Statistical field Theory lecture notes

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Statistical Field Theory R R Horgan December 4, 2014 Contents 1 BOOKS iii 2 Introduction 1 3 De nitions, Notation and Statistical Physics 2 4 The D=1 Ising model and the transfer matrix 7 5 The Phenomenology of Phase Transitions 10 5.1 The general structure of phase diagrams . . . . . . . . . . . . . 16 5.1.1 Structures in a phase diagram: a description of Q . . . 16 6 Landau-Ginsburg theory and mean eld theory 18 7 Mean eld theory 22 8 The scaling hypothesis 25 9 Critical properties of the 1D Ising model 28 10 The blocking transformation 29 11 The Real Space Renormalization Group 35 12 The Partition Function and Field Theory 45 13 The Gaussian Model 47 iCONTENTS ii 14 The Perturbation Expansion 52 15 The Ginsberg Criterion for D 59 c 16 Calculation of the Critical Index 61 17 General Ideas 68 17.1 Domains and the Maxwell Construction . . . . . . . . . . . . . . . . . 69 17.2 Types of critical point . . . . . . . . . . . . . . . . . . . . . . . . . . 69 17.3 Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 17.4 Mean Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731 BOOKS iii 1 BOOKS (1) \Statistical Mechanics of Phase Transitions" J.M. Yeomans, Oxford Scienti c Publications. (2) \Statistical Physics", Landau and Lifshitz, Pergammon Press. (3) \The Theory of Critical Phenomena" J.J. Binney et al., Oxford Scienti c Pub- lications. (4) \Scaling and Renormalization in Statistical Physics" John Cardy, Cambridge Lecture Notes in Physics. (5) \ Quantum and Statistical Field Theory" M. Le Bellac, Clarendon Press. (6) \Introduction to the Renormalisation Group of Critical Phenomena" P. Pfeuty and G. Toulouse, Wiley. ` (7) \Fluctuation Theory of phase Transitions" Patashinskii and Pokrovskii, Perg- amon. (8) \Field Theory the renormalisation Group and Critical Phenomena" D. Amit, McGaw-Hill. (9) \Quantum Field Theory of Critical Phenomena" J. Zinn-Justin, OXFORD. (10) \Phase Transitions and Critical Phenomena" in many volumes, edited by C. Domb and M. Green/Lebowitz. (11) \Statistical Field Theory" Vols I and II, Itzykson and Drou e, CUP. Note means it's a harder book.2 INTRODUCTION 1 2 Introduction A general problem in physics is to deduce the macroscopic properties of a quantum system from a microscopic description. Such systems can only be described mathe- matically on a scale much smaller than the scales which are probed experimentally or on which the system naturally interacts with its environment. An obvious reason is that systems consist of particles whose individual behaviour is known and also whose interactions with neighbouring particles are known. On the other hand ex- perimental probes interact only with systems containing large numbers of particles and the apparatus only responds to their large scale average behaviour. Statistical mechanics was developed expressly to deal with this problem but, of course, only provides a framework in which detailed methods of calculation and analysis can be evolved. These notes are concerned with the physics of phase transitions: the phenomenon that in particular environments, quanti ed by particular values of external param- eters such as temperature, magnetic eld etc., many systems exhibit singularities in the thermodynamic variables which best describe the macroscopic state of the system. For example: (i) the boiling of a liquid. There is a discontinuity in the entropy, Q S = T c where Q is the latent heat. This is a rst order transition; (ii) the transition from paramagnetic to ferromagnetic behaviour of iron at the Curie temperature. Near the transition the system exhibits large-range cooperative be- haviour on a scale much larger than the inter-atomic distance. This is an example of a second order, or continuous, transition. Scattering of radiation by systems at or near such a transition is anomalously large and is called critical opalescence. This is because the uctuations in the atomic positions are correlated on a scale large compared with the spacing between neighbouring atoms, and so the radiation scattered by each atom is in phase and interferes constructively. Most of the course will be concerned with the analysis of continuous transitions but from time to time the nature of rst order transitions will be elucidated. Con- tinuous transitions come under the heading of critical phenomena. It underpins the modern approach to non-perturbative studies of eld theory and particularly lattice-regularized eld theories. Broadly, the discussion will centre on the following area or observations: (i) the mathematical relationship between the sets of variables which describe the physics of the system on di erent scales. Each set of variables encodes the properties of the system most naturally on the associated scale. If we know how to relate di erent sets then we can deduce the large scale properties from the microscopic3 DEFINITIONS, NOTATION AND STATISTICAL PHYSICS 2 description. Such mathematical relationships are called, loosely, renormalization group equations, and , even more loosely, the relationship of the physics on one scale with that on another is described by the renormalization group. In fact there is no such thing as the renormalization group, but it is really a shorthand for the set of ideas which implement the ideas stated above and is best understood in the application of these ideas to particular systems. If the description of the system is in terms of a eld theory then the renormalization group approach includes the idea of the renormalization of (quantum) eld theories and the construction of e ective eld theories; (ii) the concept of universality. This is the phenomenon that many systems whose microscopic descriptions di er widely nevertheless exhibit the same critical be- haviour. That is, that near a continuous phase transition the descriptions of their macroscopic properties coincide in essential details. This phenomenon is related to the existence of xed points of the renormalization group equations. (iii) the phenomenon of scaling. The relationship between observables and parameters near a phase transition is best described by power-law behaviour. Dimensional analysis gives results of this kind but often the dimensions of the variables are anomalous. That is, they are di erent from the obvious or \engineering" dimen- sions. This phenomenon occurs particularly in low dimensions and certainly for d 4. For example in a ferromagnet at the Curie temperature T we nd c 1  Mh ; where M is the magnetization and h is the external magnetic eld. Then the M susceptibility,  = , behaves like h 1 1  h : Since 1, diverges ash 0. The naive prediction for is 3.  is an example of a critical exponent which must be calculable in a successful theory. The coecients of proportionality in the above relations are not universal and are not easily calculated. However, in two dimensions the conformal symmetry of the theory at the transition point does allow many of these parameters to be calculated as well. We shall not pursue this topic in this course. 3 De nitions, Notation and Statistical Physics All quantities of interest can be calculated from the partition function. We shall concentrate on classical systems although many of the ideas we shall investigate can be generalized to quantum systems. Many systems are formulated on a lattice, such as the Ising model, but as we shall see others which have similar behaviour are continuous systems such as H 0. However, it is useful to have one such model in 2 mind to exemplify the concepts, and we shall use the Ising model in D dimensions to this end, leaving a more general formulation and notation for later.3 DEFINITIONS, NOTATION AND STATISTICAL PHYSICS 3 The Ising model is de ned on the sites of a D-dimensional cubic lattice, denoted , whose sites are labelled by n =n e +:::n e , where the e; i = 1;:::;D are the 1 1 D D i basis vectors of a unit cell. With each site there is associated a spin variable  2 n (1;1) labelled by n. There is a nearest neighbour interaction and an interaction with an external magnetic eld, h. Then the energy is written as X E(fg) = J   ; J 0 n n n+ n; X H(fg) = E h  ; n n n where  is the lattice vector from a site to its nearest neighbour in the positive direction, i.e., 2 e ;:::; e ; e = (0;:::; 1 ;:::; 0) : (1) 1 D r z r-th posn Herefg is the notation for a con guration of Ising spins. This means a choice n of assignment for the spin at each site of the lattice. E is the energy of the nearest neighbour interaction between spins and the term involving h is the energy of interaction of each individual spin with the imposed external eld h. H is the total energy. The coupling constant J can, in principle, be a function of the volume V since, in a real system, if we change V it changes the lattice spacing which generally will lead to a change in the interaction strength J. Note that h is under the experimenter's control and so acts as a probe to allow interrogation of the system. We shall use a concise notation where the argumentfg is replaced by . The n partition functionZ is de ned by X X X Z = exp( H()) = exp( (E() h  )) ; (2) n   n The system is in contact with a heatbath of temperature T with = 1=kT . The way the expression forZ is written shows that h can be thought of as a chemical potential. We shall see that it is conjugate to the magnetization M of the system. The point is that the formalism of the grand canonical ensemble will apply. However, we can recover the results we need directly from (2). We shall assume that J, the coupling constant in H, depends only on the volume V of the system. Then for a given T;V and h the equilibrium probability density for nding the system in con gurationfg is n 1 p() = exp( H()): (3) Z Averages over p() will be denoted with angle brackets: X hO()i = p()O() : (4) 3 DEFINITIONS, NOTATION AND STATISTICAL PHYSICS 4 The entropy S is given by X S = k p() log(p())  X 1 = k exp( H)( H logZ) Z  = k( (UhM) + logZ); where U = hEi  internal energy; X M = h  i  magnetization: (5) n n Note that S;U;M are extensive. If two identical systems are joined to make a new system these variables double in value, that is, they scale with volume when all other thermodynamic variables are held xed. Intensive variables such as density , T;P , retain the same value as for the original system. Then we have kT logZ = U +TS +hM = F; (6) and hence 1 F = logZ (7) whereF is the thermodynamic potential appropriate forT;V andh as independent variables. From now on we shall omitV explicitly since for the Ising model it plays no r ole of interest. From (2) directly or from the usual thermodynamic properties of F we have logZ F UhM = ; M = : (8) h h T Comment: These equations are the same as apply in the grand ensemble formalism for a gas with h  and M N where  is the chemical potential and N is the number of molecules. The important point is that the external elds which we use to probe the system can be manipulated as general chemical potentials coupled to an appropriate thermodynamic observable which measures the response to changes in the probe eld. We also have X X 1 S = k p logp k p p p   X X = k p ( (Eh  ) logZ + 1): n  n Note: X X p = 1 =) p = 0: (9)  3 DEFINITIONS, NOTATION AND STATISTICAL PHYSICS 5 Now X X X U = pE ) U = pE + pE ; (10)    z PV and so we deduce the usual thermodynamic identity dU = TdS PdV + hdM : (11) We set dV = 0 from now on. In this case compare with the equivalent relation for a liquid-gas system (e.g., H 0): 2 dU = TdS + hdM Ising model or ferromagnetic system dU = TdS PdV liquid-gas system: In the latter case the density is =N=V whereN is the ( xed) number of molecules. We may thus alternatively write NP dU = TdS + d; (12) 2  and we see a strong similarity between the two systems with M . We shall return to this shortly. The system is translationally invariant and so X M = h  i = Nh i; (13) n 0 n where N is the number of lattice sites and thenh i is the magnetization per site. 0 The susceptibility  is de ned by 2 2 M F logZ  = = = kT : (14) 2 2 h h h T T T From (2) we have P X 1 (Eh  ) n n =N =  e 0 h Z  X X X kT Z = p() (  ) p() : 0 n 0 Z h  n  T But kT Z = M = Nh i; (15) 0 Z h T and so we get X =N = h  i Nh ihi 0 n 0 0 n X = (h  i h ih i) : 0 n 0 0 n3 DEFINITIONS, NOTATION AND STATISTICAL PHYSICS 6 We de ne the correlation function G(n; r) = h  i h ih i n n+r n n+r = h  i ; n n+r c where the subscriptc stands for connected. By translation invarianceG is indepen- dent of n and we can write G(r) = h  i h ih i; 0 r 0 0 X =)  = N G(r): (16) r From its de nition (16) is is reasonable to see thatG(r) 0 asr =jrj1. This is because we would expect that two spins and will uctuate independently when 0 r they are far apart and so their joint probability distribution becomes a product of distributions for the respective spins: p( ; )p( )p( ) =) h  ih ih i as r1: (17) 0 r 0 r 0 r 0 0 Consider an external eld h which depends also on n. Then the magnetic interac- n tion term is X h  : (18) n n n If we follow the same algebra as for  above we see that G(r) = h i: (19) 0 h r T Physically, G(r) tells us the response of the average magnetization at site 0 to a small uctuation in the external eld at site r. This is a fundamental object since it reveals in detail how the system is a ected by external probes. We would expect G(r) 0 as r1 since we expect the size of such in uences to die away with distance. This should certainly be true for a local theory. We shall see that in many cases we can parameterize the large-r behaviour of G by 8 1 r ; D2+ r G(r)  (20) r= e r ; : (D1)=2 (r) where  is an important fundamental length in the theory called the correlation length. The exponent (D 2 +) will be explained. The (D 2) contribution can be deduced by dimensional analysis but , which is an anomalous dimension is a non-trivial outcome of the theory. For large  1 we have, from above, Z Z X 1 1 D 2 D 2  / G(r)  d r =  d u   : (21) D2+ D2+ r r u1 u r4 THE D=1 ISING MODEL AND THE TRANSFER MATRIX 7 4 The D=1 Ising model and the transfer matrix The Ising model is only soluble in D=1,2 and theD = 2 solution is a clever piece of analysis. To discuss the concepts which will be relevant to all the models we study it is useful to investigate theD = 1 model which can be used to highlight the ideas. Note that models in D = 1 are not trivial and many models have been studied in depth. The expression forZ (2) in D = 1 can be written as follows   N1 X Y 1 Z = exp J + h ( + ) : i i+1 i i+1 2  =18i i=0 i Note that the magnetic eld term has been trivially rearranged. Now observe that this expression can be written as the trace over a product of N 2 2 matrices: X Z = W W W :::W ; (22)         0 1 1 2 2 3 0 N1  =18i i where the periodic boundary condition  =  has been used. The matrix W is N 0 identi ed by comparing these two alternative ways of expressingZ. We nd   1 0 0 0 W = exp J + h ( + ) :  2 0 Evaluating with ; =1 gives 1 z z W = ; (23) 1 1 z  z where B K  = e and z = e B = h K = J: Thus from eqn. (22)   N Z = Tr W ; and hence N N Z =  +  ; + where and are the eigenvalues ofW with  . ForN large the rst term + + dominates and we nd that N Z =  : + Hence from eqn. (23) we have 2 2 2  (2z cosh B) + (z z ) = 0 =)  q  2 2 2  = z cosh B + z sinh B + z : (24) + W is know as the transfer matrix and is very important in many theoretical analyses. In higher dimensions it is a very large matrix indeed but a similar anal- ysis goes through and the partition function is still given in terms of the largest4 THE D=1 ISING MODEL AND THE TRANSFER MATRIX 8 eigenvalue. In fact, we need know only the few largest eigenvalues to determine all the observable thermodynamic variables. However, for a very large and even sparse matrix this can be a daunting task. The free energy is then given by F = kTN log . Note that F is extensive + i.e., it is proportional to N. This is, however, only true as N1 (otherwise  contributes a term) which shows that the thermodynamic limit is necessary and N N must be large enough that ( = ) is negligibly small. + The magnetization per site M=N is given by K log e sinh B + q M=N = kT = : (25) h 2 2K 2K T e sinh B + e From now on we use M for magnetization per site, i.e., formerly M=N. In order to keep translation invariance but work with a nite but large number of spins N we shall use periodic boundary conditions. We may then consider the in nite volume limit N1. The magnetization is given by X 1 M = h i = W ::: W  W ::: W ; p     p     p p 0 1 p1 p+1 N1 0 Z  p Np N Tr (W SW ) Tr (SW ) = = : N N Tr (W ) Tr (W ) where S is the matrix 1 0 S = ; 0 1 We see explicitly that M is independent of the choice of p because of translation invariance encoded by the trace. Let W e =  e ;    P = ( e ; e ) : + Then we write  0 1 + W = P P  = ; (26) 0  and so 1 N Tr (P SP  ) M = : (27) N Tr ( ) Now N  0 N + N N N  = =) Tr ( ) =  +  : N + 0  P is an orthogonal matrix and has the form cos  sin  2K P = ; cot 2 = e sinh B : sin  cos 4 THE D=1 ISING MODEL AND THE TRANSFER MATRIX 9 Then cos 2 sin 2 1 P SP = sin 2 cos 2 and so from (27) N N (  ) cos 2 + M = cos 2 as N 1: N N  +  + This is the result we have already derived in (25). In the limit N1 1 0 N N   ; + 0 0 a result we shall use below. We now calculate G(r) and so rst look at the two spin expectation value: X 1 h  i =  W ::: W  W ::: W ; 0 r 0     r     0 1 r1 r r r+1 N1 0 Z  r Nr Tr (SW SW ) = ; N Tr (W ) where we have used translation invariance. Then immediately 1 r 1 Nr Tr (P SP  P SP  ) h  i = = 0 r N  + " r r cos 2 sin 2  0 cos 2 sin 2  0 + + Tr ; (28) r sin 2 cos 2 0  sin 2 cos 2 0 0 where the N1 limit has been taken. Then have r  2 2 h  i = cos 2 + sin 2 (29) 0 r  + We de ne  = 1= log ( = ) =) + G(r) = h  i = h  i h ih i =) 0 r c 0 r 0 r z 2 M 2 r= G(r) = sin 2e (30) which is an example of the behaviour quoted above (20) for G(r). The important point is that in most models there is a unique correlation length and it is given by  = 1= log ( = ) 1 2 where   are the two largest eigenvalues of the transfer matrix. It is possible 1 2 0 that there is more than one relevant correlation length e.g.  = 1= log ( = ) 1 3 but this depends on the physics being investigated and we shall not refer to this extension further. The mass gap m of the theory is the inverse correlation length m = 1=. We shall see that a large class of the phase transitions we will be studying are connected with the limit 1 (or m 0). This, of course, means  % , 2 1 i.e., the maximum eigenvalue of the transfer matrix is degenerate.5 THE PHENOMENOLOGY OF PHASE TRANSITIONS 10 5 The Phenomenology of Phase Transitions Statistical systems in equilibrium are described by macroscopic, thermodynamic, observables which are functions of relevant external parameters, e.g., temperature, T, pressure, P, magnetic eld, h. These parameters are external elds (they may be x;t dependent) which in uence the system and which are under the control of the experimenter. the observables conjugate to these elds are: entropy S conjugate to temperature T volume V conjugate to pressure P magnetization M conjugate to mag. eld h Of course V and P may be swapped round: either can be viewed as an external eld. More common thermodynamic observables are the speci c heats at constant pressure and volume, respectively C andC ; the bulk compressibility, K; and the P V energy density, . Equilibrium for given xed external elds is described by the minimum of the relevant thermodynamic potential: Legendre Transformation Internal energy, U for xed S,V Helmholz free energy, F for xed T,V: F =UTS; T = (U=S) V Gibbs free energy, G for xed T,P: G =F +PV; P = (F=V ) T Enthalpy H for xed S,P: H =U +PV; P = (U=V ) S A phase transition occurs at those values of the external elds for which one or more observables are singular. This singularity may be a discontinuity or a divergence. The transition is classi ed by the nature of the typical singularity that occur. Di erent phases of a system are separated by phase transitions. Broadly speaking phase transitions fall into two classes: (1) 1st order (a) Singularities are discontinuities. (b) Latent heat may be non-zero. (c) The correlation length is nite:  1. (d) Bodies in two or more di erent phases may be in equilibrium at the transition point. E.g., (i) the domain structure of a ferromagnet;5 THE PHENOMENOLOGY OF PHASE TRANSITIONS 11 (ii) liquid-solid mixture in a binary alloy: the liquid is richer in one component than is the solid; (e) the symmetries of the phases on either side of a transition are unrelated. (2) 2nd and higher order: continuous transitions (a) Singularities are divergences. An observable itself may be continuous or smooth at the transition point but a suciently high derivative with respect to an ex- ternal eld is divergent. C.f., in a ferromagnet at T =T c 1 M 1 1   Mh ;  = h : h T (b) There are no discontinuities in quantities which remain nite through the transition and hence the latent heat is zero. (c) The correlation length diverges: 1. (d) There can be no mixture of phases at the transition point. (e) The symmetry of one phase, usually the low-T one, is a subgroup of the symmetry of the other. An order parameter, , distinguishes di erent phases in each of which it takes distinctly di erent values. Loosely a useful parameter is a collective or long-range coordinate on which the singular variables at the phase transition depend. is generally an intensive variable. In a ferromagnet the spontaneous magnetization per unit volume at zero eld, M(T ), is such an order parameter, i.e., M(T ) = lim M(h;T ) h0+ thenjM(T )j = 0 for TT , andjM(T )j 0 for T T . c c 1 2 Note:=(TT ) will not do. c is not necessarily a scalar, but in general it is a tensor and is a eld of the ef- fective eld theory which describes the interactions of the system on macroscopic scales (i.e., scales much greater than the lattice spacing). The idea of such e ec- tive eld theories is common to many areas of physics and is a natural product of renormalization group strategies. Examples (1) The ferromagnet in 3 dimensions. The ferromagnet can be modelled by the Ising model de ned in the previous section. X X H = J   h  ; n n+ n n; n5 THE PHENOMENOLOGY OF PHASE TRANSITIONS 12 The order parameter is the magnetization per unit volume. X 1 3 M =  ; V = Na n V n where N is the number of sites in the lattice anda is the lattice spacing. Note, that whilst the  are discrete, M is a continuous variable in the limit N;V1. n Note: from now on we use the symbol M now to mean magnetization per unit volume Ising model in two dimensions. Well above the critical temper- p ature T = 2= log(1 + 2)  c 2:27. Note that the total mag- netization is essentially zero. The theory is paramagnetic. Ising model in two dimensions. Only just above the critical temperature T  2:27. Note c that the total magnetization is still close to zero but the sizes of the domains are larger. At the critical point there are do- mains of all sizes distributed according to a power law dis- tribution. (2) H O 2 Look at the two phases of liquid and vapour. The order parameter is the density, , which is large for the liquid phase relative to its value for the vapour phase. The properties of both systems, their similarities and di erences are best exhibited by showing the various phase diagrams.5 THE PHENOMENOLOGY OF PHASE TRANSITIONS 13 PHASE DIAGRAMS  is the chemical potential, and  =  (T ) is the line of rst order transitions in c the (;T ) plot for H 0. It corresponds to the line h = 0 in the (h;T ) plot for the 2 Ising model. The critical point terminates the line of 1st order transitions.5 THE PHENOMENOLOGY OF PHASE TRANSITIONS 14 (i) Approach along (a) gives a 1st order transition whilst approach along (b) through the critical point gives a 2nd order transition. (ii) The order parameters are: magnetization M density  The conjugate elds are: magnetic eld h chemical potential,  or pressure, P (iii) The behaviour near T =T (t = (TT )=T ) c c c (a) t 0; h = 0 t 0;  (T ) = 0 c  + + M(T )jtj  (T ) =  (T ) jtj c c   (T ) =  (T ) jtj c c Clear symmetry in curve No obvious symmetry but experimentally = + (b) t 0+; h = 0 t 0+;  = (T ) c Susceptibility   K(T ) M  =  = h K (T ) 0 T   1  K(T ) =  P T K (T ) is for ideal gas 0 Then (t)jtj 0 Note that for the Ising model with t 0; h 0+ we nd (T )jtj with 0 = . It should be remarked, howver, that 0 is not de ned for all models.5 THE PHENOMENOLOGY OF PHASE TRANSITIONS 15 (c) t = 0; h 0+ t = 0;  0+ c 1 1   Mh   ( ) c c ; ; are examples of critical exponents. (iv) Coexisting phases (a) States between the curves I and II are physical but metastable. They do not violate thermodynamic inequalities. In the PV plot this is equivalent to P 0 V T which means that the compressibility is positive. This inequality is derived from entropy being a maximum in equilibrium. However, these states are unstable against changing to the mixed system, e.g., domains in the Ising model (or ferromagnet), and liquid-gas mixture for water. The continuous curves shown are the (h;M) and (P;V ) curves for a pure phase. E.g., the Van-der-Waals equation of state:   a P + (Vb) =cT V (b) The Maxwell construction gives the true equilibrium curve taking into account the formation of the mixed system. The mixture is of the two phases A and B. The rule for nding the interpolation is illustrated in the case of H O: 2 Z B P =P ;  = )  = v dP = 0; A B A B A B A where v = V=N, and N is the number of particles. This is the equal areas rule of Maxwell.5 THE PHENOMENOLOGY OF PHASE TRANSITIONS 16 5.1 The general structure of phase diagrams A thermodynamic space, Y , is some region in an s-dimensional real vector space spanned by eld variables y ;:::;y (e.g., P;V;T;;:::). In Y there will be points 1 s of two, three, etc. phase coexistence (c.f. A and B in H O plot above), together 2 with critical points, multicritical points, critical end points, etc.. Q is the totality of such points. The phase diagram is the pair (Y;Q). Points of a given type lie in a smooth manifold, M, say. The codimension, , of these points is de ned by  = dim(Y ) dim(M): E.g., two-phase points have  = 1; critical points (points that terminate two phase lines) have  = 2. There do not exist any simple rules for constructing geometrically all acceptable phase diagrams, (Y;Q): we cannot easily construct all the phase diagrams which could occur naturally. 5.1.1 Structures in a phase diagram: a description of Q I assume that there are c components in the system, and hence there are (c + 1) external elds:  ;:::; ;T . Then dim(Y ) = (c + 1). 1 c Manifolds of multiphase coexistence. The Gibbs phase rule states that the coexistence of m phases in a system with C components has f =c + 2m where f is the dimension of the manifold of m-phase coexistence. proof: dim(Y ) = (c + 1) and hence the manifold has codimension = (c + 1f). But  = (m 1) since  external elds must be tuned to bring about m-phase coexistence. An example of structures in a three dimensional phase diagram is shown below.5 THE PHENOMENOLOGY OF PHASE TRANSITIONS 17 A tricritical point has  = 4. Its nature is most easily seen rst in three dimen- sions. This is already a special case since we can only be sure it will appear in four dimensions. We suppose we have taken the appropriate cross-section of the 4D space. this often occurs naturally since some of the parameters are naturally set to the special values necessary to show up the tricritical point: e.g., by symmetry considerations. The hatched surfaces are 1st-order surfaces: surfaces of two phase coexistence. Thus the 1st order line in 2D is really a line of triple points (three phase coexistence) in higher dimensions.

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