10.1.1.131.4735

Rep. Prog. Phys., Vol. 45, 1982. Printed in Great Britain
Commensurate phases, incommensurate phases and the
devil’s staircase
Per Bak
H C 0rsted Institute, Universitetsparken 5, DK-2100-Copenhagen 0 , Denmark
Abstract
Modulated structures with periods which are incommensurable (or high-order com-
mensurable) with the basic lattice are quite common in condensed-matter physics.
The structure may be another lattice, a periodic lattice distortion, a helical or sinusoidal
magnetic structure, or a charge density wave in one, two or three dimensions.
This review surveys recent theories on the transition between commensurate (C)
and incommensurate (I)phases, and on the properties of the ‘incommensurate’ phase.
The predictions of theories will be compared with experiments. The CI transition is
usually described in terms of wall, or soliton, formation. The nature of the transition
and the structure of the I phase are quite different in two and three dimensions. In
three dimensions the I phase seems to consist of an infinity of high-order locked C
phases, which may or may not be separated by an infinity of truly incommensurate
phases. This behaviour is known as the ‘devil’s staircase’. In two dimensions the
incommensurate phase (at T # 0) is a ‘floating’ phase without complete long-range
order, and it does not ‘lock-in’ at high-order commensurate phases. Phase diagrams
are determined by the stability of two types of ‘topological’ defects: walls, which
destabilise the C phase with respect to I phases, and dislocations or vortices which
generate paramagnetic or fluid phases. A consequence of this competition is that for
sufficiently low order of commensurability the C and I phases are separated by a fluid
phase.
The properties of modulated systems can be studied by iterating certain area-
preserving two-dimensional maps. Very recent studies indicate that, in addition to C
and I phases, there are chaotic structures which are at least metastable. The chaotic
regimes separate C and I phases and may be described as randomly pinned solitons.
The relevance of the chaotic regimes to adsorbed monolayers, pinning of charge
density waves, Peierls transitions and spin glasses is briefly discussed.
This review was received in July 1981.
0034-4885/82/060587 +43$08.00 @ 1982 The Institute of Physics 587

588 P Bak
Contents
1. Introduction
1.1. Incommensurate, commensurate and chaotic states: definitions and
1.2. Incommensurate systems in condensed-matter physics
1.3. Another simple model: the anisotropic Ising model with competing
1.4. Layout of the review
2.1. The theory of Frank and Van der Merwe (1949) (FVdM)
2.2. Coupling to the lattice
2.3. Phasons
2.4. Adsorbed monolayers: the Bak et af (1979) (BMVW) theory
2.5. CI transitions in two dimensions: the incommensurate phase
2.6. The Pokrovsky-Talapov (1980) theory
3. Global phase diagrams of periodic systems
3.1. The anisotropic 3D Ising model with competing interactions
3.2. Two-dimensional anisotropic Ising models with competing
3.3. 2D ANNNI model: the Villain-Bak theory (1981)
4. Chaotic states and the devil’s staircase
4.1. Chaotic behaviour of the Frenkel-Kontorowa and ANNNI models
4.2. Chaotic behaviour of the discrete rp4 model
4.3. Relevance of chaotic states to physical systems
Acknowledgments
References
a simple model
interactions (ANNNI model)
2. The commensurate-incommensurate transition
interactions: equivalence with the 2D x y model
Page
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623
625
626
626

Commensurateand incommensuratephases 589
1. Introduction
1.1. Incommensurate, commensurate and chaotic phases: definitions and a simple
model
Consider the model defined in figure 1. An array of atoms connected with harmonic
springs interacts with a periodic potential with period b. This could be a model of
Figure 1. The one-dimensional FVdM model. The springs represent interactions between atoms, the wavy
line the periodic potential. ( a )Commensuratestructure, (b)incommensurate structure, (c)chaotic structure.
interacting gas atoms adsorbed on a crystalline substrate. Despite its extremesimplicity
the model exhibits most of the features to be discussed in the following. The model
with a cosine potential was originally introduced by Frenkel and Kontorowa (1938),
and it has been studied by several authors, in particular by Frank and Van der Merwe
(1949),Theodorou and Rice (1978),Aubry (1979),Greene (1979)and Bak (1981~).
The Hamiltonian may be taken to be
where xn is the position of the nth atom. In the absence of the periodic potential, V,
the harmonic term would favour a lattice constant a. which; in general, would be
incommensurable with b : the adsorbed lattice forms an incommensurate ( I )structure
(figure l(6)). In a diffraction experiment one would observe Bragg spots (or sheets)
at positions Q = 27rN/ao, N integer. None of these spots coincide with the Bragg
spots of the periodic potential at positions G=2~rM/b.If the potential is strong
enough it may be favourable for the lattice to relax into a comrnensurate ( C )structure
where the average lattice spacing, a, is a simple rational fraction of the period b.
Figure l(a)shows a situation where 2a = 3b. The atoms lose elastic energy, but gain
potential energy. The diffraction pattern of the substrate and the adsorbed layer has
an infinite set of coinciding Bragg sheets.
Even in the case where the potential is not strong enough to force the chain into
commensurability, the potential will always modulate the chain. The atoms will move
towards the minima. The average period may approach a simple commensurate value,
but remain incommensurate. In the most general incommensurate structure the

590 P Bak
position of the nth atom may be written (Janner and Janssen 1977, Aubry 1979):
X n =nu + a +f(na+a) (1.2)
where a is a phase and f is continuous and periodic with period b. Here a is the
average distance between atoms (which in general is different from ao)and f represents
the modulation of the chain due to the potential. Since the energy does not depend
on a,the chain is not locked to the potential. The symmetry of incommensurate
systems, including the continuous symmetry related to a,has been described in the
elegant work of Janner and Janssen (1977). The Bragg sheets are at positions
2 7 ~ M~ I T N
Q=- +-.b a
(1.3)
The spots at non-zero N thus form satellites around the spots from the basic lattice.
Experimentally it is impossible to distinguish between a high-order C structure and
an incommensurate structure. Since the commensuratenumbers are everywhere dense
one may speculate that the I phase will always be unstable with respect to some nearby
high-order C phase. We shall see that this need not be the case for a sufficiently weak
potential.
However, the C and I structures do not exhaust the stable configurations. There
are additional chaotic structures which cannot be described by the form (1.2). The
diffractionpattern is not made up of well-defined Bragg spots. It is not difficult to
convince oneself that such structures are possible, although an accurate treatment
requires some very complicated mathematics. Consider, for instance, the situation
where the potential is very strong compared with the elastic term. Clearly there exist
metastable configurations where the atoms are distributed in a random way among
the potential minima. Figure l(c) gives an example of such a structure. The chaotic
phase is ‘pinned’ to the potential. In contrast to the I phase it is not possible to shift
the lattice without climbing a potential barrier. In this respect the chaotic phase is
similar to the C phase, although the average period is, in general, incommensurate
with the potential. If the atoms were charged, the I phase would be conducting, the
chaotic phase insulating (Aubpy 1980a, b, Bak and Pokrovsky 1981).
The model in figure 1 is a one-dimensional zero-temperature one. At non-zero
temperature the dimensionality becomes important, and the possibility of a disordered
fluid or paramagnetic phases arises. The incommensurate phases and the global phase
diagrams are qualitatively different in two or three dimensions. In general, the periodic
potential will tend to ‘lock’the system into a commensurate configuration. How does
the periodicity, a (or wavevector q = 27r/a), change as some parameter (such as, for
instance, the natural periodicity a. in (1.1))is varied? Figure 2 shows various possible
situations which we shall encounter in the following. When the parameter x goes
from x1 to x2, q goes from q1 to q2. Figure 2(a) shows a simple analytic smooth
behaviour. The periodicity passes through an infinity of C values without locking;
the system is in a so-called ‘floating’phase. We shall see that this situation may occur
in two dimensions. Figure 2(b)is an attempt to show a function which locks-in at an
infinity of C values. The value of q/27r remains constant and rational at an infinity
of finite intervals of the argument x. Of course, the stability intervals decrease rapidly
as the order of the commensurability increases. If there are incommensurate phases
between the C phases the function is called ‘the incomplete devil’s staircase’ (Aubry
1979). Figure 2(c) shows ‘the complete devil’s staircase’ where the locked portions
of the function fill up the whole range of the argument x. Even if the function assumes

Commensurate and incommensurate phases
4
IC) 41--
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591
Id)-
-
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1.2. Incommensurate systems in condensed-matter physics
Incommensurate structures generally show up in systems with competing periodicities.
In a solid-state system one of the periods is that of the basic lattice. The other period
may be that of another lattice, as in the Frank and Van der Merwe (FVdM) model
(1.1).Rare-gas monolayers adsorbed on graphite constitute a two-dimensional realisa-
tion of this situation. Such systems have been the subject of extensive experiments
(see, for instance, Kjems et a1 1976, Chinn and Fain 1977, Shaw et a1 1978, Fain et
a1 1980, Nielsen er a1 1977, 1981, Stephens et a1 1979, Moncton et a1 1981) and
theoretical studies (Bak 1979a, Bak et a1 1979, Shiba 1980, Villain 1980a, b, c,
Coppersmith et a1 1981). As the temperature and pressure is varied a large variety
of phases exists. At low densities and high temperatures the monolayers form a 2D
fluid phase. At low temperature and low pressure there may be a ‘registered’ com-
mensurate phase.

592 PBak
i o ) l b )
Figure 3. Krypton monolayer adsorbed on graphite. (a) Commensurate ‘43 structure’. The krypton
atoms occupy 5of the graphite honeycomb cells. (b)Incommensurate phase.
Figure 3(a) shows the commensurate ‘J? structure’ of krypton adsorbed on
graphite. The krypton atoms may occupy one out of three sets of equivalent honey-
comb cells of the graphite lattice, A, B or C. For sufficiently high pressure and low
temperature the phase becomes incommensurate (figure 3(6)).
Graphite intercalation compounds constitute a natural extension to three
dimensions of the rare-gas monolayers adsorbed on graphite. For recent reviews, see
Clarke (1980) and Zabel (1980). Structurally, the intercalation compounds consist
of hexagonal honeycomb graphite layers between which there are layers of, for
instance, metal ions. Figure 4 shows a ‘stage 2’ compound, like CsCz4,in which the
B
A
M --- ----
- A _ _ _ _B c A
Figure 4. Stacking of metal (M) and graphite (A, B, C) layers in stage 2 graphite intercalation compounds
such as CsCZ4.The intralayer ordering may be commensurate or incommensurate as shown in figure 3.
metal layers are separated by two graphite layers. At high temperatures there is no
long-range order in the metal system, but at low temperature there is often a transition
into a structure where the metal ions form a regular three-dimensional lattice which
may be commensurate or incommensurate with the graphite lattice. Within the layers
the ordered structures are very similar to those in figure 3. The transitions from the
disordered to the ordered phases have been analysed by Bak and Domany (1979,
1981) and Bak (1980a, b). The staging phenomenon has been analysed by Safran
(1980)using a model which, in fact, can be shown to exhibit complete devil’s staircase
behaviour (Bak and Bruinsma 1982).

Commensurate and incommensurate phases 593
Another example of a 3D system with two interpenetrating incommensurate lattices
is the mercury chain compound Hg3-,AsF6. Structurally the compound consists of a
body-centred tetragonal lattice of AsF6 molecules through which pass two non-
intersecting arrays of Hg atoms parallel to the basal plane edges. At low temperatures
the mercury chains form a regular three-dimensional lattice which is incommensurate
with the AsF6 lattice (figure 5 ) (Hastings et a1 1977).
Figure 5. Ordered structure of Hg lattice in Hg3-,AsF6 as observed in the experiment of Hastings et al
(1977). The circles indicate the projections of Hg atoms on the tetragonal basal plane. The positions along
the tetragonal t axis are as follows: 0:z = 0,O: z =a, 0:z =21’0 :t =#.
In all these systems there are two ‘incommensurate’ atomic lattices with different
concentrations of atoms, and we are speaking on ‘compositional incommensurability’.
However, the incommensurate structure need not be an atomic lattice. There exist
several compounds where the incommensurate quantity is a periodic distortion of an
otherwise regular lattice (‘displacive incommensurability’). (For reviews, see Pynn
(1979),Cowley and Bruce (1978)’ Bruce et a1 (1978) and Bruce and Cowley (1978).)
Figure 6 shows a one-dimensional array of atoms with an incommensurate modulation.
Structurally incommensurate phases were first found in insulators (NaN02,Yamada
et a1 (1963); K2Se04,Iizumi et a1 (1977)). Subsequently such transitions have also
been discovered in metallic materials. The physical mechanisms which drive the
transitions are somewhat different in the two cases. In insulators the mechanism may
be competing short-range forces. In conductors the incommensurate structures arise
from interactions between conduction electrons and the atomic lattice, the so-called
Peierls mechanism. The conduction electron density in the distorted phase is spatially
modulated, forming a charge density wave (CDW) which accompanies the periodic
lattice distortion. Examples are the ‘quasi-two-dimensional’ layered metal chal-
cogenides (TaSe2, Wilson et a1 (1975), Moncton et a1 (1975), Fleming et a1 (1980,
1981);NbSe3,Fleming et a1 (1978))’and ‘quasi-one-dimensional’charge transfer salts
such as TTF-TCNQ (Comes et a1 1975, Ellenson et a1 1976). Strictly speaking these
systems are all three dimensional.
Two-dimensional ‘CDW’exist on the surfaces of several clean metals. Low-energy
electron diffraction (LEED) experiments have revealed that the surfaces may be
reconstructed, i.e. the surface atoms are arranged in a different way from those in the
bulk material. The reconstructed surface can be formed from the regular surface by

594 P Bak
! C )
Figure 6. One-dimensional structurally modulated chain of atoms. ( a )Uniform chain, (b)commensurate
'dimerised' structure, (c)incommensurate modulation. The curves above the chains show the displacements,
U, of atoms from the positions in the uniform chain.
Figure 7. (a) Temperature dependence of modulated magnetic structure of erbium (Habenschuss et al
1974). Note the transition from the commensurate to the incommensurate phase at T = 24 K. (b)Intensities
of higher-order harmonics. 0,increasing temperature; 0,decreasing temperature.

applying a periodic lattice distortion. Of particular interest is the incommensurate
structure found by Felter et a1 (1977) on the MO[loo] surface, and the CI transition
produced by hydrogen adsorbtion on the W [loo] surface (Estrup and Barker 1980).
The transition from the high-temperature disordered phase to the low-temperature
reconstructed phase has been analysed by Tosatti (1978) and Bak (1979b, 1980a).
The most widely studied incommensurate systems are probably magnetically
ordered structures. In particular, there are a great variety of helical, sinusoidal and
conical structures with incommensurate wavevectors among rare-earth compounds.
For a review, see, for instance, Koehler (1972). Figure 7 shows the temperature
dependence of the wavevector of the magnetic structure of erbium (Habenschuss et
a1 1974). Note the CI transition at T = 2 4 K . Figure 8 shows the temperature
j*.
273 1*-
-a
'0.60
,.,$ -O:-
Figure 8. Temperature dependence of wavevector for sinusoidally modulated structure of cerium anti-
monide, CeSb (Fischer et ai 1978). 0,intense satellite; 0,weak satellite.
dependence of the wavevector for CeSb, where the wavevector jumps between a
multitude of C values. We shall return to these two cases later. The difference may
be due to a stronger effective coupling to the lattice in the latter case.
The list of experimental systems presented here is far from complete. A few typical
and well-studied examples were chosen from each group for later reference. We shall
return to the experiments to compare with theoretical predictions in the following
subsections.
1.3. Another simple model: the anisotropic Ising model with competing interactions
(ANNNI model)
We initiated this review by introducing a simple structural T = 0 model exhibiting C
and I phases. It turns out that there exists a very simple magnetic model which exhibits
phase diagrams with commensurate, incommensurate, chaotic, floating and fluid
phases: the anisotropic Ising model with competing nearest- and next-nearest-neigh-
bour interactions. Although the model is much too simple to mimic real magnetic
systems, it does reproduce most of the features encountered in experiments. The
model is defined in figure 9. There is both a three-dimensional and a two-dimensional

596 P Bak
i
II
’;
.Y 1
X f l t-J,(=J,) 4
Figure 9. The three- and two-dimensional versions of the king model with competing interactions (ANNNI
model).
version. The spins S = i l interact through a ferromagnetic nearest-neighbour interac-
tion J1.In one particular direction there is an antiferromagnetic next-nearest-neigh-
bour interaction J2. The competition between J1and J2 stabilises the various periodic
phases. The model (which has been rather controversial) is quite a challenge to those
who wish to understand modulated structures at non-zero temperatures.
1.4. Layout of the review
Much of this review will survey work on the FVdM model (and its generalisations to
higher dimensions) and the Ising model with competing interactions. The results from
these models are probably quite representative for phase diagrams of ‘incommensurate’
systems. In § 2 work on the CI transition within the continuum approximation will
be reviewed. This approximation may apply to systems wheLe only one particular
commensurate phase is of importance (as, for example, the ‘J3 structure’ in krypton
or graphite, the CI transition in TaSe2 and the CI transition in erbium). In § 3 the
global phase diagrams of systems with several possible commensurate phases will be
treated. This section contains a review of work on the Ising model with competing
interactions. It seems that in three dimensions the devil’s staircase in various versions
will show up, whereas in two dimensions there are only a finite number of C phases
separated by floating or fluid phases. Section 4 deals with recent work where incom-
mensurate systems were treated by iterating some simple area-preserving ‘maps’.
These studies clearly show the existence of the chaotic or randomly pinned metastable
states in addition to the C and I phases. The relevance of the chaotic states to several
experimental systems will be discussed.
2. The commensurate-incommensurate transition
2.1. The theory of Frank and Van der Merwe (1949) (FVdM)
The ground states of the 1D model (1.1)were found in 1949 by Frank and Van der
Merwe with the extra assumption that the discrete index n can be treated as a
continuous variable. In this section the properties of the continuum versions of the
FVdM model and related models of the CI transition will be reviewed, and in § 4 we
shall investigate the shortcomings of the continuum approximation.

The FVdM theory has an interesting history. The theory has been rediscovered
independently, in widely different contexts, many times. In 1949 FVdM presented
their solution. In 1964Dzyaloshinskii derived the theory in his study of the transition
from a ferromagnetic phase to a helical magnetic structure. In 1968 De Gennes
applied the theory to the alignment of a cholesteric liquid crystal structure in a magnetic
field. In 1976 McMillan solved the FVdM model numerically with high accuracy in
an analysis of the CI transitions in layered compounds such as TaSe2. In 1976 Bak
and Emery solved McMillan’s model analytically, again rederiving the exact solution
of FVdM.
The theory is in fact quite simple. Introducing the phase cpn by the equation
b
2T
xn = nb +- qn
and transforming to the continuum limit
the Hamiltonian becomes
1
2
H = j [;(%-a) +V(l-cospcp) dn (2.3)
with p = 1 and S =(2r/b)(ao- 6). S is the relative ‘natural misfit’ between the two
lattices. With p >1we shall see that (2.3) describes a transition to a C phase of order
p. The phase cp is the shift of the atoms relative to the potential minima. The state
q(n)=0 is thus the C phase, and the unperturbed I phase is given by the straight line
cp = Sn. The ground state which minimises (2.3) is found among the solutions to the
1D sine-Gordon equation (or pendulum equation)
d2q/dn2=pV sin pcp. (2.4)
One of the solutions to this equation is the soliton
4
P
q(n)=-tan-’ exp (pJVn). (2.5)
This solution describes a wall, centred at n = 0, which separates two commensurate
regions, one with cp = 0, the other with q = 2 ~ / p(figure 10). The wall represents an
extra atom which has been added to the C chain within a region given by the soliton
0
0 n
t-b-i
Figure 10. Single-soliton solution to the FVdM model. The soliton is a domain wall between two
commensurate regions.

598 P Bak
n
Figure 11. Regular soliton lattice solution to the sine-Gordon equation (2.4).The straight line corresponds
to an unperturbed incommensurate structure.
width lo= l / p J v . In general the solutions are regularly spaced solitons, a soliton
lattice (figure 11). The soliton lattice is a compromise between the ‘umklapp’ term
cos pcp which favours the cp =27rm/p, and the elastic energy which favours cp = Sn.
The concept of walls plays a very central role in all existing theories of CI transitions.
The average misfit between the chain and the lattice, 4 = (27r/b)(a-b), is inversely
proportional to the distance 1 between domain walls
i.e. 4 is the soliton density. The misfit 4 is given by the equation
(2.7)
where K and E are complete elliptic integrals of first and second kind and 77 is defined
by the equation
where Vcis given by
(2.8)
(2.9)
Near the commensurate phase where the soliton density is low the energy density
takes the form (Bak and Emery 1976)
4 J v 16JV ~ T J V
E = (T-s)q+-7r 4 exp (- 7).(2.10)
The first term is proportional to the soliton density and may therefore be considered
as the soliton energy. The second term decays exponentially with the distance between
solitons, and it is thus an effective repulsion between solitons. When V becomes small
enough (or S large enough) so that the soliton energy becomes negative, the C phase
becomes unstable with respect to spontaneousformation of walls. This happens when

Commensurateand incommensurate phases 599
t
Figure 12. Misfit cf plotted against 6 near the CI transition. A, continuum solution of the 1D FVdM
model. B, Pokrovsky and Talapov’s solution in two dimensions.
the relation (2.9) is fulfilled. This equation thus determines the CI transition. Figure
12(a)shows the misfit 4 near the CI transition. The soliton density has the asymptotic
form
4 --In-’ (v- v,) v>v,
with S, related to V by (2.9)
or (2.11)
4 --In (&-a)
if the natural misfit S is varied, for instance by changing the pressure. In a diffraction
experiment the strong anharmonic effects of the walls will give rise to higher-order
satellites with intensities which grow rapidly as the CI transition is approached. In
the neutron scattering experiment by Habenschuss et a1 satellites of order up to 17
were observed givingstrong indications that walls indeed play an important role (figure
7).
The lock-in at higher-order C phases with Ma =Nb can be described by a con-
tinuum Hamiltonian similar to (2.3) (Theodorou and Rice 1978)but with
(2.12)
v =VM- VM p =Ma
A phase transition occurs near any rational value of ao/b. The critical natural misfit
depends exponentially on M :
S, -JV,- vMI2. (2.13)
If there is no overlap between the C phases we therefore expect the system to lock-in
at an infinity of commensurate values. However, the total sum of the commensurate
intervals is small if V is small. In fact, it has been shown by Pokrovsky (1918) and
Pokrovsky and Talapov (1978) that the locked portion is proportional to JV. This
is the ‘incomplete devil’s staircase’ (figure l(b))! In an experiment only a few com-
mensurate phases will be observable because of the finite resolution. More and more
C phases appear if the resolution is improved.

600 P Bak
In many (if not most) experimental situations reviewed in § 1.2 only one (or no)
C phase is evident, and indeed the periodic phase appears to be everywhere incom-
mensurate outside the C phase (see, for instance, figure 7). When the potential V is
strong enough the CI transitions of high order may influence each other. In § 4 we
shall see that the I phases will disappear and the complete devil’s staircase might
replace the incomplete one.
2.2. Coupling to the lattice
Is the FVdM solution stable with respect to coupling to other degrees of freedom,
such a lattice strain? This question has been analysed by Brtice and Cowley (1978)
and Bak and Timonen (1978). It turns out that one should distinguish between two
types of strains; those which may be allowed to have spatial variations, and those
which must necessarily be homogeneous not to break up the crystal lattice. Bak and
Timonen showed that the first type will simply renormalise the soliton energy and
not give rise to qualitatively different effects. Bruce and Cowley showed that the
latter type of coupling will yield a term proportional to q2 with negative coefficient
in (2.10), which will render the transition first order.
2.3. Phasons
The spectrum of elementary excitations in incommensurate systems near the CI
transition has been calculated by McMillan (1977)and Pokrovsky and Talapov (1978).
Generally the low-lying modes in incommensurate systems are denoted phasons
(Overhauser1971)since they correspond to modulations of the phase a of the periodic
structure (1.2). Figure 13 shows the excitation spectrum in the C and I phases. In
n
4
Figure 13. Spectra of elementary excitations in the commensurate (. ..) and incommensurate phase (-)
(McMillan 1977, Pokrovsky and Talapov 1978, Novaco 1980).
the I phase the long-wavelength phasons take the form of phonons in the soliton
lattice. This branch is very soft because of the exponentially weak interactions between
walls. The short-wavelength phasons are more like ordinary phonons. In the model
in figure 1 they are simply phonons in the incommensurate atomic lattice. The two
phason branches are separated by a gap at the zone boundary wavevector of the

reciprocal lattice of the soliton lattice, q = ~ / l .At the CI transition 1 goes to infinity
and the soft phason branch disappears, leaving a gap at q =0. This gap represents
the energy of harmonic oscillations of the atoms in the commensurate periodic
potential.
The phasons give a contribution to the free energy just like the phonons in an
atomic lattice. At non-zerotemperatures the dimensionality of the system is important.
It is quite easy to generalise the FVdM model to two and three dimensions, by letting
arrays of chains (1.1)interact with each other in the perpendicular directions. In two
and three dimensions the solitons would be lines or planes, respectively. The phason
spectrum becomes (Bak and von Boehm 1980)
(2.14)
where the last term is due to the perpendicular interactions and wqllis the phason
spectrum in figure 13. Since the energy of the phasons in the I phase is lower than
that in the C phase the free energy in the I phase will be lower by an amount
(2.15)
where wqllhas been set equal to zero in the I phase. This gives a negative contribution
to the domain wall energy in three dimensions which favours the I phase. The phasons
thus narrow the C phases at finite temperatures, but we expect all the C phases to
remain stable.
In the one-dimensional model the wall energy is finite in the C phase and a finite
number of walls will be thermally excited. The system will always be incommensurate,
or rather fluid, since there is no long-range order.
The phason spectrum is also of importance in the calculation of quantum effects
(Bak and Fukuyama 1980). The zero-point energy in the I phase is lower than that
in the C phase:
where (2.6) has been used. Again, this amounts to a negative contribution to the
domain wall energy. Zero-point fluctuations destabilise high-order commensurate
phases and thus destroy the devil’s staircase in one dimension. We shall see that this
resembles the situation for two-dimensional classical systems at finite temperatures.
Generally there is a close resemblance between 2D classical systems at non-zero T,
and 1D quantum systems at T=O. By performing similar integrations in three
dimensions it can be shown that zero-point fluctuations will make the C phases
narrower but not destabilise them completely (Bak and Fukuyama 1980).
2.4. Adsorbed monolayers: the Bak et a1 (1979)(BMVW) theory
In some of the most extensively studied experimental systems the periodic potential
is two dimensional and not one dimensional as in the FVdM model. This is the case
for some layered CDW compounds (2H-TaSez; Moncton et a1 (1975, 1977)) and
rare-gas monolayers adsorbed on graphite (figure 3). The potential can be represented
by a Fourier series:
V(R)=CAGCOS G * R (2.17)
G

602 P Bak
-
Figure 14. Three intersecting walls in the I phase near the ‘43structure’ in a rare-gas monolayer (krypton)
on graphite.
where G are reciprocal lattice vectors of the basic lattice. In the case of a hexagonal
graphite substrate the walls generated near the CI transition may be aligned along
three equivalent directions forming angles of 120” with each other. In the C phase
of krypton on graphite the adsorbed atoms may occupy one of three equivalent
positions. Since the walls need not be parallel, they may cross, or rather intersect
each other, forming a domain wall pattern. Figure 14 shows three intersecting domain
walls. BMVW considered the two wall configurations shown in figure 15. Wall
- 1 -
l u ) ( b )
Figure 15. Possible domain wall structures near the CI transition in rare-gas monolayers on graphite, or
in 2H-TaSe2. ( a )The ‘striped’ I phase, (b)the ‘hexagonal’ I phase.
crossings are assumed to have a finite energy A. In case (a), where the walls are
parallel with a distance 1, the energy density (or free energy density at finite tem-
perature) near the CI transition is
(2.18)
where the first term is the domain wall energy and the second term is the repulsive
interaction between domain walls. Note that in this case the hexagonal symmetry is
broken in the I phase. In case (b)the walls form a hexagonal pattern and the energy

is
(2.19)
The last term isproportional to the number of domain wall intersections and represents
the domain wall intersection energy. When A <0 the walls attract each other and the
hexagonal I phase is stable. The transition is discontinuous and takes place for a
positive value of a. For A > O the striped phase (a) is stable. The transition in this
case is continuous and takes place when a = 0.The theory is purely phenomenological.
However, a microscopic calculation on a model (at T = 0) with a simple hexagonal
potential (Bak 1979a)gives the same result.
To conclude: if the transition is continuous, the hexagonal symmetry is broken,
while if the incommensurate phase remains hexagonal, the transition must be first
order.
How does this compare with experiments? Krypton monolayers have been studied
by specific heat (Lahrer 1978), LEED (Chinn and Fain 1977, Fain et a1 1980), x-rays
(Stephens et a1 1979)and synchrotron radiation measurements (Moncton et a1 1981).
These experiments seem to provide a counter-example. The CI transition appeared
to be continuous and there was no sign of symmetry breaking. The experiments were
performed at rather high temperatures near the melting temperature, where fluctu-
ations may be important. We shall return to finite temperature effects later. However,
very recently Nielsen et a1 (1981) performed a synchrotron radiation experiment at
lower temperatures and they found clear evidence that the transition is first order.
The transition is accompanied by hysteresis which tends to obscure the discontinuity
(figure 16). So far, fluctuations have been neglected. It will be seen shortly that they
lead to terms of different character, at least in two dimensions.
I
2000'
1000.
0.8
Pressure plp,
Figure 16. Hysteresis loop at the CI transition in krypton on graphite (Nielsen et al 1981). The intensity
of the commensurate Bragg peak in the coexistence region is plotted for increasing and decreasing pressure.
T = 39.44 K,p = 4.70 Torr.
The symmetry of the CI transition in the CDW system TaSez (figure 17) is very
similar to that of krypton monolayers on graphite and the BMVW theory is expected
to apply (Bak and Mukamel 1979). Furthermore, the system is three dimensional so
finite temperature effects are less important. In an x-ray study Fleming et a1 (1980,

604 P Bak
Figure 17. ( a ) Incommensurability as a function of temperature for TaSe2 (Fleming er al 1980). On
warming, the stripedphase is seen in the range 93-112 K. A, cooling; 0,warming. (b)Electronmicroscope
picture showing domain walls (solitons) in 2H-TaSe2 (Fung er a1 1981).
1981)have observed the striped phase in reciprocal space. The signature of the striped
phase in a diffraction experiment is a splitting of the Bragg satellite into a peak which
remains at the commensurate position and a peak which is shifted. In the hexagonal
phase the satellite moves away from the commensurate position in the symmetry
direction. As the temperature is increased the striped phase appears through a
continuous transition (figure 17). At higher temperatures there is a transition to a
hexagonal phase. When the temperature was lowered, only the hexagonal phase was
observed. The most remarkable discovery in the field is undoubtedly recent direct
observations of the solitons or discommensurations by electron microscopy by Fung
et a1 (1981) and Chen et a1 (1981). Figure 17(6)shows the soliton pattern found by
Fung et a1 in the striped phase. The overall behaviour is in agreement with the x-ray
experiments and with theory. Note the formation of dislocation of the type shown in
figure 20. The existence of such dislocations was predicted by McMillan in his original

paper (1976). The hysteresis is due to the metastability of the hexagonal phase. There
is no simple mechanism to locally change the hexagonal phase into a striped one.
Krypton on graphite and 2H-TaSe2seem to represent the two possibilities allowed
by the BMVW theory, with A <0 and A >0, respectively.
Figure 18. Irregular wall arrangement near CI transition to the 43 structure (Villain 1980a, b, c). The
broken line shows a dilation of a hexagon which does not modify the energy except for exponentially weak
terms.
In the case of a hexagonal pattern the exponentially weak interactions can be
ignored. Villain (1980a, b, c) made the interesting observation that a deformation of
the hexagon network (figure 18)costs no energy. The degeneracy amounts to l/a per
hexagonal cell and this degeneracy gives rise to an additional entropy and free energy
(2.20)= -TS = -k BT(;)?In
l a )
Fluid
Pressure Pressure
Figure 19. Theoretical phase diagrams near the CI transition to the 43 structure in rare-gas monolayers.
( a )Negative domain wall crossing energy A(A <O). A single first-order transition is predicted by BMVW
theory. ( b )A >0. At T >0 a hexagonal phase, H (Villain 1980a, b, c) and a fluid phase (Coppersmith et
al 1981) have been squeezed in between the C phase and the striped phase predicted by BMVW theory.
At low temperature the fluid phase may squeeze out the hexagonal phase completely.

606 P Bak
which dominates the (1/1)’ term for large enough 1. The transition is then always
slightly first order and there should be a narrow hexagonal phase (H) between the C
and striped phases for A >0. At low temperatures 1must be astronomical before this
term comes into play and the transition v;ould appear continuous. To get a complete
phase diagram it turns out that dislocations must be taken into account. The effects
of dislocations on the CI transition in two dimensions will be treated in § 3in connection
with the anisotropic Ising model. The consequence for the present system is that a
fluid phase appears between the H phase and the C phase (Coppersmith et a1 1981).
The resulting phase diagram is shown in figure 19. The fluid and hexagonal phases
become exponentially narrow at low temperatures. In fact, Coppersmith et a1 expect
the Villain hexagonal phase to be completely excluded at the lowest temperatures.
For comparison the phase diagram for A < O which probably applies to krypton on
graphite (Nielsen et a1 1981) is also shown.
2.5. CI transitions in two dimensions: the incommensurate phase
It can be shown that in two dimensions an incommensurate phase with long-range
order cannot exist. Instead the I phase will be a ‘floating’ phase with no long-range
order, but with an algebraic decay of correlation functions. The possibility of floating
phases was pointed out by Wegner (1979) for XY magnets and by Jancovici (1967)
for harmonic crystals. The ‘order parameter’ correlation functions take the following
forms in the various commensurate and incommensurate phases (q is the wavevector
of the phase) at long distances:
incommensurate:
floating I:
fluid:
commensurate:
(S(O)S(r)>-cos (4 * f- +cp)
(S(O)S(r))-F q cos (q e r +40)
(S(O)S(r))-exp ( - K * r )cos q ‘r
(S(O)S(r))-cos qo*r, qocommensurate, locked.
(2.21)
The floating phase, with incommensurate or accidentally unlocked commensurate
wavevector and power law decay, is believed to exist only in two dimensions. The
other phases may exist also in three dimensions. The transition between the fluid and
the floating phase is triggered by dislocations or vortices (Kosterlitz and Thouless
1973, Kosterlitz 1974, Halperin and Nelson 1978, 1979, Young 1979). The CI
transition is triggered by an instability with respect to walls. To generate phase
diagrams, both types of topological excitations must be considered.
First, we shall review the theory of Pokrovsky and Talapov (1980) for the CI
transition which takes walls into account but ignores vortices.
2.6. The Pokrousky-Talapou (1980)theory
The theory of Pokrovsky and Talapov applies to the CI transition in the case of a
one-dimensional potential (the striped case). In the PT model, walls cross from one
end of the sample to the other. Walls are not allowed to cross or turn backwards.
Figure 20 shows a ground-state configuration and a typical T f 0 configuration in the
case p = 3 where the phase shift at the wall is 21r/3. The configuration 20(c) (which
represents a vortex) is divarded. The starting point is a 2D generalisation of the

Figure 20. Typical ground-state ( a )and excited-state ( b )configurations of the Pokrovsky-Talapov (1980)
model. The lines represent walls in the case p = 3. The configuration ( c ) represents a vortex and is not
allowed. 4 d q is 2 7 along the path shown.
FVdM model:
1
2
X=j[,&--)2+T(z-S)1 dq 1 dq + V C O S ~ C ~dxdy.
(2.22)
In fact, the PT Hamiltonian is slightly more general, but (2.22) suffices to illustrate
the method. Using the transfer matrix method it can be shown that the calculation
of the free energy of the two-dimensional model amounts to the calculation of the
ground-state energy of the 1D quantum Hamiltonian a:
1A = [Cn-’+?(z1 dP -S)’+ V cos p q dx
(2.23)
with C = 1/2y. This is the 1D quantum sine-Gordon equation with a soliton ‘chemical
potential’ S. There is a phase transition in this model when the quantum soliton
energy equals the chemical potential. By translating to a fermion language Pokrovsky
and Talapov found an extra term to be added to the wall energy (2.18) at non-zero
temperature of the order of 1/13which overwhelms the exponential term for large 1.
The 1/13 term can be understood qualitatively in a simple way (Fisher and Fisher
1982) on the basis of the wandering (or diffusion) of the walls. After length n the
wandering 1, of the wall is given by (In)’ -n. For mean space 1 the density of collisions
goes as 1/12 for a single wall and reduces the entropy of the wall. The extra factor
1/1 comes from the summation over all the walls. The free energy may be written as
a 1
F =- (8-ac) +TC 3 (2.24)
1 I
where the first term has been linearised near S = 8, and minimisation with respect to
1/E leads to
1
(S,-S)fi p=’2
q - i - (2.25)

608 PBak
to be compared with (2,11), which is valid at T=O, and probably also in three
dimensions for T Z 0. In a very recent paper Fisher and Fisher (1982) argue that, in
general, p = (3-d)/2(d - 1). This is in agreement with new results by Natterman
(1981). Pokrovsky and Talapov found the power law correlation functions characteris-
tic of the floating phase (2.21). In a LEED experiment on Xe absorbed on Cu (110)
Jaubert et a1 (1981) find p = 1/2, in agreement with the Pokrovsky-Talapov theory.
Okwamoto (1980) and Schulz (1980) have solved a fermion version of (2.22) at
a special temperature and their findings are in agreement with Pokrovsky and Talapov.
Schulz also finds the exponent q in (2.21) near the CI transition to be
q = 2/p2. (2.26)
Villain has solved a discrete version of the PT model using a transfer matrix technique
developed by De Gennes (1968a). The Villain solution will not be reviewed here
since a very similar method is applied to the 2D anisotropic Ising model in 0 3.
3. Global phase diagrams of periodic systems
3.1. The anisotropic 3 0 Ising model with competing interactions
The anisotropic Ising model with competing nearest- and next-nearest-neighbour
model (the ANNNI model) is probably the simplest model in statistical physics which
exhibits transitions from a simple periodic structure to long-period commensurate or
incommensurate structures. The model (figure 9) is so simple that it cannot possibly
describe any real physical systems, but the qualitative features are similar to those of
real systems. The model seems to exhibit both incommensurate, commensurate,
floating and chaotic phases, and it is therefore a good exercise for those who want to
understand periodic structures. We shall therefore review the very extensive work
on the model in some detail. There exist no ‘exact’ theories of the global phase
diagram, neither in three nor in two dimensions. This gives room for conflicting
approximate theories, and in fact the model is quite controversial.
The model was invented by Elliott in 1961 in order to describe the modulated
magnetic structure of erbium, and it has been studied by several authors since. The
3D version has been treated by Redner and Stanley (1977a, b), von Boehm and Bak
(1979), Selke and Fisher (1979),Bak and von Boehm (1980),Fisher and Selke (1980),
Villain and Gordon (1980),Bak (1981a, b, c) and Rasmussen and Knak-Jensen (1981).
The ground state of the model is quite easy to find. For ferromagnetic nearest-
neighbour interactions, J1>0, the ground state is ferromagnetic for J2>-J1/2. When
J2<-J1/2 the ground state is an antiferromagnetic structure with two ‘up’ layers
followed by two ‘down’ layers and so on for the ++- - ++ structure. At the point
2J2+J1= 0 the ground state is degenerate and consists of successive + and - domains
separated by walls perpendicular to the direction of competing interactions. Redner
and Stanley (1977a, b) found a phase diagram consisting of three phases (figure 21):
(i) a disordered paramagnetic phase, (ii) a ferromagnetic phase and (iii) a modulated
periodic phase characterised by a wavevector q = 27r(O, 0, q). These phases are separ-
ated by a Lifshitz point P. Redner and Stanley studied the model near the phase
boundary to the disordered phase but here we shall be concerned with the properties
of the modulated phase.
At T, the wavevector changes continuously from q =0 at the Lifshitz point to
q = 1/4 (the ++- - phase) for -J2/J1+c0.At T = 0, on the other hand, the

Commensurate and incommensuratephases
0.25-
9
0.20-
0 15-
609
I I I _ I I I _
114 1/ 4
(‘) - 0.25
219 -
3/14 -
1/5 - - 0 2 0 -
3116- 2/11 3/16 ----- -______-2/11. - 3/17
116 -
Tc - 0.15- Tc -
I I I I I 1 -
-J,/J,
Figure 21. Mean field phase diagram for ANNNI model. Only the three transition lines separating
paramagnetic, ferromagnetic and modulated phases are shown.
periodicity is constant (q = 1/4) for 2J2+J1<0. The periodicity thus changes as the
temperature is lowered, and this is precisely the situation that we are interested in.
The simplest way of obtaining information on the phase diagram of a 3D system is
the mean field theory. Figure 22 shows the results of a numerical mean field calculation
for two different values of J2/J1(Bak and von Boehm 1980). The wavevector exhibits
Figure 22. Wavevector plotted against temperature as calculated numerically by Bak and von Boehm
(1980). The full ‘continuous’ curve is the result of an analytic calculation. ( a )Jl = 1, J2= -0.6; ( b ).TI = 1,
J2 = -0.7.
a stepwise behaviour. The + +- - phase is stable up to a rather high temperature;
then it changes rapidly with increasing temperature. Furthermore the curve is not
always monotonic. The overall behaviour is consistent with the ‘devil’s staircase’.
Near T, the numerical mean field calculation is not valid, but in this limit the mean
field theory can essentially be solved analytically as we shall see shortly. The stepwise

610 PBak
behaviour has also been obtained by a calculation by Rasmussen and Knak-Jensen
(1981). Monte Carlo calculations yield a similar temperature dependence, but fail to
produce the high-order commensurate phases (Selke and Fisher 1979, Rasmussen
and Knak-Jensen 1981).
How can these results be understood analytically? The mean field free energy is
given by
M,
F =1J,jMih$- T C tanh-’ (T dcr (3.1)
I ] i o
where the interactions are given in figure 9 and Miare the thermodynamic averages
of spins at sites i, M, = (Si).The free energy should be minimised with respect to the
Mi.This is not possible in general. The stability of a specific commensurate phase,
say the q = 1/4 phase, can be studied by inserting the trial function
M(r)= A exp (icp(z))exp 2 m - +cc (3.2)i 3
and allowing the phase cp to depend on z while the amplitude is kept constant. The
C phase is given by cp constant, an I phase with q = 114+S is given by cp = 2762. If
cp is not too rapidly varying the continuum approximation should be applicable:
p ( t )-c p ( ~-1)=dcp/dz. (3.3)
By inserting all of this in (3.1) and expanding to fourth order in M, we obtain the
free energy density
1
2
F = - 4 J 2 A 2 j [:(*-a) +v(cos4cp+l) dz
2 dz
(3.4)
with
8 =J1/4J2 v = -TA2/96J2 A’ = 3(4J1+2J2 - T)/T.
Compare with (2.3)! The phase function which minimises (3.4) is the soliton lattice.
The free energy near the C phase takes the form
J v 16Jv -2.rr.j~
F-[(4 ;-8) +-exp7T
(--j--)]q (3.5)
where 4 = (dcp/dz) = 27r/pl is the deviation of the wavevector from the commensurate
one. The temperature dependence of q for J 2 / J 1 = -0.7 is shown in figure 22 and
the soliton is shown in figure 23. As usual, the continuum approximation misses the
higher-order C phases. Near T, the amplitude A is proportional to (Tc- T)’”. The
lock-in to higher-order C phases is described by an expression similar to (3.5) with
v = 0, whose U, goes as A,-> so the widths of the high-order commensurate phases
in ‘2J2+J1space’ decay as ~ V , - A ( ~ - ~ ) / ~-(T,- T)‘ with 5 = ( p-2)/4. The situation
is the same as for the T = 0 FVdM model in the weak potential limit (see equation
(2.13))and the incomplete devil’s staircase is expected. Near T,, critical fluctuations
are usually important. Aharony and Bak (1981) have studied the I phase near T,
using the €-expansion technique and found that the exponents 5 are modified, but
that the staircase survives. In an experiment the I phase will look smooth since the
total width of C phases goes roughly as (T,- T)’”.

1 I
-5 5
l~l~l;,,
Z
z
I I I I
1 1 t f
Figure 23. A soliton and the corresponding spin structure calculated for J2/J1= -0.7. The ‘broken’ arrows
indicate the numerical solution.
By combining the numerical mean field calculations with the considerations above,
the global mean field phase diagram in figure 24 was constructed. The dark areas
indicate high-order Cphases with I phases in between. At low temperatures the phase
diagram includes first-order transition lines between C phases, and from the ferromag-
netic to the modulated phase. It is not clear how the various C phases disappear
when T is lowered. Bak and von Boehm (1980)suggested, on the background of the
6-
4 -
T
FM
2-
01 I I I 1 I 1 I I I
0.4 0 6
-J21J,
0 2
Figure 24. Mean field phase diagram for the 3D king model with competing interactions (Bak and von
Boehm 1980). The dark areas indicate high-order C phases with I phases between (incomplete devil’s
staircase).
numerical analysis, that an infinity of commensurate phases merge at the point T = 0,
J1+2J2= 0, in agreement with the fact that an infinity of phases is indeed degenerate
at this point. The numerical analysis, of course, is inadequate at proving this assertion.
Figure 25 shows for comparison the phase diagram derived by Aubry (1979) for the
FVdM model. Note the striking similarity between the two physically very different
cases.

612 P Bak
bl2a
Figure 25. Phase diagram for the FVdM model as suggested by Aubry (1979). The interpretation of the
various areas is the same as in figure 24. The ordinate is a measure of the strength of the potential.
Villain and Gordon (1980) and Fisher and Selke (1980) have investigated the
phase diagram near the ‘multiphase’ point where the C phases merge. Villain and
Gordon explain the first-order transitions in terms of oscillating interactions between
solitons. Fisher and Selke show, using a low-temperature expansion technique, that
only phases given by q = 1/2(21+ l), 1 integer, survive in a finite neighbourhood of
the multiphase point. There is some discrepancy between this result and the result
of Villain and Gordon. For instance, they find a first-order transition from the + + - -
phase to a high-order C phase; Fisher and Selke find a ‘quasi-continuous transition’
with 1going to infinity in the expression above. The multiphase point is quite artificial.
If a small long-range interaction is added the degeneracy can be lifted and there will
be an infinity of phases at T = 0 (Bak and Bruinsma 1982). For a detailed account
of Fisher and Selke’s low-temperature theory, see Fisher and Selke (1981a, b).
The critical behaviour near T, is that of the 3D xy model. The two components
of the order parameter correspond to the amplitude and the phase of the spin density
wave. The xy character of the critical line between paramagnetic and modulated
phases was shown by Mukamel et a1 (1976) and Mukamel and Krinsky (1976). In
two dimensions we expect similarly the critical behaviour to be that of the 2D xy
model, which makes it particularly interesting in view of recent theories of this class
of models (Kosterlitz and Thouless 1973, JosC et a1 1978, Halperin and Nelson 1978).
3.2. Two-dimensionalanisotropic Ising models with competing interactions: equivalence
with the 2 0 xy model
Using methods similar to those in § 3.1 the Ising model can be transformed to a
two-dimensional continuum model (Bak 1981b):
where cp is the phase of the spin density wave

I f I I
0 0.5 1.0 1.5
-J,IJ,
Figure 26. Schematic high-temperature phase diagram of the 2D ANNNI model as suggested by Bak
(1981b). The broken line is given by equation (3.8). No prediction about the low-temperature behaviour
is given (?).
The amplitude A has been fixed (quite arbitrarily) at A = 1, and q = 27r(q,0) is the
mean field wavevector at T,. This is the Hamiltonian used by Kosterlitz and Thouless
(1973) to discuss the xy transition. A phase transition to a disordered phase takes
place as a consequence of vortex separation at a critical temperature T,.If vortex-
energy renormalisation is ignored the critical temperature becomes
112
T,=.i[(-2J2 +<)"I8J2 2 (3.8)
which has been plotted in figure 26 (broken line).
terms of the phase of the commensurate spin density wave:
To investigate the stability of the q = 1/4 phase the Hamiltonian is expanded in
(3.9)
1Ji dcp
X = [-2J2(2-6) + (&) +$(1+COS 4cp) dx dJJ
with 6 =J1/4J2. The phase cp is now defined by (3.7)with q = 27r(a,0).
This model is identical to the 'Pokrovsky-Talapov model' (2.22). The CI transition
can be estimated by transforming to the 1D quantum sine-Gordon Hamiltonian, and
the broken line in figure 26 shows the resulting transition line. The phase boundary
of the ferromagnetic phase was calculated using a theory of Muller-Hartmann and
Zittartz (1977). Hornreich et a1 (1979) and Selke and Fisher (1980) have studied
the model using computer simulations. Selke and Fisher's phase diagram is shown in
figure 27. It includes a Lifshitz point separating the C, I and paramagnetic phases.
Their paper also mentions the observation of dislocation pairs in the Monte Carlo
simulations and discusses the relevance to modern theories of melting. Recently Selke
(1981) has studied the finite size dependence more extensively and shows that the
interpretation of the Monte Carlo data must be modified. It is interesting to note
that even lattices as big as 40 x 200 can be misleading on an apparently gross physical
feature. A Hamber (1981private communications) has made similar observations.

614
FM
P Bak
PM
- J,IJ,
Figure 27. Phase diagram of 2D ANNNI model as suggested by Selke and Fisher (1980). The phase diagram
includes a Lifshitz point as in the mean field theory. FM, ferromagnetic; PM, paramagnetic; I, incommensur-
ate. The diagram is based on Monte Carlo calculations.
The theory developed above can give, at most, a crude 'phenomenological' picture
of the phase diagram in the neighbourhood of T,. The interesting question is: what
happens at low T when 2J2+J1 is near zero? It turns out that in this regime much
more powerful methods are ava:lable.
3.3. 2 0 ANNNI model: the Villain-Bak theory (1981)
At the point J2= -J1/2 the ground state consists of stripes of alternating plus and
minus spins of width 2 or larger. The stripes are separated by walls which shift the
phase cp of the structure by 7. Figure 28 shows excited states of the model with NxNy
sites. The analysis has two steps. In the first step the walls are not allowed to turn
+ - - - - - - + t - - t t
+ - - - - - - t + - - t '
+ + - - - - r t - - - + *
( a ) + + - - - - r + - - - + -
t t - - - - + + + - - + -
+ + - - - - + + + - - + +
t t t - - - t+ + - - - -
+ + - - t + - - , + + ; - - ; + + + + t +
+ I + ' - - - t : -
+ + + - - , + f - - i + , - - , + +'-'t t +
+ + - ; - - ' + ~ - - I + _ ; - - ; ? + - 8 - t t
( b ) t + - - - +r-"-- - + + - - + + -I++ +
+ t + t + + ~ ->)+ *
_ _ ; _ - - ' + , - - - ' + + + r + 7 - - ' * t
+ + ' - - - 1 - - - - ' + + + i t t - - - + +
(0) i b ) ( C i (dl
- + t - - + + + t + +
--
_ _ _ _1 '
+ - - _ , T - - -
+ - - - + I - - - , it + + t - , + t
I
Figure 28. ( a )Ground-state configurations of the 2D ANNNI model. ( b )Excited states.

back, just as in the analysis of Pokrovsky and Talapov. The configurations (c) and
( d )are not allowed. In the second step turnbacks are allowed. They represent vortices
since the phase changes 21r along the closed path in the figure. In the first step the
position of the pth wall, x p ( y ) ,is assumed to satisfy
XP+l(Y 1-X P (Y 15 2 (3,lO)
in analogy with the ground state. Defining
5,(Y) =X P (Y1-p (3.11)
where
Q+1(Y 1-6 P(Y 15 1
the problem can be transformed into a simple fermion model. If there are v walls,
5, may assume integer values between 1 and N, - v. The partition function can be
written
Z =Tr 4" (3.12)
with the transfer matrix given by
4 = exp (--E)
(3.13)
= -c w+(&(0+Y [ C ' ( 5 ) C ( 5 +1)+c'(5 + l)c(5)1.
E
The chemical potential for the fermions is
/.I,= -2P(J1+ 2J*)
and
Y = exp (-WO) P = 1/T.
Walls in the 2D model correspond to fermions in the 1D quantum model (3.13).The
eigenvalues of (3.13) are easily found since %?is a free fermion Hamiltonian. One
obtains
(-p -27 COS k)CiCk. (3.14)
The lowest eigenvalue of E (which is the free energy) with v walls or fermions is
obtained by filling the band defined by (3.14)up to the Fermi wavevector
k
kF=Irv(N,-v)-l.
The density of walls, ij = u/N,, which minimises the energy is given by
Irij 1 Irq -p
1-4 1-q Ir 1-4 y
--cos 7- -sin -- -
1
(3.15)
The density of walls (which is twice the wavevector of the incommensurate phase) is
shown in figure 29. Near the commensurate phases ij has a square root singularity,
in agreement with the Pokrovsky-Talapov theory. The empty band corresponds to
the ferromagnetic phase, the filled band to the + + - - phase. The resulting phase
diagram is shown in figure 30.

616 P Bak
X
Figure 29. Wavevector as a function of x =-(J1+2J2)/T in the floating phase (Villain and Bak 1981).
Note the differencebetween the smooth behaviour and the staircases obtained within mean field theory.
The spin at position (x,y ) can be written
S(x, y ) = exp (icp(x,y ) )exp 21ri -x
( ,", (3.16)
where 4 is the average density of walls and cp is the fluctuation:
(3.17)
n(x, y ) is the number of walls to the left of (x,y ) . The spin-spin correlation function
-(I,+2J,l/Jo
Figure 30. Phase diagram as suggested by Villain and Bak (1981) for the 2D ANNNI model. FM,
ferromagnetic; PM, paramagnetic; PI, floating I phase.

can be calculated by the transfer matrix method:
(3.18)
The incommensurate phase is thus a floating phase according to the definition (2.21)!
Near the ferromagnetic phase, r] =3,near the ++- - phase, r] = 1/8.
The ferromagnetic phase is a commensurate phase of order p = 2, and walls produce
a phase shift 27r/2. In the case of a higher-order commensurate phase of order p ,
walls would produce a phase shift 27rlp and the value of r] near the commensurate
phase is
r] = 2/p2. (3.19)
The second step of the analysis is to include vortices or dislocations. According
to the theory of Kosterlitz and Thouless (1973) dislocations are relevant when r] >a,
and the incommensurate phase is in fact a fluid phase with exponential decay of
correlation functions. When r] < the floating incommensurate phase is stable with
respect to dislocations. Comparing with (3.18), the system is paramagnetic (fluid) for
r ] = p 4 ) 21
4 <1- 1/JZ.
The paramagnetic phase (figure 20) extends all the way down to T = 0 ,and there is
no Lifshitz point! The phase diagram is thus in severe disagreement with the phase
diagram published by Fisher and Selke (1980). Furthermore there are only two
commensurate phases compared with the infinity of C phases in the 3D model (Bak
and von Boehm 1980). The wavevector has the smooth behaviour of figure 2(a).
A direct transition from a commensurate to a floating incommensurate phase is
possible only when r] given by (3.19)is less than so when
p 2 < 8
there is a fluid phase between the C and I phases, whereas for
p 2 > 8
a direct CI transition is possible. This is in agreement with general (but very similar)
arguments by Coppersmith et a1 (1981). Coppersmith et a1 have extended the theory
to the CI transition into the '& structure' on rare-gas monolayers adsorbed on
graphite. They found that the order of commensurability is sufficiently low to squeeze
in a narrow fluid phase between the C phase and the hexagonal phase stabilised by
Villain's irregular hexagonal network entropy. The resulting phase diagram is shown
schematically in figure 19 in the case where the domain wall intersection energy is
positive. The results of Coppersmith et a1 probably do not apply to krypton on
graphite, where the clear first-order transition (Nielsen et a1 1981)indicates that A is
negative. In this case the CI transition is probably not influenced by dislocations.
Although the conclusions in this section were based on a specificmodel, the ANNNI
model, the results might well be quite general since the domain wall picture can
probably be applied to a large class of microscopic models. Figure 31 shows phase
diagrams involving C, I and paramagnetic phases for various values of the order of
commensurability, p. The diagrams have been derived by combining the present

618 P Bak
theory for the CI transition with the theory of JosC et a1 (1978), which is valid along
the broken line where the fluid and floating phases are accidentally commensurate
(8 = 0 in (3.9)). They find that for p <4 the floating phase separates the C and fluid
phases while for p >4 there is a direct transition from the C phase to the fluid phase.
Ostlund (1981)has analysed a simplep state anisotropic clock model using the method
of Bak and Villain and his findings are in agreement with figure 31.
t t 16)
'UT c c F i
,
6 6
Figure 31. Phase diagrams including paramagnetic, floating and commensurate phases of various order p
(schematic). The diagrams were constructed by combining thetheory of Bak and Villain with the theory
of JosC et a1 (1978). (a)p >4,(b)p =4,(c) p = 3, ( d )p = 2(<d8).
The theory leading to the various phase diagrams was constructed by treating the
two types of relevant topological excitations (walls and dislocations) separately. It
would be interesting to construct a unified theory considering both simultaneously.
In the fermion theory reviewed here, dislocations would correspond to many-fermion
terms in (3.13),except in the case p = 2 where a vortex can be described by two-fermion
terms of the form c'([)c+((+ l)+c(()c((+ 1). Bohr (1982) has solved this model
and finds an Ising-like transition from the C phase to the paramagnetic phase, and
not to an incommensuratephase, in agreement with the considerations above.
In fact it is possible to construct a fermion Hamiltonian which describes the ANNNI
model globally. In the very anisotropic limit the model is equivalent to a 1D quantum
Hamiltonian which, after a duality transformation, becomes a special case of the xyz
spin-; chain in a transverse field (see Rujan 1981, Emery and Peschel 1981). A
Jordan-Wigner transformation then leads to the following exact fermion rep-
resentation:
H = -1t(a:ai+l- aia:+I) +t(a:a:+, - UiUi+l) - (2a:ai - 1)(2a:+lai+l- 1)
i
1
X
+ - ( 2 ~ : ~ i-1). (3.20)
Again the fermions ai represent domain walls. The first term allows hopping to nearest
neighbours (meandering). The second term annihilates or creates pairs of walls
(back-bending). Domain walls at nearest-neighbour sites replace each other. The

Commensurateand incommensurate phases 619
quantity l/x acts as a chemical potential and t is a measure of the temperature. The
model reduces to the exact soluble xxz model in a transverse field (Lieb 1967) if we
ignore the second term (dislocations). In 2D statistical mechanics this is known as
the six-vertex model in an electric field. The Villain-Bak theory is a way to carry out
the low-temperature expansion for the xxz model.
The continuum limit form of the xxz model is well known. Indeed, one obtains
a sine-Gordon model of the form equation (3.9) (den Nijs 1981). The cos p(p operator
is generated by umklapp (Haldane 1980, Emery and Black 1981, den Nijs 1981).
The dislocation operator becomes a vortex operator (den Nijs 1981, table I and 11).
Its effects in the phase diagram are known from the work of Schulz (1980).
4. Chaotic states and the devil’s staircase
In § 2 the transition between one particular C phase and the I phase was treated
within the continuum approximation. Once the model is brought into the continuum
form, the possibiljty of locking-in at other C values has been eliminated. While some
information about global phase diagrams could be achieved in the limit of weak
periodic potential by superimposing continuum solutions of progressively higher order,
not much could be said on the situation for stronger potentials where the infinity of
C phases cannot be disentangled. For instan’ce,it is not clear whether or not incom-
mensurate phases may exist in the ‘strong’ potential case. Also, the solitons will
probably be pinned near the CI transition (Bruce 1980) even in the weak potential
limit, in contrast to the unpinned continuum solutions.
4.1. Chaotic behaviour of the Frenkel-Kontorowa and ANNNI models
To get more insight on global phase diagrams, and to elucidate the shortcomings of
the continuum approximation, it has turned out to be extremely profitable to work
directly with the discrete Hamiltonians (Aubry 1979, Bak 1981c, Pokrovsky 1981,
Bak and Pokrovsky 1981). Aubry’s starting point was the discrete Frenkel-Kontorowa
model (1.1).The configuration which minimises the energy can be found by differen-
tiating with respect to xn:
(x,,+~-x,,) - (x, -xflVl)= -V sin x,,. (4.1)
We have chosen b=2.rr. Defining W,, = X ~ - X , , - ~ we can rewrite (4.1) as a set of
recursion relations, or a ‘mapping’of (x,,, W,) on (x,,+%,Wfl+l):
This mapping has been studied in detail by Greene (1979),Aubry (1979),Pokrovsky
(1981) and Fradkin and Huberman (1981).
A similar model has been studied by the author (Bak 1981~)directly in the context
of CI transitions, and to be specific this model (which has the same properties as the
model above) will be reviewed here.
The starting point is again the mean field theory of the 3D ANNNI model, equation
(3.1). In the preceding section the transition to the q =$ phase was studied by
transforming to a continuum phase model (3.4). We shall now keep the discrete

620 P Bak
phases, and the free energy becomes
where S =J1/4J2and a V(cpi)is a periodic potential with period 2 ~ / 4 .Near T,,where
the free energy can be expanded in terms of the order parameter Mi, the potential
becomes a cosine potential:
a = -T/J2A2 (4.4)
A4
V ( & )-96cos (4%)
and (4.3) becomes identical to the Frenkel-Kontorowa model. The configurations
minimising F are the solutions to the equations
W,+I = W, - a V'(cp,)
pn+1= 40, + W,+l
(4.5)
with
V'(cp)= $[tanh-'(A cos cp) sin cp -tanh-' (A sin cp) cos cp].
Solutions to (4.5) can be generated by starting at one point (pi,Wj)and iterating
the equations. Figure 32 shows the results of computer iterations for A = 0.71 and
(a) a =3.86, (b)a = 8. The diagrams show Wnas a function of qn(mod712). Points
on the 9 and w axes were generally chosen as starting points and typically 2000
iterations were performed. Case (a) corresponds to a relatively weak potential
(temperature not too far from T, and J2--2J1 in figure 24). Several types of
trajectories are seen.
v / n ip/n
Figure 32. Computer-generated solutions to the mapping defined by equation (4.5) for the ANNNI model.
( a ) a =3.86 corresponding approximately to J1=JO, T =7.36 Jo. At the CI transition, Jz --2J0. ( b )
a = 8 corresponding to J1= 0.341Jo, T = 3 Jo, Jz = -0.375 Jo at the CI transition.
(i) Fixed point at cp =7r/4,W=O. This is the commensurate phase. There is
another fixed point at (1r/2,0),which is unphysical since it maximises F.

(ii) Smooth invariant trajectories. Starting at points (7r/4, W >0.01) the points
eventually form continuous curves, despite the discreteness of the iteration procedure.
These curves describe I phases of the form (1.2) which in the present variables reads
The existence of such curves can be proven using the Kolmogorov-Arnold-Moser
theorem (see, for instance, Arnold and Avez 1978). A diffraction experimentproduces
Bragg satellites at the positions given in the introduction, equation (1.3). Some of
the curves appear to be broken curves. This is because the iterations were stopped
before the KAM curves were fully formed. A shift of a represents a shift of the
starting point along the trajectory, but the curve and the energy remain invariant.
This is the Goldstone mode (phason) of the I phase. A gapless sliding mode exists
despite the discreteness of the Ising model, in agreement with our considerations in
the weak potential limit in 09 2 and 3. At large values of Wi,W does not deviate
much from its average value, indicating a small content of higher harmonics. When
W ibecomes smaller (but still larger than 0.01)the trajectories describe a soliton lattice
with a large density of points near (7r/4,0) and a smaller density near the phase kinks
at ( ~ / 2 , 0 . 3 ) .The closed orbits around the unphysical fixed point are energetically
unfavourable commensurate phases with an incommensurate modulation. The
existence of the continuous sliding mode despite the discreteness of the lattice has
also been found by Sacco et a1 (1979).
(iii) Chaotic trajectories. When the initial configuration is chosen sufficiently close
to (7r/4,0)the trajectories change in a dramatic way. The points do not form a closed
orbit but fill out completely a finite, but small, area in the phase space. This area is
mainly concentrated near the point (7r/4, 0) but the irregular grainy curve connecting
the point with the equivalent area around 37r/4 belongs to this ‘chaotic’ trajectory.
The recursion relations act as an information source in a way similar to a random
number generator in a digital computer (Shaw 1980). An infinitesimal shift in (Wi,a!)
changes the flow dramatically, but the area covered is essentially the same. Physically
these solutions describe a random combination of pinned solitons and antisolitons.
A diffraction experiment would show ‘smeared’ peaks corresponding to the random-
ness. Note that the random behaviour is intrinsic. There is no randomness in the
original model which is completely deterministic. Depending on the sign of S either
a series of solitons or antisolitons will be stable. The solutions which minimise the
energy eventually form a complete devil’s staircase as discussed by Aubry (1979) and
Bak and von Boehm (1980). Note: the devil’s staircase consists of commensurate
states only and no chaotic states. The chaotic ‘phases’describing random distributions
of pinned solitons are metastable. This does not mean that they are physically
uninteresting! The infinity of stable states forming the devil’s staircase are separated
by relatively high energy barriers, and a real physical system can not possibly relax
to the real ground states in a finite amount of time. The system will definitely lock
into some low-lying metastable randomly pinned configuration, a chaotic state. The
situation is similar to that proposed in several theories of spin glasses. The low-
temperature solutions found by Villain and Gordon (1980) and Fisher and Selke
(1980) can be classified as regularly ‘pinned’domain wall states. The calculation here
is valid at high temperatures where the fourth-order term dominates; it does not
provide information on the low-temperature properties.

622 P Bak
0.21
--‘I16
0 0.2
6
0.4
Figure 33. Wavevector plotted against the natural misfit S near the CI transition. Note that a ‘chaotic’
phase exists between the I and C phases. Compare with the continuum solution, figure 12(a).
Figure 33 shows the inverse average misfit 4 as a function of 6. This curve was
calculated by evaluating rf and the free energy F(4.3) of a very large number of
trajectories, and choosing for each value of 6 the one with lowest F. The wavevector
of the corresponding periodic structure is q = 2 r ( O ,O,a(l+q)). At large S the I
phase is stable. A careful numerical search in this regime reveals extremely narrow
commensurate orbits, but we shall return to this point later. When the distance
between solitons (at decreasing 8 ) becomes sufficiently large the repulsive interaction
between solitons cannot overcome the pinning to the lattice, and the soliton lattice
breaks up to form the chaotic phase. At some point, the natural misfit 6 becomes
small enough to stabilise the C phase (the soliton lattice energy becomes positive).
The critical distance between solitons, l,,, has been estimated by Pokrovsky (1981)
and Bak and Pokrovsky (1981). The pinning energy decreases exponentially with the
width lo of the soliton:
and the soliton interaction decreases exponentially with the distance 1 between solitons:
-exp (-r2l0) (4.6)
Eint -exp (-//10). (4.7)
(4.8)
All lengths are measured in units of the lattice constant, a.
In summary, for the case of a weak potential the effect of discreteness is to squeeze
in a relatively narrow chaotic or, more precisely, pinned regime between the C and
I phases. Apart from this the continuum approach is qualitatively and quantitatively
correct.
In case (b) in figure 32, the potential is stronger. In the ANNNI model this
corresponds to lowering the temperature or increasing the perpendicular coupling.
The width of the chaotic regime increases dramatically. However, invariant trajec-
tories describing I phases still exist. The maximum distance between solitons in the
unpinned phase is only -5.7 compared with 18.2 in case (a). The I trajectories near
the chaotic regime follow a quite complicated orbit.
The transition between the I and chaotic phases takes place when Epin-Eintor
2 2
lcr=7T lo.

Additional features become evident. In the chaotic regime there are series of
islands with no chaotic points inside. Near a series of islands there is a series of
limit-cycle hyperbolic fixed points describing a higher-order commensurate phase.
These C phases eventually form the devil’s staircase. There is an infinity of islands
in the chaotic phase, and the resulting structure is known as a fractal (Mandelbrot
1977).
In the incommensurate regime there are also islands which describe higher-order
C phases between the invariant trajectories. For instance, between the trajectories
through (7r/4, 0.28) and (7r/4, 0.35) there are four islands giving a C phase with
q =a(l- $)= 3/16. The C phase is given by a limit-cycle sequence between (0.2867r,
0.295) and three other points between the four islands. For Wi closer to the chaotic
phase a C phase indicated by a series of five islands is evident and q =$(l-4)=4.
This (and the other) high-order C phases in the ‘incommensurate’ regime are surroun-
ded by narrow chaotic phases of their own (see also Pokrovsky 1981). Consider the
sequence of points generated starting from (7r/4,0.21). The behaviour is clearly
chaotic: sometimes the points are below the island, sometimes they are above. The
region around ( 0 . 2 8 ~ ,0.2)in figure 32(b) is a miniature reproduction of figure 32(a)!
The I phase is thus penetrated by high-order commensurate phases, as could be
expected from the discussion in $2, and each I phase is surrounded by a narrow
chaotic structure. This forms the incomplete devil’s staircase (figure 2(b)),compared
with the complete staircase of the chaotic regime. Figure 33 shows the misfit plotted
against the natural misfit. Compare with the continuum solution, figure 12(a).
For larger values of a(> -12) no incommensurate phases exist and the staircase
is believed to be everywhere complete (Aubry 1979). The disappearance of the
incommensurate ‘KAM surfaces’ has been studied in detail by Greene (1979).
4.2. Chaotic behaviour of the discrete (p4 model
A similar model has been studied by Bak and Pokrovsky (1981). The model is the
discrete (p4 model
(4.9)
where V(A,,)is the double well potential:
V(A)= y(Ai- l)2. (4.10)
The dynamics and statistical mechanics of this model has been studied by Krumhansl
and Schrieffer (1975). The model may describe a structural phase transition from a
dimerised lattice (figure 6(b))to an incommensurably distorted lattice. The lattice
distortion at the nth site in a 1D system (or nth layer in a 3D system) would be
U, = (-1)”AoA” (4.11)
where A. is the distortion in the dimerised state. A special case is a Peierls distorted
system with half-filled bands, where the transition could be driven by adding extra
electrons, either from another band, or by doping the system. It has been suggested
that the model applies to doped polyacetylene.
In the continuum limit the model has soliton solutions
A(n) = tanh n/lo lo= ( A / ~ Y ) ” ~ (4.12)

624 P Bak
and just like in the previous model the general continuum solutions are soliton lattices.
The application of the continuum 404 theory to polyacetylene was first suggested by
Rice (1979) following the original theory of solitons in polyacetylene given by Su et
a1 (1979).
The A, which minimise (4.9) is found among an infinity of coupled difference
equations:
(4.13)4Y
( A n + l - A n ) - ( A n - A n - 1 ) =- An(Ai- 1)
A
which can be rewritten as a recursion relation for Wn= An -Anp1and An:
(4.14)
Figure 34(u) shows the results for l o = 4 lattice units. This width may be a good
estimate for the solitons in polyacetylene (Su et a1 1979, 1980). Figure 34(b) shows
similar results for lo = 1.5. The interpretationof the various featuresis quite analogous
to the interpretation in the discrete sine-Gordon-likecase in 3 4.1.
The fixed points PI and PZ= (hl,0) are the two symmetric dimerised states. The
closed orbits around (0,0) are invariant trajectories associated with incommensurate
states. In the Peierls case these states would be conducting since the soft phason
-0 4
L
-1
2 - I
0 1
h
Figure 34. Results of computer iterations of the mapping defined by equation (4.14) (Bak and Pokrovsky
1981). ( a ) lo = 1.5, (b)lo = 4. The continuous curves represent incommensurate ‘conducting’ states. The
chaotic orbits represent pinned ‘insulating’states of the lattice (p4 theory.

mode describes a translation of electrons and the associated lattice distortion through
the sample (Frohlich conductivity). A charge *2e is associated with each revolution
around (0,O) in the case where the model represents a T = 0 one-dimensional Peierls
system. Near (0,O) where the amplitude (the Peierls gap) is small the lattice distortion
is almost sinusoidal. As the amplitude becomes larger and larger the invariant
trajectories more and more take the form of soliton lattices, but the sliding mode still
exists. The islands marked 1-8 for lo= 1.5 describe a high-order commensurate state
where the distance between solitons is 4.
When the average distance between solitons exceeds a critical value, I,,, no regular
incommensuratestates exist but the iterations exhibit chaotic behaviour with randomly
pinned solitons. In the Peierls case the corresponding state would be insulating since
it is not possible to move solitons without cost of energy. We found the following
critical distances between solitons:
lo=1.5 l,, =5
lo =4 l,, -56.
(4.15)
The discrete (p4 model has very recently been studied by Bak and Hergh-Jensen (1982)
for stronger potentials near the point where the last incommensurateorbit disappears.
It was found that the system exhibits an infinite series of bifurcations (Feigenbaum
1978, 1980) characterised by well-defined exponents. In fact, the mapping defined
by (4.14) is very similar to the HCnon map which has been studied in detail by
mathematicians (HCnon and Heiles 1964).
4.3. Relevance of chaotic states to physical systems
The picture which emerges from the considerations above may be applicable to several
physical systems.
(i) Metal-insulator transitions in Peierls systems. When applied to a Peierls system
the transition from the I state to the chaotic state in the discrete (p4 model is, in fact,
a metal-insulator transition. Such a transition has been observed in doped polyacety-
lene at a concentration of about 1% (Chiang et a1 1977). If the soliton picture is
applied to polyacetylene the distance between solitons would be -100 lattice units
at this concentration, which is in fair agreement with (4.15) for lo=4. It should be
borne in mind that polyacetylene is far from an ideal crystalline system as assumed
here. Also, the dopant ions tend to pin the solitons. It has been suggested that the
transition is actually a Mott-like transition where the solitons are bound to impurities
up to a critical concentration (Chiang et a1 1977).
(ii) Adsorbed monolayers. In general, a pinned phase is expected to separate the
C and I phases. In fact, a significant broadening of the Bragg satellite near the CI
transition in krypton adsorbed on graphite has been reported (Chinn and Fain 1977,
Moncton et a1 1981). This could indicate a metastable chaotic phase. Again, other
interpretations are possible. Coppersmith et a1 (1981) argue that there should be a
fluid phase between the C and I phases for positive domain wall intersection energy,
which would also produce a destruction of the Bragg peak. The smearing could also
represent hysteresis associated with a first-order transition. Indeed, Nielsen et a1
(1981) have observed hysteresis at low temperatures (figure 16).
(iii) Pinning of charge density waves. An electric field will depin the soliton in
the chaotic regime (Bak and Pokrovsky 1981) for a CDW system. It is difficult to

626 P Bak
estimate the pinning energy since it is extremely sensitive to the soliton width. In the
CDW system NbSe3 a depinning transition at a critical field has actually been observed
(Monceau et a1 1976). The wavevector is almost commensurate (q = 2~(0,0.243,0),
Fleming et a1 (1978)),and the system could be in the chaotic or ‘pinned incommensur-
ate’ regime. Note again that if the wavevector is sufficiently close to a (low-order)
commensurate value the sliding modes cease to exist. The system is either in a
high-order commensurate state or a metastable chaotic state. In both cases the system
is pinned.
(vi) Spin glasses. The chaotic state has no long-range order, but has ‘frozen-in’
random defects. The chaotic phase is, in fact, a ‘spin glass’. This interpretation is
quite obvious, at least for the ANNNI model. Usually, in spin glasses the disorder is
caused by some quenched randomness. In the present case the randomness is intrinsic.
The chaotic states are, in fact, only metastable, but since the various states are almost
degenerate and separated by high energy barriers (frustration), they are effectively
stable. If a small concentration of impurities is added, the solitons will be pinned on
these impurities and the chaotic state may be fully stable.
(v) Superionic conductors have been described by models like (1.1) (see, for
instance, Beyeler 1981). Usually it is believed that the ionic lattice is always pinned
because of the comparatively weak interactions between ions, but in principle the
system could be in the gapless regime described by the invariant trajectories in figure
32. This could explain the high conductivity in such compounds.
(vi) Magnetic systems. In the magnetic system CeSb several commensurate phases
have been observed and no ‘smooth’ behaviour exists (Fischer et a1 1978, Rossat-
Mignod et a1 1977). This would correspond to the chaotic regime in figure 32(b)with
pinned phases only. In fact, broad ‘metastable’Bragg peaks which might indicate a
chaotic state has been observed (Lebech and Clausen 1979).
Acknowledgments
I am very grateful to M E Fisher for a careful reading and constructive criticism of
the manuscript, and to M den Nijs for supplying the paragraph on the relations
between the 2D ANNNI model and other models in one and two dimensions. I am
much indebted to J Steeds and co-workers for allowing me to reproduce their electron
microscope pictures of solitons in TaSez in figure 17(b).
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