation expansion in the
couplings (static and dynamic). In one loop order this function is Et(ft(`), w3(`)) = −f2
t /[36(1 −
w2
3)]. The asymptotic temperature dependence is
η̄(t) ∼ t−xην (89)
with xη again taken from the renormalization group theory for pure fluids. In two loop order one
obtains xη = 0.0712 [35]. The dynamical exponents fulfill the relation xη + xλ = 1 − η with the
two loop value of the static exponent η = 1/54.
Conclusion
We have shown that the critical dynamics of a magnetic liquid in an external field is analogues
to the critical dynamics of a liquid mixture at its plait point. The magnetization in the direction of
the external field is then equivalent to the concentration of the mixture. In consequence, the corre-
sponding transport coefficients behave in a similar manner although the non-asymptotic behavior
might be different due to its different background values.
Acknowledgments
This work was supported by the Fonds zur Förderung der wissenschaftlichen Forschung under
Project No. 19583-N20.
23601-13
R. Folk, G. Moser
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Стосовно класу унiверсальностi магнiтної рiдини у
зовнiшньому полi
Р. Фольк1, Г. Мозер2
1 Iнститут теоретичної фiзики, Унiверситет Йоганна Кеплера, Лiнц, Австрiя
2 Унiверситет Зальцбурга, Зальцбург, Австрiя
Коефiцiєнти часових масштабiв вiдiграють важливу роль у критичнiй динамiцi. Значення їхнiх фi-
ксованих точок керують динамiчною скейлiнговою поведiнкою полiв, динамiчними кореляцiйними
функцiями, а також гiдродинамiчними коефiцiєнтами переносу. Як приклад системи з декiлькома
коефiцiєнтами часових масштабiв ми розглядаємо магнiтну рiдину поблизу переходу газ-рiдина.
Ключовi слова: фазовий перехiд, критична динамiка
23601-14
Introduction
General considerations on dynamics
Hydrodynamics of Heisenberg fluids in an external magnetic field
Static functional of the Heisenberg fluid
Dynamical model for Heisenberg fluids near the gas-liquid critical point
Applying the field theoretic approach
Critical behavior of the hydrodynamic transport coefficients
|
| id | nasplib_isofts_kiev_ua-123456789-32092 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T16:45:22Z |
| publishDate | 2010 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Folk, R. Moser, G. 2012-04-08T15:42:22Z 2012-04-08T15:42:22Z 2010 On the universality class of a magnetic liquid in an external field / R. Folk, G. Moser // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23601: 1-14. — Бібліогр.: 35 назв. — англ. 1607-324X PACS: 64.60.Ht, 05.70.Jk, 75.50.Mm, 64.70.Fx https://nasplib.isofts.kiev.ua/handle/123456789/32092 Time scale ratios play an important role in critical dynamics. Their fixed point values govern the dynamical scaling behavior of the relevant fields and their dynamic correlation functions as well as the hydrodynamic transport coefficients. As an example for a system with several time scale ratios we consider a magnetic liquid near the gas-liquid transition. Зазначено, що коефіцієнти часових масштабів відіграють важливу роль у критичній динаміці. Значення їх фіксованих точок керують динамічною скейлінговою поведінкою полів, динамічними кореляційними функціями, а також гідродинамічними коефіцієнтами переносу. Як приклад системи з декількома коефіцієнтами часових масштабів розглянуто магнітну рідину поблизу переходу газ - рідина. This work was supported by the Fonds zur Forderung der wissenschaftlichen Forschung under Project No. 19583-N20. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics On the universality class of a magnetic liquid in an external field Стосовно класу універсальності магнітної рідини у зовнішньому полі Article published earlier |
| spellingShingle | On the universality class of a magnetic liquid in an external field Folk, R. Moser, G. |
| title | On the universality class of a magnetic liquid in an external field |
| title_alt | Стосовно класу універсальності магнітної рідини у зовнішньому полі |
| title_full | On the universality class of a magnetic liquid in an external field |
| title_fullStr | On the universality class of a magnetic liquid in an external field |
| title_full_unstemmed | On the universality class of a magnetic liquid in an external field |
| title_short | On the universality class of a magnetic liquid in an external field |
| title_sort | on the universality class of a magnetic liquid in an external field |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/32092 |
| work_keys_str_mv | AT folkr ontheuniversalityclassofamagneticliquidinanexternalfield AT moserg ontheuniversalityclassofamagneticliquidinanexternalfield AT folkr stosovnoklasuuníversalʹnostímagnítnoírídiniuzovníšnʹomupolí AT moserg stosovnoklasuuníversalʹnostímagnítnoírídiniuzovníšnʹomupolí |