Domain walls, fluctuations and the incommensurate-commensurate transition in two and three dimensions

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Domain walls, fluctuations and the

incommensurate-commensurate transition in two and three dimensions

T. Nattermann

To cite this version:

T. Nattermann. Domain walls, fluctuations and the incommensurate-commensurate tran- sition in two and three dimensions. Journal de Physique, 1982, 43 (4), pp.631-639.

�10.1051/jphys:01982004304063100�. �jpa-00209432�

Domain walls, fluctuations and the incommensurate-commensurate transition in two and three dimensions

T. Nattermann

Sektion Physik ^der Humboldt- Universität, ^Bereich 04, ¹⁰⁸⁶ Berlin, ^DDR (Reçu ^{le 4 aout} 1981, accepté le 7 dgcembre 1981 )

Résumé. ²⁰¹⁴ Nous présentons ^{un calcul de} la transition commensurable-incommensurable à basse température

pour ^un système ^{avec un seul axe} de modulation. Nous formulons la théorie en utilisant des degrés ^{de liberté des} parois ^{et des} phonons. ^{Ceux-ci ont} pour effet de renormaliser les interactions entre parois ^{en les} augmentant.

Nous calculons d’une manière self-consistante simple ^la dépendance ^en racine carrée de la densité des parois ^à

deux dimensions. Pour d ⁼ 3, ^{nous trouvons la loi} classique logarithmique contrairement à un calcul précédent

du même auteur. Ces résultats peuvent ^être interpolés ^{entre 1} d ~ 3 par ^un exposant 03B2 ⁼

3 - d / 2(d-1).

Abstract. ²⁰¹⁴ A low temperature calculation for the incommensurate-commensurate transition in a system ^with

a single ^axis of modulation is performed. ^The theory is formulated in terms of wall and phonon degrees of freedom.

Phonons renormalize the domain wall interaction to larger ^values. By ^{means of a} simple self-consistent calculation scheme the square ^root law for the domain wall density ^{in 2} dimensions is reproduced. ^{For d = 3 the classical} logarithmic ^{law is found to be} valid, ⁱⁿ contrast to previous investigations ^{of the author.} The results can be inter-

polated by ^an exponent 03B2=

3 - d / 2(d - 1)

Classification

Physics ^Abstracts

64.60F - 64.70K - 68.20

1. Introduction. ^- Condensed systems ^{with defects} in the ordered medium became of growing ^interest during ^recent years. An important example ^{are incom-}

mensurate (I) phases, ^{which are known to exist in}

many ^two- and three-dimensional systems [1, 2].

The actual order of the condensed wave in these systems is influenced by competing ^{forces - a local} Umklapp ^{term and an elastic} (Lifshitz) ^{term. The} analysis ^{of the} ground ^{state of the} I-phase ^{close to the} (IC) transition to the commensurate (C) phase ^delivers

a regular lattice of domain walls which, separate almost commensurate regions. Approaching ^the C-phase, the domain wall density 1/d goes continuously

to zero as 1/1 ex: -

In - ’ 6 -

are proportional ^{to the effective} strength ^{of the} conflicting ^{forces. For 6} 6,,, only one commen- surate domain survives.

The question ^arises, how thermal fluctuations influence this behaviour. For a two-dimensional system, Pokrovskii and co-workers [3, 4], ^Villain [1],

Okwamoto [5] ^and recently ^Schultz [6] showed, ^that

these fluctuations are important. ^In particular, they produce ^{a term} proportional ^to 1/13 ^{in the} 1/1-expan-

sion of the free energy. This leads ^{to a} non-classical

& ^1/2 behaviour of the domain wall density 11

YI )

6 1 1/2

Moreover, fluctuations suppress the C-phase ^above

a temperature To. ^{Since most of} these authors used

sophisticated methods, ^the physical origin ^{for the} change ^{of the domain wall} density ^{was not clear.}

The present ^{author tried to} attack the problem using ^a renormalization group approach ^{and obtained} essentially ^{the same} results for d > ² [7]. However,

as it became clear _now, this approach ^{is correct} only

to order 1/1 in the free _energy expansion. Although

a naive perturbation theory yields always ^contribu-

tions proportional ^to 1/13 ^{in F, it was} argued by

Villain [8], ^{that such a term should not} follow from

an exact calculation in d ⁼ 3 dimensions. Therefore,

the existence of a non-classical _square root law for

1/1 ^{cannot be} maintained in d ⁼ 3 dimensions In a series of recent papers [8-10], ^{the existence of the} 1/13-term ^{has been} explained ^{in the} two-dimensional

case by ^a loss of meander entropy of ^{walls due to} their contact interaction. According ^to [8], this mecha- nism should be uneffective for d ⁼ 3. However, ^to

our knowledge,’ ^a quantitative investigation ^{of the}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01982004304063100

632

different behaviour for d ⁼ 2 and d ⁼ 3, respectively,

is missing up ^{to now.}

It is the aim of the present paper, ^to reinvestigate

the influence of thermal fluctuations ^on the IC- transition by ^{means of a} simple ^and physically ^clear approximation scheme. In particular, ^{we are- interested}

in a comparison ^of the 2- and 3-dimensional case at low temperatures. ^{The main} steps and results of the calculation are :

(i) ^{First we} investigate ^{the energy and the inter-}

action law of walls at T ⁼ 0. The general ^{case of} arbitrary shaped walls which terminate in topological

stable defects is considered. The interaction between walls decreases exponentially with distance ^r for Kr >> 1. The screening constant K is proportional

to the inverse width of the walls.

(ii) ^Wall fluctuations ^are the most important degrees

of freedom. The remaining phonon (optical phason)

fluctuations merely ^{lead to a} renormalization of the wall parameters. ^In particular, they decrease the

screening ^{constant K.}

(iii) ^{For d =} 2, ^the T2/13-term ^{in F} is rederived

by’ ^a simple self-consistent calculation. The same treatment yields ^{for d = 3 a} contribution

which is small compared ^to the classical term e - "I

for 1 -+ oo. Thus, the classical logarithmic ^{law for} 1/1

is expected ^to be valid in the asymptotic region.

The _paper is organized ^as follows : In section 2

we rewrite the Hamiltonian in terms of wall and

phonon degrees of freedom. The statistical mecha- nics of walls in d ⁼ 2 dimensions in the incommen- surate phase ^is considered in section 3. In section 4

we extend the results to the 3-dimensional case and discuss their dependence ^on general ^d. Appendix ^A

includes an approximate ^{treatment of the} phonon

fluctuations.

2. Walls and their interaction. ^- We consider the sinus-Gordon Hamiltonian with an additional linear

gradient ^term

(2.1) ^{is one of the standard} models for the descrip-

tion of incommensurate phases ^{with a} single ^direction

of modulation [1, 2, 12]. 0 ^{is an} angular ^variable

and as such given only up ^{to a} multiple ^{of 2 n.} Actually, 0 ^{can be} considered as the phase ^{of a} complex ^order parameter ^{Y = A} ei". In this case J oc A 2 and V oc A p-2 depend ^{on the} amplitude ^{A. In the case of} physisorbed layers (d ⁼ 2) T ^describes the transla- tional order of the substrate in the direction 3. Per-

pendicular ^to this direction the substrate is assumed to be commensurate.

In the following, ^{we are} interested in the low- temperature properties ^{of the} system (2.1). ^A syste- matic _way to study ^{these is to} expand the Hamilto- nian around its ground ^{state and to treat the devia-}

tions in a perturbation series in the temperature ^T.

The configuration ^of minimal energy of (2.1)

follows from the solution of the Euler-equation

Since we use a continuum description, ^{we will assume,}

that the characteristic length ^{V- 112} p-I ^which appears in (2.2) ^is large compared ^{with the} microscopic

lattice spacing a, ^i.e. p- I V-112 >> a. Special ^solu-

tions of (2.2) have been considered by ^various

authors [11-12]. ^{For V 0 0 these} configurations

are represented by parallel straight walls of width V-1/2 which separate ^almost commensurate regions

with

- - 2

with ⁼

p p ^{n, n mteger.}

Since l/J ^{is an} angular variable, it may jump along

certain lines (d ⁼ 2) or surfaces (d ⁼ 3) by ^{2 nn.}

These (d - I)-dimensional ^surfaces Si may end ^on the surface of the system ^or may terminate in a (d - 2)-

dimensional defect line Di. The latter is determined by

its position Qi(L) ^and charge qi

t ^{denotes a (d -} 2)-dimensional ^vector specifying ^the position ^on Di. ^{The curve} Ti is assumed to enclose only ^{the defect} Di.

According ^to the usual classification of defects [13]

the lines Di ^are topological stable, ^since macroscopi- cally, ^the phase 0 ^{of the order} parameter ^{can take} all values. On the contrary, ^{in the} C-phase ^{this ma-} croscopic phase is allowed to take only ^{discrete values}

2 p 11: m

p

of dimension d - 1. However, ^since these walls appear gradually already ^{in the} I-phase, ^we may consider the configurations ^{of the} I-phase ^{to be} given by ^the distribution of walls Wi ^{and lines} Di. ^{Lines are} starting ^{and end} points ^{of walls.} Note, ^{that walls}

have nothing ^to do with the surfaces S : 0 changes gradually ^{in a} wall, ^but abruptly along ^S. The energy of a configuration depends ^{on the} shape ^{of W but}

not on S.

It is the aim of this section to consider a more gene- ral class of (approximate) solutions of (2.2). In par-

ticular, ^{we consider} arbitrary shaped ^walls terminating

in defect lines D. The interaction law between walls is then found by putting back these solutions into the Hamiltonian (2.1).

Above all, ^we perform the calculation for _p ⁼ 1 and generalize the result to arbitrary p ^at the end of this section.

We consider first the configuration ^{of a} single ^wall

which is spanned by the closed (d - 2)-dimensional

defect line D. For d ⁼ 2 this line consists of only

two points Q (t ) ^{= Q +, Q - with} opposite charges ^{± q.}

The centre of the wall will be specified by ^the position

vectors R(t), t ⁼ { tcx’ ^{a = 1... d -} 11. ^We identify

now the surface S with the centre of the wall, ^{i.e. we}

assume, that 0 jumps along R(t) by 2 aq. Then 0 ^can

be defined such that it vanishes far from the wall.

Since a general solution of (2. 2) ^{seems to be} impossible,

we determine the field configuration from its linea- rized version [1]. ^{This is a} satisfactory approximation sufficiently far from the wall

where 0 obeys ^the boundary ^conditions

From (2.2) ^follows K2 = KÕ == ^V, but since in sec-

tion 3 we will show, ^that phonons renormalize _Ko we

omit the subscript ^{0 in the} following. ^{Moreover, in this}

linear approximation ^{our results} apply ^{to aI1} periodic potentials V( l/J) ^with V"(0) ^{= K 2 and not} only ^{to the}

model (2.1) [1]. m(R(l)) denotes the normal to the

(d - I)-dimensional plane ^{S in the} point R(t ) ^{in the}

direction of increasing l/J. l/J+ ^are the field values on

both sides of S.

The problem ^{is now} analogous ^{to that of} finding ^the potential ^{of a screened} electric double layer. ^{The latter}

can be considered to consist of infinitely many infini- tesimal dipoles parallel ^{to m.} To the field ql(0) ^{at the} origin ^{of our} coordinate system ^each gives ^{a contri-}

bution

ds ⁼ m ds denotes the (d - I)-dimensional ^directed

surfaces element of S. Kv{x) ^{are the modified Bessel}

functions. In particular K:tl/2(X) =

e-x

Ko(x) £r

e-X

In Y;

and small _x, respectively. ^{The total} contribution of the wall to the field at the origin follows from the integra-

tion over S :

In order to obtain well-defined results, ^{we must leave}

out a region of size a in the neighbourhood ^{of the} boundary ^D. Actually, ^the amplitude ^{A and hence J}

vanish on D.

We have exploited ^formula (2.7) for different wall

configurations. ^In particular, ^we get ^from (2.7) ^{for an}

infinite wall at x ⁼ 0, perpendicular ^{to the} x-direction

We note, that (2. 7) ^{becomes exact for K -+ 0. In this} case q5 ^is independent ^{of the} shape ^{of S and} equal ^to

± q ^{times the} angle under which the curve D appears from the origin [14].

The _energy of the wall is now found by putting (2. 7)

into (2.1). ^{For the sake of} consistency, ^we replace V(O) = v(1 - ^cos 4) by

K 2 2

,x = I ... d})

V(d) ^{has a} non-trivial limit for K - 0

If the bent of the wall is not too large, ^we may ^use for Vo ⁱⁿ (2. 9) the result for the straight ^wall (2. 8). ^This yields

The _energy of a single wall includes therefore ^three terms : the first is proportional ^{to the total area} S ⁼

Js ds

634

/* !

the -

Js

^denotes

The additional _energy connected with the strong ^variation of q5 ^{in the} neighbourhood ^{of the} defect line D is included in the nucleation _energy E, which is of order of magnitude J per unit length ^{of D.}

We note, ^{that for K - 0 the wall} energy becomes independent ^{of the} shape of S. For d ⁼ 2 the first term in

(2.12) ^{is then} replaced by ^the logarithmic interaction between defects, which is well known from the XY- model [15]. According ^to (2.9) ^to (2.12) ^the configuration of lowest energy is that of ^a plane ^wall perpendicular

to the 8-direction. Small deviations are described by ^the transverse wall elongation f(t). Neglecting overhang configurations, (2.12) ^can be rewritten as

where ₆₀ ⁼ n2 q2 ^{xJ denotes} the bare surface tension of the wall.

Now, ^we consider the interaction between walls. We adopt again (2.4), but the total field from all walls has

now to fulfil the condition (2.5). ^{If we however assume,} that the distance between walls is large (Kr >> 1) ^the

total field is given by ^a superposition of the contributions (2. 7) from each wall. The violation of the boundary

condition for the total field is exponentially small. The energy of ^a configuration ^{with N} walls is therefore given by

This is the interaction law between walls we were

looking ^{for. Here} dsi,cx ^{is the} a-component of the direct- ed surface element ^on the i-th surface in the point Ri : dsi,,, ⁼ dsi mi,JR). ^{Summation over double} greek

indices is understood.

It is _easy to see from the matrix V(")(r ^{that the}

interaction favours the parallel orientation of the walls.

(2.14) includes the two special ^{cases considered} previously [1, 8, 12].

(i) ^For infinitely extended walls perpendicular ^to 8,

it becomes identical with the interaction law obtained from the periodic parabolic potential (K I Ri - Rj I > 1).

(ii) ^For straight finite walls in two dimensions

perpendicular ^to 8, the interaction law is that of Villain [8].

We note, ^that although the interaction between walls decays exponentially with their distance, long

range correlations arise in ^a regular lattice. These lead in d ⁼ 2 dimensions to a logarithmic interaction of defects which are separated by ^a large ^{number of}

infinite parallel ^walls [9, 15-17].

So far we considered the case p ⁼ 1 and only ^one

wall terminating ^{in a} defect. We get the results for

general p if we make the following substitution in the above formulae

In the subsequent calculation we will restrict ourselves to p ⁼ 1, the results for p > 1 follow then from (2 .15).

We note finally, ^{that the} approach used in this section has similarities with that used in a recent paper

by ^Schulz [26]. The actual calculation of Schulz

however differs considerably ^{from the} present one, in particular his model is formulated on a lattice.

3. Statistical mechanics of walls. ^- To find the

thermodynamic properties ^{of our} system, ^we express the partition ^{function Z in terms} of wall and phonon degrees ^of freedom, Ri(t) and cp = 4J - 4J(N), ^respec- tively.

The prime ^{at the} j)(N) (f)-integration ^{denotes, that the}

total number of degrees ^of freedom has to be unchang-

ed by ^this splitting. Since the interaction between walls and phonons ^is complicated ^{and not our} primary

concern here, ^we adopt ^the following approximation

scheme (for ^{a more careful treatment in d = 1 + 1}

dimensions see e.g. [18]) :

(i) ^The influence of walls on phonons ^is neglected.

Only ^the conservation of the total number of the

degrees of freedom is taken into account. This seems to be a reasonable approximation ^{if the} density ^of

walls is low, ^{i.e. at low T. For N =} 0, j)(Q)(f) = fl ^dqJ(rJ

(ra l

where { ri } denote the sites of the underlying ^lattice.

For N :0 ^{0 those modes are} excluded from the

(f)-integration, ^which correspond ^{to a} homogeneous elongation ^of the 1 oscillators across the wall.

Ka

(ii) ^The integration ^over r.p is performed in the self- consistent harmonic approximation (SCHA). ^This procedure ^is briefly ^{outlined in} Appendix ^{A. We} get

Je I 4>(N)i ^{K } is} given by (2.14), ^{but the width K-l of}

the walls is now increased by phonon fluctuations. For d ⁼ 2 we get ^from (A. 5)

Here A ⁼ 7r/a ^{denotes a} microscopic cut-off. For T >, To, ^{x =-} 0, ^i.e. only ^the I-phase ^exists. 6F(i) ^is

given ⁱⁿ (A. 7) ^{and can be} considered as a self-energy

of the wall [22]. Fo ^{is the} phonon ^free energy in the wall free state, which ^we omit in the following.

Treating ^the remaining ^{summation over the wall} configurations, ^{we found it more} convenient to work with the canonical ensemble and determine the _average number of walls from the free energy minimum.

We consider now a system ^{of infinite} walls, ^which

form at T ⁼ 0 a regular ^{lattice of} spacing I. ^{This is} the configuration ^{of the} I-phase.

Since the wall interaction is exponentially ^small except for walls in contact (see (2.11)), ^Villain [8] ^has replaced ^{the mutual} repulsion ^{between walls} by ^the

condition fi’ l’ for each wall separately. ^This procedure ^delivers the lowest order term in a low T

expansion ^{of In Z. With the defect line D now on the}

surface of the system ^and using ^{the hard} repulsion

condition f 2 1 ^we get ^from (2. 13), (3.1) ^and (3. 3)

for the partition function : Z(N) = (Z (1) )N,

L is the linear dimension of the system. D f ⁼ n ^qK1t df(ri) ^where ( r§ ) ^{denotes the set of the} (L/a)d lattice sites

M

along ^{the centre of the} plane ^{wall. In} (3.5) only ^walls with q ^{= 1 and the favoured} orientation have been kept,

since these give the main contribution at low T. For 1 ^-+ oo f2 > diverges ⁱⁿ d :5 ³ dimensions, ^{i.e. the} roughen- ing transition of the present ^{model occurs at T = 0. For a} three-dimensional system, ^{where the} roughening

transition temperature TR ^{is in} general ^{non-zero, our results} apply ^{for T} > TR.

In the following, ^we proceed ^{in a somewhat different} way : ^we regard the effective repulsion between the walls by adding a ^{harmonic term}

in the exponent ^of (3.5). ^The f integration ^{is extended to} infinity. With other words, the wall is moving ^{in a}

harmonic potential ^{with a} 1-dependent ^{force constant a} ml (1). m(l ) ^is subsequently determined from the condition

f2 >m ^{= l2. This} condition substitutes the sharp cut-off for f ⁱⁿ (3. 5). ^This procedure ^is clearly approximative

and not necessary in 2 dimensions, where the transfer matrix method can be applied ^to (3. 5) [8], ^but it allows the

comparison ^{of the cases d = 2 and d = 3. Moreover, it is a} priori ^not obvious, ^{that such a treatment should}

deliver a less reliable description of the actual repulsion between walls than the use of the condition f 2 ^{d 2.}

Thus

The partition ^function (3.6) has been considered in [21]. The non-linear terms in the exponent renormalize the surface tension ^Q to lower values. Approaching ^the disordered phase, ^{Q vanishes} as -(d- 1) where denotes the bulk correlation length. ^{On a scale A} ç the wall is fuzzy [21]. ^{Since we are at low} temperatures, ^we may neglect

these effects and keep only ^{the term} (Y f)2 ^{in the expansion} of the square ^root in (3.6). ^The free energy of ^a single

wall F(1) ^{is then}

The denominator of the logarithm arises from the wall self-energy 6F(i), 9 ^{denotes a} (d - I)-component ^vector.

636

For d ⁼ 2 we get from the condition

With dq ^{and "4 K « .4 one obtains} finally ^{for d = 2}

Here we have used the definitions of _Qo and To.

The effective surface tension is lowered and vanishes at Tc = ^{To. Since} the deviation from the exact

4

result T c ⁼ To, which is known for d ⁼ 2 [23], ^is small, ^we conclude the validity ^{of our} approximation

scheme at low T. The phase boundary between the I- and the C-phase follows from d ⁼ 1, ^{i.e. for}

The second term on the r.h.s. of (3.9) ^has been inter-

preted by ^a loss of meander entropy ^{due to the wall} collisions [8], [9]. The total free _{energy F} from N ⁼ L/1

walls includes therefore a term oc ¡- 3 :

In order to include the case T ⁼ 0 we have added the classical expression ^{for the} repulsion ⁱⁿ (3.12).

Minimizing ^{F with} respect ^to l yields ^{for 1 -+ oo and}

T =A 0

To is twice the Luther-Okwamoto temperature T,.

Comparison ^{with the} calculation of Haldane and Villain [10] shows, ^{that our} coefficient of the 1- 3-term in F is smaller by ^{a factor}

3864

that a better description ^{of the contact} interaction

by considering higher ^{order terms in} f 2 ^{in the} exponent of (3.6) ^{would lead to an} increase of this coefficient.

Likewise, the consideration of the terms C£f)2n (n > 1)

in (3.6) would renormalize _co toward lower values and hence increase this coefficient. These effects could become important ^{for a} comparison ^with experiments performed ^{at T - Z} Tc.

4. The three-dimensional case. ^- The extension of the results of the previous ^{sections to the case}

d ⁼ 3 is straightforward ^and partly considered already

there.

The main difference between d ⁼ 2 and d ^{= 3}

arises from the different expressions for ( (p2 >0 and f2 >m’ ^{For d = 3 one} gets ^from (A. 5)

Thus, walls have a finite width at all temperatures

T T,o ^where TeO ^is the transition temperature of the Landau-theory and defined by ^the vanishing

of To, A 2 ^oc T,,o - ^{T. The} self-consistent calculation of the mass m(l) ^{from the} condition f2 )m ^{= 12} yields

Repeating the calculation after (3. 6) ^{for d = 3} (the

momentum q ⁱⁿ (3.7) ^{becomes now a} 2-dimensional

vector) ^one gets ^{a free energy F}

B ⁼ B(m) stands for logarithmic corrections in m(l),

which are omitted, ^since they ^cannot be calculated from the self-consistent scheme used here. The most

important feature of (4. 3) is, ^{that the contact inter-}