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J. Phys. C : Solid State Phys., Vol. 6 , 1973. Erinted in Great Britain. @ 1973

Ordering, metastability and phase transitions intwo-dimensional systems

J M Kosterlit7 a n d D J Thouless

Depar tment of Ma them atical Physics, University of Birmingham , Birmingham B15 2TT,UK

Received 13 November 1972

Abstract. A new definition of orde r called topological order is proposed for two-dimension alsystems in which no long-range order of the conventional type exists. The possibility of a

phase transition characterized by a change in the response of the system to an external

perturbation is discussed in the context of a mean field type of approxim ation. The cr it ical

behaviour found in this model displays very weak singularities. The application of these

ideas to the x y model of magnetism, the solid-liquid transition, and the ne utral superfluid

are discussed. This type of phase transition cannot occur in a superconductor nor in a

Heisenberg ferromagnet. for reasons that are given.

1 . Introduction

Peierls (1935) has ar gue d that therm al mo tion of long-wavelength pho nons will destroythe long-range order of a two-dimensional solid in the sense that the mean squaredeviation of an atom from its equilibrium position increases logarithmically with thesize of the system, and the Bragg peaks of the diffraction pattern formed by the systemare bro ad instead of sharp. Th e absence of long-range ord er of this simple form has beenshown by M erm in (1968) using rigorous inequalities. Similar argum ents can be used t oshow that there is no spontaneous magnetization in a two-dimensional magnet withspins with mo re than one degree of freedom (Merm in and Wa gner 1966) and that theexpectation value of the superfluid order parameter in a two-dimensional Bose fluidis zero (Hohenberg 1967).

On the other hand there is inconclusive evidence from the numerical work on atwo-dimensional system of hard discs by Alder and Wainwright (1962) of a phasetransition between a gaseous and solid state. Stanley and Ka pla n (1966) found tha t high-temperature series expansions for two-dimensional spin models indicated a phasetransition in which the susceptibility becom es infinite.The evidence for such a transitionis much stronger for the xy model (spins confined to a plane) than for the Heisenbergmodel, as can be seen from the pap ers of Stanley (1968) and M oo re (1969). Low-tem-pera ture expansions obta ined by Wegner (1967) and Berezinskii (1970) give a m agnetiza-tion proportional to some power of the field between zero and unity, and indicate thepossibility of a sharp transition between such behaviour and the high-temperatureregime where the m agnetization is prop ortion al to the applied field.

In this paper we present arguments in favour of a quite different definition of long-range order which is based on the overall properties of the system rather than on the

1181

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1182 J M Kosterlitz and D J Thouless

behaviour of a two-point correlation function. In the cases we consider, the usualco rre latio n function, such as the spin-spin correlation function, vanishes at any finitetemperature in contrast to the corresponding cases in three dimensions. This type oflong-range or der, which we refer to as topological long-range orde r, may exist for thetwo-dimensional solid, neutral superfluid, and for the xy model, but not for a super-

conductor or isotropic Heisenberg model. In the case of a solid, the disappearance oftopological long-range order is associated with a transition from an elastic to a fluidresponse to a small external shear stress, while for a neu tral superfluid it is associatedwith the instab ility of persistent cu rre nts . Recently, Berezinskii (1971) has put forwardsimilar arguments, but there are some important differences in our results. A briefaccoun t of this theor y ha s already been given (Kosterlitz and Thouless 1972).

The definition of long-range ord er which we ado pt arises naturally in the case of a

solid from the dislocation theo ry of melting (N ab ar ro 1967).In this theory, it is suppos edthat a liquid close to its freezing point has a local structure similar to that of a solid,but that in its equilibrium configurations there is some concentration of dislocations

which can m ove to the surface unde r the influence of an arb itrarily small shear stress,and so produce viscous flow. In the solid state there are no free dislocations in equi-librium, an d so the system is rigid. This theo ry is much easier to apply in two dimensionsth an in three since a dislocation is associated with a point rather tha n a line.

Although isolated dislocations cannot occur at low temperatures in a large system(except near the boundary) since their energy increases logarithmically with the size ofthe system, pairs of dislocations with equal and opposite Burgers vector have finiteenergy and must occur because of thermal excitation. Such pairs can respond to anapplied stress and so reduce the rigidity modulus. At sufficiently high temperatures, thelargest pairs become unstable under an applied shear stress and produce a viscousresponse to the shear.

The presence or absence of free dislocations can be determined in the followingma nne r. We suppose tha t the system ha s a fair degree of short- rang e orde r so that alocal crystal stru cture ca n be identified. To be definite we take the crystal str uctur e to bea simple sq uare lattice with spacing U . Using the local order, we attem pt to trac e a pathfrom a tom to at om which, in the perfect crystal, would be closed. Mermin (1968) hascommented that the thermal motion does not necessarily destroy the correlation inorie nta tion of the crystal axes at large distance s, but even if this is destroyed, the direc tiontak en in one region c an be defined in terms of tha t taken in a previous region in the sam eneighbourhood. Local defects can be avoided by small deformations of the path. Thi5procedure is possible provided there are no grain boundaries. If there are free disloca-

tions present, the num ber of dislocations contained within the region surro unde d by theconto ur will be pro portio nal to the area of the region. Since the Burgers vectors of theindividual dislocations can point in two possible directions, the average total Burgersvector will be proportional to the square roo t of the a rea. The path will fail to close b>

an amount proportional to L, he length of the path . If there a re only pairs of dislocationspresent, only those pairs which are cut by the con tour will con tribute to the total B urgersvector. The number of pairs cut by the path is proportional to Lr where r is the meanseparation of the pairs. Averaging over the possible orientations of the individualBurgers vectors, we see that the path will fail to close by an amount proportional to(Lr)”’ . This allows us to determine the presence or absence of topological long-rangeorde r in the system.

W e can ob tain estimates of the transition tem pera ture for the systems discussed byarg um en ts similar to those used by Thouless (1969) for a one-dimensional Ising model



Metastability and phase transitions in two-dimensional systems 1183

with an interaction falling off at large distances as r - ’ . At large distances from a dis-location (or vortex in the case of the x y model and neutral superfluid) the strain producedis inversely proportio nal to the distance and so the energy of a single dislocation de pendslogarithmically on the size of the system. In fact, the energy of an isolated dislocationwith Burgers vector of magn itude b in a system of area A is (Friedel 1964)

vb2(1 + z) AE = In-

4n A0

where v and z are the two-dime nsional rigidity modu lus an d P oisson’s ratio respectively,and A , is an area of the order of b2 .The entropy associated with a dislocation alsodepends logarithmically on the area, and, since there are approximately A/Ao possiblepositions for the dislocation, the entro py is

where kB is the Boltzmann constant.Since both energy and entropy depend on the size of the system in the same way,

the energy term will dom ina te the free energy at low temperatures, an d the p robabilityof a single dislocation appearing in a large system will be vanishingly small. At hightemp eratures , dislocations will appe ar spontaneously when the entropy term takes over.The critical temperatu re at which a single dislocation is likely to occur is the tem peratu reat which the free energy changes sign, namely (K osterlitz and T houless 1972)

vb2(1 + z)

471k B r =

Identical considerations for the

k B q = 715

xy model give

where J is the spin-spin coupling constant and

for the neutral superfluid where p is the density of particles per unit area and m is theeffective atom ic mass which is not necessarily the sam e as the ato mic m ass for a particlemoving on a substrate.

That these estimates are upper bounds can be seen by the following argument.Although the formation of isolated dislocations will not occur at low temperatures,there can always be productio n of a pair of dislocations with equal and oppo site Burgersvector, since the strain produced by such a pair falls off sufficiently rapidly at largedistances so that the energy is finite. The critical temperatures as calculated above arethe temperatures at which a pair of dislocations (or vortices) will dissociate, ignoringthe effect of other pairs in the system. These other pairs will relax in the field of the firstpair thereby renormalizing the rigidity modulus etc. downwards and consequently

reducing the critical temp erature. M ost of the rest of the paper is devoted to a detailedstudy of this phenom enon in the systems discussed.



1184 J M Kosterlitz and D Tlzouless

2. A model system

Th e statistical problem we are faced with is essentially tha t of a two-dimensional gas ofparticles with charges + q interacting via the usual logarithmic potential, the number ofparticles being constraine d only by the requirem ent tha t the system has overall electrical

neutrality. The only essential modification necessary to treat the dislocation problemis to allow for the two different possible orientations of the Burgers vector. The hamil-tonian for such a system is

H ( v , . . . r N )= 5 li(l v i - v j / )i # j

where

= o r < r o .

Here q i and Y, are the charge and position of the ith particle, 2p is the energy requiredto create a pair of particles of equal and opposite charge a distance r o apart , and ro issome suitable cutoff to avoid spurious divergences at small separations. We expect thatthe cutoff ro is of the order of the particle diameter or, for a lattice, the lattice spacing.In such a lattice near the critical point, we will be able to replace sums over the latticesites by a n integral over the whole system since the im porta nt contr ibution s to the sumwill come from the long-range part of the interaction, that is 1 Y, - y j I % ro .

To obtain a trac tab le theory, we further assume tha t the chemical potential issufficiently large that there are very few particles present in the system. Since we havesuch a dilute gas, the configurations of least energy are those where equal and oppositecharges are closely bound in dipole pairs, well separated from one another. At low

temperatures, therefore, the fluctuations of charge within a given region are restricted.while at high te mp eratu res charges may occur in isolation so m uch larger fluctuationsmay occur. For such configurations, we see that the mean square separation of theparticles making up a dipole pair (ignoring for the present interactions between theDairs) is

,

I,",d r r3 exp{ - pq2 In ( r / r o ) }

1 d r r exp{- pq2 In (r /ro)}

Pq2 - 1( r 2 > = = ro-

pq2 - 2( 7 )

where ,!?= l jk ,T The probability of finding a pair within a given area is found bysumming exp( - 811- 2Pq2 in ( vi - r j / y o ) > over a ll values of vi and r j in the area . Thedouble sum can be replaced by a dou ble integral if we normalize by a factor of order

r i 4 , so that the m ean s epar ation d between such pairs is given in the same app roxim ationby

e - m1-d 2 T j 2 r exp{ -2/3q2 In ( r / r o ) }+ O(e-48p)

Here we can see the necessity for a large chemical poten tial p, since already in calculatingd L 2 . any terms beyond first order in exp(- 0p) become intractable. Th us



Metastabil i ty mid pha.c.c transitions in two-dimensional sys tems 1185

We ca n already see tha t a phase tra nsition to a conducting s tate will take place a t atemperature T, given by

since, in this simple approximation the polarizability, which is proportional to ( r ’ ) ,

diverges. Th e closely b ou nd dipole pairs will separa te to give a uniform two-dimensionalplasma of oppositely charged particles. The free energy of an isolated charge is

F = E - 7s z q 2 In (R’lr;) - kBTln ( R 2 / r i ) (11)

where R is the radius o fth e system. Thu s we see tha t isolated charges can app ear sp ontan e-ously when the temperature reaches T, as given by equation (10). We are thereforeparticularly interested in values of pq2 near 2.

Hau ge an d He mm er (1971) have investigated the two-dimensional C ou lom b gasunder very different conditions. In the limit of vanishing particle size, they find a transi-t ion at k T = q2 (in our units), the tem pera ture at w hich pairs begin t o form as k T + q 2

from above. We look at a different tem pera ture range a nd find a transitio n near kT= $q2 .when th e largest pairs dissociate as the transition is appr oach ed from below. The m aindifference between the two models is the presence of a finite cutoff in this paper. whichmakes the potential well behaved at small distances, while Haug e and H em me r allow thepotential to be singular at the point of closest approach.

The divergence of the polarizability as the transition tem pera ture is app roac hed from

below suggests that the most important effect of the interactions between differentpairs may be described by the introduction of a dielectric constant. The field of a pairseparated by a small distance of the ord er of ro does not extend much beyond the distanceof separation of the two charges and from equation (8 ) it is very unlikely that there willbe anoth er pair in their imm ediate vicinity. Thus the dielectric constant app rop ria te

for the expression of the energy of such a pair is unaffected by the polarizability of theother pairs. It is only for charges sepa rated by an am ount greater than the mean separa-tion d of pairs that the energy is modified by the presence of other smaller pairs lyingwithin th e range of the field. Th e effective dielectric c ons tan t, therefore, becomes largeras we consider larger and larger pairs, so that the problem becomes a sort of iteratedmean field app roxim ation. We ar e therefore faced with problems similar to those whichled to rescaling in the paper of Anderson et a1 (1970).

Within the range of a pair with large separation r % yo it is expected that there willbe other pairs with sepa ration up to r exp( -8p). Th e effect of these will be to reduce theinteraction energy of the large pair by introducing an effective dielectric constant ~ ( r ) .

To calculate this dielectric co nsta nt we consider pairs with sep ara tion s lying in a smallinterval between r and r + dr wi th d r 4 , and consider terms to first order only in dr.Using the standard methods of linear response theory, we apply a weak electric field E

at an angle 0 to the line joining the two charges The polarizability per pair is given by

ap ( r ) = q- r co s 0)

8E Z = O

where the average is taken over t h e annulus r < r’ < r + d r with Bolt7mann factor- 2 p q 2 / e ( r i

exp { - ~ u ( i . ) $ =k) ex p ( p E q r cos 0 ) . (13)

Although formally the factor ex p(pE rq cos 0) will cause a divergence in the integrals ofequ ation (12), we can a rgue tha t we c an carry out the differentiation with respect t o E



1186 J M Kosterlit: arid D J Tliouless

and take the limit E -+ 0 before integrating as follows. Inside the medium the effectivefield is set up by the tw o ch arges, an d falls off rapidly at distances greater tha n the separa-tion between the charges and so has a finite range. Moreover, since the dipole pairswithin this field have a sepa rat ion very much less th an the se paration of the two charg esof interest, the field experienced by the dipole pairs is effectively constant over them.

Thu s, our m ethod of calculating the polarizability of a given pair from eq uation (12) isjustified. We easily obtain

p ( r ) = $42'2. (14)

The density of such pairs is, by the same argu me nt that leads to eq uation (8),

dn(r) = I j 2 f f d Q l + d rr ' r' exp { -bUeff(r ')} + O(ev4flp) (15)0

where Ueff(r)s the energy of two charges separated by r in such a dielectric medium.The terms O (e-48p)corre spon d to the fact that in the normalization of the probabilityfor seeing such pairs there are terms corresponding to seeing no pairs, one pair etc, allof which we ignor e except for the no- pa ir term. Another com plication arises at this pointbecause the energy Ueff(r)s given by

As it stands, we cannot evaluate equ ation (16) because of the unknow n function E ( r ) .

However, assuming that de- '(r)/dr is sufficiently small, we can write

and check the consistency of this assum ption a posteriori. We finally obta in the suscepti-bility du e to these pairs

Lastly, we change variables to

to obtain the differential equatio n

which is subject to the boundary conditions

where E is the macroscopic dielectric constant. Since, as discussed previously, whenx = 0, r = r 0 so that the interaction of a pair separated by r0 is unaffected by a ny ot he rpairs, so that E (X = 0) = 1 . Similarly, as x -+ a . he effective interaction is reduced by a

factor E(X = a ) , hich must be the same as the dielectric constant as measured byapplication of an external field to a macroscopic region of the medium. We can simplify



Metastabilitv nrid phase transitions in two-dimensional systems 1187

equation (19) further by noticing th at, since the derivative dy/dx is pro po rtio nal toe-2 8p , the difference y ( 0 ) - y(a3) for y(0) > y,(O) must also be very small (in fact y(0) -y(co) < O(e-OP))so th at we can replace the factor (y + 4)2 in equa tion (19) by (y(0) + 4)2.

Th e problem is now reduced to solving the equation

= - exp(-Rjj)y"-d3where we have rescaled x = (ne-pp((y(O)+ 4))- ' R and y = (ne-Bp(y(0)+ 4))y" toeliminate the constant multiplying the exponential. This equation does not possess ananalytic solu tion bu t the solution s fall into tw o classes (see Appendix)

"2

2 > 0~-

(i) kBTE(T)f o r T < T ,

We interpret this sudden change in the behaviour of E(T) s Tpasses through T , as aphase transition to the conducting s tate where the pairs become u nbo und . Th e criticaltem pera ture is given by

an d th e critical value of the dielectric cons tant at the transition tem pera ture is given by

"2

__ - 2E(T,)= 0 .k B T ,

To determine the values of T , and E , and to investigate the nature of the phasetransition, it is necessary t o find an appro ximate solution of eq ua tion (21). This can bedone in two limiting cases, T < T , and T 5 T,. In the first case, y(0) z (c0) > 0 so wecan immediately integrate equation (21) by replacing y(x) by y(0) in the exponential toobtain

In the second case T 5 T,, equ ation (21) may be solved approximately by the methods

described in the Appendix to o btain y",(O) = 1.3 so that

and

This singularity in E(T) s of a most unusual type but, since we have used a mean fieldtheory, this form of the singularity is unlikely to be the correct one. Note in particular

that E(T,)s finite so tha t the susceptibility does n ot diverge at the critical temp eratu re asone would expect from the simple arguments presented earlier.




We ar e now in a position t o check our assum ption t hat dE-'(r):dr is sufficiently smallto do the integration in equ ation (16). Clearly, for T 6 T,, the replacement of E - ' ( 7 )by acon sta nt is valid. Th e mos t unfavourab le case is for the critical trajectory when y ( x ) = 0and dE-'(r)/dr is largest. The co ndition for the replacement to be valid is

1___ < _ _ - .e- ' ( r )d r r + J j r l n r

Th e critical solu tion of equ atio n (21) is, as c an be verified by direct substitution

21112 In In Ey"(x) N ~ - + . . .

x + x 2 2

Remembering that R cc In r /ro , this gives

1 In In r 1N ~ _ _- .e- ' ( r )

d r r l n r l n r r l n r

The free energy ha s a singularity at the critical tem pera ture but it is so weak that anyobservable except the dielectric constant is finite and all derivatives are finite andbounded for T < T, . Not surprisingly, this is in agreement with the results ob tain ed byAnderson an d Yuval (1971) for the one-dimensional Ising model with inverse squareinterac tion, which can be treated with ou r meth ods to o btain m ost of their results. Theenergy of a pair of charges is

qzs: d x x ~ - l ( x ) e x p { 2 ( p y ' ~ - ~ ( x ) 1)x)U = ~~

s dx exp{ -2(Pq '~- ' (x) - l)x}

where we have substituted x = ln( r/r o) . Since the specific heat C is proportional todU /db , any singularities in C will sho w up in integrals of the form

where we have used the fact that P q 2 c - ' ( x ) - 1 % const > 0 fo r T < T,. This integralwill certainly be finite except possibly at T = T, , in which case we want the asymptoticform of y ( x ) as given by e qu atio n (29), since it is clear th at d y ( x ) /d y ( O) s singular only atinfinity.

Differentiating equ atio n (21) with respect t o p(0)an d using equ ation (29) we obtain

which has the solution

Th e integrals of the form (32) in the expression for C are thus all finite since the eCaxfactor ensures convergence. We can similarly show that all derivatives of C of finite

orde r are finite and b oun ded at the critical temp eratu re, which strongly indicates thatC is analytic there.



Metastabil i ty and phase transitions in two-dimensional systems 1189

It must be borne in mind that mean field approximations are notoriously bad inpredicting the form of the singularities in specific heats etc, since such a the ory ignores

fluctuation s in the internal field. We therefore expect tha t ou r value of is an over-estimate and that in an exact treatment there may be a weak singularity in C.

Just above the critical temperature, the largest pairs will dissociate and be able to

carry charge so that the medium will conduct. Assuming that the mobility of a freecharge behaves smoothly at T,, the behaviour of the DC conductivity as T+ T, ro mabove will be determined by that of the density of dissociated pairs 6 n ( T ) .This numbermay be estimated by calculating the number of pairs whose separation is less than R ,where R is defined by y ( R )= 0. When the separation r increases so that y ( r ) < 0, thetrajectory y(x) (x = In (r/r,,)) alls off rapidly to minus infinity (see figure Al) . We in terpretthis failure of the theory as the complete dissociation of such a pair. We find that R isgiven by (see Appendix)

In -R - exp{(ln-)”’] 1

ro r - F,‘ T - T ,

(35)

where we have taken E ( r ) z 1 thr ou gh ou t the region of integration . Th is is justified here

because {Pq2 /E ( r ) }-

1%

1 nearT,

for all r . Thus, the density of dissociated pairs&(T) s given by

where 2n is the total density of charges. Also

so that

and

(38)

(39)

Thus, the DC conductivity tends to zero with all derivatives zero as the critical tempera-ture is approached from above .




To conclude this section on the model system, we would like to point out that theassumption of a very dilute system (e-2Bp< 1) is not necessarily valid in a real system.However, we expect that the qualitative ,arguments will go through even in such a caseand the general form of the results will be unchanged. We can imagine increasing thecutoff r , to some value R, such that the energy of two charges a distance R , apart is

2jc(R,) where exp { - 2 p ( R 0 ) P ) < 1. For charges further apart than R,, we can use thetheory as outlined previously. The boundary conditions given by equation (20) will bechanged to

-2

with E(R , ) an unknown function. The critical temperature and the dielectric constantwill now be determined in terms of E (R , ) nd p(&). To determine these two quantities,a more sophisticated treatment is required, but we expect that the behaviour of thedielectric constant and specific heat at the critical temperature will be unchanged.

3. The two-dimensional xy model

The two-dimensional xy mod el is a system of spins constrained to ro tate in the plane of

the lattice which, for simplicity, we take to be a simple square lattice with spacing a .

The h am iltonian of the system is

where J > 0 and the sum ( i j ) over lattice sites is over nearest n eighbours only. W e havetaken I Si = 1 and q5 i is the angle the ith spin makes with some arbitrary axis. Onlyslowly varying c onfigurations, th at is, those with a djacent angles nearly eq ual, will giveany significant contribu tion to the partition function so that may expand the hamiltonianup t o terms qu adra tic in the angles.

It has been shown by m any a uthors (M ermin and Wagner 1966, Wegner 1967.Berezinskii 1970) that this system do es not have any long-range o rder as the g roun dstate is unstable against low-energy spin-wave excitations. However, there is someevidence (Stanley 1968, M oo re 1969) that this system has a phase transitio n, but it

cannot be of the usual type with finite mean magnetization below T,.As we shall show.there exist metastable states co rresp ondin g to vortices which a re closely bound in pairsbelow some critical temperature, while above this they become free. The transition is

characterized by a su dden change in the response to a n applied magnetic field.Expan ding about a local minimum of H

where A den otes the first difference oper ator, +(U) is a function defined over the latticesites, an d the sum is tak en over all the sites. If we consider the system in the co nfigurationof figure 1, its energy is, from equation (43)

RH - E , d l n - (44)

U

where R is the rad ius of the system. Th us we have a slowly varying co nfiguration. whichwe shall call a vortex, whose energy increases logarithmically with the size of the system.




\ \ - A / /

Figure 1. An isolated \ortex in th e xy model

From the arguments of the Introd uction , this suggests tha t a suitable description of thesystem is to app roxim ate the hamiltonian by terms qua dra tic in A+(r) and split this upinto a term corresponding to the vortices and another to the low-energy excitations(spin w aves).

W e extend the dom ain of +(Y) to -m < + ( r )< cc to allow for the fact that, in theabsence of vortices, ((+(Y) - ( Y ‘ ) ) ~ ) increases like In ( 1 Y - u ’ l ) (Berenzinskii 1971).Thu s, at large separations, th e spins will have g one throu gh several revolutions relative

to one another. If we now consider a vortex configuration of the type of figure 1, as wego round some closed path containing the centre of the vortex, +(Y) will change by 271

for each revolution. Thus, for a configuration with no vortices, the function + ( r ) willbe single-valued, while for one with vortices it will be many-valued. This may be sum-

marized by

E A+ ( 4 = 2 w q = 0, * 1, * 2 . . . (45)

where the sum is over some closed contour on the lattice and the number q defines thetota l strength of the vortex d istri but ion cont ain ed in the con tou r. If a single vortex of thetype shown in figure 1 is contained in the contour, then q = 1.

Let now +(Y) = $(Y) + $(Y), where $(r ) defines the angular distribution of the spinsin the configuration of the local minimu m, and $(Y) the deviation from this. The energyof the system is now

H - E , = J c A$(4 )2 + J c A&Y))2 (46)r r

where

a A$(v) = 0 and A$(r) = 2719. (47)

The cross term vanishes because of the c ond ition (47) obeyed by $(v). Clearly the con-figuration of abso lute min imu m energy correspo nds to q = 0 for every possible contour

when $(Y) is the same for all lattice sites. We see from equ ation (45) tha t, if we shrink thecontour so tha t it passes throu gh only four sites as in figure 2, we will obtain the strength




I

1

\

f

\ /

Figure 2. Contour a r o u n d cen t re of Lortex

of the vortex whose centre we can take to be located on a dual lattice whose sites r* lieat the cen tres of the squ ares of the original lattice (Berezinskii 1971). Th is procedu reenables us to define the vortex distribution function p ( r * ) given by

(48)( V * ) = 2 4 e h*i.e*e

Go ing now to a co ntin uu m no tatio n for convenience, using equ ation (47). we caneasily find the equation obeyed by & v )

VZ$(r) = 271p(r). (49

In terms of p , the energy of the system in a given configuration is

R+ 2nJ [[ 2 rd 2 r ' p ( v )p ( r ' ) In -

Y O

where R is the radius of the system, ro is a cutoff of the o rder of the lattice spacing a and

g( r ) is the Green function of the square lattice defined so that g(0) = 0 (Spitzer 1964).The last term of equ atio n (50) requires that C q , = 0 which is the condition that the totalvorticity o fth e system vanishes, and corresponds to the requirement ofelectrical neutralityin the mo del system. Th e first term produces the spin-wave excitations and is responsiblefor destroying long-range order. and the second term is the interaction energy of thevortices which cause the phase transition.

If we ignore al l vortex config uratio ns, we obta in for the spin-spin correla tionfunction

where (Spitzer 1964)

and

e-Y- -a 2\12

(?;= Euler's constant)



Meta stability and phase transitions in two-dimensional system s 1193

and we can see that the spin-wave excitations are responsible for destroying any long-range order in the system, but have nothing to do with any phase transition which mayoccur .

Th e energy E , of the vortex configuration is now simply

r . - r .E , % - 2715 1 iqj ln

l r , - v l , f a i # j

(54)

where q1 is the strength of the i th vortex whose centre is located at rL. We no longerdistinguish between the original and dual lattices. This asymptotic expression for theenergy is in fact a very good estimate even down to 1 Y 1 = a (Spitzer 1964) so we can useit for all 1 Y I> a. Since in ou r appro xim ation . the spin-waves an d vortices do no t interact

with one anoth er, the problem is now reduced to that o f the model system with thevortices playing the r61e of the charged particles and

E , = - 2715 1 i4 j ln for 1 yi - y j > ai # j a

= o otherwise . ( 5 5 )

The chemical potential p of a single vortex of unit strength is

l i = - 2715 In- = 2nJ(g + 1 In 2) ( 5 6 )

which is to be com pare d with the exact expression for the energy of two vortices separated

by a single lattice spacing

r0

a

/L = z2J (57 )

which is obtai ned from equ atio n (50) using the exact value of g ( a ) = b. The criticaltemperature T, at which the trans ition takes place is given by the solution of the equa tion

% 0.12.

Below T,, the vortices will be bound in pairs of zero total vorticity, while above T , they

are free to move t o the surface und er the influence of an a rbitra rily weak ap plied m agneticfield, thereby causing a sudden change in the form of the response to the applied field.

O n e way of seeing the effect of the transition is to cons ider the spin-spin correlatio nfunction (Si. S j ) . Ta kin g the vortices in to acco un t, this is modified to

(59)

Since the vortices an d spin waves do not interact in our ap proxim ation. the two averagesare taken indepen dently. The average (exp(i($i - +hj))) is taken over all possible valuesof $i. and is given by equation (53) and (exp{i($i - $ j ) } ) is taken over all possiblevortex d istributions. In o ur me an field app rox ima tion . we can estimate the latter termas follows

<Si*Sj> <exp{i($i - $j)})<exp(i($i - j ) } > .

- -(exp{i($i - $j)}) = ( S i .S j ) K ( T ) (60)

I V i - r , 1 -+ r




where sis the spin on the ith site in the absence of spin-wave excitations. Thus.

K ( T )= O2 (61)

where 0 is the mean magnetization of the metastable configurations without the spin-wave excitations. Thus, 0 corresponds exactly to e - ' of the model system so that

K ( T )= O(1) fo r T < T, and vanishes discontinuously at T = T, according to equation

Since the spin waves and vortices do not interact. the system of vortices alone is

equivalent to a system of freely moving, straight parallel current-carrying wires whosenumber is not conserved, located at the centres of the vortices. The sign of the forcesbetween the wires is reversed, and I i = qi$nJ, Ii eing the current in the ith wire insuitable units. The chemical potential = n2J is the energy required to 'create' a wire

with no c urre nt. Th e sum of the cu rren ts in the w ires is contained to be zero, corres-ponding to the condition C q , = 0. This analogy tells us that, on applying a magnetic

field in the plan e of the system. the vortices will tend to move at right angles to the field.

the direction determined by the sign of q i .

(27) .

4. Two-dimensional crystal

Fo r simplicity, we consider a crystal with a squ are lattice of spacing a, nd when possible

as a continuum with a short distance cutoff y o of the order of the lattice spacing a. s

far as any critical phen om ena are concerned, this will make n o essential difference. The

sho rt-di stan ce behaviour will affect qu ant ities like the effective chemical potential. bu tfor our purposes we require only that this is sufficiently large. We can therefore usestan dard linear elasticity th eory t o describe our crystal (L anda u and Lifshitz 1959).

The stress oij s related to t h e strain u i j for an isotropic medium by

oii= 2vuij + 3.6iju,, (62)

and the interna l energy by

As mentioned in the Introduction. there is no long-range order of the usual type inthis system, exactly as for the xy model of the previous section. Writing the energy in

terms of the displacement field u(v).which describes the displacement of the lattice sitesfrom their equilibrium positions, and expanding to terms quadratic in u(J*),we can showthat the mean squ are deviation of a site from its equilibrium position increases logarith-mically with the size of the system (Peierls 1935. Berezinskii 1970). This destruction oflong-range order is caused by the low-energy phonon modes, and the phase transitionfrom the solid to the liquid s tate by the dislocation configurations. Since in our app rox i-mation, the phonons and dislocations do not interact, we may treat them separately.As in the magnetic case, we ma y decom pose the displacement field as

(64)(v)= v (v )+ Is(v)

where$u(v)dl = 0 and @(v)dl = b



Metastabil i ty and phase transitions in two-dimensiorial sys tem s 1195

where the integral is taken rou nd s om e closed con tour a nd b is the tot al Burger's vectorof the dislocation distribu tion containe d within the c ontou r. v ( v ) is the displacem ent fieldof the phonons a nd Iz(v) that of the dislocations. If the contou r in equation (65) is takenround only one dislocation it gives the Burger's vector b of that dislocation. F or simpli-city we shall assu me tha t 1 b I is the sa m e for all dislocations an d the m ost likely value of

1 b 1 will be the smallest possible. that is I b I = a .The most convenient way of treating a medium with dislocations is to introduce a

stress function ~ ( v ) hich is related to the stress oi j (v ) y (Friedel 1964, Landau andLifshitz 1959)

wherec 1 2= + 1 , c Z 1= -1, e i j = Ootherwise.

We c an then define a source function y ( v ) describing the distribu tion of dislocations so

that ~ ( v )beys the equation

4v(v + A)

2v + 3.K = = 2v(l + 7 ) .

y(r) is given by (Friedel 1964)

(69)

where and b(') ar e the position an d Burger's vector respectively of the cxth dislocation.

Having set up such a formalism, we can investigate the response of a mediumconta ining dislocations to an applied stress in exact analogy t o the electrostatic case. Inou r case, crij and ui jco rre spo nd respectively to th e electric field E and displacement fieldD . Th e strain energy of the medium due to the dislocations is

ay(v) = a 11 J axja)

. .b(?) $ 2 ) ( v - r (a ) )

U = $ [ d 2 r ( v )(u )

d2v d2v' y ( v ) g( v - v ' ) y(v') + 02 U

where y(u) is the Green function of equation (67)

and

g(0) = 0.

Using tho se equations, we ca n imm ediately see that th e energy of an isolated dislocationincreases logarithmically with the area of the system. The strain energy of a pair of

dislocations with equal but oppositely directed Burger's vectors is easily found to be(Friedel 1964)



1196 J M Kosterlitz and D Thouless

where I r 1 is the separation of the two dislocations, 0 the angle between b and r , and 2 p

the energy required to create tw o d islocations one lattice spacing ap art . This resultimplies that, at low temperatures, the dislocations tend to form closely bound dipolepairs. Clearly, the condition Cb'")= 0 must be satisfied so that at low temperatures theenergy of the system is finite, corresponding to the condition of electrical neutrality in

the model system.Th e next step is to identify a qu antity analogous to the dielectric consta nt. Consider

the stress function ~ ( " ' ( r )ue to the clth dipole with source function q("'(r), cf Jackson1962)

where r(") enotes the centre of the ath dipole. Assuming the dipoles are very small, weexpand in powers of r' for 1 Y' 1 < 1 r - 4') 1

since the linear term vanishes for a dipole. We can now define the macroscopic stress

function by averaging over a region AA such that ( r ' ) AA < A , where ( r ' ) is themean s qua re separation of the dipole and A is the area of the system.

@g(v - r ')~ ( r ) K d2v'y(r')g(r - i ) K d2v'Cij(v ' )s s axpx;.

where

d2r'y("'(r')6(r - Y ( ' ) ) ) ~ ~ ~ ( Y )

(75)

where n(r) s the density of dipoles.

the applied stress aijbySimple tensor analysis shows that, for an isotropic medium, C,, is given in terms of

(77)

where C, and C, are to be determined by a model calculation. From equation (75) ,wefind after an integration by parts

Cij = ~ i k ~ k L C I a k l C 2 6 k~ m m ) 0(o2)

d2r'q(r')g(p - i ) .X ( r )= 1 - K ( C , + C ,)

Thus, the effect of the dislocation pairs is to renormalize K - ' ---f K - ' - C , + C,) ,and we immediately see that the quantity corresponding to the dielectric constantE(r) is E ( r ) = 1 - K { C , ( r ) + C,(r ) ) . Using simple linear response theory as in the modelsystem, we ob tain

C , ( r )= - @b'n(r2>

C, ( r ) = & /?b2n(r2 os 26) (79)



Metastability and phase transitions in two-dimensional system s 1197

where the averages are to be taken with Boltzmann factor exp (-fiUe ff(r)} and

Provide d we choose the cutoff ro sufficiently larg e so that our cont inuum approxima-tion holds, an d that p is large eno ugh , we only have to show tha t the average over 8 doesnot change the analogy with the m odel system of 3 2. We can easily show tha t eq uation(19) is modified to

with

fib2y ( x )= - 4

471K4x)

and Z,(z) is the vth-order m odified Bessel function (A m bra m ow itz an d Stegun 1965).Provided / L is sufficiently large, y ( x ) is very small for all x, so that the Bessel functionsoccurring in eq uation (81) are essentially cons tant. Thu s we can imm ediately apply allthe results of 9 2 to this case.

5. Neutral superfluid

The same type of argument can be applied to a neutral superfluid in two dimensions.All that is required is that Bose conde nsatio n should o ccu r in small regions of the system,so that locally a cond ensa te wavefunction can be defined. Th e argum ent of Ho hen berg

(1967) shows that the phase of the cond ensa te wavefunction fluctuates over large distancesso that the type of order defined by Penrose and Onsager (1956) cannot occur. If acon de nsa te wavefunction ca n be defined in a local region, it should be possible to explorethe va riation of its phase from o ne region to a neighbouring region. Th e total vorticitywithin a region is found by the change of phase along the boundary, divided by 271.

Just as the energy of a vortex in the ma gnetic system o r dislocation in the crystal, so th eenergy of a superfluid vortex increases logarithmically with the size of the system. Atlow temperatures, there will be no free vortices, only clusters of zero total vorticity.Neglecting the interaction between clusters, the critical tem per atu re is given by equa tion(5). Th e analogy with the model system is close as there is only on e type of vortex which

may have either sign. As in this model, we expect the in tera ctio n between vo rtex clustersto lower the critical temp eratu re from the estimate of equatio n (5). The lattice Bose gashas been discussed in detail by Berezinskii (1971) who finds the parameters of the Bosegas to be related t o those of the m agnetic system by

where c is the speed of sound. Using these parameters, we can estimate the value of T,from eq uation (58) and the superfluid density p,(T,)which is nonzero in contrast to theresult of Berezinskii.

Below the critical temperature, for a lattice Bose system with periodic boundary

conditions, superfluid flow is stable in the therm ody nam ic limit, since a sta te with flowis one in which the phase of the ord er para me ter changes by a multiple of 271 round the




system. The system can change from o ne such state to an othe r by the creation an d sepa ra-tion of a pair of vortices, on e of which passes right r ou nd th e system before recom biningwith the other, but there is a high energy barrier which prevents this process. Thedifferent states of superfluid flow are topologically distinct states which do not maketransitio ns between one anot her at low temperatures. Th e phase transition is char acter -

ized by these states becoming mutually accessible above the transition temperature, sotha t the flow states are no longer metastable.

Exp erim ents on the onset of superfluid flow for thin films-see, for exam ple, Sym ond set al(1966), Chester et al(19 72) and He rb and Da sh (1972)-show a stron g depression ofthe temperature as the film becomes thinner. According to the considerations of thispap er the critical tem pera ture should be given by

where ps s the density of superfluid particles per u nit ar ea an d m x is the effective ma ss ofthe helium atoms in the film. If ps is taken to be the bulk value multiplied by the filmthickness and m* is taken to be unaffected by the substrate, an d so equal to the atomicmass, this formula gives to o high a value for T,, so the effect of the b ou nd arie s in reducingps must be importan t. The value of p, at the onset temperature must be nonzero: fora film thickness 1.5 nm at 1.5K the superfluid density given by equation (81) is abo ut0.22 times the p article d ensity, if m* is taken equal to th e atom ic mass. It is not surprisingtha t the specific heat m axim um occurs at a higher tem pera ture than the onset of super-fluidity, since a conside rable de gree of shor t-ra ng e ord er is necessary before the con -siderations of this paper have any relevance. D e Genn es has made a relevant commenton this mat ter (see Symonds et a1 1966).

Fo r a charged superfluid (sup erconductor) the argu men t cannot be carried through

because, as a result of the finite penetra tion de pth A , the energy of a single flux line isfinite. The circulating current density inside a thin superconducting film of thicknessd 6 A is (Pearl 1964)

where r is the distance from the flux line. The repulsion energy between two vorticeswith opposite circulation falls off like r - ’ instead of increasing as In r for large separa-

tions. T he self energy of a single flux line is thus determ ined by the cu rre nt near the fluxline and is approximately (D e Ge nne s 1966)

where is the ‘hard core’ radius of the flux line.

6. Isotropic H eisenberg model in two dimensions

The isotropic Heisenberg model for a two-dimensional system of spins is quite differentfrom the xy model. To show the nature of this difference. we consider a large system



Meta stnhility and phase trnnsitions in two-dimensional system s 1199

with periodic bo unda ry conditions, but similar considerations a pply to other boundaryconditions. In the xy model a t low tem peratu res, the direction of magnetization in aregion is defined by a single angle $ which varies slowly in space. Althoug h the ang le 4fluctuates by a large am ou nt in a large system, the nu mb er of multiples of 271 it changesby on a path that goes completely round the system is a topological invariant, so that

are num bers defining a particu lar metastab le state. Tran sitions can only take place fromone metastable state to another if a vortex pair is formed, the two vortices separateand recombine after one has g one right rou nd the system. Such a process will cause achange of one in either n, or ny , but there is a logarithmically large energy barrier toprevent such a transition.

In the case of the isotropic Heisenberg model, the direction of magnetization isdefined by tw o pola r angles 8 and $.A quan tity such as

is not a topological invariant. A twist of the angle $ by 271 across the system can bechanged continuously i nto no twist by changing the other polar angle 8 (which we taketo be the same everywhere) from 371 to zero. Th ere is in fact a single topological in var ian tfor the Heisenberg model in two dim ensions, which is

If we regard the direction of magnetization in space as giving a mapping of the spaceon t o the surface of a unit sphere, this invarian t measures the nu mb er of times the ma pof the sp ace encloses the sphere. This invariant is of no significance in statistical mech an-ics, because the energy barrier separating configurations with different values of N isof order unity. To show this, we consider how a configuration with N equal to unitycan be transformed continuously into on e with N equal to zero.

A simple example of a configuration with N = 1 is one in which 8 is a continuousfunction of r = (x' + yZ)'l2, equal to 7c for r greater than some value a, and equal tozero at the origin. Th e angle $ is assumed t o be equal to tan -'(y/x). Th e energy of aslowly varying configuration is proportional to

3jf(V8)' + sin' 8(V$)2) dx dy

which, for the configuration described above

= 71 jo' (g)'+ f sin'e} r d r (87)

Even if 8 varies linearly between the origin and r = a, this integral is finite and inde-pendent of a. The configuration can be changed continuously into a configuration withN = 0 by letting a tend to zero. Of course, for sm all values of a , this expression fo r the

energy is invalid, but the nu mb er of spins in a disc of radius a is then small so that anyenergy barrier is small.



1200 J M Kosterlitz an d D J Thouless

We must conclude that there is no topological long-range order in the Heisenbergmodel in two dimensions. Supposing that the occurrence of a phase transition is reallyconnected with the existence of long-range order we would conjecture that the x y

model has a phase transition but the Heisenberg model does not. Such a conjecturedoes not seem to be in conflict with the evidence provided by the analysis of power

series for the mo dels by Stanley (1968) an d M oo re (1969).

Acknowledgments

The authors would particularly like to thank Professor T H R Skyrme for the solutionof the differential equation and other members of the Department of MathematicalPhysics for many useful and illum inating discussions, especially Dr R K Zia and D r J G

Williams. They would also like to thank Professor P C Martin of Harvard Universityfor drawing their attention to the work of Berezinskii, and for a helpful discussion.

Appendix. Solution of dyidx = - CXy

\

Figure A l . Trajectories of dydx = e-X’ .

From the method of isoclines, we see that the trajectories y(x) behave as plotted infigure A1 with the singular point at infinity. It is easy to see that the solutions fall intotwo classes

(i ) Y ( W )3 0 Y(0)3 Y m(ii) Y@) -+ -cc Y ( 0 )< Y m (A.1)

correspond ing to equ ation (22).For convenience, we mak e a transf orm ation of variablesto bron g the singular point to the origin by defining

( A 4




Figure A2. Tra jec to r ies i n t rans fo rmcd var iab les .

so that

dw 1 1- - - -co th+w.dZ z2 z

04 .3 )

The trajectories w (z )have the form displayed in figure A2, where the trajectories corres-ponding to y ( 0 ) B y,(O) are those with w(z) 2 0. We can find approximate solutions inthe three regions

0) w - 0 ) z - 0

(ii) w - + c o , z - 0

(iii) w ---f 03, z + 03

and m atch the solutions at w = 2. In region (i)

dw 1 2

dz z2 wz- - .e _ -

Changing variables to 2t = ( l / z ) - w we obtain

z ( t ) el2 r' - s 2dsJ t

where t , -, CO as y ( 0 ) -+ y,(O) from above. I n regions (ii) an d (iii)

('4.6)1

w(z) % - + In z + 2 In yZ

where y = y(m) in (ii) an d y = y ( 0 ) n (iii).We ca n mak e the best match a t w(z) = 2, where the gradients in the three regions are

equal. How ever, the numb ers a re not too good because coth 1 % g,while we have takencoth iw = 1 in regions (ii) and (iii). T he p oints at w hich we m atc h a re defined by theintersections of the curves

(A.7)




The ma tch between regions (i) an d (ii) corresponds to the intersection at t -, CO, that is

In t ,t % t , - .

2tl

At the other intersection, we put t = t , + 6, where t , is defined b y

Numerical calculations give t , % - 0.84. Expanding z(t) o first order in 6 we obtain

Using equations (A.6) to (A.10), we obtain

~ ( c o ) Jz t , e-"

(A.10)

(A.11)

where

y:(O) = 2( tc + 1) e-''. % 1.7. (A.12)

Taking logarithms of equation (A.11) eliminating t , , we obtain the leading singularity

in Y(CO)fo r Y(0)2. A O )

(A. 13)

Substituting the expressions for ~ ( c o ) nd y ( 0 ) given by equation (20), and ignoring allbut the mo st singular terms, we immediately obtain eq uation (27) .Just ab ove the crit ical tempera ture where y (0) < y,(O), the a ppropriate solution of

equation (A.4) ism

z(t)= e" 1 e-s2 ds + K ef2 (A.14)

where K 2 0. In the region w -, - CO, -, + CO

(A.15)

where X is defined by y(X) = 0. We can now carry out exactly the same procedure

as above to estimate the behav iour of X or y ( 0 )+ y,(O) when K -,O f by matching thesolutions (A.6), (A.14) and (A.15) at 14= + 2. We find

1w(z)% - - n z - 2 In X

Z

(A.16)

Using th e expressions for y ( 0 )and y,(O) of equation (20),we immediately ob tain equ ation

(35).

References

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