MODULATED STRUCTURES OF ONE-COMPONENT ORDER PARAMETER

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MODULATED STRUCTURES OF

ONE-COMPONENT ORDER PARAMETER

V. Klepikov

To cite this version:

V. Klepikov.

MODULATED STRUCTURES OF ONE-COMPONENT ORDER PARAMETER.

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JOURNAL DE PHYSIQUE

Colloque C8, Suppl6ment a u no 12, Tome 49, dbcembre 1988

MODULATED STRUCTURES OF ONE-COMPONENT ORDER PARAMETER

V. F. Klepikov

Institute of Physics €5 Technology, the Academy of Sciences of Ukrainian S S R , Kharkov 310108, U.S.S.R.

Abstract. - Studied are the magnetic systems that has one-component order parameters and a Lifshits point. We show four different phases t o converge a t this point: symmetrical, non-symmetrical, and two modulated phases with a different symmetry. The symmetry and exact solutions of the Euler-Poisson differential equation for the order parameter distribution are investigated. The regions of phase stability are determined and the phase transitions in this system are studied.

Symmetry of the majority of the magnetic crystals with modulated structure (MS) does not admit the invariants linear by the order parameter (OP). If OP is one-component, then the modulated along the axis OZ structure ought to arize as a prcblem solution on

thermodynamic potential minimum @ (cp)

,

having the

form [l-41:

+

( ~ 1 2 )

v4

+

(X/3) cp6] dz, (1) where p (z) - OP; p: = d p (z) /dz; g, y, T , s, X -

the material constants (depending on the temperature, magnetic field, pressure etc.), T-MS period.

OP distribution satisfies the differential equation

(DE) :

We shall consider further that y

>

0, X = 1, g-

may have any sign. Scale transformation p (z) =

fif

( z f i ) = &@ (x)

,

r = qy2, s =

m

leads @ to a form:

+qf2

+

( ~ / 2 )

f 4

+

(113) @6] dx. (3)

As it is known in a case of homogeneous OP there are

two phases in such system in a plane (p, q) : $Jo = 0

and f: = (112) [(p2 - 49) 'I2

-

p]

,

which for p

>

0 are divided by a line of phase transition (PT) of the second kind (q = 0)

,

and for p < 0 are divided by a line of P T of the first kind (q = (3116) p2) . Let's search for

the OP distribution in a system as (q

>

0) :

p (x) = a [cos kx

+

a2 (A1

+

a2A2

+

...)I

,

₍₄₎

where A1 = crl cos 3 t x , A;-the harmonics of odd-

numbers. We obtain an expression to @ :

By minimizing (5) by a1 and k we obtain:

From (6) we see that under (3p

+

g)

>

0 P T of

the second kind occurs on a line q = 114 from phase

a = 0 (cp = O...) to MSI phase (a

#

0)

.

In a phase MSI

r T

(f ( x ) ) ~ =

T-'

/

$J (x) dx = 0, in other words the

J 0

symmetry relatively to parity (+ --+ -4) is "locally" broken, but "globally" is reconstructed owing to the fact that the average value by a period OP equals zero. To this, MS is a stationary analog of the Goldstone mode, arizing at a spontaneous breaking of contin- ious group and just restoring the broken symmetry. In this concrete case the spontaneous symmetry breaking

(SSB) takes place for discrete group ( 4 -+ -@) but due

to spontaneous breaking of a translational symmetry the Goldstone mode arises all the same.

The tri-critical point for @ (6) in the plane (p, q)

has the coordinates q = 114, p = -913. Let's study,

for example, a special case, p = -g. Then @ leads to

From (7) it is followed that if g

5

0 (p

>

0) then P T of the second kind occurs at q = 1/4. Ig g

>

0 (p

<

0) the P T of the first kind takes place on the line

(3)

C8 - 1806 JOURNAL DE PHYSIQUE

in a plane (p, q)

.

As under p

<

0 the phase boundary for $ = const is a parabola q = 3/16 p2 then it is possible to demonstrate that a general ' icture of phase transitions look like: if 0

>

p

2

-&,

the P T of the first kind occurs from the phase II, = 0 to MSI on the line (8). If

-m

>

p

>

-m

then on a line q = 3/16 p2 takes place the phase transition of the first kind (PTI) from the phase $ = 0 to phase

1/2

$ = l / J Z [ ( p 2 - 4 q ) 1 / 2 - p l ,If - , / E p > p >

L J

-m

then occurs PT1 from phase $ = 0 to MSI on the line (8).

One may conclude, that under the condition (p

+

g) = 0 @ (6) does not contain the addends as. Therefore in order to make, in this case, an analysis of a system behaviour at p

<

-@

it is necessary to take into account in (1) the other non-linear terms like cp2m (cpl) 2n

Let's study the stability of the phase cp(x) = const #O relatively to the formation of MS phase of the second kind (MSII, (cp (z)),

#

0)

.

We shall consider the most simple case, that is in (1) g = 0, X = 0, S = 1, y

>

0, r = -qy2 (q

>

0). fjcale transformation cp (z) = y$ ( z m = y$ (a) leads @ to a form

"7'

@, = ,),7/2@ = Y7/2T-1

+1/2+41dz. (9) To study phase $ =

f i

stability let's consider a new function $ (x) =

f i

(1

+

f (x)) : then

We shall search OP distribution of

f (a) = a [cos kx

+

a (A1

+

aA2

+

...)I

,

(11) where

A1 =

p

+

a1 cos 2kx, A2 = a3 cos 3kx.

Putting (11) in (10) and minimizing @ by a;,

P

and lc

we obtain

(12) As the terms N a4 at q

>

0 are negative, it is ap-

parently that P T from phase $ =

&

into phase MS I1 is P T of the first kind. In order to define the line equation of this P T (q = qo

>

1/8) it is necessary to analyse multicritical behaviour of @ taking into tac- count the other non-linear terms

(N

cp2mcp'2n) in the highest approximations N (a2')

(e

>

3 ) .

Differential equation (2) in a case of g = 0, X = 0, y

>

0, s

>

0 has a non-trivial exact solution in ele-

sh z mentary functions. This solution cp ( z ) =

-

takes

ch2 z place only when -q = r/y2 = -11/100, and is con- ditioned by a hidden conformal symmetry of DE at r/y2 = -11/100. MS cp (z) is a correlation function of stationary OP fluctuations along the axis Oz. It is known [5] that the presence of conformal symmetry of fluctuations gives the additional information on the correlation properties of the system (in our case, this information is an exact solution for cp (z) in ele- mentary functions) and takes place only in a point of P T 11. Therefore, the line T = -lly2/100 in the plane (r, y) one may consider as a line of continuous P T of

1

the I1 kind MS I --+ MS 11; r = -y2 is a line of P T I1 4

cpo + MSI (Fig. 1).

Analysis given in this paper shows that the modulated phases are the intermediate state for the magnetic transitions from the symmetry phase to a non- symmetry one. They appear as a response to SSB and seek to weaken the symmetry breaking. The reasons of the appearing of this two types of MS are not only of energetic nature, but symmetrical ones.

Fig. 1. - Phase diagram (g = 0, X = 0 , ~

>

0, s

>

0)

.

[I] Izyumov, Yu. A., Syromyatnikov, V. N., Phase

transitions and crystals' symmetry (M. Nauka) 1984.

[2] Buzdin, A. I., Tugushev, V. V., Zh. Eksp. Teor. Fiz. 85 (1983) 735.

[3] Mertsching, J., Fischbeck, H. J., Phys. Status So- lidi (b) 103 (1981) 783.

[4] Buzdin, A. I., Menshov, V. N., Tugushev, V. V., Zh. Eksp. Teor. Fiz. 91 (1986) 2204.