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 ...
J.
Physique 43 (1982) 631-639
1982,
AVRIL
631
Classification Physics Abstracts 64.60F - 64.70K - 68.20
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 )
Nous présentons un calcul de la transition commensurable-incommensurable à basse température 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
Résumé. pour
2014
un
=
=
3 - d / 2(d-1).
Abstract. 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 inter2014
=
polated by
an
exponent 03B2=
3 - d / 2(d - 1)
for 1
d ~ 3.
1. Introduction. Condensed systems with defects in the ordered medium became of growing interest during recent years. An important example are incommensurate (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 a local systems is influenced by competing forces 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 - ’bc6 - 1 . Here 6c and 6
sion of the free energy. This leads to
a
non-classical
&
1/2
YI6 1 1/2 )
behaviour of the domain wall density 11 oc
.
Moreover, fluctuations suppress the C-phase above
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 contributions 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, a
proportional to the effective strength of the conflicting forces. For 6 6,,, only one commen- the existence of a non-classical square root law for surate domain survives. 1/1 cannot be maintained in d 3 dimensions In a series of recent papers [8-10], the existence of the The question arises, how thermal fluctuations influence this behaviour. For a two-dimensional 1/13-term has been explained in the two-dimensional system, Pokrovskii and co-workers [3, 4], Villain [1], case by a loss of meander entropy of walls due to Okwamoto [5] and recently Schultz [6] showed, that their contact interaction. According to [8], this mecha3. However, to these fluctuations are important. In particular, they nism should be uneffective for d in the to a term our produce proportional 1/13 1/1-expanknowledge,’ a quantitative investigation of the are
=
=
=
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01982004304063100
632
2 and d different behaviour for 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 ICtransition 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 :
following, we are interested in the lowtemperature properties of the system (2.1). A systematic way to study these is to expand the Hamiltonian around its ground state and to treat the deviations in a perturbation series in the temperature T. The configuration of minimal energy of (2.1) follows from the solution of the Euler-equation
(i) First we investigate the energy and the interaction 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.
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 solutions 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 7rn’ with n mteger. (n, n’ n, integer).
=
=
=
(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 (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 constant
K.
=
=
which is small compared to the classical term e - "I for1 -+ 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 rewrite the Hamiltonian in terms of wall and phonon degrees of freedom. The statistical mechanics of walls in d 2 dimensions in the incommensurate 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. we
=
2. Walls and their interaction. sinus-Gordon Hamiltonian with
-
an
We consider the additional linear
gradient term
In the
- - 2pnn 2p =
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 macroscopically, the phase 0 of the order parameter can take all values. On the contrary, in the C-phase this macroscopic phase is allowed to take only discrete values and the topological stable defects are walls Wi p
2p1:m
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.
(2.1) is one of the standard models for the description 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 translational order of the substrate in the direction 3. Perpendicular to this direction the substrate is assumed to be commensurate. =
=
It is the aim of this section to consider a more general class of (approximate) solutions of (2.2). In particular, 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 =
633
defect line D. For d
2 this line consists of
only 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 two
points Q (t )
=
=
=
=
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 linearized version [1]. This is a satisfactory approximation sufficiently far from the wall
m ds denotes the (d ds surfaces element of S. Kv{x) =
the modified Bessel
e-x and e-X and Ko(x) £f - In Y; for large
functions. In
Ko(x) £r
I)-dimensional directed are
particular K:tl/2(X) =
and small x, respectively. The total contribution of the wall to the field at the origin follows from the integration over S :
where 0 obeys the boundary conditions
From (2.2) follows K2 = KÕ == V, but since in section 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 infinitesimal dipoles parallel to m. To the field ql(0) at the origin of our coordinate system each gives a contribution
V(d) has a non-trivial limit for K
If the bent of the wall is not too
-
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
K 222
V(O) = v(1 - cos 4) by 2 02 in (2. 7) and Gauss’ integral theorem ,x = I ... d})
R.
Using (2.5),
we
get
(b = { ba,
0
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 s
of
634 !
/*
the -
Js denotes ds
the area of the projection of the wall on a plane perpendicular to the,&-direction.
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 XYmodel [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 =
n2
q2 xJ denotes the bare surface tension of the wall. 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 where 60
=
Now,
we
by
This is the interaction law between walls we were looking for. Here dsi,cx is the a-component of the directed 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 =
however differs considerably from the present one, in particular his model is formulated on a lattice. 3. Statistical mechanics of walls. To find the 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-
thermodynamic properties tively.
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]. 1 and only one So far we considered the case p wall terminating in a defect. We get the results for general p if we make the following substitution in the above formulae =
=
The prime at the j)(N) (f)-integration denotes, that the total number of degrees of freedom has to be unchanged by this splitting. Since the interaction between walls and phonons is complicated and not our primary concern here, we adopt the following approximation 1 + 1 scheme (for a more careful treatment in d 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 =
(ral
In the subsequent calculation we will restrict ourselves 1, the results for p > 1 follow then from (2 .15). to p 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 =
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 Ka1 oscillators across the wall.
635
(ii) The integration over r.p is performed in the selfconsistent harmonic approximation (SCHA). This procedure is briefly outlined in Appendix A. We get
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,
given
=
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
T >, To,
=
7r/a denotes a microscopic cut-off. For 0, i.e. only the I-phase exists. 6F(i) is
x =-
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, oo f2 > diverges in d :5 3 dimensions, i.e. the roughensince these give the main contribution at low T. For 1 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 L is the linear dimension of the
=
=
-+
=
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
dq and "4 K
With
« .4
one
obtains finally for d
Here we have used the definitions of Qo and To. The effective surface tension is lowered and vanishes at
Tc =4 To. Tc
2
The extension 4. The three-dimensional case. 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 (p2 >0 and f2 >m’ For d = 3 one gets from (A. 5) -
=
Since the deviation from the exact
To, which is known for d 2 [23], is conclude the validity of our approximation small, scheme at low T. The phase boundary between the I- and the C-phase follows from d 1, i.e. for
result
=
=
=
we
=
=
for (
=
The second term on the r.h.s. of (3.9) has been interpreted 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 : =
0 we have added In order to include the case T the classical expression for the repulsion in (3.12). Minimizing F with respect tol yields for 1 -+ oo and
Thus, walls have
a
finite width at all temperatures
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 T
=
T =A 0
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
Repeating
=
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 0.4. We note however, n6
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.
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 interB
=
637
action between walls
ex - 4 7raoTI’
tional to 1
produces
now a
term propor-
which vanishes faster for
than the classical wall interaction. Therefore, contrary to the case d 2 (and to our previous statement [7]) the classical law for the average distance between walls -+
oo
=
which agrees, besides of the
logarithmic
terms with
(4. 3). Some remarks are in order : First we note, that 3 dimension the coefficient of the AF(,)-term in d is indeterminate in the present calculation. For instance, in computing ( f’ > we could have chosen K instead of A as the upper integration cut-off. This would result in a decrease of AF(i in (4.7) by a factor ... 10 -3 for typical a2/,,2, which is of the order is connected with Another uncertainty examples. the logarithmic corrections of the type =
10
remains valid in the asymptotic region even for T i= 0. This difference follows from the fact, that the stiffness of the 2-dimensional wall is larger than that of a 1-dimensional. Qualitatively, one can obtain the same result from the following consideration. If we assume the wall to be rigid on a scale K the entropy loss per wall
which appear in (4.3). We believe, that reliable expressions for these corrections cannot be obtained from the approximate treatment used here. The existence of such corrections is however not unexpected, since d 3 can be considered as a lower critical dimensionality of the model defined by (3.5). For d _ 3 and T =1= 0 walls always touch each other 12) whereas for d > 3 f2 > 12 at low T. Increasing T we reach a region where f’ > -- l’ and it is conceivable, that this change of the behaviour is separated by a sharp phase transition. The essential non-linearity is due to the condition f ’ l’ and the situation is reminiscent of the non-linear a-model for d >_ 2 [24]. On the other hand, d 3 is the upper critical dimensionality for the non-classical behaviour of the domain wall density 1/1. An extension of the above discussion to arbitrary dimensions yields =
collision is -
bSo = - (KLKLd-1 0
0 denotes So. 0 So
the
entropy of the free wall :
(( f 2
Fluctuations of a finite piece of the wall of size C lead to a contact between walls, if ( f2 >r > 12. Ford
=
2fl>r.- Tt 21 i.e. the average number JY’ (10 n
201320132013
=
of wall collisions is
for 1 The change of the free energy of a single wall due to the loss of entropy is therefore
AF(l)
2/n2
with
which agrees, besides of
a
constant factor
surate
d
3 and T # 0. For
phase is suppressed
at
d _ 1 the
commen-
T # 0. For d >, 3 the
classical result 71 0 applies. Let us now comment on the approximation scheme we used for the actual repulsion between walls. If we would approximate the latter by its expression for large distances, the partition function of a single wall becomes =
(3 . 9). In d
=
3 the
mean
square fluctuation increases
only logarithmically with C
f2 >r
= exponential-2 In At.
na 0
The number of wall collisions is therefore ly small
and hence
The self-consistent for m(1)
analysis of (4. 8) yields
in this
case
where ( f I > has
to be calculated from (3. 6). The 2. further calculation gives åF(1) oc T2/1 for d For d 3 the classical domain wall repulsion e-Kj =
=
638
is
replaced
by ex - Kl 1
+
TT-1 . 7
Since
2 expression contradicts the exact result, conclude, that this approximation overestimates
the d we
now
=
the effective wall interaction. Unfortunately, the theoretical result for d 3 equation (4.5) seems to contradict at present the experimental findings. The latter show for 6 -+ 6, a less steep descent of the domain wall density than predicted by the classical logarithmic law. The data can be roughly fitted by a square root behaviour p 1/2 [25, 27]. It should be noted however, that the classical behaviour predicted by the theory for d 3 should be only observable very close to the IC-transition, where the width of the walls is small compared with their distance. Moreover, the susceptibility data of a variety of improper incommensurate ferroelectrics seems to support the classical law = 0 [28]. For a detailed comparison it would be necessary to regard other mechanisms like the influence of impurities or amplitude fluctuations, =
=
=
too.
We mention finally, that an alternative way of treating (3.1) would be to map the wall system on a lattice. Including defects, our analysis led however to a rather general case of the sixteen vertex model, which is, to our knowledge, not solved so far. After this paper has been submitted for publication, we received two preprints which address essentially the same points as discussed in sections 3 and 4. Lederer and Moudden [29] find from a model similar to ours (eq. (3.5)) a non-classical behaviour with p 1/2 for d 3 ! However, the approximations they use are unclear to us. The results of the second paper by M. E. Fischer and D. S. Fisher [30] agree with those presented above, as far as there is over=
=
lapping. The author thanks Prof. Acknowledgments. J. Villain for illuminating discussions and a helpful correspondence. He also thanks Prof. W. Ebeling for his support of this work and Dr. S. Trimper for a critical reading of the manuscript. Appendix A. In the SCHA iC is given by [19, 20] -
-
The phonon free energy F 0 (N) for a system with N walls and the average ( ... )o are calculated with the trial Hamiltonian
The variation
of k yields
From the assumption of periodic tions for 9 (cp(0) (p(L)) follows
boundary
condi-
=
If the system includes walls, we have to subtract from (A. 6) the contribution 6F(i), which corresponds to homogeneous elongations of the 1/xa (f)-oscillators across the wall. This degree of freedom is described by f. E.g. for a d 2 system with a wall with Q + - Q- i, there are rla such configurations, =
From (A. 2) follows ( cos T >0 exp(- -L (p2 )o). According to point (i) of our approximation scheme (see section 3) we treat now the phonon spectrum and thence 92 )o in the wall free state, i.e. we replace in (A. 3) cos O(N) by 1. Then we get =
=
which
are
classified
by the
Thus, the phonon free The
by
phonon free energy in the wall free state is given
wave-vector
energy
F 0 (N)
is
kjj
=
qT T
639
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[3]
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