# Critical ultrasonic propagation in magnetic fields

A. Pawlak<EMAIL_ADDRESS>Faculty of Physics, Adam Mickiewicz University,
Uniwersytetu Pozna skiego 2, 61-614 Poznań, Poland

###### Abstract

Effect of an external magnetic field on the critical sound attenuation and
velocity of the longitudinal wave is studied in ferromagnets. We derive a
parametric model that incorporates a crossover from the asymptotic critical
behavior to the Landau-Ginzburg regular behavior far away from the critical
point. The dynamics is based on the time dependent Ginzburg-Landau model with
non conserved order parameter (model A). The variations of the sound
attenuation coefficient and velocity have been obtained for arbitrary values
of the magnetic field and reduced temperature. The scaling functions are given
within the renormalization group formalism at one-loop order. Using MnP as an
example, we show that such parametric crossover model yields an accurate
description of ultrasonic data in a large region of temperatures and magnetic
fields around the critical point.

Keywords

critical behavior, ultrasonic attenuation, sound dispersion, Ginzburg-Landau
model, RG method, parametric model

††preprint: APS/123-QED

##  I Introduction

As the sound wave strongly couples with the order parameter fluctuations the
critical dynamics of sound is a research field where we can test modern
concepts of the phase transition theory such as the universality of critical
exponents, scaling or the crossover to another universality class etc. Sound
attenuation and velocity in magnetic systems have been extensively studied
both experimentally and theoretically Luthi81 -Paw11  . Whereas, the
temperature behavior of these quantities has been relatively well recognized
during last decades lut04 ; Deng87 ; Paw98 ; Paw00 ; Erdem04 ; PawFech06
using mean-field theories, scaling theory and renormalization group framework
a relatively little attention has been paid so far to the problem of critical
attenuation in the presence of the ordering magnetic field Kom74 ; Tach74 ;
Paw09 ; Paw12 ; Paw20 ; Paw22 .

The singular behavior of the sound attenuation coefficient is connected with
very strong fluctuations of the magnetic order parameter near the critical
temperature. These fluctuations give rise to a characteristic attenuation peak
whose position is correlated with a dip in the sound velocity lut04 . In
metallic magnets the critical anomalies in the sound attenuation coefficient
are of different types than in magnetic insulators. The difference in the
critical exponents describing these anomalous behaviors are usually explained
by the occurrence of different kinds of magnetoelastic coupling in the two
classes of magnets mentioned Kaw68 . It was also shown Paw98  that a very
important role is played by the ratio of the spin-lattice relaxation time to
the characteristic time of spin fluctuations. It is a crucial parameter
determining whether the sound attenuation coefficient reveals a strong or a
weak singularity in a given material. In this paper we will focus on the class
of metallic ferromagnets which are characterized by strong anomaly of the
sound attenuation coefficient (sometimes called Murata-Iro-Schwabl singularity
Mur76 ; IroSchwabl83 ) connected with short spin-lattice relaxation times
Paw98 ; Paw11 . In these magnetic materials the maximum in the sound
attenuation coefficient occurs in the ordered phase for vanishing magnetic
field due to the domination of the magnetic analogue of the Landau-Khalatnikov
sound damping near the superfluid transition of liquid 4He LK54 . Such
mechanism can be explained as a result of relaxation of the average order-
parameter and should be distinguished from fluctuation contribution described
by relaxation of a non-conserved variable which is proportional to the square
of the order parameter. The Landau-Khalatnikov term is the only one which
contributes to the ultrasound attenuation in the mean-field theory. However in
the scaling region (close to the critical temperature), as was noted by
Dengler and Schwabl Deng87  both terms combine just to one single scaling
function as there will be cancelations between the various contributions HH77
. When a magnetic field is applied the maximum in the sound attenuation
coefficient is shifted towards higher temperatures and can be located even at
temperatures much higher than the Curie temperature Kom74 ; Tach74 ; Paw20 .
Another characteristic feature of the ultrasonic attenuation in magnetic field
is that the increase in the sound attenuation in high temperature phase is
much faster than in the low temperature phase as we approach the critical
point. The steeper rise in the high temperature region may be explained by the
increase of the magnetization induced by the field. However, finding a single
expression to describe the ultrasonic attenuation in a broad range of
temperatures and magnetic fields is a very difficult problem that has not been
fully resolved so far. As was shown recently Paw20  away from the critical
point a sixth order term in Landau-Ginzburg energy may be also of importance.

It is the aim of this paper to describe the magnetic field and temperature
dependence of ultrasonic attenuation and sound velocity in ferromagnets in
terms of a parametric crossover model Paw22  which incorporates both the
asymptotic critical behavior near the Curie temperature as well as the
Ginzburg-Landau behavior (with sixth-order term included in the free energy)
as we cross over to large magnetic fields or for the temperatures not very
close to the Curie temperature. We illustrate our findings with ultrasonic
data in manganese phosphide MnP. A good agreement is achieved for a wide range
of temperatures and magnetic fields. Our results successfully explain the
rapid increase in the sound attenuation in the high temperature phase and the
shift of the maximum of attenuation towards the higher temperatures as the
magnetic field increases. The magnetic field as well as frequency dependence
of the hight of this maximum can be understood within this model.

The paper is organized as follows. In Sec. II we define the model and recall
the equivalent functional form of the equations of motion. The results of the
dynamic renormalization group for the acoustic self-energy are presented in
Sec. III. We discuss the Landau-Ginzburg approximation in Sec. IV. Sec. V
presents the details of our parametric crossover model where we have included
the sixth-order term in Landau-Ginzburg expansion of the free energy density.
We also discuss the sound attenuation coefficient and sound dispersion scaling
functions in this representation. In Sec VI we compare the theoretical results
found in this work with the experimental data obtained by Komatsubara at al.
Kom74  and Ishizaki et al. Ishi77  for longitudinal sound wave propagating in
MnP. Finally, the conclusions are summarized in Sec. VII.

##  II  MODEL

###  II.1 Statics

We consider Ising-like ($n=1$) continuous order parameter on a $d$-dimensional
elastic solid Paw98 :

$\mathcal{H}=\mathcal{H}_{\mathrm{OP}}+\mathcal{H}_{\mathrm{el}}+\mathcal{H}_{\mathrm{int}},$
(1)

where $\mathcal{H}_{\mathrm{OP}}(S)$ is the Ginzburg-Landau part of the free
energy for the one-component order parameter $S(\mathbf{x})$:

$\mathcal{H}_{\mathrm{OP}}=\int
d^{d}\\!x\left[\frac{1}{2}r_{0}S(\mathbf{x})^{2}+\frac{1}{2}(\mathbf{\nabla}S)^{2}+\frac{u}{4}S(\mathbf{x})^{6}+\frac{v}{6}S(\mathbf{x})^{4}\right]$
(2)

and

$\mathcal{H}_{\mathrm{el}}=\frac{1}{2}\int
d^{d}\\!x\left\\{B\varepsilon_{ii}(\mathbf{x})^{2}+2\mu\left[\varepsilon_{ij}(\mathbf{x})-\frac{1}{d}\delta_{ij}\varepsilon_{ll}(\mathbf{x})\right]^{2}\right\\}$
(3)

is the elastic contribution in the harmonic approximation, with
$\varepsilon_{ij}(\mathbf{x})$ denoting the strain tensor related to the
displacement vector components $u_{i}(\mathbf{x})$ by

$\varepsilon_{ij}(\mathbf{x})=\frac{1}{2}(\nabla_{i}u_{j}+\nabla_{j}u_{i}).$

The first term in Eq.(3) describes the contribution due to the volume changes
($\varepsilon_{ii}(\mathbf{x})$ is proportional to the density fluctuation)
and the second term is due to the shear distortions bergman ; luben . $B$ and
$\mu$ are the (bare) bulk and shear modulus (the unitary mass density and
$k_{B}T_{C}=1$ have been assumed), respectively. For simplicity, we have
assumed the solid to be isotropic. Finally, the interaction Hamiltonian is
given by

$\mathcal{H}_{\mathrm{int}}=f_{0}\int
d^{d}\\!x\varepsilon_{ii}(\mathbf{x})S(\mathbf{x})^{2},$ (4)

which describes the volume magnetostriction with the coupling constant
$f_{0}$. In the magnetic part $r_{0}=at$ is linear in the reduced temperature
$t=\frac{T-T_{C}}{T_{C}}$ and the parameters $u$ are $v$ are coupling
constants. It is well known that $v$ is irrelevant for the critical behavior.

A given elastic configuration $\varepsilon_{ij}(\mathbf{x})$ can be separated
larkin ; Paw89  into a homogenous deformation and the constant-volume phonon
part:

$\varepsilon_{ij}(\mathbf{x})=\varepsilon_{ij}^{0}+\frac{1}{2V}\sum_{\mathbf{k}\neq
0}[k_{i}u_{j}(\mathbf{k})+k_{{}_{j}}u_{i}(\mathbf{k})]\exp(i\mathbf{k\cdot
x}),$ (5)

where $V$ is the volume of the system at equilibrium. We decompose the
displacement vector $\mathbf{u}(\mathbf{k})$ into longitudinal and transverse
parts defined via

$\mathbf{u(k)}=\widehat{\mathbf{k}}Q_{\mathrm{L}}(\mathbf{k})+\mathbf{Q}_{\mathrm{T}}(\mathbf{k}),$
(6)

where $Q_{\mathrm{L}}(\mathbf{k})=\widehat{\mathbf{k}}\cdot\mathbf{u(k)},\
\mathbf{Q}_{\mathrm{T}}(\mathbf{k})=\mathbf{u(k)-}\widehat{\mathbf{k}}Q_{\mathrm{L}}(\mathbf{k})$
and $\widehat{\mathbf{k}}=\mathbf{k/}\left|\mathbf{k}\right|.$ Only the
longitudinal mode $Q_{\mathrm{L}}$ is coupled, in this model, to the order
parameter fluctuations so we integrate over the transverse modes as well as
over homogenous deformations $\varepsilon_{ij}^{0}$ in the partition function.
This is equivalent to considering the system under a fixed external pressure
nattermann . As a result we get a new Hamiltonian whose elastic part takes a
simple form

$\mathcal{H}_{\mathrm{el}}=\int\frac{d^{d}\\!k}{(2\pi)^{d}}k^{2}c_{0}^{2}\left|Q_{\mathrm{L}}(\mathbf{k})\right|^{2}$

with $c_{0}^{2}=B+2(d-1)\mu/d$ as the velocity square of the longitudinal
sound mode of the non-interacting system. The interaction part is transformed
into:

$\mathcal{H}_{\text{int}}=f_{0}\int\frac{d^{d}\\!k}{(2\pi)^{d}}kQ_{\mathrm{L}}(\mathbf{k})S_{-\mathbf{k}}^{2},$
(7)

where
$S_{\mathbf{k}}^{2}=\int\frac{d^{d}\\!p}{2\pi}S(\mathbf{p})S(\mathbf{k}-\mathbf{p})$.

###  II.2 Dynamics

In an isotropic solid, which is considered here for simplicity, the transverse
sound decouple from the order parameter and will be neglected. We consider
only the longitudinal sound and order-parameter modes which are mutually
coupled to each other. On the grounds of the hydrodynamics and renormalization
group arguments, the following system of Langevin equations Paw98 , is
considered:

$\dot{S}_{\mathbf{k}}=-\Gamma\frac{\delta\mathcal{H}}{\delta
S_{\mathbf{-k}}}+\xi_{\mathbf{k}},$ (8)
$\ddot{Q}_{\mathbf{k}}=-\frac{\delta\mathcal{H}}{\delta
Q_{\mathbf{-k}}}-\Theta k^{2}\dot{Q}_{\mathbf{k}}+\eta_{\mathbf{k}},$ (9)

where the index ’longitudinal’ for the elastic modes has been omitted. The
Fourier components of the Gaussian white noises $\xi$ nd $\eta$ have variances
related to the bare damping terms $\Gamma$and $\Theta k^{2}$ through the usual
Einstein relations. Here, $\Theta k^{2}$ is responsible for the noncritical
sound dumping and $\Gamma$ is a relaxation coefficient of the order parameter.

It is convenient to represent the model in terms of the equivalent functional
form Janss79 ; Domin77  with an Onsager-Machlup functional
$\mathcal{J}(S,Q;\widetilde{S},\widetilde{Q}),$ where auxiliary ”response”
fields $\widetilde{S}\ $and $\ \widetilde{Q}$ are introduced:

$\displaystyle\mathcal{J}$ $\displaystyle=$
$\displaystyle\int_{\omega}\sum_{\mathbf{k}}\left\\{\Gamma\tilde{S}_{\mathbf{k},\omega}\tilde{S}_{-\mathbf{k},-\omega}+\Theta
k^{2}\widetilde{Q}_{\mathbf{k},\omega}\widetilde{Q}_{-\mathbf{k},-\omega}-\widetilde{Q}_{\mathbf{k},\omega}\left[(-\omega^{2}+i\Theta
k^{2}\omega)Q_{-\mathbf{k},-\omega}+\frac{\partial\mathcal{H}}{\partial
Q_{\mathbf{k},\omega}}\right]\right.$ (10)
$\displaystyle\left.-\tilde{S}_{\mathbf{k},\omega}\left(i\omega
S_{-\mathbf{k},-\omega}+\Gamma\frac{\partial\mathcal{H}}{\partial
S_{\mathbf{k},\omega}}\right)\right\\}.$

The magnetic linear response function, for example, is defined as the
derivative $\langle S(t)\rangle$ over the external field $h(t^{\prime})$:

$\frac{\delta{\large{\langle}}S(t){\large{\rangle}}}{\delta
h(t^{\prime})}=\Gamma{\large{\langle}}S(t)\tilde{S}(t^{\prime}){\large{\rangle,}}$
(11)

where angular brackets denote nonequilibrium average Janss79 ; Domin77

${\large{\langle}}O(S,\tilde{S},Q,\widetilde{Q}{\large{)\rangle}}=\frac{1}{Z}\int{\large{\mathcal{D[}}}i\tilde{S}{\large{\mathcal{]}}}{\large{\mathcal{D[}}}S{\large{\mathcal{]D[}}}i\widetilde{Q}{\large{\mathcal{]}}}{\large{\mathcal{D[}}}Q{\large{\mathcal{]}}}\
O[S,\tilde{S},Q,\widetilde{Q}]\
\exp\mathcal{J}\\{\tilde{S},S,Q,\widetilde{Q}\\}.$ (12)

In this formalism we have two kinds of propagators: the free response
propagators and the free two-point correlation functions IroSchwabl83 ; Deng87
; Paw98 .

Next, using dynamic Gaussian transformations we decouple the sound mode from
the order parameter getting an expression for the acoustic self-energy
IroSchwabl83 ; Deng87 ; Paw98

$\Sigma_{ac}(\emph{k},\omega)=2f_{0}^{2}\emph{k}^{2}\Pi(k,\omega),$ (13)

where the four-spin response function
$\Pi(k,\omega)\equiv\langle\Gamma\widetilde{S^{2}}_{\mathbf{-k},-\omega}S_{\mathbf{k},\omega}^{2}\rangle^{\mathcal{J}_{A}}$
is calculated with the effective, phonon-free dynamic functional
$\mathcal{J}_{A}$ of the model A in the classification of Halperin and
Hohenberg HH77 ; Paw98 . Here
$\widetilde{S^{2}}_{\mathbf{k}}(t)=\frac{1}{\sqrt{V}}\sum_{\mathbf{k}_{1}}\tilde{S}_{\mathbf{k}_{1}}(t)S_{\mathbf{k}-\mathbf{k}_{1}}(t)$
plays the role of reaction field coupled to the square of spin. The sound
attenuation coefficient is given by the imaginary part of the acoustic self-
energy

$\alpha(\omega)=\frac{1}{2c_{0}\omega}\mathop{\mathrm{I}m}\Sigma_{ac}(\frac{\omega}{c_{0}},\omega)\simeq\frac{f_{0}^{2}\omega}{c_{0}^{3}}\mathop{\mathrm{I}m}\Pi(0,\omega),$
(14)

and the sound velocity by the real part

$\frac{c(\omega)-c_{0}}{c_{0}}\simeq-\frac{f_{0}^{2}}{c_{0}^{2}}\mathop{\mathrm{R}e}[\Pi(0,\omega)],$
(15)

where $c_{0}$ is the bare sound velocity. In Eqs. (14) and (15) the ultrasonic
wave vector $\mathbf{k}$ has been put to zero in $\Pi(k,\omega)$ as in the
ultrasonic experiments the wavelength is much greater than the correlation
length $\xi$ so we can take $k\xi=0$. In Eq.(14) the noncritical term
$\Theta\omega^{2}/2c_{0}^{3}$ has been omitted.

In this model we have considered the isotropic solid for simplicity. Although
most of the crystals are anisotropic but, as was shown by Schwabl and Iro
IroSchwabl83 , the acoustic self-energy is still given by Eq. (13) provided
the isotropic coupling constant $f_{0}$ and bare velocity $c_{0},$ are
replaced by $f_{0}(\hat{k},\lambda)$ and $c_{0}(\hat{k},\lambda),$
respectively, where $\hat{k}=\vec{k}/k$ is the direction of propagation and
$\lambda$ the polarization of the sound mode. In high symmetry directions of
propagation the coefficient $f_{0}(\hat{k},\lambda)$ is different from zero
only for the longitudinal modes Deng87 .

##  III Critical attenuation

Putting $S=M+\delta S,$ where $M=\langle S\rangle$ is the equilibrium value of
magnetization and $\delta S$ is the spin fluctuation, we can write the four-
spin response function as Deng87

$\Pi=\langle(\Gamma\widetilde{S}\delta S)(\delta S\delta
S)\rangle+2M\langle(\Gamma\widetilde{S}\delta S)\delta
S\rangle+M\langle\Gamma\widetilde{S}(\delta
S)^{2}\rangle+2M^{2}\langle\Gamma\widetilde{S}\delta S\rangle.$ (16)

The last term known also as a relaxational term is the analogue of the Landau-
Khalatnikov sound damping LK54  which is the only one which contributes to
sound attenuation in the mean-field theory. The first term in Eq.(16) is known
as a fluctuation contribution to the sound attenuation and the other two terms
are sometimes called the mixing contribution Foss85 . In the high-temperature
phase only the four-spin response function $\langle(\widetilde{S}\delta
S)(\delta S\delta S)\rangle$ has to be obtained. The critical exponent of this
’energy’ response function is equal to the specific-heat exponent $\alpha.$
The theory becomes more complicated as there is non-zero magnetization.
Considered separately, the three contributions are characterized by different
critical exponents and for example the Landau-Khalatnikov term diverges with
the critical exponent equal to $2\gamma-2\beta$ Foss85 , where $\gamma$ and
$\beta$ are the susceptibility and order parameter exponents, respectively.
However, as was noted by Halperin and Hohenberg HH77  in the scaling region
there should be cancelations between different contributions and the critical
sound attenuation exponent should be the same as in the disordered phase. It
was pointed later explicitly by Dengler and Schwabl Deng87  that this was the
case. To evaluate $\Pi$ in the critical region several RG methods are at
disposal. It is well known that calculations of dynamical functions are rather
complicated because we need to exponentiate singularities at both large and
small arguments Nelson76 . Many alternative approaches can give different
results depending on the exponentiation procedure and the region of the
parameters $\omega,~{}\ t$ and $h$. We use here the direct perturbational
method based on the $\epsilon=4-d$ expansion Wilson72 ; Comb75  with sharp
cutoff $\Lambda=1$ and coupling constants chosen equal to their fixed-point
values $u=u^{\ast}=$O$(\epsilon)~{},v=0$. The one-loop diagrams contributing
to $\Pi$ are shown in Fig. 1a.

Figure 1: a) One-loop diagrams for the acoustic self-energy. Full lines
represent the spin response function $\langle\widetilde{S}\delta S\rangle$ and
the dashed lines represent the correlation function $\langle\delta S\delta
S\rangle$. The arrows represent the static magnetization $M$ and the black
points the coupling constant $u$. b) The two-loop approximation for
$\Sigma_{ac}$ in the disordered phase. c) One-loop diagram for the self-energy
of the spin response function.

Exponentiation of logarithms Wilson72 ; Comb75  gives

$\Pi=\chi^{\alpha/\gamma}f(y,s),$ (17)

where $\chi(t,h)$ is the static susceptibility and $\ f$ is a scaling function
depending on the reduced frequency $y=\omega/(\Gamma b\chi^{-z\nu/\gamma})$,
and on a scaling variable $s=2tm^{-1/\beta}$ , where $m^{2}=2u^{\ast}M^{2}$
and $z=2+c\eta$ is a dynamic exponent for the universality class A (with a
constant $c=$O$(1))$ HH77 . Here $\beta,\gamma,\nu$ and
$\eta=\mathrm{O}(\epsilon^{2})$ are usual static critical exponents. For
calculating the attenuation scaling function we used a popular approximation:
$f=A_{\text{rel}}f_{\text{rel}}+$ $A_{\text{fluc}}f_{\text{fluc}}$ where
$A_{\text{rel}}$ and $A_{\text{fluc}}$ are critical amplitudes and the
relaxational and fluctuation scaling function are found to first order in
$\epsilon$:

$f_{\text{rel}}(y,s)=P(s)^{2\beta/\nu}\frac{1-6u^{\ast}\unit{K}_{d}\pi_{1}(y)}{1-iy-9u^{\ast}\unit{K}_{d}\pi_{2}(y,s)},$
(18) $f_{\text{fluc}}(y)=\frac{iz\nu}{2\alpha
y}\left[\left(1-\frac{iy}{2}\right)^{1-\frac{\alpha}{z\nu}}-1\right]+O(\epsilon^{2}),$
(19)

where $\pi_{1}(y)=-\frac{i}{y}(1-\frac{iy}{2})\ln(1-\frac{iy}{2})$ and
$\pi_{2}(y,s)=-(\frac{2}{s+3})[-\frac{1}{2}+\frac{iy}{8}+\frac{i}{y}(1-\frac{iy}{2})\ln(1-\frac{iy}{2})$
with $\unit{K}_{d}=2^{-d+1}\pi^{-d/2}/\Gamma(d/2)\ $. The function
$f_{\text{rel}}$ corresponds to the Landau-Khalatnikov relaxational
contribution. The factor $P(s)^{2\beta/\nu}$ in Eq. (18) is a consequence of
$M^{2}$ factor in the last term in Eq. (16). The function $\pi_{2}(y,s)$
describes contribution from the second diagram in Fig. 1a which is responsible
for renormalization of the bare magnetic response function by the only
O($\epsilon$) diagram contributing to the magnetic self-energy shown in Fig.
1c. The forth diagram in Fig. 1a give an O($\epsilon$) correction to the
Landau Khalatnikov theory which is proportional to the function $\pi_{1}$ in
Eq.(18). In the order O($\epsilon$) considered here, there is no need to
exponentiate logarithms like $\ln(1-\frac{iy}{2})$ in $f_{\text{rel}}(y)$ due
to the fact that with increasing $y$ the leading Landau-Khalatnikov term
vanishes more rapidly than the first-order correction Deng87 . In order to
obtain the function $f_{\text{fluc}}(y)$ also a two-loop diagram shown in Fig
1b needs to be considered. The most important in the exponentiation procedure
is that in order to obtain the correct asymptotic behavior for
$y\rightarrow\infty$ the exponent $\epsilon/12$ has been replaced by
$\alpha/2z\nu$ Deng87 .

The normalization factor $b(s)=1+\frac{\epsilon}{8}P(s)^{2\beta/\nu}$, is
introduced here for the normalization reason and follows from the low $\omega$
expansion of the dynamic order-parameter susceptibility Calab ; PawErd11 ;
PawErd13 . The function
$P(s)=m^{2}\chi^{2\beta/\gamma}\simeq(\frac{2}{s+3})^{\nu}[\frac{s}{s+3}+\frac{3}{s+3}(\frac{2}{s+3})^{\frac{\nu}{2\beta}-\nu}]$
may be thought of as a crossover profile which is equal to $0$ for
$s\rightarrow\infty$ (in the paramagnetic phase) and $1$ in the coexistence
region ($s=-1$) and changes to $\left(\frac{2}{3}\right)^{\nu/2\beta}$ for the
critical regime, $s\rightarrow 0$. Note that this O($\epsilon$) correction
vanishes for the disordered phase $s\rightarrow\infty.$

##  IV Landau-Ginzburg theory

If the fluctuations of the order-parameter are neglected only the relaxational
term remains:

$\Sigma_{ac}(k,\omega)=4f_{0}^{2}k^{2}M^{2}\chi(\omega,t)=4f_{0}^{2}k^{2}M^{2}\frac{\chi_{GL}}{1-iy_{GL}},$
(20)

where $y_{GL}=\omega\tau_{GL}$ is the reduced frequency with
$\tau_{GL}=\chi_{GL}/\Gamma$ as a characteristic relaxation time and$\
\chi_{GL}$ as the static susceptibility in the Landau-Ginzburg theory. In
order to calculate the equation of state as well as the susceptibility,
$\chi_{GL}=(\partial^{2}\Phi_{\mathrm{GL}}/\partial M^{2})^{-1},$ the Landau-
Ginzburg functional is postulated in the form of regular expansion

$\Phi_{\mathrm{GL}}(M)=\frac{1}{2}r_{0}M^{2}+\frac{u}{4}M^{4}+\frac{v}{6}M^{6}+~{}...~{}-Mh$
(21)

In many phenomenological approaches the sixth and higher-order terms in Eq.
(21) are omitted, however as was shown by Kuz’min Kuz08  the sixth-order term
is essential for finding an appropriate equation of state for ferromagnets,
which is valid for an arbitrary magnetic field $h$ and reduced temperature
$t$. At thermal equilibrium $\Phi_{\mathrm{GL}}$ must be minimum

$\frac{\partial\Phi_{\mathrm{GL}}(M,h,t)}{\partial M}=0,$

so the equation of state is written as

$h=M(at+uM^{2}+vM^{4}),$ (22)

and the susceptibility as $\chi_{GL}=(at+3uM^{2}+5vM^{4})^{-1}$. A log-log
plot of Eq. (22) for $h=0$ shows that the effective exponent $\beta~{}$crosses
over from typical Landau theory value $1/2$ to $1/4~{}$as the temperature
decreases from $T_{C}$ Kuz08 . The proximity to the tricritical point ($u=0$)
explains $\beta\approx 1/3$ observed in many ferromagnets Kuz08 . Recently we
have shown Paw20  that the sixth-order term is also crucial for understanding
of the ultrasonic attenuation in manganese phosphide MnP. It was demonstrated
that the characteristic quotient $Q=v/u$ Kuz08 , which determines how far
below the Curie temperature the Landau asymptotic behavior for magnetization
$M\sim|t|^{1/2}$ still holds ($M\sim|t|^{1/4}$ relation is observed for the
tricritical regime), has also dramatic effect on dynamical properties Paw20
and large values of $Q$ favor the shift of the maximum of the ultrasonic
attenuation to higher than $T_{C}$ temperatures as is seen also in the
ultrasonic data in MnP Kom74 ; Tach74 . The imaginary part of $\
\Sigma_{ac}(k,\omega)$ is the source of ultrasonic attenuation and from Eq.
(20) we obtain

$\alpha(\omega,t,h)=\frac{2f_{0}^{2}\omega^{2}}{c_{0}^{3}\Gamma}\frac{M^{2}(t,h)\chi_{GL}(t,h)^{2}}{1+y_{GL}^{2}(\omega,t,h)}.$
(23)

The product $M^{2}\chi_{GL}^{2}$ determines the sound attenuation exponent:
$\alpha\sim|t|^{-x_{\pm}}$ in the hydrodynamic regime, where
$\omega\tau_{GL}\ll 1$. The index $+$ refers to $T>T_{C}$ end $-$ to
$T<T_{C}$. This exponent determines the slope of the attenuation peak in the
log-log plot. It is clear by a simple inspection of Eq. (23) that $x_{-}=1$
for the ordinary Landau theory behavior ( $Q=0$) and $x_{-}=3/2$ for the
tricritical behavior ( $Q=$ $\infty$). So, in the Landau-Ginzburg theory the
effective sound attenuation exponent $x_{-}^{eff}$ Riedel74 ; Paw09  in the
low-temperature phase may take an intermediate value between $1$ and $3/2$
depending on the value of $Q$ and the range of the reduced temperature. It is
obvious that $x_{+}=0$ for $h=0$ in this theory as $M=0~{}$ for $T>T_{C}$. For
non-zero magnetic field however $M\neq 0$ even for $T>T_{C}$ and much larger
attenuation exponent $x_{+}=4$ is obtained. It is consequence of the relation
$M=\chi_{GL}h$ for small magnetization, where the Landau-Ginzburg
susceptibility obeys $|t|^{-1}$ power-law in both: Landau theory and the
tricritical limits. So for finite magnetic field the sound attenuation
coefficient curve has much steeper slope for $T>T_{C}$ than in the low-
temperature phase as was shown in Fig. 2. From Fig. 2 one can also observe
that the displacement of the sound attenuation peak towards the higher
temperatures is monotonic with respect to the value of the characteristic
quotient $Q.$

Figure 2: The typical ultrasonic attenuation coefficient temperature
dependence in the Landau-Ginzburg theory for three values of $Q$. Non-zero
magnetic field $h=1$ Oe is assumed. The continuous lines correspond to
$T>T_{C}$ and the dashed lines to $T<T_{C}$. The hydrodynamic sound
attenuation exponent $x$ is also shown.

A crossover from mean-field to tricritical behavior explains the ultrasonic
data in MnP for moderate and high magnetic fields Paw20  but does not explains
the data in the region near the critical point called also the fluctuation
region. For example, $\alpha(T>T_{C},h=0)$ is equal zero in Landau-Ginzburg
theory contrarily to what is observed in experiment. The fluctuations has to
be taken into account in MnP at least in the region of low magnetic fields and
temperatures very close to the critical point.

##  V Crossover parametric model

As regards the explicit representation of the critical behavior it is
frequently more convenient to express everything in the parametric
representation of an asymptotic equation of state. The reason for that is that
many scaling functions are given by various approximations of renormalization
group theory and therefore are sometimes very cumbersome and inconvenient for
practical use Aga01 . Usually the starting point is a simple linear model
originally introduced by Schofield SchLitHo69 ; Sch69 ; Jos69 . It is defined
by:

$h=h_{0}r^{\beta\delta}\theta(1-\theta^{2})\equiv
h_{0}r^{\beta\delta}l(\theta),$ (24) $t=r(1-b^{2}\theta^{2})\equiv
rk(\theta),$ (25) $M=M_{0}r^{\beta}\theta$ (26)

where $h_{0}$, $M_{0}$ and $b^{2}$ are constants and $r$ and $\theta$ are
abstract parametric variables chosen such that $\theta=\pm 1$ correspond to
the two branches of the coexistence curve, $\theta=1/b$ to the critical
isotherm and $\theta=0$ to the critical isochore. The parameter $r$ in non-
negative and measures a ”radial” distance from the critical point and $\
\theta$ (pseudoangle) specifies the location on the contour of constant $r$.
In this representation all critical singularities are incorporated as power
laws in the variable $\ r$, while the dependence on $\ \theta$ is kept
analytic. The great advantage of the linear parametric representation is that
it generates a closed form expressions for all thermodynamic functions. The
resulting parametric expressions for a number of thermodynamic properties are
given in Appendix A. For example the susceptibility is given by

$\chi=\frac{M_{0}}{h_{0}}r^{-\gamma}\frac{k-\beta
k^{\prime}\theta}{kl^{\prime}-\beta\delta k^{\prime}l}.$ (27)

The quantities with a prime indicate the derivative with respect to $\theta$.
The constants $h_{0}$ and $M_{0}$ are determined by fitting the experimental
data. According to the universality of the critical phenomena the constant $b$
is assumed to be universal and may be fixed by matching the ratios of critical
amplitudes obtained within the parametric representation to the known ratios
obtained, for example, from the renormalization group theory or Monte-Carlo
simulations. A satisfactory fit to the amplitude ratios is provided by taking
$b^{2}=1.30$ Borjan  and this value is adopted within present study. The
linear model appears to be consistent with the asymptotic renormalization
group theory to second order in $\epsilon$ WallZia74 .

Various improvements of the linear model has been proposed Aga01 ; FishZinn .
Many of them concentrate on the critical region with inclusion of the Wegner
correction–to-scaling contributions Weg72  and the others on the crossover
from the asymptotic critical region to the regular classical (mean-field)
behavior far away from the critical point Luett92 . In the approach presented
by Agayan et al. Aga01  a crossover parametric transformation similar to the
one deduced for the renormalization-group theory Nic81 ; Nic85 ; Nic86 ; Chen1
; Chen2  was proposed. According to non-asymptotic renormalization group
procedure and implementing a matching point proposed by Nicol and co-workers
Nic81 ; Nic85 ; Nic86  Chen et al. Chen1 ; Chen2  put forward the following
crossover expression for the singular part of the actual Helmholtz free energy
in the Ising system:

$\Delta
A_{s}=\frac{1}{2}tM^{2}\mathcal{TD+}\frac{\overline{u}u^{\ast}\Lambda}{4!}M^{4}\mathcal{D}^{2}\mathcal{U}-\frac{1}{2}t^{2}\mathcal{K},$
(28)

where $\overline{u}=u/u^{\ast}$ is a rescaled coupling constant and
$\mathcal{T}$,$\mathcal{D}$,$\mathcal{U}$ and $\mathcal{K}$ are rescaling
functions defined by

$\mathcal{T}=Y^{(2\nu-1)/\Delta_{s}},~{}\mathcal{D}=Y^{-\eta\nu/\Delta_{s}},\mathcal{U}=Y^{\nu/\Delta_{s}},$
(29)

and

$\mathcal{K}=\frac{\nu}{\alpha\overline{u}\Lambda}(Y^{-\alpha/\Delta_{s}}-1).$
(30)

Note that in Eq. (28) the substitution $at\rightarrow t,$ $v=0$ and
$\frac{u}{4}\rightarrow\frac{\overline{u}u^{\ast}\Lambda}{4!}$ was performed
compared to our starting Landau-Ginzburg Hamiltonian
$\mathcal{H}_{\mathrm{OP}}$. The parameter $\Lambda$ introduced here for later
convenience is to be interpreted as a dimensionless wave number related to the
actual cutoff wave number Nic81 ; Aga01 . The crossover function Chen1 ; Chen2
$Y\equiv\frac{u(l)-u^{\ast}}{u-u^{\ast}},$where $l$ is a rescaling parameter,
calculated at the matching-point $l=l^{\ast}$

$\Lambda e^{-l^{\ast}}=\kappa,$ (31)

is given by the relation Chen1 ; Chen2 ; Aga01 :

$1-(1-\overline{u})Y=\overline{u}(1+\frac{\Lambda^{2}}{\kappa^{2}})^{1/2}Y^{\nu/\Delta_{s}},$
(32)

with $\kappa$ (a parameter proportional to the inverse of correlation length,
which is a measure of the distance to the critical point) defined by

$\kappa^{2}=t\mathcal{T}+\frac{1}{2}\overline{u}u^{\ast}\Lambda
M^{2}\mathcal{DU}.$ (33)

When $\kappa$ is small the crossover function $Y$ approaches zero like
$(\kappa/\overline{u}\Lambda)^{\Delta_{s}/\nu}$ and the asymptotic power-laws
are recovered from Eq. (28) including the leading Wegner corrections to
scaling. The classical limit corresponds to
$\overline{u}\Lambda/\kappa\rightarrow 0$ or $Y\rightarrow 1$ and the Landau-
Ginzburg expansion is easily recovered from Eq.(28) as the rescaling functions
$\mathcal{T},\mathcal{D},\mathcal{U}$ tend to unity while
$\mathcal{K}\rightarrow 0$. In Eq.(28) higher-order terms can be also
included, for example the contribution from the $\frac{v}{6}M^{6}\mathcal{\ }$
in Eq.(21) term transforms to
$\frac{v}{6}M^{6}\mathcal{D}^{3}\mathcal{U}^{3/2}$ in the crossover Helmholtz
free energy Chen2 ; Luett92 .

The asymptotic parametric models (including described here the simplest linear
model) are valid only in the vicinity of the critical point. In order to built
a crossover parametric model able to describe the crossover from critical to
mean-field behavior Agayan et al. Aga01  related the distance parameter $r$ in
parametric representation to the inverse of correlation length by equation

$\kappa^{2}(r)=arY^{(2\nu-1)/\Delta_{s}},$ (34)

which reflects the observation that the parameter $\kappa^{2}$ plays a similar
role to the distance variable $r$ ($\kappa\rightarrow 0$ near the critical
point and $\kappa\rightarrow\infty$ far away from the critical point). It is
easy to see from Eq. (34) that $\kappa\sim r^{\nu}$ near the critical point
and $\kappa\sim r^{1/2}$ far away from the critical point. The crossover
function is again given in the implicit form by Eq. (32) and can be rewritten
as

$1-(1-\overline{u})Y(r)=\overline{u}(1+\frac{\Lambda^{2}}{arY(r)^{(2\nu-1)/\Delta_{s}}})^{1/2}Y(r)^{\nu/\Delta_{s}}.$
(35)

The crossover function $Y$ is only a function of $r$ and is independent from
the angle variable $\theta.$ It also depends on two parameters $\overline{u}$
and $\Lambda^{2}/a$ called also crossover variables Aga01  which may be
related to the Ginzburg number $N_{G}$:

$N_{G}=n_{0}g=n_{0}\frac{(\overline{u}\Lambda)^{2}}{a},$ (36)

where $n_{0}$ is a constant. In the asymptotic critical limit
$Y\rightarrow(r/g)^{\Delta_{s}}$ where $g=(\overline{u}\Lambda)^{2}/a$ is
again a crossover parameter proportional to the Ginzburg number. Far away from
criticality $Y\rightarrow 1$.

To completely define the crossover parametric model one need to specify the
equations for $h,$ $t$ and $M$. For this purpose we modify the linear model
equations (24) as

$h=h_{0}g^{\beta\delta-3/2}r^{3/2}\theta[Y^{(2\beta\delta-3)/2\Delta_{s}}(1-\theta^{2})+vr\theta^{4}],$
(37) $M=g^{\beta-1/2}M_{0}r^{1/2}Y^{(\beta-1/2)/\Delta_{s}}\theta$ (38)

while the dependence of $t$ on $r$ and $\theta$ is left unchanged (Eq. (25)).
Note that near the critical point ($r\rightarrow 0$,
$Y\rightarrow(r/g)^{\Delta_{s}})$ the term $vr\theta^{4}$ in the square
brackets in Eq. (37) is very small and can be neglected so the linear model
equations are recovered. The virtue of this parametrization is that in the
other (classical) limit, $r\rightarrow\infty$, these equations reduce to

$h=\tilde{h}_{0}r^{3/2}\theta[(1-\theta^{2})+vr\theta^{4}],$ (39)
$M=\tilde{M}_{0}r^{1/2}\theta$ (40)

which in turn leads to the very simple Ginzburg-Landau expression for the
equation of state:

$h=M[at+u^{\prime}M^{2}+v^{\prime}M^{4}],$ (41)

with $\ \ a=\frac{\tilde{h}_{0}}{\tilde{M}_{0}},$
$\tilde{h}_{0}=g^{\beta\delta-3/2}h_{0}$, $\
\tilde{M}_{0}=g^{\beta-1/2}M_{0}$, $\tilde{v}=vg^{3/2-\beta\delta}$, $\
u^{\prime}=\frac{(b^{2}-1)\tilde{h}_{0}}{\tilde{M}_{0}^{3}}$ and $\
v^{\prime}=\frac{v\tilde{h}_{0}}{\tilde{M}_{0}^{5}}$. Similar classical
expressions are obtained for the susceptibility and other thermodynamic
quantities (see Appendix B).

Recalling the definitions (13-19) crossover parametric attenuation coefficient
and dispersion are given by

$\alpha(\omega,r,\theta)=A\omega
Y(r)^{-\alpha/\Delta_{s}}\mathop{\mathrm{I}m}\left[f_{\text{rel}}(y,\theta,r)+\frac{A_{\text{fluc}}}{A_{\text{rel}}}f_{\text{fluc}}(y)\right],$
(42)
$c^{2}(\omega)-c^{2}(0)=2Ac_{0}^{3}Y(r)^{-\alpha/\Delta_{s}}\mathop{\mathrm{R}e}\left[f_{\text{rel}}(y,\theta,r)-f_{\text{rel}}(0,\theta,r)+\frac{A_{\text{fluc}}}{A_{\text{rel}}}(f_{\text{fluc}}(y)-f_{\text{fluc}}(0))\right]$
(43)

with

$f_{\text{rel}}(y,\theta,r)=\tilde{P}(\theta,r)\frac{1-6u^{\ast}p(r)\unit{K}_{d}\pi_{1}(y)}{1-iy-9u^{\ast}p(r)\unit{K}_{d}\pi_{2}(y,\theta)},$
(44)

where the function $p=u(l)/u^{\ast}$ is given in parametric representation by

$p(r)=1-(1-\overline{u})Y(r),$ (45)

and
$\pi_{2}(y,\theta)=\theta^{2}X(\theta,r)[-\frac{1}{2}+\frac{iy}{8}+\frac{i}{y}(1-\frac{iy}{2})\ln(1-\frac{iy}{2})]$.
As we approach the critical point, $r\rightarrow 0,$ the parameter $p\ \ $goes
to unity and the renormalized coupling constant $u(l)$ approaches the fixed
point value $u^{\ast}.$ On the other hand for $r\rightarrow\infty$ it takes
the initial value $u$. In Eq. (44) the function $\widetilde{P}(\theta,r)$
corresponds to the factor $P(s)^{2\beta/\nu}$ in Eq. (18). It plays important
role as it includes all dependence on $\theta$ in the static case ($y=0$).
Remembering also that in the statics the four-spin response function $\Pi$ is
proportional to the specific heat we took in this paper

$\tilde{P}(\theta,r)=\tilde{C}(\theta,r)-\tilde{C}(0,r)$ (46)

where $\tilde{C}(\theta,r)$ is an auxiliary angular function of the specific
heat Aga01  which is given in the parametric crossover model by

$C_{h}=Y^{-\alpha/\Delta_{s}}\tilde{C}(\theta,r),$ (47)

with

$\tilde{C}(\theta,r)=\frac{[(1-\alpha
Y_{1})\tilde{s}+r\frac{\partial\tilde{s}}{\partial
r}|_{\theta}](l^{\prime}+5vrY^{\frac{2\beta\delta-3}{2\Delta_{s}}}\theta^{4})-[(\frac{3}{2}+(\beta\delta-\frac{3}{2})Y_{1})l+\frac{5}{2}vrY^{\frac{2\beta\delta-3}{2\Delta_{s}}}\theta^{5}]\tilde{s}^{\prime}}{k(l^{\prime}+5vrY^{\frac{2\beta\delta-3}{2\Delta_{s}}}\theta^{4})-k^{\prime}[(3/2+(\beta\delta-\frac{3}{2})Y_{1})l+\frac{5}{2}vrY^{\frac{2\beta\delta-3}{2\Delta_{s}}}\theta^{5}]},$
(48)

where $\tilde{s}$ is an angular function of the entropy (see Appendix B) and
$\frac{\partial\tilde{s}}{\partial r}|_{\theta}$ denotes the radial derivative
of $\tilde{s}$ whereas the primes denote the derivatives with respect to
$\theta$ (at constant $r$). In this equation $Y_{1}$ is a logarithmic
derivative of the crossover function

$Y_{1}(r)\equiv\frac{d\ln Y}{d\ln
r^{\Delta_{s}}}=\frac{1}{\Delta_{s}}\frac{r}{Y}\frac{dY}{dr},$ (49)

introduced by Agayan et al. Aga01 . This function approaches unity in the
asymptotic critical limit $r\rightarrow 0$ whereas $Y_{1}\rightarrow 0$ far
away from the critical point.

The reasons we have chosen the ”shape” function $\widetilde{P}(\theta,r)$ in a
form of specific heat are as follows. First, in the limit of low frequency the
response function $\Pi$ transforms into the specific heat. The four-spin
response function is sometimes called the frequency-dependent specific heat.
It is well known that it plays the essential role in calculations of sound
attenuation and velocity shift in magnets HH77 ; Kaw68 ; Paw98 ; Paw11 .
Secondly, Eq. (46) correctly describes also the mean-field limit as for
$Y\rightarrow 1,$ $Y_{1}\rightarrow 0$ we see that
$\tilde{C}(\theta,r)-\tilde{C}(0,r)\rightarrow\frac{\theta^{2}}{(1-b^{2}\theta^{2})+3(b^{2}-1)\theta^{2}+5vr\theta^{4}}.$
Multiplying both the numerator and denominator by $r$ the equation (23) is
recovered.

It should be noted that also the reduced frequency is changed to

$y=\omega\tau_{0}b(\theta,r)r^{-1}Y(r)^{\frac{1-z\nu}{\Delta_{s}}}X(\theta,r),$
(50)

where $\tau_{0}$ is a bare relaxation time setting the time scale and
$X(\theta,r)$ is an angular function of susceptibility (Eq. (84)). The
frequency normalization factor $b(\theta,r)$ PawErd13 ; Calab  replaces $b(s)$
from Sec. II:

$b(\theta,r)=1+\frac{\epsilon}{8}p(r)\theta^{2}X(\theta,r).$ (51)

##  VI  Application to manganese phosphide MnP

As an illustration, we show here how the crossover parametric model can be
applied to ultrasonic attenuation data obtained by Komatsubara et al. Kom74
for MnP. We have fitted the experimental data to Eq. (42) using the critical
temperature $T_{C}\ $and the attenuation ($A$), time ($\tau_{0}$), temperature
($a$) and magnetic field ($h_{0}$) scales as well as
$\frac{A_{\text{fluc}}}{A_{\text{rel}}}$ , $v$, $\overline{u}$ and $\Lambda$
as adjustable parameters. The values obtained for MnP are presented in Table
I.

Table 1: Parameters for MnP from the experimental ultrasonic data Kom74 . $\ \ T_{C}$ [K] |  $\ \ A~{}\ $[dBcm-1] |  $\ \ \ \ \tau_{0}$ [s] | $\ \ \ a$ |  $\ h_{0}$ [MOe] | $\ \frac{A_{\text{fluc}}}{A_{\text{rel}}}$ | $\ \ \ \ v$ | $\ \ \ \ \overline{u}$ | $\ \ \ \ \Lambda^{2}$
---|---|---|---|---|---|---|---|---
$\ ~{}290.76$ | $\ \ \ 0.0994\ \ $ | $9.08\cdot 10^{-12}$ | $~{}2.7\ \ $ | $\ \ \ \ 0.70\ $ | 0.183 | $\ 0.75\ \ $ | $\ 0.089\ \ $ | $\ 0.078\ \ $

The resulting value of the crossover parameter is

$g=6.25\cdot 10^{-4}.$ (52)

We have adopted the critical exponents for the tree-dimensional Ising-like
systems from the work of Pelisseto and Vicari Peliss :

$\alpha=0.110,\ \ \beta=0.326,\ \ \gamma=1.237,\ \ \delta=4.789,\ \
\Delta_{s}=0.53.$ (53)

Figure 3: Ultrasonic attenuation in MnP. The dashed lines correspond to
$T<T_{C}$ and the continuous lines to $T>T_{C}$. The data are taken from Ref.
Kom74 .

A comparison of Eq. (42) with experimental sound attenuation data in MnP is
shown in Fig. 3 using a log-log scale. Six different values of magnetic field
were shown.The ultrasonic data points are experimental results obtained by
Komatsubara at al. Kom74  in uniaxial ferromagnet MnP (for longitudinal sound
wave of 90 MHz propagating along the $b$ axis) for the magnetic field applied
along the easy $c$ axis. The dashed lines correspond to the $T<T_{C}$ and the
solid lines represent the $T>T_{C}$ phase. For large and moderate magnetic
fields one can observe a characteristic region where the sound attenuation in
the high-temperature range exceeds the corresponding attenuation value in the
low-temperature phase (for the same reduced temperature $|t|$). As was shown
earlier in Paw20  this characteristic region (called a region II in Paw20 ) is
a manifestation of a shift of maximum of attenuation towards the higher
temperatures. What can also be easily recognized from Fig. 3 is that the
curves show much steeper slope in the hydrodynamic regime ($\omega\tau\ll 1$)
for $T>T_{C}$ than in the low temperature phase. It follows from the fact that
for the high-temperature phase the relaxational term is proportional to
$h^{2}$ and is much larger than the fluctuation term. So the exponent for the
sound attenuation is equal to $\ \
x_{+}^{\mathrm{eff}}=z^{\mathrm{eff}}\nu^{\mathrm{eff}}+\alpha^{\mathrm{eff}}+2\Delta^{\mathrm{eff}}$
where $\Delta^{\mathrm{eff}}=\beta^{\mathrm{eff}}+\gamma^{\mathrm{eff}}.$ Here
the effective exponents Paw09 ; Riedel74  which depend on the temperature and
magnetic field has been used. For example the order-parameter exponent
$\beta\simeq 0.326$ in the critical region but $\beta^{\mathrm{eff}}=1/2$ in
MFA regime or $\beta^{\mathrm{eff}}=1/4$ in the tricritical range. As was
shown by Kuz’min Kuz08  in many ferromagnets the equation of state can be
described quite well by relatively simple Landau Ginzburg theory with the
functional (2). This means that the critical fluctuations are confined to much
narrower interval around the critical temperature than was thought before and
the single critical exponent $\beta$ should be rather replaced by the
effective exponent in order to take account of the crossover from the
asymptotic critical regime to the Landau theory region. It follows from that
analysis that the proximity to the tricritical point should be also taken into
account. It was confirmed recently Paw20  that away from the critical point
the sound attenuation in MnP can be well described by the Landau Ginzburg
theory with the sixth-order term in the free energy playing the crucial role.
The crossover model presented here Paw22  has an advantage on the previous
approach that it satisfactorily describes the ultrasonic attenuation
coefficient also in critical regime i.e. for small magnetic field and the
reduced temperature. It should be noted that the Landau Ginzburg theory gives
$\ \alpha(h=0,T>T_{c})=0$ and $\alpha(h=0,T<T_{c})\overset{T\rightarrow
T_{c}}{\longrightarrow}0$ , which is contrary to the experiment. From Fig. 3
one can observe that the displacement of the sound attenuation peak is
monotonic with respect to the intensity of the magnetic field. The maximum
moves towards higher temperatures.

Figure 4: Sound velocity change in MnP versus the reduced temperature $t$. The
continuous lines correspond to the parametric model and the dashed lines
contain also phenomenological term proportional to the square of magnetization
in order to improve the fit. a) 10 MHz longitudinal wave along the $a$ axis
Ferry . b) 30 MHz longitudinal wave propagating along the $b$ axis Ishi77 . c)
10 MHz longitudinal wave propagating along the $c$ axis Ferry .

It would be interesting to see if the parameters obtained from fitting our
model to the attenuation data lead also to sensible predictions for the sound
wave velocity. In Fig. 4 we show the relative sound velocity changes

$\frac{\Delta
c}{c_{0}}\equiv\frac{c(t,\omega)-c_{0}}{c_{0}}=Ac_{0}Y(r)^{-\alpha/\Delta_{s}}\mathop{\mathrm{R}e}\left[f_{\text{rel}}(y,\theta,r)+\frac{A_{\text{fluc}}}{A_{\text{rel}}}f_{\text{fluc}}(y)\right]$
(54)

calculated for our model and compared with experimental data obtained by
Ishizaki et al. Ishi77  and by Ferry and Golding Ferry . It is known that in
MnP the easy, intermediate and hard axes of magnetization in the ferromagnetic
phase are the $c$, $\ b$ and $a$ axes, respectively. For the 30 MHz
longitudinal sound wave propagating along $b$-axis (Fig. 4b) we have used the
same coupling constant $A_{(b)}$ as for the sound attenuation coefficient
predictions presented in Fig. 3, but of course it is obvious that the
magnetostrictive coupling is weakest in MnP along the $a$ axis Ferry  and
strongest along the $c$ axis so different coupling constants have to be used
for all three directions of propagation. In Fig. 4 we compare our predictions
with the measurements of velocity changes $\ \Delta c/c_{0}$ for 10 MHz
longitudinal waves propagating along the $a$ and $c$ axes (Fig. 4a and 4c) for
zero field Ferry . It is generally observed that the depth of the velocity
dip, the location of the extremum and the shape of the curve in the high-
temperature phase is correct and only the width of the theoretical curve in
the ordered phase is too wide for all theoretical curves. To improve the
fitting we added to the critical velocity a phenomenological term proportional
to the square of magnetization (the dashed curves). Such terms may appear in
ferromagnets as a result of higher order magnetoelastic couplings like
$\varepsilon^{2}S^{2}$ ($\varepsilon$ is the elastic strain) in the Landau-
Ginzburg interaction functional lut04  , or from other mechanisms Pov98 ;
Rou81 .

Figure 5: The magnetic field dependence of the sound velocity change in MnP
for several temperatures. The points show the measurements for the 30 MHz
longitudinal sound wave propagating along the $b$ axis Ishi77  with magnetic
field applied along $c$ axis.

In Fig. 5 we show the magnetic field behavior of the sound velocity change

$\frac{\Delta c(h)}{c_{0}}\equiv\frac{c(t,h)-c(t,0)}{c_{0}},$ (55)

as a function of the magnetic field. For the longitudinal wave of 30 MHz
propagating along the $b$ axis we took the same fitting parameters as in Fig.
3. The velocity change shows a quadratic increase in low fields and decreases
in high fields after showing a broad peak. Similar behavior was reported for
$\ \Delta\alpha(h)$ in our earlier work Paw20 .

We present the height of the ultrasonic attenuation peak observed in the field
dependence of the sound attenuation in Fig. 6, where the Landau-Ginzburg
estimation of this height is also shown. From Eq. (23) it follows that at
large reduced temperatures,

$\alpha(\omega,t,h)\simeq\frac{2f_{0}^{2}\omega^{2}}{c_{0}^{3}\Gamma}M^{2}(t,h)\chi_{GL}(t,h)^{2},$
(56)

Figure 6: The magnetic field dependence of the sound attenuation peak height
$\Delta\alpha^{\text{max}}$. The continuous line is an estimation from Landau-
Ginzburg theory, Eq. (56), and the dashed lines are the asymptotic power-law
behavior $\Delta\alpha^{\text{max}}\propto h^{-2/3}$ correct for $Q=0$ or
$\Delta\alpha^{\text{max}}\propto h^{-6/5}$ correct for $Q=\infty$. The points
show the results in MnP crystal for 90 MHz longitudinal sound wave propagating
along the $b$ axis Ishi77 . The magnetic field is directed along the easy
$c$-axis.

because the reduced frequency $y$ is very small. Differentiating
$M^{2}\chi_{GL}^{2}$ with respect to $h$ we find the extremum condition:

$at^{\text{max}}=3uM_{\text{max}}^{2}+15vM_{\text{max}}^{5},$ (57)

and inserting it into the equation of state (22) we obtain

$4uM_{\text{max}}^{2}+16vM_{\text{max}}^{5}=h/M_{\text{max}}.$ (58)

The last equation gives us $M_{\text{max}}(h)$ and setting it into Eq. (56) we
obtain the function $\Delta\alpha^{\text{max}}(h).$ Simple, power-law
expressions are obtained for the characteristic quotient $Q$ equal to $\infty$
and $0$. Thus, for the tricritical regime ($u=0)$
$\Delta\alpha^{\text{max}}(h)\propto h^{-6/5}$ as well as
$\Delta\alpha^{\text{max}}(h)\propto h^{-2/3}$ for the Landau theory ($v=0$).
In Fig. 6 the Landau-Ginzburg estimation of $\Delta\alpha^{\text{max}}(h)$ is
compared with experimental data obtained by Ishizaki at al. Ishi77  for the
ultrasonic 90 MHz wave propagating along $b$-axis in MnP. The plot indicates
that neither of the asymptotic regimes can be associated with the experimental
behavior in MnP in this narrow range of observations.

Figure 7: The frequency dependence of the sound attenuation at
$T_{\mathrm{max}}$. The straight lines are power-law fits of the data (dashed
line with exponent $x=0.82$ and dot-dashed line with $x=0.91$) . Solid line
corresponds to Eq. (42) . The points show the measurements for the
longitudinal sound wave propagating along the $a$ axis Suz82  .

It would be of interest to see the dependence of the sound attenuation on
frequency. We consider the height of the ultrasonic attenuation peak observed
in the temperature dependence of the attenuation for vanishing magnetic field.
The critical attenuation shows a maximum at $T_{\text{max}}$, below the
critical temperature, which shifts to lower temperature with increasing
frequency. Such measurements were performed in MnP by Suzuki and Komatsubara
Suz82  for the longitudinal waves propagating along $a$ axis. Comparing the
height of the attenuation peaks for both propagation directions at 90 MHz
Kom74 ; Suz82  we estimated the ratio of the critical attenuation amplitudes
as $A_{(a)}/A_{(b)}\simeq 4/15$. This value was also applied in Fig. 4a and
using this value we calculated the heights of the sound attenuation maxima for
several frequencies and compared it with the experimental data from Suz82 . In
Fig. 7 it is seen that in the considered range of frequencies
$\alpha_{\max}(\omega)$ shows the power law behavior and our fit (the solid
line) is only slightly worse than the original experimental fit from the work
of Suzuki and Komatsubara Suz82  (dashed line), where the relation
$\alpha_{\max}(\omega)\propto\omega^{x}$ with $x=0.82$ was used. In our
theory, the low-frequency asymptotic behavior,
$\alpha_{\max}(\omega)\propto\omega^{1-\alpha/z\nu},$ is characterized by the
exponent $\ x=1-\alpha/z\nu\simeq 0.91.$

##  VII Summary

In this article we have performed calculations of the sound attenuation
coefficient and velocity within the parametric model containing sixth order
term. This model is valid not only the near the critical point but also
accounts for the crossover to simple Landau-Khalatnikov behavior or to
tricritical behavior. The constructed acoustic self-energy is valid in a broad
range of temperatures and magnetic fields. Crucial to the success of this new
approach is a feature that the MFA models usually lack the ability to handle
high value of the characteristic quotient $Q$ Kuz08 . Among other properties
strongly affected by large value of the characteristic quotient $Q$, one
should mention the field-induced displacement of the sound attenuation peak.
It has been demonstrated that such displacement is monotonic with respect to
the intensity of the magnetic field as well as with respect to the value of
$Q$. Expressions obtained for the sound attenuation coefficient and sound
velocity are in fair agreement with available experimental data Kom74 ; Ishi77
; Ferry ; Suz82  in manganese phosphide MnP.

##  Appendix A Linear Parametric Representation

In the asymptotic parametric representation the variables $r$ and $\theta$ are
defined in Eqs. (24-25) in the text and the scaled equation of state in terms
of these variables is Eq. (26), which, despite its simplicity, gives a good
experimental data approximation. Eq. (26) gives the attractively simple
equation and the name of the linear model obviously stems from it. For the
linear model the singular Helmholtz free energy can be obtained analytically
SchLitHo69 ; HB72

$A_{s}=h_{0}M_{0}r^{2-\alpha}f(\theta),$ (59)
$f(\theta)=f_{0}+f_{2}\theta^{2}+f_{4}\theta^{4},$ (60)
$f_{0}=(1/2b^{4})[\delta-3-b^{2}\alpha(\delta-1)][(\delta+1)(\alpha-1)\alpha]^{-1},$
(61)
$f_{2}=-(1/2b^{2})[\beta(\delta-3)-b^{2}\alpha(1-2\beta)][(\alpha-1)\alpha]^{-1},$
(62) $f_{4}=-\frac{1}{2}(1-2\beta)\alpha^{-1}.$ (63)

The specific heat is given by

$C_{h}-C_{B}=\left(\frac{h_{0}M_{0}}{2b^{4}(1-\alpha)\alpha}\right)r^{-\alpha}c_{h}(\theta),$
(64)

with

$c_{h}(\theta)=\frac{(1-\alpha)l^{\prime}s-\beta\delta
ls^{\prime}}{l^{\prime}k-\beta\delta lk^{\prime}},$ (65)

where $C_{B}$ represents an analytic fluctuation-induced background
contribution Aga01  which is a smooth function of $\ t$ and the functions
$l(\theta)$ and $k(\theta)$ are defined in Eqs (24) and (25). The singular
part of entropy satisfies:

$S_{s}=\left(\frac{h_{0}M_{0}}{2b^{4}(1-\alpha)\alpha}\right)r^{1-\alpha}s(\theta)$
(66)

with the angular function $s(\theta)=\bar{s}_{0}+\bar{s}_{2}\theta^{2}$ where

$\bar{s}_{0}=\beta(\delta-3)-\alpha\beta(\delta-1)b^{2},$ (67)
$\bar{s}_{2}=(\alpha-1)(\delta-3)\beta b^{2}.$ (68)

General expression for the susceptibility is

$\chi=\frac{M_{0}}{h_{0}}r^{-\gamma}X(\theta),$ (69)

with

$X(\theta)=\frac{k-\beta k^{\prime}\theta}{kl^{\prime}-\beta\delta
k^{\prime}l}.$ (70)

The critical exponents obey the usual scaling relations,
$2-\alpha=\beta(\delta+1)=\gamma+2\beta$ Stan72 . The coefficients $h_{0}$ and
$M_{0}$ are system-dependent constants related to the critical amplitudes.

##  Appendix B Crossover Parametric Representation

The definition of the crossover model is:

$h=h_{0}g^{\beta\delta-3/2}r^{3/2}\theta[Y^{(2\beta\delta-3)/2\Delta_{s}}(1-\theta^{2})+vr\theta^{4}],$
(71)

$t=r(1-b^{2}\theta^{2})\equiv rk(\theta),$ (72)

$M=g^{\beta-1/2}M_{0}r^{1/2}Y^{\beta-1/2}\theta.$ (73)

The radial crossover function $Y$ and its derivative are given by Aga01

$1-(1-\bar{u})Y(r)=\bar{u}(1+\frac{\Lambda^{2}}{arY(r)^{(2\nu-1)/\Delta_{s}}})^{1/2}Y(r)^{\nu/\Delta_{s}},$
(74)
$Y_{1}(r)=\frac{1}{\Delta_{s}}\frac{r}{Y}\frac{dY}{dr}=\frac{1}{\Delta_{s}}\frac{c_{1}\kappa^{2}}{1+c_{1}c_{2}},$
(75)

with

$\kappa^{2}=arY(r)^{(2\nu-1)/\Delta_{s}},$ (76)

$c_{1}=\frac{\Lambda^{2}}{2\kappa^{4}}\left(1+\frac{\Lambda^{2}}{\kappa^{2}}\right)^{-1}\left[\frac{\nu}{\Delta_{s}}+\frac{(1-\bar{u})Y(r)}{1-(1-\bar{u})Y(r)}\right]^{-1}$
(77)

$c_{2}=-(\frac{2\nu-1}{\Delta_{s}})\kappa^{2}.$ (78)

The thermodynamic functions are given by:

$A_{s}(r,\theta)=\tilde{h}_{0}\tilde{M}_{0}r^{2}[Y^{-\alpha/\Delta_{s}}f(\theta)+\frac{1}{6}vr\theta^{6}],$
(79)
$S=\tilde{h}_{0}\tilde{M}_{0}rY^{-\alpha/\Delta_{s}}\tilde{s}(\theta,r)-C_{B}rk(\theta),$
(80)

$C_{h}=\tilde{h}_{0}\tilde{M}_{0}Y^{-\alpha/\Delta_{s}}\widetilde{C}(\theta,r)-C_{B},$
(81)

$\tilde{s}(\theta,r)=-\frac{(2-\alpha
Y_{1})\tilde{f_{0}}+[1-(\alpha+2\beta-1)Y_{1}\tilde{f_{2}}\theta^{2}-(\alpha+4\beta-2)Y_{1}\tilde{f_{4}}\theta^{4}}{1-b^{2}[1-(1+(2\beta-1)Y_{1})\theta^{2}]},$
(82) $\widetilde{C}(\theta,r)=\frac{[(1-\alpha
Y_{1})\tilde{s}+r\frac{\partial\tilde{s}}{\partial
r}|_{\theta}](l^{\prime}+5vrY^{\frac{3-2\beta\delta}{2\Delta_{s}}}\theta^{4})-[(3/2+(\beta\delta-3/2)Y_{1})l+\frac{5}{2}vrY^{\frac{3-2\beta\delta}{2\Delta_{s}}}\theta^{5}]\tilde{s}^{\prime}}{k(l^{\prime}+5vrY^{\frac{3-2\beta\delta}{2\Delta_{s}}}\theta^{4})-k^{\prime}[(3/2+(\beta\delta-3/2)Y_{1})l+\frac{5}{2}vrY^{\frac{3-2\beta\delta}{2\Delta_{s}}}\theta^{5}]},$
(83)
$\chi=\frac{\tilde{M}_{0}}{\tilde{h}_{0}}r^{-1}Y^{(1-\gamma)/\Delta_{s}}X(\theta,r),$
(84)
$X(\theta,r)=\frac{k-[1/2+(\beta-1/2)Y_{1}]k^{\prime}\theta}{k(l^{\prime}+5vrY^{\frac{3-2\beta\delta}{2\Delta_{s}}}\theta^{4})-k^{\prime}[(3/2+(\beta\delta-3/2)Y_{1})l+\frac{5}{2}vrY^{\frac{3-2\beta\delta}{2\Delta_{s}}}\theta^{5}]}.$
(85)

## References

  * (1) B. Lüthi , [in:] G. K. Horton, A. A. Marudin (eds), Dynamical Properties of Solids, V. 3, North-Holland, Amsterdam, 1980.
  * (2) B. Lüthi, Physical Acoustics in the Solid State, chap. 6, pp. 93–108, Springer, Berlin, 2005. DOI: https://doi.org/10.1007/b138867
  * (3) K. Kawasaki, Phys. Lett. A 29, 406 (1968). DOI: https://doi.org/10.1016/0375-9601(69)90330-2
  * (4) J. O. Fossum, J. Phys., C 18, 5531 (1985). DOI: https://doi.org/10.1088/0022-3719/18/29/006
  * (5) F. Schwabl, Phys. Rev. B 7, 2038 (1973). DOI: https://doi.org/10.1103/PhysRevB.7.2038
  * (6) K. K. Murata, Phys. Rev. B 13, 4015 (1976). DOI:https://doi.org/10.1103/PhysRevB.13.4015
  * (7) R. Dengler, F. Schwabl, Z. Phys. B 69, 327 (1987). DOI: https://doi.org/10.1007/BF01307292
  * (8) A. Pawlak, Eur. Phys. J., B 4, 179 (1998). DOI: https://doi.org/10.1007/s100510050367
  * (9) A. Pawlak , Phys. Rev. B 68, 094416 (2003). DOI:https://doi.org/10.1103/PhysRevB.68.094416
  * (10) A. Pawlak, Critical sound propagation in magnets, in: Magnets: Types, Uses and Safety, Ed. T. Akitsu, Nova Science Publishers, 2011\.
  * (11) A. Pawlak, Acta Phys. Pol. A 98, 23 (2000). DOI: https://doi.org/10.12693/APhysPolA.98.23
  * (12) A. Pawlak, B. Fechner, phys.stat.sol. (c) 3, 208 (2006). DOI:https://doi.org/10.1002/pssc.200562471
  * (13) R. Erdem, M. Keskin, Phys. Lett. A 326, 27 (2004). DOI: doi: 10.1016/j.physleta.2004.04.030
  * (14) T. Komatsubara, A. Ishizaki, S. Kusaka, and E. Hirahara, Solid State Commun. 14, 741 (1974). DOI: https://doi.org/10.1016/0038-1098(74)90876-X
  * (15) A. Ishizaki, T. Komatsubara, S. Kusaka, and E. Hirahara, Sci. Rep. Tohoku Univ. Ser. 1, vol. LX No 1, p 1 (1977).
  * (16) M. Tachiki, S. Maekawa, Prog. Theor. Phys. 51, 1 (1974). DOI: https://doi.org/10.1143/PTP.51.1
  * (17) A. Pawlak, Acta Phys. Polon. A 115, 229 (2009) DOI: 10.12693/APhysPolA.115.229
  * (18) A. Pawlak, Acta Phys. Polon. A 121, 1185 (2012) DOI: 10.12693/APhysPolA.121.1185
  * (19) A. Pawlak, J. Magn. Magn. Mater. 513, 167177 (2020) DOI: https://doi.org/10.1016/j.jmmm.2020.167177
  * (20) The preliminary findings about our parametric model was presented at conference paper: A. Pawlak, J. Magn. Magn. Mater. 543, 168617 (2022). DOI: https://doi.org/10.1016/j.jmmm.2021.168617
  * (21) H. Iro , F. Schwabl, Solid State Commun. 46, 205 (1983). DOI: https://doi.org/10.1016/0038-1098(83)90611-7
  * (22) L. D. Landau , I. M. Khalatnikov , Dokl. Akad. Nauk SSSR 96, 469 (1954); reprinted in Collected Papers of L. D. Landau, ed. D. ter Haar (Pergamon, London, 1965).
  * (23) P. C. Hohenberg, B. I. Halperin., Rev. Mod. Phys. 49, 435 (1977). DOI: https://doi.org/10.1103/RevModPhys.49.435
  * (24) D.J. Bergman and B.I. Halperin, Phys. Rev. B 13, 2145 (1976). DOI:https://doi.org/10.1103/PhysRevB.13.2145
  * (25) P.M. Chaikin and T.C. Lubensky, Principles of Condensed Matter Physics, Cambridge University Press, Cambridge 1995. DOI: https://doi.org/10.1017/CBO9780511813467
  * (26) Larkin A.I., Pikin S.A., _Ż.E.T.F._ 56, 1664 (1969).
  * (27) Pawlak A., J. Phys. CM: Condens. Matter 1, 7989 (1989). DOI: https://doi.org/10.1088/0953-8984/1/43/001
  * (28) T. Nattermann, phys. stat. sol. (b) 85, 291 (1978). DOI: https://doi.org/10.1002/pssb.2220850132
  * (29) H.K. Janssen, [in:] C.P. Enz (ed.), Proceedings of the International Conference on Dynamic Critical Phenomena, Springer, Berlin 1979.
  * (30) C. De Dominicis , L. Peliti , Phys. Rev. B 18, 353 (1978). DOI: https://doi.org/10.1103/PhysRevB.18.353
  * (31) D.R. Nelson Phys. Rev. B 14, 1123 (1976). DOI: https://doi.org/10.1103/PhysRevB.14.1123
  * (32) K. G. Wilson, Phys. Rev. Lett. 28, 548 (1972). DOI: https://doi.org/10.1103/PhysRevLett.28.548
  * (33) M. Combescot, M. Droz, and J. M. Kosterlitz, Phys. Rev. B 11, 4661 (1975). DOI: https://doi.org/10.1103/PhysRevB.11.4661
  * (34) P. Calabrese, V. Martin-Mayor, A. Pelissetto, and E. Vicari, Phys. Rev. E 68, 016110 (2003). DOI: https://doi.org/10.1103/PhysRevE.68.016110
  * (35) A. Pawlak, R Erdem, Phys. Rev. B 83, 094415 (2011). DOI: 10.1103/PhysRevB.83.094415
  * (36) A. Pawlak, R Erdem, Phys. Lett. A 377, 2487 (2013). DOI: 10.1016/j.physleta.2013.07.033
  * (37) M. D. Kuz’min, Phys. Rev. B 77, 184431 (2008). DOI: https://doi.org/10.1103/PhysRevB.77.184431
  * (38) E. K. Riedel, F. J. Wegner, Phys. Rev. B 9, 294 (1974). DOI: https://doi.org/10.1103/PhysRevB.9.294
  * (39) V. A. Agayan, M. A. Anisimov, and J. V. Sengers, Phys. Rev. E 64, 026125 (2001). DOI: https://doi.org/10.1103/PhysRevE.64.026125
  * (40) P. Schofield, Phys. Rev. Lett. 22, 606 (1969). DOI: https://doi.org/10.1103/PhysRevLett.22.606
  * (41) P. Schofield, J.D. Litster, and J.T. Ho, Phys. Rev. Lett. 23 1098 (1969). DOI: https://doi.org/10.1103/PhysRevLett.23.1098
  * (42) B.D. Josephson, J. Phys. C2 1113 (1969). DOI: https://doi.org/10.1088/0022-3719/2/7/302
  * (43) Z. Borjan and P. J. Upton, Phys. Rev. E 63, 065102(R) (2001). DOI: https://doi.org/10.1103/PhysRevE.63.065102
  * (44) D. J. Wallace and R. K. P. Zia, J. Phys. C 7, 3480 (1974). DOI: https://doi.org/10.1088/0022-3719/7/19/008
  * (45) M. E. Fisher, S.-Y. Zinn, and P. J. Upton, Phys. Rev. B 59, 14533 (1999). DOI: https://doi.org/10.1103/PhysRevB.59.14533
  * (46) F. J. Wegner, Phys. Rev. B 5, 4529 (1972). DOI: https://doi.org/10.1103/PhysRevB.5.4529
  * (47) J. Luettmer-Strathmann, S. Tang, and J. V. Sengers, J. Chem. Phys. 97, 2705 (1992). DOI: https://doi.org/10.1063/1.463060
  * (48) J. F. Nicoll and J. K. Bhattacharjee, Phys. Rev. B 23, 389 (1981). DOI: https://doi.org/10.1103/PhysRevB.23.389
  * (49) J. F. Nicoll and P. C. Albright, Phys. Rev. B 31, 4576 (1985). DOI: https://doi.org/10.1103/PhysRevB.31.4576
  * (50) J. F. Nicoll and P. C. Albright, Phys. Rev. B 34, 1991 (1986). DOI: https://doi.org/10.1103/PhysRevB.34.1991
  * (51) Z. Y. Chen, P. C. Albright, and J. V. Sengers, Phys. Rev. A 41, 3161 (1990). DOI:https://doi.org/10.1103/PhysRevA.41.3161
  * (52) Z. Y. Chen, A. Abbaci, S. Tang, and J. V. Sengers, Phys. Rev. A 42, 4470 (1990). DOI: https://doi.org/10.1103/PhysRevA.42.4470
  * (53) A. Pelissetto, E. Vicari, Phys.Rep. 368, 549 (2002). DOI: https://doi.org/10.1016/S0370-1573(02)00219-3
  * (54) B. Ferry and B. Golding, AIP Conference Proceedings 24, 290 (1975). DOI: https://doi.org/10.1063/1.30093
  * (55) V. Yu. Bodriakov, A. A. Povzner, and S.A. Nikitin, Eur. Phys. J. B 4, 441 (1998). DOI: https://doi.org/10.1007/s100510050401
  * (56) J. Rouchy, P. Morin, E. du Tremolet de Lacheisserie, J. Magn. Magn. Mat.  23, 59 (1981). DOI: https://doi.org/10.1016/0304-8853(81)90068-8
  * (57) M. Suzuki and T. Komatsubara, J. Phys. C 18, 5459 (1982). DOI: https://doi.org/10.1088/0022-3719/15/21/020
  * (58) P.C. Hohenberg and M. Barmatz,Phys. Rev. A 6, 289 (1972). DOI:https://doi.org/10.1103/PhysRevA.6.289
  * (59) H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford, New York 1971.