Spaces of states for heterophase systems

Physica ll0A (1982) 247-256 North-HoUand Publishing Co.

SPACES OF STATES FOR H E T E R O P H A S E SYSTEMS V.I. YUKALOV Department of Theoretical Physics, University of Oxford, Ox[ord, OXI 3NP, UK Received 27 May 1981

By constructing the spaces of physical states and realizing their decomposition into mutually orthogonal subspaces, we show that our statistical theory of heterophase fluctuations~), is applicable to a large class of systems exhibiting configurational and magnetic transitions. We demonstrate that this theory can describe not only transitions between a phase with a discrete symmetry and another phase with a continuous symmetry, as in the case of a crystal-liquid transition or a ferromagnet-paramagnet one, but also transitions between phases, corresponding to point symmetry groups such as polymorphictransitions or reorientational magnetic transitions.

I. Introduction In a previous paper ~) a statistical approach to treat systems with heterophase fluctuations has been formulated. After averaging over these fluctuations 2) we come to a method representing such systems as heterophase mixtures. The first technical question which arises when dealing with a particular heterophase model is how to construct the space of its physical states. An actual construction of the space and the possibility to decompose it into mutually orthogonal subspaces would check the applicability of the method. An investigation of these questions for different models is just the aim of the present article. We examine systems with configurational transitions in section 2 and show that our theory t'2) is applicable to transitions such as crystal-crystal, ideal crystal-non-ideal crystal, crystal-glass as well as to crystal-liquid phase transitions3'4). Magnetic transitions are considered in section 3, where we check that the theory discussed can depict heterophase fluctuations in systems exhibiting reorientational magnetic transitions, ideal magnet-disordered magnet, magnet-spin glass and magnet-paramagnet transitions. Section 4 is devoted to continuous systems with a Bose-Einstein condensate; arguments are given that in such systems heterophase fluctuations are impossible, at least in the framework of the method used.

0378-4371/82/0000-0000/$02.75 0

1982 North-Holland

248

V.I. YUKALOV

2. Configurational transitions W e shall c o n s i d e r here a solid s y s t e m in an infinite v o l u m e f o r which there exists a set A = {a} of m e a n positions a of particles. W e call A a lattice and a a lattice v e c t o r e v e n w h e n this lattice is not regular. If there are j particles near e a c h lattice point with the v e c t o r a, then we can define a Hilbert space ~

= { b ~ ( r , . . . ri)}

(!)

of normalized c o m p l e x f u n c t i o n s d,, with a scalar p r o d u c t (b,, d~9 = f ~b*(rl... rj) ~b~(rl... r ~ ) d r l . . , drj and a n o r m

114'all = (4'o, 4'o)= 1. A direct p r o d u c t ~A = @ ~ . = {hA = @ & . ( r , . . . ri)} a

(2)

o

is a space o f p h y s i c a l states f o r a f i x e d l a t t i c e A. T h i s space is the H i l b e r t space w i t h a scalar p r o d u c t

(6A, '~;,)= ]-I (6o, '/'g)a

If the s y s t e m u n d e r c o n s i d e r a t i o n is translationally invariant, then the total space of all its possible states is a direct sum (direct integral)

= ® ~A

(3)

A

o v e r all regular and irregular lattices. In order to define in the space :2 a scalar p r o d u c t of f u n c t i o n s d~A and ~bB, c o r r e s p o n d i n g to lattices A = {a} and 15 = {b}, respectively, we need to put into c o n g r u e n c e e v e r y pair of v e c t o r s a and b. W e shall a s s u m e f u r t h e r on that the n u m b e r of particles per unit cell j is the same for A as for 13. L e t a c o n g r u e n c e b e t w e e n two v e c t o r s a and b be established b y m e a n s of a relation b = a + ~,

(4)

in which g = 0 w h e n 15 coincides with A. T h e n a scalar p r o d u c t of &A and d,B can be written as

(~,A, 6~) = [-I (~o, ~'~+~). a

(5)

SPACES OF STATES FOR HETEROPHASE SYSTEMS

249

Statement. If the set C = (~, U B)---. (~,n B)

(6)

is denumerably infinite then ~ a is orthogonal to ~B.

Proof. It follows from the Schwartz inequality and the normalization of the functions ~ba that we can find a positive y such that ](O., ~bb){--
(a # b).

(7)

For a finite volume with N >> 1 lattice sites we can get a set CN, whose connection with C is through the thermodynamics limit: C = lim CN.

(8)

N -~oo

If C is denumerably infinite, then the number of sites in CN is N(CN)=cN

(0
(9)

where c represents a mean concentration of non-coincidences between the lattices A and B. Eq. (9) means that the lattices do not coincide in the macroscopic volume, therefore I(~bA, qbs)l < T~N~O

(N ~o¢),

(10)

which concludes the proof. This result permits us to decompose the space (3) into mutually orthogonal subspaces and, consequently, to apply the theory of refs. 1-4 to systems with various configurational transitions, some examples of which will be given later on. We should emphasize that the space decomposition, mentioned above, is a necessary but not sufficient condition for the existence of heterophase fluctuations. Their actual existence depends on particular characteristics of a model. Some general connections between the properties of interparticle interactions and the coexistence of phases have been studied by IsraelS), Pirogov and Sinai6-s), Ruelle 9,j°) and Daniels and van Entern'~2). Now we proceed to the enumeration of examples when heterophase fluctuations are possible. This is for instance the case for crystals with structural transitions. If there are polymorphic fluctuations of one lattice in the other one, then the total space of states for a mixed system is "~ = " ~ 1 ( ~ 2 ,

~1 =

~A,

"~2 = ~B.

The case when one of lattices is a sublattice of another is included here.

250

V.I. Y U K A L O V

H o w e v e r , a formal process of lattice dilatation which is the basis of renormalization-group methods 13-18) does not c o r r e s p o n d to a change of spaces, because the n u m b e r of particles per unit cell is renormalized when scaling a lattice. It is interesting that the states of a crystal with any finite density of defects f o r m a space which is orthogonal to the space corresponding to an ideal crystal. Thus, the a p p e a r a n c e of lattice defects is a kind of phase transition. The previous example m a k e s it clear that defining an ideal lattice A and a lattice B of randomly positioned lattice sites, we are able to describe c r y s t a l glass phase transitions in the presence of heterophase fluctuations. Finally we m a y r e m e m b e r 3'4) that crystal-liquid and glass-liquid mixtures correspond to a space ~ = ,~ (~ ~2, when

B#A

The set A is an ideal lattice for a crystal or a random lattice for a glass. The space 3T2 is the real space f r o m which a fixed lattice A is picked out. As we see, a system described by a translationally invariant hamiltonian can exhibit three principally different physical states: ideal crystal, non-ideal crystal (including crystals with defects and glasses) and liquid. This conclusion is in agreement with the results of Kastler and Robinson19), who classified invariant states with respect to the discrete eigenstates of the representation of the space-translation group. They found that any invariant state has a unique decomposition into three classes of states. In any state of the first class there is one and only one discrete eigenstate, in a state of the second class there is an infinite n u m b e r of discrete eigenstates occurring with a lattice structure and in a state of the third class there is also an infinite n u m b e r of discrete eigenstates but they occur in a dense manner making it impossible to define a lattice distance. We can interpret the first kind of states having a h o m o g e n e o u s nature as a liquid, the second kind of states with a periodic nature as a crystal and the third kind of states, representing a solid in which there is no minimum non-zero distance between any pairs, as an a m o r p h o u s solid (crystal with defects, glass). When describing a system with heterophase fluctuations we should, in principle, take into consideration all three kinds of states, and in the solid state examine all possible lattices. H o w e v e r , in order to make a problem solvable, we always have to restrict ourselves to several states, either using an a priori knowledge or checking by estimates which states are the most probable ones.

SPACES OF STATES FOR HETEROPHASESYSTEMS

251

3. Magnetic transitions

It seems clear that if a first-order transition takes place in a magnet, metastable nuclei of a competing phase can appear on both sides of the phase-transition point. The existence of first-order magnetic transitions is shown in many experiments2°-26). The change of the transition order from the second to the first one can be explained by several reasons: by the influence of phonons27-3~), by strong fluctuations in the presence of an anistropya2), by a quadrupole exchange 33-36) or other many-spin interactions37-39). Magnetic transitions of the first order are also possible in exciton-ferromagnet models4°-42). Generally speaking, a second order magnetic transition turns to a first-order one, when there are some additional hidden variables in a spin system43). Heterophase fluctuations could act just as a kind of hidden variables. If a magnetic transition is a second-order one, then heterophase nuclei are labile but not metastable. In any event, whether these fluctuations are metastable or labile, they must be taken into account, because they can strongly change the behaviour of the thermodynamic characteristics. For instance, the presence of hidden variables renormalizes critical exponents44), sometimes making even the thermodynamic potential singular at the critical temperature45). There are experimental indications that an assumption about the existence of nuclei of competing phases is crucial for the interpretation of a magnetic-excitation spectrum in ferromagnets46'47). Let us now construct the space of states for a magnet with heterophase fluctuations. This construction is quite similar to the one given in section 2 for systems with configurational transitions. We can attach to each lattice point a E A a Hilbert space =

( l l g d = 1)

ill)

of normalized spin functions g~. The rotational symmetry becomes broken when we fix in real space a direction corresponding to a spherical angle O. A product eo = Q e~ a

(lle21l =

1)

(12)

of spin functions ea, each of which is associated with the fixed direction, is called a generating vectorS). Selecting from all possible sets of functions ga those which belong to a class of equivalence defined by an inequality I(e$, ga) - II < ~, a

(13)

252

V.I. YUKALOV

we introduce functions

(14)

go = (~° ga a

forming a Hilbert space

~ o = ® o ~ . = {go}

(15)

a

with a scalar product O

(go, g~) = Ia]

(ga, g~).

The superscript 0 near the products @~ and II~° means that only those functions ga are multiplied which f o r m a get corresponding to the inequality

(13). These deviations of go f r o m eo which pertain to the class of equivalence (13) describe spin waves in a magnet. The space (15) is called an incomplete N e u m a n n product. A complete N e u m a n n product, which is a direct sum (direct integral)

5~ = ~) ~o,

O E [0, ~r],

(16)

0

c o r r e s p o n d s to the total space of states of a rotational-invariant system. Statement. If the conditions

O ~ 0 ' , O ~ 0 ' + ~r

(17)

are simultaneously satisfied, then the spaces ~o and ~(o, are mutually orthogonal. Proo[. This can be p r o v e d in complete analogy with the statement of section 2. A simple illustration is given in ref. 49. The p r o o f is based on a relation

(go, go') ~ [-I (ga, ga) cos(O - O'),

(18)

a

f r o m which, using the normalization [[gall = 1, we have (go, go,) ~ lira cosN(O - 0 9 = 0, N~o~

if eq. (17) holds.

(19)

SPACES OF STATES FOR HETEROPHASE SYSTEMS

253

In this way we have verified that the method of papers 1, 2 can depict reorientational magnetic transitions, when the direction of the magnetization changes by a finite angle. Such a transition is the analogue of a structural transition in a crystal. The space of a magnet with non-magnetic fluctuations can be written as a direct sum ~l @ ~2, in which

O:#0,~

It is obvious that ~1 is orthogonal to ~2. Other examples of states having mutually orthogonal spaces are also similar to those of section 2; they are: ideal magnet and disordered magnet, magnet and spin glass, paramagnet and spin glass. The space of states for a ferromagnet is not orthogonal to that for an antiferromagnet, as follows from eq. (19). Although ferromagnetic states are separable from antiferromagnetic ones by means of eq. (13). We discussed above magnets on a rigid lattice. When the lattice is soft, then the corresponding space of states is

~spin,

= O~conf @

where "~confis to be constructed according to section 2 and .~spin according to section 3. If a sequence of phase transitions happens in a system, then their algebraic properties may be described in terms of powers of the Boolean algebra. For uniform spin systems, for example, liquid magnets, the index " a " in eqs. (l) and (2) must mean an integer, so that to.f- @ ~ . a

(a

1,2 . . . . .

Generally, a scalar product of functions 4) = ®4),,, a

4)' = ®a 4)',~

with a continuous measure /~, may be defined as (4), 4)')= exp[/ln(4)a, 4)'~) d/~, 1,

(20)

where In is one of the branches of Ln. In this instance two functions 4) and 4)' are mutually orthogonal if f In(~, ~ ) d ~ = - o~.

254

V.I. YUKALOV

In case of the discrete m e a s u r e the e x p r e s s i o n (20) t r a n s f o r m s to a c o n v e n i e n t form

(0, 4,3 = ~ (4,~, +'). a=l

A great n u m b e r of different physical s y s t e m s can be rewritten in quasi-spin r e p r e s e n t a t i o n s , for example, superconductorsS°), J a h n - T e l l e r systemsS~), ferroelectrics52), semiconductors53), Dicke models~4'55), lattice gauge fields -~'57) and others58). S p a c e s of states for e a c h s y s t e m can be c o n s t r u c t e d in a w a y similar to that c o n s i d e r e d in section 3.

4. Bose condensation T h e B o s e - E i n s t e i n c o n d e n s a t i o n gives us an e x a m p l e of a p h a s e transition, near which h e t e r o p h a s e fluctuations are p r e s u m a b l y impossible. In order to s h o w this we shall use a c o n s t r u c t i o n due to Araki and W o o d s 59) and Hugenholtz6°). The total space of states of a c o n t i n u o u s gauge-invariant s y s t e m can be written as "~ :

'~conf (~) "/~,

where

a=!

~a(') being a space-integrable function, and 2~r a=O

is the Hilbert space of square-integrable f u n c t i o n s on the unit circle with respect to the n o r m a l i z e d L e b e s q u e m e a s u r e dal2~r. The f u n c t i o n s e ~m~ with m = 0, _+ 1, _+2 . . . . f o r m a c o m p l e t e set of v e c t o r s in M. The B o s e - E i n s t e i n c o n d e n s a t e c o r r e s p o n d s to a b r o k e n gauge s y m m e t r y w h e n a is fixed. L e t us fix a = 0. T h e n the space of states is ~1 = ,~co°f ® ~o. Cutting the circle at a = 0 and defining a space ~ ' = @ . ~ = d~ --. d~o, ~0

SPACES OF STATES FOR HETEROPHASE SYSTEMS

255

we get

~2 = ~o~f ® ~t'. The space 3~ could be presented as a sum ~ (~Y(2, but the space ~ and if(2 are not mutually orthogonal. Thus, in the f r a m e w o r k of the present method ~'~) there are no heterogeneous fluctuations in a system with a Bose-Einstein condensate.

Acknowledgements I am very indebted to Dr. D. ter H a a r for reading the manuscript and a n u m b e r of comments. I would also like to thank Prof. N.N. Bogolubov Jr., Y.A. Kostikov, Dr. A.S. S h u m o v s k y , Dr. R.B. Stinchcombe and Prof. I.R. Y u k h n o v s k y for discussion. I express my thanks to the British Council for a grant.

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