On the phase transitions in abelian Higgs models

Nuclear Physics B200[FS4](1982) 1-19 © North-Holland Publishing Company

ON THE PHASE TRANSITIONS IN ABELIAN HIGGS MODELS I. D. LAWRIE

Department of Physics, The University, Leeds, LS2 9JT, England Received 22 June 1981

A study is reported of the phase transitions in n-component abelian Higgs models, of euclidean dimensionality d = 4 - e. The occurrence of a fluctuation-induced first-order transition for appropriate values of n and of the coupling constants is confirmed by explicit construction of the coexistence curve and the equation of state, at order e. An exponentiation of the equation of state is obtained which: (i) is analytic in n away from thermodynamic singularities; (ii) describes either a first-order or a continuous transition depending on the value of n; (iii) incorporates canonical singularities induced by Goldstone modes on the coexistence curve, and by gauge field fluctuations in the disordered phase.

1. Introduction The aim of this article is to elucidate the mechanism which drives the phase transitions in the gauge-invariant model defined b y the reduced hamiltonian density (or, in field theory, the euclidean lagrangian density)

+½1D, ,°f'+ltotO°l'+

-'

#o(t4,°l')

(l.l)

with v,,.

=

-

D~, = O~ + ieoA~,.

(1.3)

T h e field C is a complex scalar field,

whose index a is summed in (1.1) from 1 to ½n, while # and r are summed from 1 to d, the spatial dimensionality. At this stage, n is a positive, even integer, but correlation functions m a y he continued analytically to all real n. Variation of the gauge-defining parameter ~ yields the set of Fermi gauges; our discussion is restricted mainly to the transverse gauge, ~ = 0, which is generally adopted in applications to condensed matter physics.

2

I.D. Lawrie / Phasetransitions

Special cases of this model arise in various contexts. With a Lorentz metric, the case n = 2 is scalar electrodynamics, and was studied by Higgs and others [1-4] as a prototype of spontaneous symmetry breaking in gauge theories. In statistical mechanics, it serves as a model of superconductivity [5], of certain liquid crystals [6, 7], or of magnetic systems with annealed disorder [8, 9]. Brereton and Shah have argued [10] that the limit n--,0 provides a description of topological entanglements in polymers. Since we are concerned with general features of the model, rather than with specific physical applications, we adopt the simplest thermodynamic terminology, regarding ~ ( x ) as an n-component spin field, whose expectation value is a magnetization density M, and t o as a temperature-like variable. The main question we wish to resolve was first raised by Halperin, Lubensky and Ma [7], who investigated the phase transition which occurs as t o passes through a critical value, toc(Uo, eo). Applying a renormalization group analysis at first order in e = 4 - d, they discovered, for sufficiently large n, an infrared-stable fixed point, indicating the occurrence of a continuous transition, which can be studied by standard methods. However, for n < n c ~- 365.9, their renormalization group equations possess no stable fixed point. They interpreted this as signaling a first-order transition, and supported this interpretation by evaluating the free energy for n = 2 and d = 3, using the mean field theory corrected for the leading gauge field fluctuations. Their result has the form

F( M ) = ½to M2 + ~--(.uoM' - clM~',

(1.5)

with p = 3 and c positive. More recently, Lovesey [9] has calculated a free energy of this form using first-order perturbation theory, and has also found a continuous transition in a variational approximation, which should be equivalent to the large, n limit. Now, the form (1.5), taken at face value, certainly implies a first-order transition. Unfortunately, however, it cannot be taken at face value, for two reasons. Firstly, the exponent p generalises to 4 - e for d ~ 3, and such exponents, obtained directly from perturbation theory, are generally unreliable in the neighbourhood of a phase transition. Secondly, for general n, the coefficient c remains positive at n --- n c, while p is independent of n. Consequently, (1.5) cannot account for the possibility of a continuous transition above n¢. A more detailed analysis of the two-component model has been given by Coleman and Weinberg [11], for the "naively massless" theory, i.e. with t o adjusted so that the susceptibility at M = 0 diverges. They found, for d = 4, a stable minimum of the free energy (or effective potential) at non-zero M, which indicates the occurrence of a first-order transition at a slightly higher temperature. As in (1.5), this minimum arises from a competition between zero- and one-loop terms in perturbation theory, and the result of such a competition is not in itself reliable. However, Coleman and Weinberg present a renormalization group argument to show that the minimum

I.D. Lawrie / Phase transitions

3

persists for values of the effective coupling constants for which perturbation theory can be trusted. In this work, we seek to clarify the mechanism of the transition, by making explicit the relation between the renormalization group structure and the equation of state where, in perturbation theory, the special value n = n c does not appear to play any distinguished role. In sect. 2, we obtain the coexistence curve, M ( t ) , and the transition temperature, t c, for all n, at first order of the e expansion, and confirm that the transition is always first order for n < n c. In sect. 3, we study the coexistence singularity, using a method developed recently [12] for the neutral model, e 0 = O. In sect. 4, we exhibit an exponentiated form of the equation of state which, although it is analytic in n away from the thermodynamic singularities, yields either a first-order or a continuous transition according as n < n c or n > n c.

2. The coexistence curve

The first step towards obtaining the coexistence curve is to find a renormalized form of the hamiltonian (1.1). We adopt the minimal subtraction scheme [13], whereby fields are rescaled according to

dp~. ~ Z~/2tk~,

A , ~ ZJ/2A~,

(2.1)

and renormalized mass and coupling constants t, u, e are defined so as to subtract all poles at e = 0 from the renormalized vertex functions. In general, the gauge-defining parameter must also be renormalized by a factor Z A (see, e.g., ref. [14]). However, we have been able to carry out the explicit calculations we require only in the transverse gauge, ~ = 0. Evidently, this gauge is invariant under renormalization, and we adopt it immediately. At one-loop order, the results are

Z~ = 1 + 3eZS/e,

(2.2)

ZA = 1 - ne2S/6e,

(2.3)

K-~u0 = u +

(n+8)u2S/6e-6e2uS/e+

18e4S/e-6e4S,

(2.4)

K-'e~ = e 2 + ne4S/6e,

(2.5)

r - 2 t o = t[1 + (n + 2 ) u S / 6 e - 3e2S/e],

(2.6)

where, as usual, ~ is an arbitrary parameter with the dimensions of inverse length, ensuring that u, e and t are dimensionless, and S = 2~rd/2/(2~r)dF(½d). The final term in (2.4) is not required by the minimal subtraction scheme, and has no significant effect on the renormalization group structure. It is included here in order to remove an inessential term from the equation of state.

4

I.D. Lawrie/ Phasetransitions

To derive the equation of state, we introduce an ordering(magnetic) field which, without loss of generality, may be taken to couple to ~] and contributes a term

-x3-~/ZH~k~

(2.7)

to the hamiltonian, the factor 1¢3 - e / 2 ensuring that H is dimensionless. Denoting the dimensionless magnetization by M = g,/2-,(q,~ ),

(2.8)

epll(x ) = x'-~/2M + o( x ),

(2.9)

we define shifted spin fields by

q~t9(x) = ~r(x),

(2.10)

(2.11) and determine H as a function of u, e, t and M by requiring (o(x))= 0. At one-loop order, retaining leading terms in the expansion in powers of e, we obtain

H / M = t + ~ u M 2 +¼Su(t+½uM2)ln(t+½uM 2) + ~ ( n - - 1)Su(t + ~uM2)ln(t + ~uM 2) + 3Se4M21n(e2M2).

(2.12)

To locate the coexistence curve in the ordered state, We require a solution of H = 0 with M ~ 0. In the case of a continuous transition, with u > 0, this is possible only for negative t, and the solution M(t) approaches zero continuously as t --, 0 - . For a first-order transition, this solution, which corresponds to a minimum of the free energy F(M, t), must persist to positive values of t, where a second minimum, corresponding to the disordered phase, M = 0, appears. The transition then occurs when the two minima yield equal free energies. Thus, to investigate the possibility of a first-order transition, one must understand how a non-trivial solution of H = 0 can occur for positive t. At the mean field level, where only the first two terms of (2.12) are retained, there is no such solution. When the first-order loop corrections are included, a suitable solution may be possible, since the logarithms are negative for small M and c However, such a solution is unreliable, since one is balancing terms of different order in perturbation theory, and there is no reason to believe that higher order terms, involving further logarithms can safely be neglected. We obtain a more reliable estimate of the coexistence curve by a renormalization group method which parallels, in part, the argument of Coleman and Weinberg [11].

I.D. Lawrie/ Phasetransitions According to the standard procedure, the renormalization group equation for H, which we give below, may be solved by the method of characteristics, replacing u, e, t and M by characteristic functions u, e, t, M of a free parameter, ~,. By a suitable choice of A, as a function of u, e, t, M, one may effectively eliminate the loop corrections, transferring their effect into the characteristic functions. If, for this choice of ~, the effective coupling constant ~ becomes negative, then H = 0 can be satisfied for positive t and t. Now, however, the required cancellation takes place between terms which both appear at leading order of the loop expansion for H, and a perturbative treatment of higher order corrections should be adequate. Specifically, we require the renorrnalization group equation

0

+d--YM ]

=0,

(2.13)

where

h = H/M,

(2.14)

and the coupling constant v=u[1 +

½Suln(u/3e2)]

(2.15)

has been introduced for later convenience. The coefficients in this equation are determined in the usual way (see, e.g., ref. [15]) from (2.2)-(2.6) as

Wv=-ev+(~-)Sv2-6Se2v+18Se /1

4,

4

"6=2-(-~--~)Sv+3Se 2, 7m=2-e-3Se

2,

(2.16) (2.17) (2.18) (2.19)

at this order. We now obtain the coexistence curve in the form

t= teoex(v,e, M).

(2.20)

Perturbatively, one obtains from (2.12) the expression to~.x = -~vM2[1 +

(½Sv + 9Se4/v)ln(e2M2)],

(2.21)

6

I. D, Lawrie / Phase transitions

while setting h = 0 in (2.13) yields

[l/K~-~+l'Ve~e2-MZO----~-+5,t]tcoex=O,oM z

(2.22)

where the dots indicate division by "/M. Solution of this equation by the method of characteristics gives the relation

tcoex(V,e,M ) =P(X)t¢o~x(g(X),~(X),M/~t/2),

(2.23)

with auxiliary functions defined by the characteristic equations aT = W~(g,~),

g(1) = v ,

(2.24)

X b~z = lg'e(~ ),

~(1) = e ,

(2.25)

a-~aP = . ? t ( 6 , ~ ) p '

P(1) = 1,

(2.26)

and the logarithmic term in (2.21) is eliminated by choosing h to satisfy

~2MZ/)~ =

1.

(2.27)

The equation of the coexistence curve then becomes t=

61 ~g(X) :(a)

P(X).

(2.28)

For a given positive t, this may be satisfied if the initial values of v and e are such that g(X) becomes negative as M and h are reduced. At this stage, it is useful to enumerate the fixed points, at which 14,'~ and l//e vanish. For e = 0, we recover the usual gaussian and Wilson-Fisher fixed points of the neutral model, with v = 0 or Sv* = 6e/(n + 8) + O(e2), respectively. For e > 0, and sufficiently large n, there is a pair of fixed points with S(e*) 2 =

Sv*+ =

6e/n + O(e2), 3 ( _ ~ 8 ) ( 1 + ~36)

(2.29) "q- eO"1/2 "of-O ( g 2 )

-Tj

(2.30)

I.D. Lawrie / Phase transitions

7

The latter exist only for positive o, that is, for n > n c = 365.9032 .... + O(e),

(2.32)

which reproduces the result of ref. [7]. In fig. 1, we plot renormalization group trajectories in the (e 2, v) plane, determined from (2.24) and (2.25) for n > n c and n < n c. Evidently, for any n, there is a region of initial points in this plane for which ~ becomes negative as ~--, 0. For n < n c, however, this occurs for all real e. We observe that all trajectories for which e ~= 0 approach the line e = e*, and to obtain explicit results, we now specialise to this case, at the expense only of inessential corrections. For this case, solution of the characteristic equations yields (2.33)

S e = Svmi n -- e o l / 2 t a n h O ( X, v ) ,

P = ~a[cosh0(k, v)]("+2)/("+ 8),

(2.34)

where

( n + 14) ( a= 1 +4(n+8)1+

_3n.)6_

e + O(e2),

(2.36)

Iit

2/ ?//

,U / o

ff

vi

(a)

v"

~o 0

V~ v

(b)

Fig. 1. Renormalizafion group trajectories in the (e2,v) plane, computed at O(e) with e = I, and (a) n - - 5 0 0 > no; (b) n = 200 < n c. Fixed points are represented by filled circles, and arrows indicate the direction ot flows as ~, --, 0.

I.D. Lawrie / Phase transitions

8

and

3

(1+36 .

(2.37)

locates the minimum of l~o(v, e*). Note that, since tanh0 is odd and analytic, the functions v and P are, in fact, analytic in n at n = no, unless ?~= 0. To illustrate the essential features of the transition, we consider the special case v = vmi~, which leads to a continuous transition for all n < n c. For this case, the equation of the coexistence curve reads r/

t = ---~m

2at

/cosn~tm)jxl(n+2)/(n+8)[[ (n + .,~z

8)

1+

]

--al/2tanhO(m) , (2.38)

with

O(m)=e(n~--~)ol/21n(m2),

(2.39)

and m = e*M. For n > n c, this gives a critical point at t = 0. The coexistence curve then has the usual form, t ~ - c o n s t • m l/a,

(2.40)

as m ~ 0, with the exponent given by 1[ (~2 2 fl=~ 1+ °1/2

n+14 ( ~ ) ) 4(n+8) 1+

] e+O(e2) .

(2.41)

+ Io If: tan0] ,

(2.42)

For n < n c, a is negative, and (2.38) becomes

t=--~6m2"[cosOI(n+2)/(n+s)[3(l+3~6n)~ with

(2.43) In this case, tan0 can become negative and arbitrarily large as m ~ 0, and the coexistence curve persists to positive t, indicating a first-order transition. However, the transition temperature t I and magnetization M 1 at which the ordered phase

I.D. Lawrie/ Phasetransitions

9

becomes unstable is not given directly by (2.42). To locate t I, we need the auxiliary condition

F( u, e, M, t ) = fogtt( u,e, M', t )dM' = O,

(2.44)

so that the two phases have equal free energies. A reliable estimate of t I must be obtained from the renormalization group equation

Since (2.20) and (2.44) are to be satisfied simultaneously, consider the quantity

G(v, e*, M) = M -4F(v, e*, M, t~x(V, e*, M)).

(2.46)

Using (2.22) and (2.45) we obtain

[Wo~--)'MM2~-~+ d - 2YM]G = 0, and setting

(2.47)

G(v, e*, Mi) = 0 yields 0v = - M 2 "

(2.48)

On integrating, we have

(~---28)]o['/2eln(e2M2):-tan-t( Sv-svmi~'o[,/2e ] + C,

(2.49)

where the constant of integration, C, may be found by evaluating M 1 for any convenient value of v. Taking v = 0, we have the perturbative result

G(O,e*,M): - ~ S ( e * ) ' [ 1 + 21n(e*2M2)],

(2.50)

which gives n+ 8

z/2

~ ) .

(2.51)

O,:e(~28)loJ'/~ln(e*2M2)=C,

(2.52)

Finally, setting v = vmi~ in (2.49) gives

which, together with (2.42) yields the transition temperature t t for this special case.

10

I.D. Lawrie / Phase transitions

0

n

-1

1 -1

0

1

Fig. 2. Coexistence curves calculated at O(e) with e= 1 for several values of n. For n < n c, the dotted curve represents the variation with n of the first-order transition temperature and the magnetization discontinuity. Fig. 2 shows the coexistence curves calculated in this way, for several values of n, with e = 1, and t normalized so that t ( m = 1) = - 1. It will be seen that the plots are restricted to fairly large values of n. The curve for n = 10 looks somewhat unusual, and for n = 2 which, unfortunately is the case of greatest physical interest, the transition temperature t t cannot be located. The reason is that, as n is decreased, the value of 0 l, eq. (2.52), eventually passes through -½rr. F o r e = 1, this occurs when n = 8.940-... At this point, the effective coupling constant ~ is negative and infinite, and the first order perturbative treatment, even as improved b y the renormalization group, becomes meaningless. We cannot rule out the possibility that this signals some new effect, b e y o n d the reach o f perturbation theory. It seems more likely, however, that the qualitative behaviour is unchanged at small n, even though the numerical results of the e expansion, at least at the order we have considered explicitly, b e c o m e unreliable at e = 1. F o r smaller e values, n can be further reduced; to obtain a result for n = 2, we would require e < 0.28. 3. The coexistence singularity F o r the neutral n-vector model, we have given elsewhere [12] a renormalization group treatment of the singularity induced b y Goldstone modes at the coexistence

11

L D, Lawrie / Phasetransitions

curve. In this section, we prepare the way for an exponentiation of the equation of state by describing a similar treatment of the present model. We introduce rescaled longitudinal and vector fields, s( x ) = moo(X ) ,

(3.1)

a~,( x ) = moA~,( x ) ,

with (3.2)

m~ = ½Uo(Olt) z,

these being the fields which remain massive after spontaneous symmetry breaking. The fixed point hamiltonian which controls the coexistence singularity is obtained by taking the limit m~ --, oo, and renormalized quantities r~, fi, 5, are defined in such a way that the correlation functions involving s ( x ) and a~(x) remain finite in the limit ~ --, oo. The calculation is similar to that described in ref. [12], and we quote results for the coefficients in the modified renormalization group equation:

#,=-*~+( n+s g

'~ ) s~ ~-6s~(a-3~ ~) ( 1 (~+~,~) '~ ) , ~3 (~--~) (3.3)

rVe=-,~2+("6 31 (a +,~rh2) ) S~4'

=2(n+83 2) 6

2 (a+rh2)

Sfi+3SeZ

(3.4)

(

l-6-ff

1

(a+rh2)

.

(3.5)

The constant a is arbitrary, since only the large rh behaviour is important, and may indeed be assigned different values in the various terms in these equations. Inspection of (3.3) and (3.4) reveals an infrared-stable fixed point, in the limit r~ -, oo, with Su** = 6 e / ( n - 1) + O(e2),

(3.6)

S(e**) 2 = 6 e / ( n - 2) + O(e2),

(3.7)

Ym-**= 2 - e.

(3.8)

As in the neutral case, the last result should be exact. This follows from the gaussian character of the fixed point hamiltonian, which is most easily seen [12] in the bare theory, The steps required are: (i) make the substitution (3.1) in the hamiltonian (1.1);

12

I.D. l_xavrie / Phase transitions

(ii) introduce a source term for s(x), which is conveniently written as

(3.9)

-- 2( 3/Uo) t/2rt( x )s(x ) ; (iii) take the limit m 0 ~ oo, and shift the longitudinal field according to

g=s+½(½Uo)'/2(er 2 +~ri~i')-- 2(3/Uo)l/2~+ B,

(3.10)

where the counterterm B represents the difference between t o and m 2. Up to terms which are independent of the fluctuating fields, we obtain the gaussian hamiltonian 2

~(2mo,~ )

--1

(a~a~)

2 .~½[~,//..~ /,~ 2--

\1/2

t.~eo/Uo)

]2

a~j

(3.11) The effect of the first term here is ambiguous. Our perturbative treatment, in the transverse gauge, corresponds to taking the limit m ~ ~ 0, which is equivalent to the condition ~ a , = 0. With this condition, the coupling between ~r and a~ is only via a total divergence, ~,(~r%), which may be assumed irrelevant for bulk thermodynamic behaviour, and one obtains the effective hamiltonian (3.12) with g and % completely decoupled. This is identical with the result of ref. [12]; the longitudinal source couples to the operator (~r 2 + ~ri~Iri~),whose scaling dimension in the gaussian ensemble is given by (3.8), and leads, for example, to a divergence of the longitudinal susceptibility, X L ~ -~/2

(3.13)

at the coexistence curve. It would appear, however, that this result does not respect the usual Higgs mechanism [2], according to which, the Goldstone field ~r(x) should be absorbed into the longitudinal component (in coordinate space) of the massive vector field. This is, of course, a feature of the transverse gauge. If the steps taken in this section were repeated with any non-zero ~, one would instead take the limit rnZo~--, oo. The first term of (3.11) then vanishes, and a shift in a~ eliminates ~r from the second term. For ~i = 0, one is then left with a set of (n - 2) Goldstone fields, as required by the Higgs mechanism, and as suggested by (3.7). To avoid the embarrassing term ,l~r2, one should then couple ~/ to Iq~J[, rather than s, make the substitution ~t = i~l lexp(iO) ' and eliminate 0 by a gauge transformation [2]. It is not clear to us how serious this ambiguity is, or how it should best be resolved. In particular, the apparently exceptional behaviour of the transverse gauge

I.D. Lawrie / Phase transitions

13

is worrying, since many features of spontaneous symmetry breaking have been shown to be gauge invariant (see, e.g. [14, 16, 17]). The calculation of the next section is restricted to the transverse gauge, and our equation of state incorporates a coexistence singularity characteristic of (n - 1) massless modes.

4. Exponentiation of the equation of state In sect. 2, we verified the assertion of Halperin et al. [7] that a continuous transition is not possible for n < n c. When a continuous transition does occur, critical exponents such as that given in (2.41) contain singularities at n --- n c. These results appear surprising when one considers perturbative expressions for thermodynamic functions, such as the equation of state (2.12), which give no indication of any special behaviour at n--no. We now seek to clarify this situation by obtaining an exponentiation of (2.12) in which the role of the special value n c becomes explicit. To accomplish this most effectively, we require an approximate solution, in closed form, of a somewhat complicated renormalization group equation. It should therefore be emphasized that our primary purpose is to understand the mechanism of the transition; we do not attempt either to maintain full generality, or to satisfy all detailed requirements of analyticity. As in ref. [12], the exponentiation is achieved by means of a mass-dependent renormalization scheme analogous to that of sect. 3. We define a new set of renormalized parameters, w, ~, #, ~ by

u:w[, _ I t=~'l-¼Swln

w_6:) ~ De

)

-¼S(w-6e2)ln(3+am~)-7--~

,,., l+m2

,

(4.2) [ M 2 = #2[1 -~Se21n(3 + 2m 2)

9 Se41n( 3+2m2)] + ~ --~-

l+m 2

H =(#/M)~,

'

(4.3) (4.4)

where

m 2 = -6e2t/u + O(1

loop)

(4.5)

is proportional to the square of the spontaneous magnetization. We have incorrectly omitted to introduce a new charge, analogous to ~, which led to (3.4), since we wish to simplify the renormalization group equation by setting e = e*. As a result of this

14

I.D. Lawrie / Phase transitions

omission, the fixed point value e** will not appear in the coexistence limit as it should. However, (3.3) and (3.5) are insensitive to the actual value of e** in the limit rh ~ oo, and we therefore lose, at most, some inessential corrections to the leading singularity. For fixed, non-zero u and e, the new parameters ~', # and ~ are related to t, M and H by factors which depend only weakly on m and are analytic at m = 0. In terms of these new variables, the equation of state reads

h=~/l~-,r+lwl.t2+tSw(~.q_½w~2)ln( 3e2/~2-m2 )3+2m 2 +(~21)Su(~.+~wlz2)ln(6e2#2-m2)+3Se4#21n( e2/~------~ 2 ) l+m 2 '

(4.6)

at this order. On setting e = e*, we have the renorrnalization group equation

[

w w + v , , ~0

__~bt/12+._~

1]h=O,

(4.7)

with

~

,~[,+~°( ~. 3+~m~)]~w+~["+~6 3+~m~m~ }'~w'~ (4.8)

3+2m 2 '

7~ = 1 - ~

,(n+23

2o2 )sw

(4,9)

3+2m 2

~=, ~[,+36(,

2°2 )] 324~ 2°2 3+2m 2

--~-[

3 + 2m 2

m2 ](e2/Sw).

1-I-m E

(4.10) This equation has the solution

h(.r,l~,W) ----~h(~, ~ , ~ ) ,

(4.11)

with h arbitrary, and characteristic functions defined by X-~-=a~ Ww(~, ~ ) , a~

~(1) = w,

(4.12) (4.13)

I.D.Lawrie/ Phasetransitions

x-~:-½~(~,~)~,

~(1):~,

~=m(~,~,e*).

15 (4.14) (4.15)

The desired exponentiation is now achieved by choosing h so that (e,)2/~2 _ ~2 = 1.

(4.16)

With this condition, (4.6) and (4.11) yield

h =X(~+-~2).

(4.17)

Provided that solutions of the characteristic equations can be found, (4.16) and (4.17) provide an implicit form of the equation of state. While approximations to the solutions of such equations can be found systematically in various limits [18], our illustrative purposes are better served by exhibiting approximate closed form solutions. The following may be verified to satisfy (4.12)-(4.15) up to corrections which can be regarded as O(e2), and reproduce the types of limiting behaviour found in earlier sections:

S~=eQ(~)(3+2~2)~/2[ [ 3----~ ,n+ 81(I+--~)+(-o)'/2tan(O+Oo)}, (4.18) ? = z~A- IQ(~)(3 + 2~2)A [cos 0o/cos(0 + 0o)] ("+2)/("+8),

Q(m):[(n--~ ) +[n- 81 l l(3+ 2m2)"/2] ~2

= m2/X + O(e),

~: ~(,?~__~8)(_o),:1oi~(3 ÷ ~)1,

(4.19)

(4.21) (4.22)

(4.,_3)

Oo=..-,[( " (1 A ; (n + 2)B/2(n + 8),

(4.25)

B = ~(1 + 36/,,)~,

(4.26)

C = 108(n + 8)/n 2.

(4.27)

I.D. Lawrie / Phase transitions

16

These solutions, obtained largely by guesswork, are certainly not unique. The value of O0 implies a specific choice for the initial value of w, and multiplicative factors of the form (1 + O(e)) which are required to satisfy the initial conditions have been absorbed into ~"and At. The particular form of (4.20) has been chosen in anticipation of an analytic continuation of the equation of state into the disordered phase. Thus, the last term of (2.12) indicates a singularity at M ~ At= 0 induced by the gauge field for any positive t. An analysis similar to sect. 3 shows that this singularity is canonical, and leads to a free energy which is essentially of the form (1.5) with p = 4 - e, above the transition temperature. Using (4.16) and (4.17), the equation of state may now be written as [(~_~_..~) + ( ~n_- _1 ~ ) ( 3~2 2 m2 ) e / 2 ( ~ 9

2

)--e/2](~)

/ Kcos(O+Oo)

+~ ( 3~22)-m2 n ) + (-o)'/'tan(o+o0)

,

(4.28) where

),

(4.29)

00 is given by (4.24) and/~ has been rescaled by a factor of e*. The quantities ~ and differ from H and M of (2.12) only by non-singular factors. The spontaneous magnetization, m(T) is determined by the condition that ~ --- m when (~//~) = 0, and one recovers the coexistence curve of sect. 2, with a = 1 + B - A.

(4.30)

Below the transition temperature, (4.28) therefore gives the desired exponentiation, and describes either a first-order or a continuous transition depending on the value of n. However, the analytic properties of the trigonometric functions guarantee that it is analytic at n = n c, away from the thermodynamic singularities. In the case of a first order transition, the equation of state does not directly give the transition temperature, which must be located by the subsidiary calculation of sect. 2. In the case of a continuous transition, the equation of state should admit analytic continuation to positive t or r. To obtain such a continuation, some further

I.D. Lawrie / Phase transitions

17

manipulation is necessary. We consider the asymptotic critical region, where z,/~ and m are small. In this region, we have m 2 _ (_z)2t3 = - ¢ + O(e),

(4.31)

and it is consistent with our O(e) approximation to retain only the leading term. Of course, this step changes the analytic structure of (4.28), and is equivalent to a modification of the renormalization scheme (4.1)-(4.4), appropriate only for n > n c, in which m 2 is replaced by (-~'). The usual scaling variables, x and y, are defined, up to inconsequential scale factors, by y = ~ / # 8 ~ H I M 8,

(4.32)

x = "r/# '/t~ ~ t / M 1/p,

(4.33)

where fl is given by (2.41) and , ----3 + 2 B = 3 + ( 1 +

-36) - e + O(e2). n

(4.34)

The scaling relation dp = fl(6 + 1) then gives 2p= 1 +

[(n~_~)

o I/2 q

n2 + 2 n - - 2 1 6 ] -4n(~2~-8-~ e + O ( e 2 ) ,

(4.35)

in agreement with the result of Halperin et al. [7]. Asymptotically, the equation of state may then be written in the form • + •

y = x K t V - ' ) / v + [1 + C - C K t~/v ] K i v -

2,8)/y (4.36)

with y =- 8 ( 8 - 1), P /v =

+

(4.37) (4.38)

The function K ( x ) , obtained from our earlier equations is K ( x ) =-½(3 + x) + O(e).

(4.39)

However, some desirable analytic properties can be built into (4.36) by requiring K

18

I.D. Lawrie / Phasetransitions

to satisfy x K (v-l)/~' + 3K (v-2#)/v = 2K,

(4.40)

which reproduces (4.39) when expanded. The coexistence curve is now given by y=0,

x= -1,

K = 1,

(4.41)

and the quantity (1 + x) is an analytic function of the two powers y a n d y 1-~/2. The functions K ( x ) and y ( x ) are analytic at x = 0, as they should be. However, for large x the function h'(x-2#) = x -'tK(x)

(4.42)

is analytic in x-2~, and we have y = x r g ( x -2~, x(~-2)tJ),

(4.43)

with g analytic in both its arguments. As a consequence, for fixed, positive t, we have H = M~( M 2,

[MI2-~),

(4.44)

as exemplified by the first derivative of (1.5), the singular terms arising from fluctuations in the gauge field of mass M. Eq. (4.36) differs from (4.28) in the critical region by corrections which are nominally of higher order than that to which we are working, but are nevertheless essential to the analytic structure. We do not know how to produce a representation of the equation of state which both exhibits a first-order transition for n < n c and also has the correct analytic properties for n > n c, although a sufficiently ingenious parametric representation might achieve this.

5. Conclusions When a phase transition is predicted by mean field theory to be second order, the absence of a stable fixed point of the renormalization group is commonly held to indicate a fluctuation-induced first-order transition. While this is probably true, little information about the behaviour of the system can be obtained from the renormalization group alone. To understand the mechanism of the transition, one must study the actual form of the thermodynamic functions. In the model we have considered, a detailed interpretation of the renormalization group results is particularly interesting because a stable fixed point appears only when the number of components exceeds a critical number, nc, which appears to play no special role in perturbative expressions for the thermodynamic functions. To clarify this situation, we have used renormalization group methods to obtain exponentiations of the coexistence curve (2.38) and

L D. Lawrie / Phase transitions

19

the equation of state (4.28) in which the occurrence of a first order transition and the role of n c become explicit. It is presumably well understood that the existence of a stable fixed point does not guarantee a continuous transition. In our case, even for n > n c, the transition is first order when the initial value of v is sufficiently small. At first order of the e expansion, this happens for v / e 2 < v * / ( e * ) 2. The line v / e 2 = v * / ( e * ) 2 in fig. 1 marks the border of the domain of stability of the fixed point (e*, v~_ ). It is, in fact, a locus of multicritical points, controlled by the fixed point (e*, v* ), which attracts trajectories starting on the special locus, and whose exponents may be obtained from (2.41) and (4.34) by the replacement o 1/2 ~ - 0 1 / 2 . Finally, we remark that the possibility of fluctuation-induced first-order transitions is by no means restricted to the model we have studied. It is well known to occur, for example, in models of ferromagnets with large cubic anisotropy [19-21]. This example has been studied in some detail by Rudnick [21], using methods which bear some resemblance to ours. After completion of this work, we learned that Yamagishi [22] has also given a reformulation o f the argument of Coleman and Weinberg in terms of renormalization group flows. The questions I have attempted to answer arose during conversations with S.W. Lovesey, concerning his work reported in ref. [9].

References [1] P.W. Higgs, Phys. Rev. Lett. 13 (1964) 508 [2] P.W. Higgs, Phys. Rev. t45 (1966) 1156 [3] F. Englert and R. Brout, Phys. Rev. Lett. 13 (1964) 321 [4] G.S. Guralnik, C.R. Hagen and T.W.B. Kibble, Phys. Rev. Lett. 13 (1964) 585 [5] E.M. Lifschitz and L.P. Pitaevskii, Statistical Physics, part 2 (Pergamon Press, Oxford, 1980) ch. V [6] B.I. Halperin and T.C. Lubensky, Solid State Comm. 14 (1974) 997 [7] B.I. Halperin, T.C. Lubensky and S. Ma, Phys. Rev. Lett. 32 (1974) 292 [8] G. Toulouse, Comm. Phys. 2 (1977) 115 [9] S.W. Lovesey, Z. Phys. B40 (1980) 117 [10] M.G. Brereton and S. Shah, L Phys. AI3 (1980) 2751 [I 1] S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888 [12] I.D. Lawrie, J. Phys. A14 (1981) 2489 [131 G. 't Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189 [14] J.S. Kang, Phys. Rev. DI0 (1974) 3455 [15] D.J. Amit, Field theory, the renormalization group and critical phenomena (McGraw-Hill, London, 1978) [16] N.K. Nielsen, Nucl. Phys. BI01 (1975) 173 [17] J. Illiopoulos and N. Papanicolaou, Nucl. Phys. BI05 (1976) 77; B i l l (1976)209 [18] D.J. Amit and Y. Goldschmidt, Ann. of Phys. 114 (1978) 356 [19] D.J. Wallace, J. Phys. C6 (1973) 1390 [20] E. Domany, D. Mukamel and M.E. Fisher, Phys. gev. BI5 (1977) 5432 [21] J. Rudnick, Phys. Rev. BI8 (1978) 1406 [22] H. Yamagishi, Phys. gev. D23 (1981) 1880