On quantum spin chains and liquid crystal films

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On quantum spin chains and liquid crystal films

T.J. Sluckin, Timothy Ziman

To cite this version:

T.J. Sluckin, Timothy Ziman. On quantum spin chains and liquid crystal films. Journal de Physique,

1988, 49 (4), pp.567-576. �10.1051/jphys:01988004904056700�. �jpa-00210731�

E 567

On quantum spin chains and liquid crystal ^films

T. J. Sluckin (1, ^and Timothy ^Ziman (1, 2)

(1) Institut Laue Langevin, ^B.P. 156X, 38042 Grenoble Cedex, ^France

(2) Department ^of Physics ^and Astronomy, Rutgers University, ^{P.O. Box 849,} Piscataway, ^New Jersey ^08855- 0849, ^U.S.A.

(Requ ^{le 16} juillet 1987, accepté ^sous forme définitive le 14 dgcembre 1987)

Résumé.

Nous avons étudié un modèle bidimensionnel de spin, introduit par Korshunov d’une part ^et par Lee et Grinstein d’autre part, ^décrivant l’apparition de l’ordre dans des molécules soumises à des interactions

nématiques ^et ferromagnétiques antagonistes. ^{Utilisant un} formalisme de matrices de transfert nous avons

établi une correspondance ^{avec un modèle} quantique unidimensionnel de spin ^{2. Celui-ci a été} diagonalisé

pour des chaînes de longueur délimitée. Nous trouvons alors un comportement ^{cohérent avec le} diagramme ^de phase prédit : ^une phase planaire ^{à basse} température séparée ^d’une phase désordonnée de haute température

par ^une ligne de transitions du type Kosterlitz-Thouless, ^et séparée ^d’une phase nématique ^{à basse} température par ^une ligne ^de transitions du type Ising. On s’attend à ce que la transition nématique- paramagnétique ^{soit du} type Kosterlitz-Thouless. Ici, les résultats sont moins clairs, ^mais cependant ^nous

discutons des topologies cohérentes avec nos résultats.

Abstract.

A two dimensional spin model introduced by Korshunov and by ^{Lee and} Grinstein, ^which describes ordering of molecules with competing nematic and ferromagnetic interactions, has been studied.

Using the transfer matrix formalism, ^we have derived an equivalent spin ² quantum ^{model in one} spatial

dimension. This has been diagonalised for chains of finite length. We find behaviour consistent with the

predicted phase diagram ; ^{a low} temperature planar phase separated ^{from a} disordered highs temperature phase by ^a line of transitions of the Kosterlitz-Thouless type, ^{and from a low} temperature ^nematic phase by ^a

line of Ising transitions. The nematic to paramagnetic transition has also been predicted ^to be Kosterlitz- Thouless-like. Here the numerical evidence is less clear ; ^{however we discuss} topologies consistent with our

results.

J. Phys. ^{France 49} (1988) ^{567-576 AVRIL} 1988,

Classification

Physics ^Abstracts

75.10H - 75.10J

05.70J

61.30

1. Introduction.

Many ^{of the most} interesting problems in statistical mechanics are concerned with competing ordering

mechanisms. The resolution of the competition

often leads to an interesting ^low temperature phase diagram. In this paper ^we study ^{such a} problem, ^the

« mixed » model introduced independently by ^Kor-

shunov [1] and Lee and Grinstein [2]. ^{This is a two-}

dimensional lattice model, ^{with X-Y} spins ^{at each}

lattice point. ^The spins ^at neighbouring ^lattice points

(*) ^{Permanent address :} Department ^of Mathematics, University ^of Southampton, Southampton ^S095NH, Great-Britain.

(**) (Lady ^Davis fellow) Department ^of Chemistry,

Technion-Israel Institute of Technology, ³²⁰⁰⁰ Haifa, Israel.

JOURNAL DE PHYSIQUE. - T. 49, ^N° 4, AVRIL 1988

are coupled, firstly by ^a ferromagnetic interaction

favouring parallel orientation and secondly by ^a

« nematic ^» interaction favouring ^either parallel ^or antiparallel orientation. These two tendencies com-

pete ^{once we include} entropy and it has been

predicted ^{that this} gives ^{rise to a} variety ^{of low} temperature phases : ^we summarise these predictions

at a later stage in the article. This Hamiltonian may describe the physics ^of liquid crystal ^{films, in which}

the molecules composing ^the liquid ^are ferroelectric and therefore have ^a polarity. ^The Hamiltonian we

shall discuss does correctly ^include the short range effective potential implied by ^{such a weak} polarity although ^{if the} ferroelectricity ^were sufficiently ^de- veloped it would be essential to include the additional long-range electromagnetic interactions that lead to domain formation.

The formal analogy between the partition ^function

in statistical mechanics and the Lagrangian ^{in the}

37

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01988004904056700

path integral formulation in quantum mechanics is

by ^now well-known [3]. It allows a one-dimensional

problem ⁱⁿ statistical mechanics to be mapped ^onto

the quantum mechanics of a single particle, ^and by extension, maps d-dimensional statistical mechanics

problems ^onto (d -1 ) dimensional quantum prob-

lems. This permits mathematical techniques ^and experience developed ^for understanding quantum mechanics to be brought ^{to bear in} statistical mech- anics and vice versa.

In this paper ^we exploit ^this analogy ^{to transform}

the mixed model into an equivalent one-dimensional chain of quantum spins. ^This quantum spin ^{chain is}

similar to others that have been studied in the past

[3-6], ^{and we can} profit ^{from this} experience. ^We perform ^{an exact} diagonalisation of the Hamiltonian of finite periodic chains of n spins with n between 4 and 10, ^and extrapolate ^{to infer} properties ^{of infinite}

chains.

It turns out that to provide ^a simple description ^of

the nematic phase the minimum quantum spin ^chain

has spin 2 ; ^{in the} past intensive studies have been carried out on spin 1 , ¹ and 3 , each of which has

2 ’ 2

rather different properties [5, 6]. ^{This chain has}

nematic as well as ferromagnetic interactions and as

such is of some intrinsic interest. Indeed Andreev and Grischuk have postulated that solid He3 is ^a spin

nematic [7] ^and Papanicolau has studied the ground

state properties ^of spin 1 nematics [8].

This paper is organised ^{in the} following way. In section 2 we discuss in more detail the mixed model.

Then in section 3 we describe how we derive the

equivalent quantum spin problem, ^and explain ^the

limitations to the approach. ^{In section 4 we solve the} spin problem using ^a superposition approximation ^as

an Ansatz to the ground ^{state wave} function. This

approximation ^{has a} theoretical status more or less

equivalent ^{to mean field} theory ^{and we} shall refer to it as such. In section 5 we discuss the results obtained from exact diagonalisation of finite length ^chains.

Finally in section 6 we draw some general ^con-

clusions.

2. The mixed model.

First we describe the mixed model studied by

Korshunov [1] and Lee and Grinstein [2] ; ^{then we}

discuss some related models. The Hamiltonian of the mixed model is

where Jl, J2 . ^{0 and the} angle Oi is defined at each site i of a two-dimensional _square lattice, ^{and the} pairs [ij] are ^nearest neighbours. ^This Hamiltonian

can also be rewritten in terms of the spins ^cri

(cos 0 j, sin 0 i ) ^{in which case} it contains ^terms

quartic ^{as well as} quadratic ^{ins ;}

It is convenient to define a variable ^a by

A point ^{in the} phase diagram ^{is now} defined in ^terms of the variables (T

T/J, a ). ^{T is the scaled}

temperature ^{and we can without} ambiguity drop ^the

tilde. The degree ^{of «} nematicity ^{» is} expressed by

a ; ^a

0 corresponds ^to the pure XY model, ^and

a =1 corresponds ^{to a} purely nematic version of the XY model.

The authors of references 1 and 2 used ^a number of techniques ^to define the phase diagram. ^The

a

0 line is already well-known [4] ; ^{there is a low}

temperature phase ^with algebraic ^{order defined} by

At T = 0, TJ 1 (T) = 0 implying ^true long range order. As T is increased q ¹ increases until at the Kosterlitz-Thouless temperature q - 1 ₄ beyond

which order decays exponentially ^{with distance} rij. ^A quantity of interest in describing ^the phase diagram ^{as a whole is} TJ2(T), ^defined by

However so long ^{as a}

0 the Gaussian result

’TJ 2 = 4 ’TJ ¹ should hold.

The ^a =1, ^nematic line is also easily ^understood by making ^the transformation I/J i = 2 P i, ^which provides ^a mapping ^{to the a}

0 line. In the low temperature phase exp i (oi - Oj) ^{decays ex onen-}

tially ^but ’TJ 2 is well-defined and less than 4. ^For

tially ^{but r} 2 is well-defmed and less than 4 For intermediate a we shall show that finite-size studies

can be used to find the phase boundaries. For

a :s:: 1 we expect ^two transitions as first ’q 2 ^{and then}

,q 1 become finite, ^with decreasing temperature. ^The

latter transition is expected ^{to be} Ising like in that above it (gii) ^{is well defined,} implying Pi) ^is

defined only ^mod (-fr), whereas below it Pi) ^is

defined mod (2 7T ). ^In between there will be a

multicritical point ^{where the two} transitions join ^and

near it some or all of the phase transition lines _may be of first order. The hypothetical phase diagram ^is

shown in figure ^1.

It is the equivalent quantum version of this model

on which we concentrate our attention. It is never-

theless of some interest to mention some related

models, all of which in some way pit magnetic

interactions against ^{the effect of nematic} liquid

crystallinity. ^A simple generalization ^{is to allow}

569

Fig. ^1.

Expected phase diagram of the Hamiltonian (1)

of Korshunov, Lee and Grinstein.

J1 ^or J2 ^{to be} negative [9]. ^If J2 ^{0 the} ground ^state

becomes multiply degenerate ^{over a} certain range of

a and there are a wide variety of ordered phases.

One might also consider the usual mixed model on a

three dimensional cubic lattice. There have been recent predictions [10] ^that although the transition to a (now long-ranged) ^ordered phase is continuous for

a close to 1 and 0, ^{in a} limited range of ^a there is a

first order transition. Alternatively ^one might ^con-

sider a Hamiltonian of the form (2) but suppose that the ( cr ) ^{are now} Heisenberg spins ; ^this corresponds

to competition ^between Heisenberg and classical nematic order. Then in two dimensions, using ^the spin ^wave arguments ^of Polyakov [11] ^{there are no}

low temperature ^ordered phases. However in three dimensions there is ^a small ^a low T ferromagnetic phase separated ^{from a} high a low T nematic phase by ^an Ising transition, ^{which is itself} separated ^from

the high ^T disordered phase by ^a continuous tran- sition. Arguments ^{based on Landau} theory suggest that there will be a tricritical point beyond ^{which the}

transition is first order. A number of authors, indeed, have studied a magnetic model with multis-

pin interactions [12] ^and found such first order transitions.

3. The equivalent quantum spin system.

In this section we follow the method of Stoeckly ^and Scalapino [3] who discussed the mapping ^between

the planar model, ^a continuum version of the X-Y

model, ^{and an} equivalent quantum system.

The strategy ^{is as} follows. We first write down the free energy appropriate ^{to a} continuum model with the same order parameters ^as the mixed model.

This, ^we believe, ^{will have a} phase diagram topologi- cally equivalent ^to the mixed model. We then shall take an anisotropic limit in which the y-direction ^is

discretized. The partition function is then equivalent

to the path integral formulation in quantum ^mech- anics if the x-direction is identified with imaginary

time (it). ^{This enables us to} identify ^the generating quantum mechanical Hamiltonian, ^{which now} applies ^{to a} chain of sites. Finally ^we carry ^{out some} further approximations and truncations which leave

an effective spin ^chain.

A system ^described by ^the Hamiltonian has two relevant order parameters. ^{These are a vector} mag- netisation m, where

where a is a Euclidean subscript, ^{and a traceless}

nematic tensor Q ^where

It is convenient to introduce complex order par- ameters

The angles qi and § give the orientation of the local

magnetic and nematic order parameters respectively,

and the numbers I m I and I Q I give ^their respective magnitudes.

We suppose the system anisotropic ^{and write}

down the Landau free energy in ^terms of the order parameters m ^and Q :

where A1 ^and A2 ^{are the} purely magnetic ^and

nematic free energies respectively :

(there ^{is no cubic term} because there is no third order invariant in two dimensions), ^and A3 ^{is the} coupling ^term

We now suppose the y direction ^to be discretized, ^so

that the free energy ^now refers to chains in the x-

direction, coupled ^{to their} neighbours, ^{and a distance}

8y apart. ^The gradient ^terms in the y direction ^now

become

where the i subscripts ^{are labels} referring ^{to the} (one-dimensional) chain array.

The quantities a,

ai (T - Tern), ^a2

a2 (T - Ten) identify ^{the mean} field transition temperatures Tcm, Ten ^to magnetic ^{and nematic states} respectively.

At low temperatures ^{there are} four order parameters

I m I, qi, I Q I ^and ç. However from equation (10) ^it

can be seen that fluctuations in I m I , I Q I ^and

5 = ç - 2 qi ^cost free energy locally and may be

ignored ; ^the important fluctuations, corresponding

to what would be Goldstone modes if there were true long range order, ^{are those in} qi, ^{and on} those

we concentrate. The varying ^{terms in the} free energy functional are now given by

The mapping ^{to the} quantum system is effected by observing ^{that the} partition ^function

and its derivatives can be found via the properties ^of

the transfer matrix defined in the x direction.

Making ^the mapping x ^{= it} (h

1 ) the transfer matrix can be written as a quantum-mechanical

Hamiltonian of variables gii i

where T = /3 ^and Boltzmann’s constant is taken to be 1. This is now the quantum mechanical Hamilto- nian of a chain of sites i. We investigate ^the ground

state properties ^{of the} Hamiltonian, ^{as a} function of the parameters ^{D and} {3, ^where

and identification is made between the parameters ⁱⁿ equations (14) ^and (15). ^{Now D is the} temperature- like variable and K2 ^and Kl ^{tend to} induce nematic order and magnetism respectively.

We can identify on-site and inter-site terms in the Hamiltonian : the operator ^{= - I} a _has

a(pi eigenstates [ pi)

exp ipgii, ^{we define} operators

subject ^to the commutation relations

In terms of these operators the Hamiltonian is ^now

and ^we observe that bt , bi- ^are raising ^and lowering

operators ^with respect ^{to the levels} Ipi) ^though

without the Clebsch-Gordon coefficients which occur

in angular momentum ; bi ^is analogous ^{to the z-} component ^{of the} spin.

This Hamiltonian without the K2 ^{term has been}

much studied [3-6] ; the fundamental features of the

ground ^{states can} be delineated by restricting ^the

basis set to p

0, ^{± 1} and now p

^± 2 states. Thus within our approximation

That the analogy ^is good, ^{can be seen} by comparing

the limiting ^{cases of} K2

⁰ and p restricted to 0,

± 1 and Ki = 0 and p restricted to 0, ⁺ 1, ^{± 2. In the}

latter case the p

^{± 1 states are not} coupled ^{to the} ground ^{state even} indirectly, and may be ignored.

Defining operators ci - - (bt-)2, and ci = 1/2 bi,

the Hamiltonian for the limiting ^{case can be rewrit-}

ten

The properties ^{of the} operators {cj ^{in this} problem

are now identified to those of the operators

(bi) ^for K2 = 0 and p restricted to 0, ^{± 1. We}

conclude that the minimal basis set for understanding ground ^{states in the} Kl

^{0 case is} the p ^{= 0 ± 2 set.}

To study ^the competition ^if Kl =1= ⁰ the p ^{= ± 1 must}

also be included.

Before studying the Hamiltonian (18) ^{some com-}

ments on the transformation from the original equation (1) ^are in order. In reference [3], ^{there is a}

discussion of the same transformation for the simple

X-Y model which explains why fluctuations in

I m I ^{can be} ignored ^{in that case. In our} problem I Q I fluctuations are analogous ⁱⁿ being amplitude

fluctuations. There remain fluctuations in the two

phase variables ç ^and qi. They ^are coupled ^{with two}

inequivalent ^but equally likely ^minima corresponding

571

to, respectively, )

2.p and § = 2 (gi + w ) [0 -- 6 -- ² 7T ]. ^We ignore coupling between these

minima ; ^{in the} quantum language ^{this occurs} by tunnelling and could be calculated by ^{the WKB}

method. This tunnelling corresponds, ^{in the} original

model to the ^« strings ^» joining spin 1/2 vortices of

opposite parity predicted ^to exist in the low T nematic phases. ^These strings ^cost energy in the low T planar phase (and ^can presumably ^be ignored ⁱⁿ

the classical to quantum transformation for that

reason) ^{but are} soft in the nematic phase ^and perhaps should be considered explicitly.

We shall seek in subsequent ^{sections an} analogue

to the low T nematic phase ^{in the} quantum ^model.

This phase will possess spatial correlation functions

corresponding ^to the existence of ^« soft » strings ⁱⁿ

the time direction. As the correlations are Lorentz invariant this implies time correlation functions of the same form. Thus although the transformation breaks the symmetry ^{of the two} lattice directions this symmetry ^is restored for long wavelengths. ^The string tension in the space direction, though initially finite, ^is renormalised to zero. We return to this

point briefly in the conclusions.

4. Mean field theory.

It is in general ^{difficult to find the} ground ^{state of}

many body quantum systems. ^{Even in one dimension} only special Hamiltonians may be integrated by ^the

Bethe Ansatz and those that can be are not necess-

arily representative. ^{The first} approximation ^{for a} qualitative understanding ^{is the} superposition ap-

proximation. ^Quantum fluctuations are simplified

rather as mean field theory simplifies thermal fluctu- ations in classical statistical mechanics. In low dimen- sional systems this may be dangerous ; nevertheless with proper ^care the results may be useful.

We use a superposition approximation ^{to construct}

the ground ^state of the Hamiltonian (18) following

closely ^the argument ^of Solyom ^{and Ziman} [5]. ^The

form of the ground ^{state does not} depend ^{on the}

overall magnitude ^of JCQ. ^{There are two} dimension-

less parameters ^which change the form of the ground

state. Defining

where

the parameters ^are D (henceforth ^we drop ^the tilde)

and _y, which is analogous ^{to a} in section 2.

The ground ^{state Ansatz}

where

where

and 80, 81, 82 ^{are real} quantities.

This state is invariant under time reversal as

appropriate ^for planar ^or singlet ground ^{states. The}

energy per spin ^{is found} using the variational prin- ciple

and yields

Minimization of E with respect ^{to the} variational parameters allows the following possibilities :

(1) So ^{= 1,} 81= 8 2 = 0. ^{This is} singlet ground

state, corresponding ^{to a} disordered phase ^{of the} equivalent thermodynamic system :

and

This state obtains for large ^D, which favours the

10 > ^as against the I -t 1 ) , ± 2 ) ^states.

(2) 8 0, 81 ^{1 and} 52 =1= 0; 0 ¹ arbitrary ^but

0 1 - (J 2

⁰ (mod 7T). ^This corresponds ^{to a state}

with planar ^{and nematic} order, ^the equality ^of 0 1 and 9 2 indicates that the orientation of the planar

and nematic order parameters ^{is the same,} although

the nematic order parameter ^{can reverse in direction} without change ^{n the state. The state is} degenerate

in the x-y plane ^since 01 ^is arbitrary. ^The planar

order parameter (see Eqs. (8)) ^is

and the nematic order parameter ^is

This state obtains for low values of y and D.

(3) 8 0, 5 2 =A 0, 5 1 ^{0 .} 9 2 ^{= 0} arbitrary. ^{This is a}

nematic state with m = 0 and Q # ^{0 and obtains for}

low D and high ^{y. The full} phase diagram, ^{which can}

be obtained by ^a Landau-like analysis ^of equation (26) ^{is shown if} figure ^{2. We note the} identity ^{of the a}

^{0 and} a 7r ^lines, and that for

2

any ^a - 2 , ^for sufficiently ^{low D the} planar ^{state is}

favoured. The phase boundaries are of continuous transitions.

Fig. ^2.

Mean field phase diagram for the Hamiltonian

Eq. (18).

The superposition approximation implies long-

range order which, ^for phases ^of continuous broken symmetry, ^{must be} destroyed by quantum ^fluctua-

tions. We discuss the numerical diagonalization ^of

finite systems ^{in the next} section ; ^this allows calcu- lation of the details of the algebraic order that may

replace ^true long range order.

5. Analysis of finite size systems.

We consider chains of spins ^with periodic boundary

conditions. A chain of N spins ^{is a} system ^of 5N states that can be diagonalised exactly. ^In practice

exploiting symmetries reduces the size of the mat- rices to be diagonalised. ^We extrapolate properties

of finite systems ^{as N} increases in order to predict

the properties of the infinite system. ^{In our calcu-} lations 4 , N -- 10. Information is sought ^{as to the}

nature of the ground ^{state as D and} y are varied. It is necessary ^to identify ^the phases ^and properties thereof, locate the phase transitions between differ-

ent phases, and estimate the critical exponents ^and

related properties associated with the phase ^tran-

sitions.

The eigenstates of the model can be classified by ^a

few quantum ^numbers [5]. ^{These are M} = ¿ hi, ^the

total spin ^{in the} z-direction, ^{the momentum k and}

p, which is ± 1 according ^to whether the state is even or odd under left-right reflection. For M

0 there is also the time reversal operator cr.

The phases ^can be defined from the degeneracy ^of

the ground ^{state in the} thermodynamic limit. As in the mean field approximation ^three phases ^are

discerned.

a) ^The singlet phase, in which the ground ^{state is a} non-degenrate ^M

0, ^k

0, cr = ⁺ 1, p

⁺ 1 state, ^for large ^D predominantly ^the configuration (0, 0, 0, ... 0) .

b) ^The (normal) planar phase, again ^{as for the} spin 1 planar ferromagnet [5] ^{there are for k}

^0,

-

IN- degenerate ^{states for an} N-spin system ; ^these include levels with M

0, ± 1, ± 2, ^etc.

c) ^{The nematic} planar phase, ^{which is} similar, ^but

the degeneracy ^{has been} partially ^{lifted. The} degen-

erate states have M

0, ± 2, ± 4, ^etc.

We concentrate on two quantities, AEi = Eo(M = 1) - Eo ^and AE2 = EO(M = 2) - . Eo, ^where Eo ^{is the} ground ^{state and the brackets}

denote orthogonal subspaces ^{of the} quantum ^num- bers cited. We recall that in general (to leading order), whereas in the singlet ground ^state AYE [] ]

remains finite and in a fully ^{ordered state} AE [N ] -- exp - AN, ^{in an} algebraically ^ordered phase

AE[N] - N-1.

In fact it is more convenient to examine the

Callan-Symanzik {3 ^-function

where X

D or y is a coupling ^constant. Roomany

and Wyld introduced [13] ^{a finite} system approxim-

ant for {3 i

where i

573

We also recall that, ^{in the} singlet ^{state, AE is the} quantum analogue of §-1 , where 6 ^{is the} correlation

length. ^{Close to the} phase transition to an ordered

x - -1

p hase ^at Xc, ^AE- X - Xc ) ^{and jS =} I ^{X 1-1}

phaseatXc,IlE-IX-Xclvand13= xc 1

In figure ^{3 we} plot j3i against ^{tan y for D}

^0.

Along ^this line 16 2

^{0 to} within the accuracy of the

plot (± 0.02). ^The results, extrapolated ^{to N}

^oo,

are consistent with the conclusion that j31 = ^{0 for}

y l’ c = tan-l (1.3) with v = 1.25 ± 0.25. This

identifies, ^as expected, ^{the usual} planar phase ^{in the} region ^y yc, and identifies these phases ^as having algebraic order. For the exponent v i along ^{the line}

D

0 we find _v = 1.0 ::!:: 0.25 consistent with the

Ising ^value v 1=1.

Fig. ^3.

The beta function fJfN’ (1’) along ^{D = 0.}

N’

N + 2, ^N

⁶ dot-dash, ^{N = 8} dash, N = 10 full line.

In the algebraically ^ordered phases ^the exponents

1 l and ’TI2 ^are of interest. These are calculated

following ^{a method} originally postulated by ^Luck [14] and shown by Cardy [15] ^{to be} justified by

conformal invariance. This was applied ^to quantum spin ^chains by Schulz and Ziman [6] ^and indepen- dently ^{to a} quantum problem ^{of different} symmetry by ^von Gehlen et al. [16].

where vi ^{is the} velocity ^{of sound} corresponding ^to

the relevant Goldstone mode. This can be calculated from the energy difference AEi (N, k ) between the

ground ^{state which has k}

0 and the lowest excited state of wave number k

2 ’TT / N yielding

vi = N âEi(N, k)/2 ’TT.

We identify ^the phase boundary along ^the (well- understood) planar ^line ( y

0), using ^evidence

from the behaviour of f3 ⁱ and qi. These results are

shown in figure ^{4. We} identify ^the phase boundary

between the planar disordered phases ^{from the} point

at which 1 4,ql = 1. ^{n i}

₄ ⁷⁷²

^{For D--0.27 dif-}

ferent size systems give ^stable estimates of _{qi i} and the relation TJ 2 = 4 TJ ^{1 is} satisfied. At D = 0.27 ,q 1 - 0.23. Different ^measures of _{q i} suggest ^the

Fig. ^4.

fl£’’(y ) ^and "f1m/m2, ^m

1, ² along ^y

^0.

Curve 1: (31’ ^curve 2: 132 (scale ^{on the} right) ; ^N

^6,

N=8. Curve a : q i (N = 8 ) ; ^b: q i (N = 6 ) ; ^c:

"f12/4 (N = 8); d: T,,/4(N=6).

phase transition occurs at D

0.28 ± 0.02. Beyond

this point application ^of equation (31) ^to systems ^of different sizes gives increasingly inconsistent results, and the relation 11 2

4 11 is ^no longer good. ^{All this}

is consistent with the high ^D phase ^no longer being

critical in which case formula (31) ^no longer applies.

The behaviour of Qi ^{1 and} 13 2 is consistent with

13i=O ^for D Dc and /3i- (D-Dc)’+a ^with

Dc

^{0.25 ±} 0.01, ^{and a}

^0.5 ± 0.1; the Koster- litz-Thouless result would be cr = 0.5. The various different criteria for identifying ^the phase boundary give ^{close but not} identical results.

We show in figure ⁵ phase boundaries over the whole range of D and 0, constructed using ^the

criteria described above. The general form of the

phase diagram ^is unambiguous, ^{and is as} predicted

from the mean fied picture, although fluctuations

cause the ordering transition to occur at lower D. A number of features are particularly noteworthy.

First we comment on the low D behaviour close to y

7T /2. ^{Mean field} theory predicts ^that this will be

a planar phase, with nematic behaviour only ^occur- ring ^for high D. The finite size analysis ^{shows that}

Fig. ^5.

^Phase diagram for the Hamiltonian Eq. (18).

Full line : limit of planar ^ordered phase ^from

,B = 0 (N

6). Dashed line : limit of nematic phase

from IB2=0 (N = 6 ).

for y > tan-1 (1.3 ) ^{at D}

⁰ planar order is des-

troyed ^and only ^nematic order remains. Secondly,

there is a strong suspicion ^{of reentrant behaviour in}

the planar-nematic phase boundary ; ^{in the} inter-

mediate _y regime ^{the low D} phase appears ^to be the nematic phase, ^{and as} D increases one passes

successively through planar ^and disordered phases.

This is somewhat contrary ^to intuition and mean

field theory, for which the (more ordered) planar phase ^{occurs at} lower D than the (less ordered)

nematic phase. Finally, ^{close to what is} presumably

the multicritical point of coexistence of the dis- ordered, planar ^{and nematic} phases there appears ^to be anomalous behaviour. The phase boundary ^be-

tween the nematic and the nematic and disordered

phases, ^as determined by ^the calculated 132, appar-

ently ^{occurs in a} region ^which is, according ^{to the}

calculated 13 1, unambiguously ^{in the} planar ^ordered regime. ^{However as} earlier discussed one believes in

general ^that planar ^order enforces nematic order. To cast further light ^{on this} apparent contradiction, ^we

show in figure ^{6 some} results for 8 1 and {32 ^as

functions of D in this region ^{of the} phase diagram.

For D

0.65 and D

0.7 the results on 132 ^are

consistent with ^a low D nematic ordered phase, although ^{the exact} position of the transition is somewhat ambiguous. ^The 61 ^{I results are more}

difficult to interpret. However the N dependence suggests ^{that for} large ^N limiting behaviour in which for D D,l f31 -- (Del - D), ^for Del ^D Dc2 /3

0, ^and for D > De2 81 - (D - De2)3/2 ^{would be}

consistent with the available data. The apparent

Fig. ^6.

i3 m (D) ^for (i) ^y

^0.65 7T/2 (ii) ^y

^0.7 7r/2.

Curves (a) N = 4, N’ = 6, m = 1 ; (b) N = 6, N’ = 8,

m = 1 ; (c) ^N

^{4, N’ = 6, m}

^{2 ;} (d) ^N

^{6, N’ = 8,}

m = 2.

inconsistencies in results are a result of the proximity

to the multi-critical point, ^{and the} consequent ^slow convergence of 8 NN’.

6. Conclusions.

We have demonstrated that the classical mixed XY model may be studied in the critical phase by mapping ^{it onto a} quantum spin ^{chain of} spin ^2.

While it is appealing ^{to relate} properties ^of liquid crystals ^{to those of} quantum magnets, ^one might ^ask

is there any real practical advantage ⁱⁿ doing ^{this ?}

In fact there are several. As is frequently ^{the case}

when analogies ^{are found new} approximations ^are suggested : ^{here we have} displayed ^{a mean-field} theory ^{in the} spin language which is distinct from the molecular field theory of the classical picture. ^Fur-

thermore these are numerical consequences : ^while it is possible, and indeed has been found useful, ^to study the statistical mechanics of the model by

Monte Carlo on the model as defined directly, ^the mapping ^{onto a} spin ^chain suggests different numeri- cal procedures that may give supplementary ^infor-

mation. By restricting ^the expansion ^{to the lowest}

five levels, ^the phase space of ^a single spin ^{is reduced}

from an infinite to finite dimension, ^{with no} change

in the universal scaling properties. Within this re-

stricted space of fluctuations ^{we are} able to diagonal-

ise the transfer matrix exactly for up ^to 10 spins. ^This corresponds ^to finding ^{the exact} partition ^function

for periodic strips infinitely long ^{in one direction and}

up ^to 10 wide in the other. Furthermore this geometry ^{is a} particularly propitious one, for confor- mal invariance [15, 17] provides ^{one with a detailed} scaling theory ^for the finite size corrections. Nat-

urally this could have been done without reference

to the spin 2 truncation but reformulation in this

language ^{allows us more direct} comparison ^with

other parallel work. The details of solving ^the

transfer matrix for such a system would involve

solving ^a complicated integral equation ^{rather than a}

matrix problem ^{that is} easily manageable. ^{There are}

consequences ^{of the} operator algebra ^{that have not} yet ^been fully exploited in the work reported ^here

that _{may be} expected ^to increase the accuracy of the results in the future. With the procedures ^followed

to date we have not unambiguously ^{defined the} topology ^{close to the} multicritical point. ^{For some}

classes of spin ^{chains it was found to be} possible ^to identify multicritical points ^{to various} spin ^Hamilto-

nians which are exceptionally integrable ^{via the}

Bethe Ansatz [6, 18], ^{it is to be} hoped ^{that the same}

program will clarify ^the multicriticality ^{here. It is} probably ^true that until this point ^is fully ^understood

that numerical results, ^{both of the kind} presented

and those found by ^{Monte Carlo} studies of the

original statistical mechanical model will defy ^clear

interpretation.

575

Having made the translation to the spin language

one way ^to understand the critical phases ^is by

fermion representation ^{of the} spin operators, ^as

proposed by Timonen and Luther [19] ^{for the} spin

one chain. Calculation of correlation function func- tions by such methods checks the equivalence ^{of the} spin chain and the original ^lattice formulation for

long wavelengths. ^For example, ^{while use of the}

transfer matrix breaks the symmetry ^{between the}

two lattice directions of the original model, ^{in the} massless phases ^this symmetry is restored. Numeri-

cally this is confirmed by the fact that both phases

have finite spin ^wave velocities (although the veloci- ties are calculated with respect ^to excitations of different symmetry). The fermion representation,

valid in the perturbative regime, gives ^{critical corre-}

lations that are explicitly ^Lorentz invariant. In other words the correlations could be made isotropic simply by rescaling the time direction with the calculated velocity.

Although the details of such a continuum theory

remain to worked out, ^{we can} already ^{make some}

remarks useful for comparison ^to other results. For the spin-one chain with large negative anisotropy ^D

there is ^a transition, ^discussed by ^{one of us and}

Schulz [6] from the normal planar phase (XY 1) ^to

one with effective spin 1 (XY 2). ₂ ^{The transition}

between these two massless phases, like that from

ferromagnetic ^to nematic in the present study, ^is

believed to be Ising-like and the mechanism must be similar : in that case the (S+)2 ^{operator remains}

massless while the S+ acquires ^a gap. This ^occurs because only ^the single power includes ^a contribution from the disorder parameter ^{of the} Ising part ^{of the} Hamiltonian. This leads to exponential decay ⁱⁿ

correlation functions S+ S- but not (S+ )2 (S- )2.

The Ising ordering, ^{or the} disordering ^{of its dual, is} analogous ^{to the} vanishing ^{of the} string ^{tension in}

the classical language of Lee and Grinstein. There

are important differences that limit the resemblance to the present situation, however. The XY 2 phase

has the correlations of an effective spin 1 ₂ ^Hamilto-

nian and is unstable to formation of an ordered

antiferromagnetic phase ^{at a value of} 11 2 = 1, ^{i.e. the}

same value taken at the boundary of the normal

planar phase. ^Therefore the transition to the antifer-

romagnetic ^{state can be} considered to be driven by

the same vortex operators that disorder the planar

state. In contrast, the nematic phase ^{has the corre-}

lations of an integer system and is unstable for

’q 2- 1 ^{This means} that in the continuum limit of 4

this model there must be a different operator driving

the transition. This half-vortex operator ^{can be} identified from the comparison ^{to the a = 1 line and}

the transformation of section 2 from operators ^{b to}

c. It is distinct from the vortex operator ^that becomes relevant in the ferromagnetic-paramagnetic boundary, just ^as geometrically ^the half-integer

vortices are distinct from the integer ^vortices.

Acknowledgements.

We would like to thank John Chalker for useful conversations and communicating unpublished ^re-

sults. We are grateful ^{to the} Lady Davis Foundation for supporting ^{T.J.S. at the} Technion. We thank Ph.

Nozieres and the Theory college of the ILL for

facilitating this collaboration. Computer programs

are based on those developed previously ^{with H. J.}

Schulz to whom we are most grateful. ^{We thank J.}

Solyom ^for helpful discussions. This work was sup-

ported ⁱⁿ part by the National Science Foundation under Grant No. DMR-85-20190 at Rutgers ^and by ^a grant from the New Jersey State Commission on

Science and Technology supercomputer ^fund.

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