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Rep. Prog. Phys., Vol. 45, 1982. Printed in Great Britain
Commensurate phases, incommensurate phases and the devils staircase
Per Bak H C 0rsted Institute, Universitetsparken 5, DK2100Copenhagen 0 , Denmark
Abstract
Modulated structures with periods which are incommensurable (or highorder com mensurable) with the basic lattice are quite common in condensedmatter 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 highorder locked C phases, which may or may not be separated by an infinity of truly incommensurate phases. This behaviour is known as the devils staircase. In two dimensions the incommensurate phase (at T # 0) is a floating phase without complete longrange order, and it does not lockin at highorder 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 twodimensional 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.
00344885/82/060587 + 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 condensedmatter 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 PokrovskyTalapov (1980) theory
3. Global phase diagrams of periodic systems 3.1. The anisotropic 3D Ising model with competing interactions 3.2. Twodimensional anisotropic Ising models with competing
3.3. 2D ANNNI model: the VillainBak theory (1981) 4. Chaotic states and the devils staircase
4.1. Chaotic behaviour of the FrenkelKontorowa 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 commensurateincommensurate transition
interactions: equivalence with the 2D x y model
Page 589
589 591
595 596 596 596 600 600 60 1 606 606 608 608
612 614 619 619 623 625 626 626
Commensurate and incommensurate phases 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 onedimensional FVdM model. The springs represent interactions between atoms, the wavy line the periodic potential. ( a ) Commensurate structure, ( b ) incommensurate structure, ( c ) chaotic structure.
interacting gas atoms adsorbed on a crystalline substrate. Despite its extreme simplicity 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 nonzero N thus form satellites around the spots from the basic lattice. Experimentally it is impossible to distinguish between a highorder C structure and an incommensurate structure. Since the commensurate numbers are everywhere dense one may speculate that the I phase will always be unstable with respect to some nearby highorder 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 diffraction pattern is not made up of welldefined 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 onedimensional zerotemperature one. At nonzero 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 x 2 , 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 socalled floating phase. We shall see that this situation may occur in two dimensions. Figure 2(b) is an attempt to show a function which locksin 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 devils staircase (Aubry 1979). Figure 2 ( c ) shows the complete devils 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 

   I z; , ;  42
:: .L .y ~ 591
Id)  
 I I 

42t
1.2. Incommensurate systems in condensedmatter physics
Incommensurate structures generally show up in systems with competing periodicities. In a solidstate 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). Raregas monolayers adsorbed on graphite constitute a twodimensional 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 P Bak
i o ) l b )
Figure 3. Krypton monolayer adsorbed on graphite. (a) Commensurate 43 structure. The krypton atoms occupy 5 of 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 raregas 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 longrange order in the metal system, but at low temperature there is often a transition into a structure where the metal ions form a regular threedimensional 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 devils 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 bodycentred 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 threedimensional 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 onedimensional 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 shortrange forces. In conductors the incommensurate structures arise from interactions between conduction electrons and the atomic lattice, the socalled 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 quasitwodimensional layered metal chal cogenides (TaSe2, Wilson et a1 (1975), Moncton et a1 (1975), Fleming et a1 (1980, 198 1); NbSe3, Fleming et a1 (1978)) and quasionedimensional charge transfer salts such as TTFTCNQ (Comes et a1 1975, Ellenson et a1 1976). Strictly speaking these systems are all three dimensional.
Twodimensional CDW exist on the surfaces of several clean metals. Lowenergy 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. Onedimensional 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 higherorder harmonics. 0, increasing temperature; 0, decreasing temperature.
Commensurate and incommensurate phases 595
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 hightemperature disordered phase to the lowtemperature 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 rareearth 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 wellstudied 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 nextnearestneigh 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 threedimensional and a twodimensional
596 P Bak
i
I I ;
.Y 1 X f l t J, (=J, ) 4
Figure 9. The three and twodimensional versions of the king model with competing interactions (ANNNI model).
version. The spins S = i l interact through a ferromagnetic nearestneighbour interac tion J1. In one particular direction there is an antiferromagnetic nextnearestneigh bour interaction J2. The competition between J1 and 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 nonzero 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 devils 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 areapreserving 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 commensurateincommensurate 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.
Commensurate and incommensurate phases 597
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 1964 Dzyaloshinskii 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 McMillans 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(lcospcp) dn (2.3)
with p = 1 and S = (2r /b)(ao 6). S is the relative natural misfit between the two lattices. With p > 1 we 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 sineGordon equation (or pendulum equation)
d2q/dn2 = p V 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. Singlesoliton 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 sineGordon 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 Vc is 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 1 6 J V ~ 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 spontaneous formation of walls. This happens when
Commensurate and 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 Talapovs 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 higherorder 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 giving strong indications that walls indeed play an important role (figure 7).
The lockin at higherorder 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 = M a 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 i s no overlap between the C phases we therefore expect the system to lockin 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 devils 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 devils 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 lowlying modes in incommensurate systems are denoted phasons (Overhauser 1971) 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 longwavelength phasons take the form of phonons in the soliton lattice. This branch is very soft because of the exponentially weak interactions between walls. The shortwavelength 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
Commensurate and incommensurate phases 601
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 nonzero temperatures 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 wqll is 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 wqll has 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 onedimensional 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 longrange order.
The phason spectrum is also of importance in the calculation of quantum effects (Bak and Fukuyama 1980). The zeropoint 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. Zeropoint fluctuations destabilise highorder commensurate phases and thus destroy the devils staircase in one dimension. We shall see that this resembles the situation for twodimensional classical systems at finite temperatures. Generally there is a close resemblance between 2D classical systems at nonzero T, and 1D quantum systems at T=O. By performing similar integrations in three dimensions it can be shown that zeropoint fluctuations will make the C phases narrower but not destabilise them completely (Bak and Fukuyama 1980).
2.4. Adsorbed monolayers: the B a k et a1 (1979) ( B M V W ) 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 (2HTaSez; Moncton et a1 (1975, 1977)) and raregas monolayers adsorbed on graphite (figure 3). The potential can be represented by a Fourier series:
V(R)=CAG COS G * R (2.17) G
602 P Bak
 Figure 14. Three intersecting walls in the I phase near the 43 structure in a raregas 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 raregas monolayers on graphite, or in 2HTaSe2. ( 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
Commensurate and incommensurate phases 603
is
(2.19)
The last term is proportional 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), xrays (Stephens et a1 1979) and synchrotron radiation measurements (Moncton et a1 1981). These experiments seem to provide a counterexample. 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 xray 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 striped phase is seen in the range 93112 K. A, cooling; 0, warming. ( b ) Electron microscope picture showing domain walls (solitons) in 2HTaSe2 (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 xray 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
Commensurate and incommensurate phases 605
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 2HTaSe2 seem 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 raregas monolayers. ( a ) Negative domain wall crossing energy A ( 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 1 must 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 3 in 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 longrange order cannot exist. Instead the I phase will be a floating phase with no longrange 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, qo commensurate, 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 PokrouskyTalapou (1980) theory
The theory of Pokrovsky and Talapov applies to the CI transition in the case of a onedimensional 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 groundstate configuration and a typical T f 0 configuration in the case p = 3 where the phase shift at the wall is 2 1 r / 3 . The configuration 20(c) (which represents a vortex) is divarded. The starting point is a 2D generalisation of the
Commensurate and incommensurate phases 607
Figure 20. Typical groundstate ( a ) and excitedstate ( b ) configurations of the PokrovskyTalapov (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(zS) 1 d q 1 d q + 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 twodimensional model amounts to the calculation of the groundstate energy of the 1D quantum Hamiltonian a:
1 A = [ Cn+?(z 1 dP S)+ V cos p q dx (2.23) with C = 1/2y. This is the 1D quantum sineGordon 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 nonzero temperature of the order of 1/13 which 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) f i p = 2 q  i  (2.25)
608 P Bak
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 PokrovskyTalapov 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 nextnearestneighbour model (the ANNNI model) is probably the simplest model in statistical physics which exhibits transitions from a simple periodic structure to longperiod 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 KnakJensen (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 incommensurate phases
0 .25 
9 0.20
0 1 5 
609
I I I _ I I I _ 114 1 / 4
()  0.25
219  3/14 
1 /5   0 2 0  3116 2 / 1 1 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+ J 1 < 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 J 2 / J 1 (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 devils 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 P Bak
behaviour has also been obtained by a calculation by Rasmussen and KnakJensen (1981). Monte Carlo calculations yield a similar temperature dependence, but fail to produce the highorder commensurate phases (Selke and Fisher 1979, Rasmussen and KnakJensen 1981).
How can these results be understood analytically? The mean field free energy is given by
M ,
F = 1 J,jMih$  T C tanh (T dcr (3.1) I ] i o
where the interactions are given in figure 9 and Mi are 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) +exp 7T (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 higherorder C phases. Near T, the amplitude A is proportional to (Tc T). The lockin to higherorder C phases is described by an expression similar to (3.5) with v = 0, whose U, goes as A,> so the widths of the highorder commensurate phases in 2J2 + J1 space 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 devils 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).
Commensurate and incommensurate phases 61 1
1 I 5 5
l~l~l;,;,;~~,~l~l , Z
z I I I I 1 1 t f
Figure 23. A soliton and the corresponding spin structure calculated for J 2 / J 1 = 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 highorder C phases with I phases in between. At low temperatures the phase diagram includes firstorder 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 highorder C phases with I phases between (incomplete devils 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
b l2a
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 firstorder transitions in terms of oscillating interactions between solitons. Fisher and Selke show, using a lowtemperature 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 firstorder transition from the + +   phase to a highorder C phase; Fisher and Selke find a quasicontinuous transition with 1 going to infinity in the expression above. The multiphase point is quite artificial. If a small longrange 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 Selkes lowtemperature 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 x y 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. Twodimensional anisotropic 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 twodimensional continuum model (Bak 1981b):
where cp is the phase of the spin density wave
Commensurate and incommensurate phases 613
I f I I 0 0.5 1.0 1 .5
J,IJ,
Figure 26. Schematic hightemperature 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 lowtemperature 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 x y 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 +
614
F M
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 VillainBak 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   ,i t + + t  , + t I
Figure 28. ( a ) Groundstate configurations of the 2D ANNNI model. ( b ) Excited states.
Commensurate and incommensurate phases 615
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 1 5 2 (3,lO) in analogy with the ground state. Defining
5, (Y ) = X P (Y 1  p (3.11) where
Q + 1 (Y 1  6 P (Y 1 5 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  2 7 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, i j = u/N,, which minimises the energy is given by
Irij 1 Irq p 14 1  q Ir 14 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 i j has a square root singularity, in agreement with the PokrovskyTalapov 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 difference between 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 spinspin 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.
Commensurate and incommensurate phases 617
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 higherorder 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 ) 2 1
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 raregas 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 firstorder 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 specific model, 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 simple p state anisotropic clock model using the method of Bak and Villain and his findings are in agreement with figure 31.
t t 16)
'\U T 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(
Commensurate and 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 sixvertex model in an electric field. The VillainBak theory is a way to carry out the lowtemperature expansion for the xxz model.
The continuum limit form of the xxz model is well known. Indeed, one obtains a sineGordon 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 devils 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 lockingin 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 instance, 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 FrenkelKontorowa 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). Aubrys starting point was the discrete FrenkelKontorowa 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 1 9 8 1 ~ ) 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 = J1/4J2 and 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) A 4
V ( & )  96 cos (4%) and (4.3) becomes identical to the FrenkelKontorowa 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 c p ] . Solutions to (4.5) can be generated by starting at one point (pi, W j ) 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 Wn as a function of qn(mod 712). 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 J22J1 in figure 24). Several types of trajectories are seen.
v / n ip/n
Figure 32. Computergenerated 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, J z  2J0. ( b ) a = 8 corresponding to J 1 = 0.341 Jo , T = 3 Jo, J z = 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.
Commensurate and incommensurate phases 621
(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 KolmogorovArnoldMoser theorem (see, for instance, Arnold and Avez 1978). A diffraction experiment produces 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 Wi becomes 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 devils staircase as discussed by Aubry (1979) and Bak and von Boehm (1980). Note: the devils 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 devils 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 lowlying 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 fourthorder term dominates; it does not provide information on the lowtemperature properties.
622 P B a k
0 . 2 1 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 r f 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 (  r 2 l 0 ) (4.6)
Eint  exp (  / / 1 0 ) . (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  Eint or 2 2 lcr= 7T l o .
Commensurate and incommensurate phases 623
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 limitcycle hyperbolic fixed points describing a higherorder commensurate phase. These C phases eventually form the devils 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 higherorder 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 limitcycle 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) highorder 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 highorder 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 devils 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 ( p 4 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 halffilled 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 ) = A n ( A i  1) A
which can be rewritten as a recursion relation for Wn = An  Anp1 and 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 interpretation of the various features is quite analogous to the interpretation in the discrete sineGordonlike case 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.
Commensurate and incommensurate phases 625
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 onedimensional 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 18 for lo = 1.5 describe a highorder commensurate state where the distance between solitons is 4.
When the average distance between solitons exceeds a critical value, I,,, no regular incommensurate states 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 HerghJensen (1982) for stronger potentials near the point where the last incommensurate orbit disappears. It was found that the system exhibits an infinite series of bifurcations (Feigenbaum 1978, 1980) characterised by welldefined 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) Metalinsulator 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 metalinsulator 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 Mottlike 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 firstorder 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 (loworder) commensurate value the sliding modes cease to exist. The system is either in a highorder commensurate state or a metastable chaotic state. In both cases the system is pinned.
(vi) Spin glasses. The chaotic state has no longrange order, but has frozenin 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 coworkers for allowing me to reproduce their electron microscope pictures of solitons in TaSez in figure 17(b).
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