HAL Id: jpa-00209062
https://hal.archives-ouvertes.fr/jpa-00209062
Submitted on 1 Jan 1981
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.
Splitting of commensurate-incommensurate phase transition
V.L. Pokrovsky
To cite this version:
V.L. Pokrovsky. Splitting of commensurate-incommensurate phase transition. Journal de Physique, 1981, 42 (6), pp.761-766. �10.1051/jphys:01981004206076100�. �jpa-00209062�
In the last few years, the C-I phase transition has become the subject of keen interest for experimenters
as well as for theoreticians [2]. The theoretical approach
was mainly based on a continuous approximation.
The effects of discreteness of a lattice was first consi- dered by Aubry [1]. He established a rather compli-
cated equilibrium arrangement of atoms of a chain interacting via next-neighbour forces and placed into
an external periodic field. In particular, he found a very irregular sequence of C-I phase transitions near any rational ratio of a chain (overlayer) and external field (substrate) period. Similar conclusions have been obtained in the works by the author and
Talapov [3]. Bak and Boehm [4], Bak [5] and Villain
and Gordon [6] applied the Aubry’s idea for a descrip-
tion of real systems.
The main purpose of this work is to show that the C-I phase transition splits into two phase transitions.
In the first of them, solitons arise spontaneously;
however, they are pinned rigidly to sites of a substrate
lattice. The second phase transition unpins solitons turning them into a periodic superstructure moving
without resistance.
Analogous conclusions have been obtained recently by Aubry [7, 8]. Unfortunately, these publications
have not been available to the author until the present work was. completed. We believe, however, that a repeated consideration of this difficult problem is justified, especially because we rederive the statements of the works [1, 7, 8] in a less formal, more simple and
quantitative way. We represent a mathematical formu- lation of the problem, which can be easily comprehend-
ed and used by physicists.
1. Solutions of the equilibrium equation. - Let us
consider the simplest problem of an equilibrium configuration of a chain consisting of atoms connected
with strings and placed in an external periodic field.
It can be described by the potential energy :
where xn is the coordinate of the nth atom, q is the length of a free string, V(x) is a periodic potential with
the period ao. Introducing displacements,
from minima of the periodic potential, the equation
of equilibrium arrangement is
For the sake of simplicity, we shall consider further the case V(qJ) = V(1 - cos qJ). Introducing the first difference, cvn = qJn+ 1 - qJn, the second order diffe-
rence equation (2) can be turned in the usual way into
two first order difference equations :
761
LE JOURNAL DE
PHYSIQUE
Splitting of commensurate-incommensurate phase transition
V. L. Pokrovsky
Landau Institute for Theoretical Physics, Academy of Sciences of the U.S.S.R., Chernogolovka 142432, U.S.S.R.
(Reçu le 17 novembre 1980, accepté le 23 Jevrier 1981 )
Résumé. 2014 Nous montrons que la nature discrete du réseau a pour effet de séparer la transition de phase com-
mensurable-incommensurable (C-I) en deux transitions de phases. La première conduit à un état avec une sur-
structure de solitons accrochés au réseau. La deuxième rend les solitons mobiles, leur donnant un degré de liberté
continu.
Nous reformulons l’approche d’Aubry de façon à en simplifier le traitement mathématique.
Abstract. 2014 The discrete nature of a lattice is shown to manifest itself in a splitting of the commensurate-incom-
mensurate (C-I) phase transition into two phase transitions. The first of them brings an overlayer to a state with
a pinned soliton superstructure. The second phase transition unpins the soliton superstructure supplying it with a
continuous degree of freedom.
The treatment by Aubry [1] is rederived to simplify its mathematical formulation.
J. Physique 42 (1981) 761-766 JUIN 1981, 1
Classification -
Physics Abstracts
63.75
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01981004206076100
762
Equation (3) can be treated as a mapping T of the
2-dimensional vector space (wn, qJn) into a 2-dimen-
sional space (wn + 1, 9,,, 1). We shall rewrite it, omitting
the indices
This approach was proposed by Aubry [1]. The general properties of similar mappings were investigated in a rigorous way by Birkhoff [9]. We present here a non- rigorous, but more quantitative and visual treatment.
We consider the invariant curves of the transformation T (3’). Suppose cv = (o(g) is the equation of an inva-
riant curve. Then after the transformation T :
cv ... 4,wf(a), (p), (p - qJ’(w, (p), the points (o, 9’ have to satisfy the same equation. In other words
Substituting cv’, (p’ from (3’), we obtain the func-
tional equation for the function w(qJ) :
We shall search for analytical solutions of equa- tions (5), assuming y to be small. Expanding w in a
power series in y
one obtains for Wo :
The only analytical solution of this equation is
cvo = const. The equation in the first approximation
is :
Its solution has the form :
The first approximation term in equation (6) is small, compared to the zero approximation one, provided
cvo lies out of bands of width - / stretched along the
lines wo = 2 nm, where m is an integer. We shall call them first order dangerous bands.
For the second approximation, one has the equation
with the solution
The term proportional to 7’ in equation (6) is small, compared to a linear term, if cvo lies out of the first order and second order dangerous bands. The latter
are bands of the width - y stretched along the lines
w = 03C0(2 m + 1).
One can easily check that in the nth approximation,
a new denominator sin (rruo/2) appears resulting in the
exclusion of the nth order dangerous bands stretching along the lines w = 2 03C0m/n with the width _ yn/2.
No secular terms arise (1). The total width of the
dangerous bands is proportional to joy at small y.
Hence, for small y, most of the invariant curves are
slightly modulated straight lines. Now the question
is what the invariant curves look like in the dangerous
bands. We consider first the simplest case I OJ I 1.
Returning to equation (5) and expanding it in powers of oi and .JY, we obtain the equation of the first
approximation :
Its solution is :
where C is a constant. For C > 1, solution (12) represents a periodic curve. For - 1 C 1, it represents a closed curve degenerating into a point
qJ = (2 m + 1) rc for C = - 1. The points 9 = m03C0,
w = 0 are also the fixed points of transformation T.
The value C = 1 presents special interest. It corres- ponds to a separatrix connecting two fixed points.
The total picture of invariant curves near the first
dangerous band is shown in figure 1.
Fig. 1. - Invariant curves near the first dangerous band.
The physical meaning of-a separatrix solution is quite clear. It represents a single soliton bringing one
extra particle or hole. As long as a separatrix connects
two fixed points, a continuous degree of freedom can
be attributed to a single soliton. Indeed, one can start
from an arbitrary point of the separatrix and moving right or left obtain the values 9 = 2 z or 0, respecti- vely, at n - ± oo. Geometrically, this degree of free-
(1) The best way of checking the absence of secular terms is the
investigation of the initial equation (2) which is equivalent to (5), assuming n to be a continuous variable.
dom is the coordinate of the centre of mass of a
soliton. It means that a single soliton in this approxi-
mation is not pinned to a lattice. Going to the next approximation, we represent co as a sum
with W(l) satisfying equation (12). The equation for
W(2) is : .
The solution of equation (13) should be chosen in a way to avoid singularities at C 1. No new constants
enter the solution :
For small y the correction (14) does not change the picture of invariant curves qualitatively. The nth approximation is proportional to 03B3nl2 and represents
a sum of sines and cosines of kcp with k varying from 1
to n/2 for even n and (n + 1)/2 for odd n (2). Thus, the qualitative picture of invariant curves is not changed by any power correction to equation (12). The same
consideration is valid for any first order dangerous
band.
Considering the second order dangerous bands
near the lines co = (2 m + 1) z, we obtain an equation similar to equation (11) for the function
with the substitution of y by y2 and sin (9 + co) by
sin 2(qJ + w). We shall not repeat the previous consi-
deration. The result is geometrically obvious. Near
the modulated curve of the first approximation
a secondary structure of closed and unclosed curves
separated with separatrices arises. Its period is two
times smaller than the period of modulation and the
period of the 1 st dangerous band structure. Its width is proportional to y.
Now we can imagine the whole picture of the
invariant curves. It consists of dangerous bands filled
by the corresponding separatrix structures and conti- nuous , curves between them. This picture is shown schematically in figure 2.
I
The points of crossing of separatrices are fixed points of the transformations T, T2, T3, T 4 etc. From
the physical point of view these points correspond to
different commensurate phases. Separatrices corres- pond to solitons near a given commensurate phase.
(2) As in the previous case, the best way of checking it is to ana- lyse equation (2).
Fig. 2. - General view of invariant curves.
With the precision of any power of y, every sepa- ratrix connects two fixed points. In other words,
solitons are not pinned with any power-like precision.
The unclosed curves correspond to periodic solitori
structures while the closed ones correspond to oscillat- ing arrangements of atoms. The general picture of
invariant curves is the same as that in the Kolmogorov-
Arnold-Moser theory (see the review by Arnold [10]).
2. Pinning of solitons and stochastic region. - The power-like approximation appears to be insufficient to
yield the soliton pinning phenomena, which seem physically to be quite unavoidable. The reason is that the series (6) does not converge at any y. It is an
asymptotic series. It is already clear from the fact that
by changing y only slightly we can turn the closed
curves into unclosed ones, and vice-versa. Thus, we
can obtain a very sharp change in the solution co(g)
of equation (5). We can estimate the precision of the
power approximation by the following consideration.
The term of the nth order contains sin n(g + co),
We expanded this term in powers of w. However, the correction is small only in the case mv « 2 03C0. Other- wise, our method fails. For the first dangerous band
w ~ *03B3. Hence, the series does not work at
n > 2 7r/,,/-y. Assumin the nth term of the series to have the form n ! y/2 a)" at large n, we obtain the
minimum of this value just at the n indicated above where it is equal to exp( - 2 n/JY). Certainly, our
estimate is very crude. A more accurate estimate will be
given later.
The main change in the picture is related to the separatrices. We can define a separatrix locally as an
invariant curve entering or going out of the hyperbolic
fixed point, say, w = 0, qJ = 0 (see Fig. 1). However, there is no reason for the separatrix to reach precisely
the next fixed point. We have shown them to reach it in any power approximation. Therefore, their deviation from the next and other fixed points is exponentially
small. They begin to walk randomly between fixed
points located in an exponentially narrow band of
width - exp( - const./,fy-), along the « old » sepa- ratrices. Their behaviour is shown schematically in
764
figure 3. The computer simulation of invariant curves
demonstrates a stochastic behaviour of trajectories forming a narrow separatrix band of width 10-4 at y = 1/6.
Fig. 3. - Random behaviour of separatrices.
We consider now two neighbouring fixed points A
and B (Fig. 4). Two separatrices, one going from A and
another coming to B, have to intersect generally. Foi
a rather simple case of Hamiltonian systems with a small perturbation violating conservation of the action integrals, the rigorous proof of the separate crossing and qualitative estimates of their splitting
was given by Mel’nikov [11]. Unfortunately, his esti-
mate cannot be applied to our case since the effect oi
crossing is exponentially small. Qualitatively, it results
in the pinning of soliton.
Fig. 4. - Intersection of separatrices.
Indeed, if two separatrices intersect at some point Oo, they have to intersect in an infinite discrete set of
points 0 ± 1, O ± 2... which can be obtained from 0, with the repeated transformation T or T -1. This set tends asymptotically to points A and B. Therefore, the only way to come from A at n = - oo to B at n = + oo
is to get precisely at this set of points. The only degree
of freedom for a single soliton on a discrete lattice is the choice of the « initial » point from this discrete set.
It corresponds to a translation of a soliton by the
lattice constant. Other solutions of equation (2) with
initial points on a separatrix correspond to the picture of randomly distributed solitons and anti-
solitons, as is shown schematically in figure 5.
Fig. 5. - Typical solution in the random region.
However, there exist quasiperiodic curves out of the
stochastic region which do not intersect, having no
width and stochasticity. One can start from any point
of these curves resulting in the continuous degree of
freedom for a periodic soliton structure.
For the closed invariant curves, the transformation T is equivalent to a rotation through a small angle
0 - y1/2. Applying T approximately y-1/2 times,
one can obtain new fixed points on closed curves
which in turn take on the separatrix structure. The whole picture becomes more complicated. It is shown schematically in figure 6. Certainly, one can continue ,
this procedure to obtain fixed points of the next order,
etc. Fortunately, these tiny details are not relevant to
the physical problem and are presented here only for
better comprehension of the situation.
Fig. 6. - Fine structure of invariant curves.
Since the exact proof of the intersection and sto- chastic behaviour of separatrices is not known so far, computer simulation is important for our work.
Therefore, we have to check the possible influence
of computer errors on the previous results. For this reason, we considered the potential with the deri- vative :
V’(qJ) = 4 arctan sin qJ(tanh v ry r /2)2
. ( 15)
1 + (tanhjY/2)2 cos (p
At small y it becomes asymptotically equal to y sin qJ.
However, for this potential there exists an exact
separatrix solution of equation (2) :
tan (qJJ4) = C eV1n (16)
with an arbitrary value of the constant C. It means that separatrices do not intersect and there is no stochastic
region. Indeed, the computation did not show any width or stochastic behaviour of curves for more than a
thousand repetitions of the transformation while they
can be seen with less than a hundred steps for V’(qJ) = y sin qJ at the same value of y = 1/6.
3. Estimate of the pinning energy. - We apply the
Poisson summation method for estimating the pinning
energy of a soliton at a zero initial misfit Ô = a - ao.
Starting with equation (1), one can obtain :
where
The continuum approximation for ~n in equation (2)
is equivalent to solution (9) and gives :
The integration constant in (19) is chosen in a way to
correspond to the single soliton solution. We estimate
only the term with m = 1 in (17), since the term with
m = 0 gives the continuum approximation result and
the rest of the terms are small. Inserting equations (18)
and (19) into this term, one obtains an integral :
Taking into consideration the smallness of y the function can be approximately written as follows :
The integral (20) with the function f (n) from equa- tion (21) can be exactly calculated :
Asymptotically at small y we obtain for the pinning
energy
The large preexponential factor 32 03C02 is strongly suppressed by the small exponential factor. In our
numerical calculations the width of the stochastic
region was - 10-4 at y = 1/6.
The nature of the pinning energy is the same as the nature of Peierls barriers for dislocations. Moreover, it is just the Peierls force for the Frenkel-Kontorowa model of dislocation.
4. Pinned soliton and incommensurate phases. -
So far we investigated, say, the most probable solu-
tions of equation (6) (or Eq. (2)) (3). However, the
absolute minimum of the energy may correspond to a
very improbable curve. The simplest example is a
commensurate phase, corresponding to a hyperbolic,
and consequently, unstable fixed point.
(3) We can introduce the probability of a curve as a measure of
its initial point on the plane (w, ~).
In the continuum approximation, the energy of a
single soliton is equal to
At some critical value of 6
the creation of solitons becomes energetically favou-
rable and only their repulsion stops their further multiplication. The pinning energy shifts slightly
the critical value of 6, but does not change anything in principle. Let 6 be very close to 6,. Then the distances between individual solitons are very large. The energy Bint of interaction between two solitons separated by
the distance much larger than the width of an indi-
vidual soliton 10 = y-1/2 a can be easily calculated
in the continuum approximation. The result is well known
Comparing this result with the pinning energy (23),
two different situations can be distinguished. The
first occurs at lla > ¡r;2Iy. Then the pinning energy
prevails. In this case, we can consider solitons as particles interacting one with another according to equation (26) and situated in the sites of a linear lattice. This is the problem of a one-dimensional lattice gas with an exponentially decreasing inter-
action at zero temperature. It was solved exactly
in the works of Hubbard [12] and Pokrovsky and
Uimin [13]. The resulting arrangement of atoms can be described by a continued fraction and for any
given concentration of solitons
it is defined unambiguously. We refer to the original
works [12, 13] for details. A most essential feature of this solution is the absence of acoustic excitations, i.e., the absence of a continuous group of transfor- mations. Nevertheless, the whole picture changes continuously with n in a definite way. This means that
a small variation bn of n causes changes in arrange- ments at distances larger than (bn)-1. In terms of the
initial misfit 6, this region is 03B4 - 03B4c exp( - 03C02 y - ).
In the second region 6 - 03B4c > exp(- 03C02 y- 1), the
soliton repulsion prevails and solitons are not pinned.
Strictly speaking there should be no pinned soliton phase in the one-dimensional case, since the Peierls barriers form a periodic potential for a soliton.
According to a quantum mechanical consideration, the periodic potential does not interfere with the free motion of a particle and results only in limiting the possible energies for a particle. The situation is quite
different in two- and three-dimensional cases. Solitons
are linear or plane defects in these cases. Therefore, the soliton energy is proportional to linear size L of a
system or its cross-section area. In the thermodynamic
