HAL Id: jpa-00209379
https://hal.archives-ouvertes.fr/jpa-00209379
Submitted on 1 Jan 1982
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
The commensurate-incommensurate transition and fluctuations in two and three dimensions
T. Nattermann, S. Trimper
To cite this version:
T. Nattermann, S. Trimper. The commensurate-incommensurate transition and fluctuations in two and three dimensions. Journal de Physique, 1982, 43 (1), pp.23-29. �10.1051/jphys:0198200430102300�.
�jpa-00209379�
The commensurate-incommensurate transition and fluctuations in two and three dimensions
T. Nattermann
Sektion Physik der Humboldt-Universität, Bereich 04,1086 Berlin, DDR
and S. Trimper
Sektion Physik der Karl-Marx-Universität, 7010 Leipzig, DDR
(Reçu le 9 avril 1981, révisé le 28 juillet, accepté le I S septembre 1981)
Résumé. 2014 Nous calculons à l’aide de l’approximation harmonique autocohérente le diagramme de phase d’un système de sine-Gordon à deux et trois dimensions. Pour d = 2, nous trouvons trois phases avec, en général, des
transitions discrètes entre les deux phases incommensurables (IC) A et B et la phase commensurable (C). Nous
montrons que la phase incommensurable A existe dans un domaine de température plus grand que celui estimé par Saito [20]. Pour d = 3, seules existent la phase incommensurable A et la phase commensurable. La disconti- nuité à la transition commensurable-incommensurable est proportionnelle au rapport du paramètre du réseau à la
largeur des parois des domaines. A suffisamment haute et basse température, la thermodynamique du système est
donnée par la théorie classique. Les fluctuations cependant diminuent la stabilité de la phase commensurable.
Abstract.
2014The phase diagram for the two- and three-dimensional sine-Gordon system with incommensurability
is investigated by means of the self-consistent harmonic approximation. For d = 2 we find three phases with in general discontinuous transitions between the two incommensurate (IC) phases A and B and the commensurate
(C) phase, respectively. It is argued that the IC-phase A extends over a larger temperature regime than anticipated by Saito [20]. For d = 3 only the IC-phase A and the C-phase are found. The discontinuity of the IC-C transition is
proportional to the ratio of the lattice parameter to the width of the domain walls. For sufficiently low and high temperatures the thermodynamics is given by the classical theory. Fluctuations decrease however the stability region of the C-phase.
Classification
,Physics Abstracts
64.60
-64.70
1. Introduction.
-Many physical systems exhibit modulated phases, where the variation of the order parameter (e.g. magnetization, polarization, charge density, mass density, etc.) is not in registry and in general not commensurate with the underlying lattice (for a review of systems exhibiting incommensurate
phases see e.g. Villain [1], Dvorak [2], Przytawa [2]).
The actual modulation of the ground state is the result of the competition between elastic terms which favour the incommensurate (IC) phase and local Umklapp
terms which favour the commensurate (C) phase.
If the coupling to the lattice is strong enough, the
latter may even drive the system to undergo an IC-C
transition to the commensurate phase.
A refined Landau theory of this transition has been considered by many authors [3-8]. In particular, it was shown, that close to the IC-C transition the IC-phase
consists of large, almost commensurate regions, seperated by narrow domain walls where the phase of
the order parameter changes rapidly. The density n
of these domain walls approaches zero continuously
n oc In - 03B4c] as the C-phase is reached. 6 is defined in equation (2.1).
The influence of fluctuations on this picture has
been considered so far for a 1-D quantum system [9]
and 2-D classical systems [10-14]. The authors map the problem onto a 1-D interacting spinless Fermion
model or the 2-D Coulomb-system in an imaginary field, respectively. It is then possible to solve the problem exactly at least at a special temperature
T = 2 To, where n oc (b _ ðc)1/2, contrary to the classical theory. Nattermann [15] has tried to attack
the problem by using the renormalization group (RG) approach and obtains the same result for n in d = 2 + c dimensions. The RG-approach gives probably the
correct expression for the single domain wall energy
(see also [16]). Its application is however a delicate
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198200430102300
24
problem and has been criticized in its present form
(Villain [17] and private communications). We men-
tion here only two points, which make the problem so
difficult and do not arise in a RG-treatment at ordinary
critical points : (i) the condensed mode is described by
a large (or infinite) number of critical wave-vectors and (ii) the fluctuation spectrum of the IC-phase depends on the profile of the condensed mode and is
even complicated at zero temperature (including a
non-trivial density of wave-vectors). Further, Villain [18] concluded from general arguments, that for d = 3 the classical theory should be essentially correct,
a statement which probably cannot be proved by approximative methods. Nevertheless, it seems to be
useful to test other calculation schemes in treating
this problem.
In this paper we consider the influence of thermal fluctuations on the IC-C transition by means of two
versions of the self-consistent harmonic approximation (SCHA), known from the theory of the anharmonic
crystal [19]. The first version corresponds to the more general variational treatment, but is unfortunately impracticable. However, it is possible to find non-
trivial results for the second one. After the main work of this paper has been done, we were informed about
a paper by Saito [20], who used essentially the same
calculation scheme as our version II. Since Saito [20]
restricts himself to the case d = 2 and moreover we
do not agree in all conclusions with Saito’s results, this paper seems to have its own right.
The main results of our approach are :
(i) For d = 2 we find three phases : a C-phase,
a low-temperature IC-phase (ICj and a high-tempe-
rature IC-phase B (ICB) with discontinuous IC-C tran-
sitions, in agreement with Saito [20]. In [20] the IC- phase B has been called a liquid phase, which is misleading, since dislocations are excluded from the very beginning (we thank the Referee for pointing
out this fact). The appearance of two IC-phases is
therefore probably an artefact of the SCHA. More
likely, the ’CA-’CB transition line reflects a strong
quantitative change in the properties of a unique IC-phase which appears in an exact treatment. Con- trary to Saito [20] we do not find a continuous ’CA-’CB
transition at 2 To for all values of the parameter 6.
Here 6 is defined in equation (2.1) and To denotes
the ICB-C transition temperature for 6 = 0. Moreover
we argue, that the ’CB-C transition is restricted to
T=To.
(ii) For d = 3 only the C- and the ICA-phases are
found. The IC-C transition is again discontinuous.
However, the deviations from the classical theory (and hence the discontinuity also) are of the order of magnitude TB,ITO A. Here To is proportional to
the strength of the elastic interaction.
denotes the critical value of 6, at which the IC tran-
sition appears. 03B4c-1; 1 is approximately the width of the
domain walls close to the IC-transition. A denotes a
microscopic cut-off A = nla where a is the lattice
spacing. Since we adopt a continuum version, the
ratio ðcl A = a/7rb - ’ is assumed to be small throughout
the calculation. Thus TbclTo A - 0 for very low and very high temperatures, respectively and the classical
theory remains valid besides of the renormalization of the phase boundary bc(T).
Despite of the fact that first order transitions are
usually considered to be artefacts of the SCHA we
believe that the influence of fluctuations in two and three dimensions is quite correct described in calcu-
lating the phase boundary and the broadening of the
width of the domain walls. Moreover, an estimation of the jump of the order parameter in K2Seo4 shows good agreement with experimental results [21].
The paper is organized as follows. In section 2 we
outline the two versions of the calculation scheme of the SCHA and give a qualitative discussion of the solution of version I. Section 3 is devoted to the analy-
sis of the version II of the SCHA and to the discussion of the phase diagram.
2. Self-consistent harmonic approximation.
-To
be specific, we consider in this paper the IC-C tran-
sition of a system which can be described by a sine-
Gordon model with an additional linear gradient term [5-8]
The Hamiltonian (2.1) can be understood :
(i) as the truncated form of a Ginzburg-Landau
Hamiltonian allowing an incommensurate phase [7].
Close to the IC-C phase boundary, the amplitude A
of the complex order parameter T = A exp(io) is
here assumed to be spatially constant [5]. However A
and hence To oc A 2 and v oc A p- 2 are temperature dependent A 2 oc (Tc - T) where Tc is the transition temperature to the disordered phase in the framework of the Landau- theory ;
(ii) as a continuum description of incommensurate ordered chains imbedded in a (two- or three-dimen-
sional) matrix. In this case 0 describes the deviations of the chain atoms from the positions of the commen-
surate ordering [9]. In treating (2.1) we will disregard
the fact, that 0 as a phase variable is given only up to
a multiple of 2 n, i.e. we exclude the existence of vortex
configurations, dislocations etc...
The true free energy F corresponding to (2.1)
follows from
where { xi } is the set of sites of the underlying lattice
with the spacing a. Actually we will not consider the limit a - 0, but assume a’ 1/v. Since V-1/2 is the
characteristic length for the modulation of 0, this is just the condition for the applicability of the conti-
nuum description.
Instead of F we calculate here the free energy of the
SCHA from [19]
with the trial Hamiltonian
where 4>(x) = §s(x) + (p(x) and
... >0 denotes the thermal average with Ho
The value of the function os(x) and the mass m(x) are
now determined from the variational treatment, which minimizes F1 :
In order to calculate (2. 7) we rewrite ( H - if 0 > 0 as
The terms in the last bracket vanish, since
From (2. 7) we find then, assuming a modulation only
in the xl = x-direction
Both equations take the same form as in the ground
state but with v replaced by the spatially varying effective coupling parameter [22]
The relations (2.3)-(2.5) and (2.8)-(2.10) determine
the calculation scheme of the SCHA-I.
Since all averages depend crucially on the fluc-
tuation spectrum of (p, a comment about the relation between an exact treatment and the SCHA is impor-
tant. In an exact treatment, which uses the bare (or
also a proper renormalized potential rn(x) for the (p- fluctuations as the starting point) nKx) is a periodic
function of x with the period l (Fig.1 a). As it has been shown in [9], in this case the excitation spectrum consists of a gapless acoustic branch and an optic branch, separated by a gap at ) k 1 I = nll (see Fig. 1&).
The gapless part of the spectrum leads to a diver-
gence of qJ2 >0 for d 2. Ford > 2 ( qJ2 >0 is finite,
Fig. la.
-Effective potential for the ~-fluctuations in an exact
treatment (solid line), in the SCHA-I (dashed line) and in the SCHA- II (dot-dashed line).
Fig. lb. - Resulting fluctuation spectrum for the three potentials
depicted in figure la.
26
but reaches its maximum at the positions of the walls.
In the SCHA-I the excitation spectrum agrees with that of the exact treatment for T = 0, since
For T "# 0 we did not succeed in finding a solution
of the calculation scheme I. However, qualitatively
we expect the following picture : since the fluctuations
qJ2(X) )>o at the position of a wall are larger than in
between two neighbouring walls, the potential for the (p-fluctuations decreases in magnitude most strongly
at the walls (Fig. la). As a result we suppose that due to the SCHA the small k-spectrum acquires a gap
(see Fig. 1 b). In this case there is also a non-trivial solution for the IC-phase in d 2 dimensions, since
qJ2 > 0 remains now finite. A finite ( (p2 )o was
however also a supposition for the existence of a
non-zero mass m2(x). Thus we conclude, that there is
a non-trivial solution for our calculation scheme I, corresponding to an intrinsic IC-phase A for d = 2, 3.
Besides of this, there exist two other solutions with
spatially constant m : the first one corresponds to an
incommensurate phase B with 4>s = bx, m(x) =- 0.
The second solution describes the C-phase, where 4>s = 0, + 2 03C0, ... and m2 = v exp( - ! qJ2 >0)’ The
corresponding free energies will be calculated in section 3.
In order to have a more tractable version of the
SCHA, we introduce now an additional approximation replacing cos ~s(x) in equation (2.10) by its spatial
average
(see Fig. la). This implies, that f, qJ2 >0 and m become
position-independent and the fluctuation spectrum consists of only one branch (see Fig. lb). Approaching
the IC-C phase boundary, the gap of the approximated spectrum increases and reaches the value of the C-
phase at the boundary. This approximation scheme,
which we denote SCHA-II and indicate by omitting
the tilde, could be also obtained by assuming m as spatially constant from the very beginning. It is
therefore the same as that used by Saito [20].
The free energy F, can be decomposed into three
parts
where Fs and Fo denote the free energy of the rigid
wall lattice and the harmonic fluctuations around this
configuration, respectively. We note, that in the case
where the SCHA-I scheme delivers a spacially varying mass ax), the SCHA-I leads to a lower free energy
F1 Fl, since it satisfies a more general variational
treatment.
The analysis of the calculation scheme II, the main
advantage of which is its relative simplicity, will be performed in the following section.
3. Analysis of the SCHA-11 scheme.
-In this
section we exploit the calculation scheme II derived in section 2 in two and three dimensions. To this aim it is convenient to introduce a further, as we believe
not very essential approximation, namely we replace everywhere W(oo = 1
-cos os by the parabolic potential W(oo = 2 a2 ~2, - 7r qls 03C0 with the
same periodicity W(~s + 2 nn) = W(~s) [1, 4, 7].
In this way one avoids the use of elliptic functions.
Below wo choose a = 8/n2 such that for T = 0 the
phase transition occurs at the same 6 as for the original
cosine potential. Then (2.9) and (2.10) become finally
which yield [7]
§£x) = n sinh (aKx)/sinh (al/2)
With (3. 3) and the replacement of the potential, we get for the free energy of the walls
Calculating Fo and FR, we have to distinguish the
cases d = 2 and d = 3.
We start with the consideration of the two-dimen- sional system. From
we obtain with (2.11)
Using equations (3.4) and (3-6) m can be eliminated
and we find, neglecting exponentially small terms
Kc- 1 is proportional to the effective width of a wall,
which is broadened by thermal fluctuations and
n = rx1f,z 12 Kc 1 denotes the fraction’ of the system which is occupied by the walls. Next we calculate Fo
as
where the constant means m-independent terms which
are not of interest here. Since the walls were assumed to be broad Kc ’ 1 A, we neglect terms of the order
m2IA 2. From (2.15), (3.6) and (3-9) we get then
Finally, from (3. 5), (3.7), (3. 8) and (3.10) we find the following expansion for the reduced free energy density
The dots denote higher order terms. In writing (3.11)
we have omitted terms which are equal in all three
phases, to be discussed below. In order to find the actual value of 11(t, 6), one has to minimize 11(n; t, 6)
with respect to n (or to l). At t = 0 equation (3.11)
reduces itself to the result of the classical theory.
At finite t, however, the SCHA leads to an attractive interaction between the walls if I is large. Decreasing 6 a second minimum of fi at n = 0 appears in the region
ðc 6 ðc. The transition becomes discontinuous in this case. The upper limit for the stability of the C-phase
is in agreement with the result from the renorma- lization group calculation [10, 15, 16]. The ICA phase
reaches its stability limit at
where the order parameter jumps from
to zero, t 1. The discontinuity of the transition is however probably an artefact of the SCHA.
In order to show, that fl(n) is actually bounded from
below for large n in the IC-phase, it is convenient to follow [20] and to consider f, as a function of both n and K :
The minimum condition f1/ðK = 0 agrees with the result from the variational treatment (see Eqs. (3.4)
and (3. )). In accordance with SCHA-I and Saito [20]
we find three phases :
(i) The C-phase, where n = 0, K = Kc and
Thus 11,C changes its sign at 6 = b 1 = 2( 1 - t) Kc.
(ii) An incommensurate phase B with m = x = 0,
n = 2 6/nK,, and 11,n = 0.
(iii) An IC-phase A with K # 0 and n # 0. The minimum condition OflIOn = 0 has now the same
form as for T = 0 (but K2 0 v) and leads to a non-zero
value for n-’, a fact, which cannot read off directly
from (3.11). Despite the simple form of (3.13) we
are not able to find a closed expression of f, as a
function of t and 6. However, contrary to [20] we do
not see a change of the sign of fl,A at t = 1/2 close
to the transition to the C-phase. This is connected with the fact, that Saito [20] considers the case of large incommensurability (the module of the elliptic func-
tions vanishes). Moreover, we remind on the fact, that in the intrinsic IC-phase 11 f, and therefore its
stability region will be larger than even a correct
calculation of f 1 would predict. From the discussion of different special cases for fl,A(t, b) and from the result of [10, 13] we anticipate a phase diagram as given in figure 2. The ICB-C transition is then restricted to the temperature T = To only.
We come now to the discussion of the three-dimen- sional case. Since the calculation goes along the same steps as for d = 2, we will be brief. From
we get
28
Fig. 2. - Expected phase diagram for the sine-Gordon model with incommensurability in d = 2 dimensions.
Finally, with
we obtain for the reduced free energy
From the minimum conditions we find only two phases :
(i) The C-phase with n = 0, m = x = Kc and
(ii) The IC-phase A with n > 0, m > 0 and K K,,.
Also here we are unable to determine 11,IC(t, 6) expli- citly.
’
Eliminating K from 11 we get for n 1 the following expansion :
The negative sign in front of the n2, n3, ... terms lead again to a first order transition between the C and the
IC-phase. There is no IC-phase B solution with m = 0.
It is interesting to note, that for t, KclA --+ 0, 0
vanishes and therefore
In this case follows a continuous IC-C transition at
where vanishes as in the
P-2
classical theory. Note, that V1/2 oc A 2 is itself temperature-dependent and vanishes at T = T c (p > 2).
Since Vl/2 lA, i.e. the ratio of the basic lattice constant to the width of the domain walls at T = 0, was assumed
to be small, we conclude that, for T To and T > To,
0 - 0. Then practically fluctuations only increase the width Kc ’ 1 of the walls. The critical behaviour n(b)
is the same as in the classical theory. This is in agree- ment with the result from general considerations [18].
In the intermediate region T To where
. ,,
