PHYSICAL R E V I E W D V O L U M E 1 2 , N U M B E R 6 1 5 S E P T E M B E R 1 9 7 5

Infrared effect of the Goldstone boson and the order parameter*

H. Matsumoto, N. J. Papastamatiou, and H. Umezawa Department of Physics, Universiry of Wisconsin-Milwaukee, Milwaukee. Wisconsin 53201

(Received 19 May 1975)

We present a physical argument to show that the influence of infrared Goldstone bosons precludes the appearance of spontaneous breakdown in one space dimension, or in two space dimensions when the temperature is not zero. The argument includes the case when gauge fields are present and a Higgs mechanism is operating.

The most s t r iking manifestation of sys tems with spontaneous breakdown of symmetry i s their high degree of o rder (e.g., ferromagnet , c rys ta l ) . F r o m this point of view the designation "ordered state" which is given to such sys tems by solid- s ta te physicists i s very appropriate. The t e r m "spontaneous breakdown," on the other hand, tends to emphasize the disappearance of the or igi- nal dynamical symmetry and i s popular in high- energy physics. A more ap t designation i s per- haps "symmetry rearrangement ," s ince it bridges the two aspec t s of the phenomenon by focusing on the change of the dynamical symmetry into the observable symmetry of the o rdered state.'. '

Spontaneous breakdown of symmetry i s accom- panied by the appearance of m a s s l e s s Goldstone b o ~ o n s , ~ which provide the long-range force nec- e s s a r y f o r maintaining the ordered s tate . Spe- cifically, Goldstone bosons of very low energy (i .e. , infrared Goldstone bosons) were found to be responsible f o r the rea r rangement of symmetry mentioned above.4' Th is a r t i c le i s concerned with yet another effect due to these infrared Gold- stone bosons: When their influence becomes too s t rong, the o rdered s ta te i s destroyed, i.e., spon- taneous breakdown i s made impossible. This situation occurs part icular ly in sys tems of low (one o r two) space dimensions.

An argument that the one-dimensional super - conductor i s impossible because of such infrared effects of Goldstone bosons was presented in Ref. 6 . Also, Coleman has given a rigorous proof7 that spontaneous breakdown in (I + 1)-dimensional spacet ime i s impossible when there a r e no gauge fields. In the following we shall demonstrate in detail how the presence of infrared Goldstone hosons makes the ordered s ta te unstable, and will extend the argument to include situations where gauge fields a r e present . (It i s generally believed among solid-state physicists that one- and two- dimensional superconductors a r e impossible. O u r conclusion supports this viewpoint.)

T o study the stability of the spontaneous-break-

down solution we shal l proceed a s follows: F i r s t , we shall assume that such a solution does exis t ; i t i s character ized by the fact that the vacuum ex- pectation value of a cer tain operator i s nonzero. This expectation value will be called the order parameter, in accordance with sol id-state t e r - minology. Then, symmetry requ i rements will be used to determine the existence of the Goldstone boson and the low-energy behavior of Green 's functions. Since the order parameter i s de te r - mined self-consistently in any given model, we can then est imate the effect of infrared Goldstone bosons on the o r d e r parameter and determine the conditions under which these effects a r e too s t rong and cause the o r d e r parameter to vanish.

F o r definiteness, we will examine a model of a self-interacting fermion field Q(x) with a ch i ra l - phase-invariant Lagrangian (Nambu model):

The nonvanishing o r d e r parameter i s given by

This model w a s examined exhaustively in Ref. 2. It was shown there that the s e t of in fields contains a t l eas t a massive fermion @'"(x) and a m a s s l e s s Goldstone boson x '" ( x ) :

The matr ix elements of the relevant Heisenberg opera tors in the in-particle Hilbert space a r e given by the in-field expansions (dynamical maps)

where the dots stand for normal products of in fields which a r e bilinear o r higher. As mentioned above, Eqs . (4) should be understood a s equalities of matr ix elements of e i ther s ide in the in-particle

12 - I N F R A R E D E F F E C T O F T H E G O L D S T O N E B O S O N A N D T H E . . . 1837

Fock space. In writing (4b), we assumed that induces the chiral phase t ransformation of the there is no bound s ta te corresponding to Jl,(x); i t s Heisenberg field existence i s not dictated by the symmetry and does not affect the argument to be presented below. O ~ ( X ) - exp(2ys @)@,(x) (10) (In the pa i r approximation, such a bound s ta te with a m a s s 2m does appears; higher-order correct ions in the limit f ( ~ ) -

may make i t unstable, however.) Therefore the in-field expansion of $,(x) must

The in-field expansion of the S matrix and the have the f o r m

fermion Heisenberg operator i s given by the func- tional integrals2 F[@'",BX'"]:. (11)

where (F[@]) denotes the functional average [the E t e r m ie$(x)+(x) i s included to specify the d i rec - tion of spontaneous breakdown; the l imit E - 0 i s to be taken a t the end of computationz]:

(6)

and

a = (d4x[z . - 1 1 2 ~ m ~ ~ ) ( i y . a - rn)p(x)

+ Z,-1'2$(x)(-iy 5 - m ) ~ ' " (x)

where F is a functional of the indicated flelds and ax1" s tands f o r any derivat ives acting on ~ ' " ( x ) .

T o investigate the effect of in f ra red Goldstone bosons, we spl i t ~ ' " ( x ) into an infrared par t x:(x) and a remainder x?(x). By "infrared partJ ' we mean contributions of momentum smal le r than a cer tain infinitesimal cutoff value. F o r example, x ; (x) can be defined a s

Indeed, the F o u r i e r decomposition of (12) i s

+ ~ , ~ 1 1 z x x ' n ( x ) ( - ~ 2 ) ) x ( x ) ] . (7) which in the limit 7- 0 tends to

The basic symmetry requirement (1) l eads to the 17 Ward-Takahashi identityz

(14) = -i,Z,-"' I d4r(-a:):( x(L), p(x)e-"): (8)

Since XI" (x) appears in F[+'", ax1"] only through derivat ives, the contribution of ,yLi; in F is neg-

which can be used to show that the in-field t r a n s - ligible and (11) can be written a s formation

(-a?,f(x) = 0 (9b) where Fs i s independent of x:. Then Eq. (2) gives

1838 H . M A T S U M O T O , N. J . P A P A S T A M A T I O U , A N D H . U M E Z A W A 12 - where

D , ( O ) = ( O ~ X ~ ( X ) ~ ~ 0)

fo r a spacet ime of dimension n + 1. If the sys tem under consideration i s a t finite temperature T, (17) i s changed into

where p = (kT)-'. These expressions show that D,(O)-.o fo r n = l and a l so f o r n = 2 when T#O. On the other hand, D ,(O) - 0 f o r n 2- 3 . To appre- c iate the significance of this resul t , notice that ii,, which i s the o r d e r parameter without the in- f ra red contribution, i s a finite quantity after r e - normalization. Therefore, (16) shows that, owing to infrared effects ,

0 = 0 f o r n = l and n = 2 , T z O . (19)

Spontaneous breakdown i s impossible in 1 + 1 spacet ime dimensions always, and in 2 + 1 dimen- s ions for T # 0.

A s another illustration of this resu l t , we con- s i d e r a phase-invariant model of a self -inter - acting complex boson field @(x) (Goldstone model). The o r d e r parameter is defined by

.C'=(OI @.(x)IO). (20)

The in-field expansion of @,(x) i s given by2

where

@,(X;X~...)=D~+Z~~'~~~(X)+.-. . (22)

Therefore, in this model - - u=2', .

To proceed, we must examine the self-consis- tency condition which determines E,. If the Heisenberg equation of motion for @,,(x) i s

t (-a?q.(x) = - C L ~ ~ @ ~ ( X ) +~@H(x)QH(x)@H(x) (24)

this condition r e a d s

g,"D, =h(O/ @H(X)@;(X)@H(X)~ 0 ) . (25)

Substituting (21) into (25), we have

After renormalization, p,".ii, and h ( 0 ( @,+J@,j 0) will become finite; however, D,(O) i s infinite f o r a one-dimensional space, and a l so for two space dimensions if T # 0. Therefore, again we obtain B = 0 in th i s model.

It is natural to ask whether the s a m e resul t holds when gauge fields a r e present in the model. We examine this question in the Nambu model extended to include an axial gauge field A, (x). The Lagran- gian h a s the following symmetry propert ies:

(i) =C[eiq5q(~),A,, (41 = G[$(x), A,, (XI] ,

(ii) ~ . [ e ' ~ ~ ( * ) ~ s @ ( x ) , Ap (x) + a , h(x)] = c[@(x), A, (x)]

In the Lorentz gauge, the quantization i s p e r - formed in t e r m s of the modified Lagrangiang

C,(x) = Z(x) + B(x)aPA,(x), (27)

with B(x) a supplementary field to f ix the gauge. The o r d e r parameter i s given by

0 = (01 TH(x)@,(x)I 0 ) . (28)

It i s found that the Hilbert space i s constructed from the following minimal se t of in fields.''

( i ) A massive fermion OIn(x): (iy . a -m)@In(x) =o .

(ii) A m a s s l e s s Goldstone boson xLn(x): (-a2)x1"(x) = 0.

(iii) A m a s s l e s s negative-norm s c a l a r field bln(x): ( - a 2 ) b ' n ( ~ ) = 0 .

(iv) A mass ive vector field U;"(X): (-a2 --mV2)u;"(~)=0, a M u , ( x ) = o .

The asymptotic f o r m s of the S matr ix and various Heisenberg opera tors a r e the following:

H e r e a(x'" - bLn) s tands fo r any derivat ives of (XI" - bln). Since any physically real izable s ta te (say, 1 a) ) i s required to sat isfy the condition

the Goldstone boson x'" and the ghost par t ic le bin d o not participate in the S matr ix. However, the infrared Goldstone boson s t i l l influences the order parameter , because it i s the agent of gauge and phase t ransformations. Also, it w a s shown in

12 - I N F R A R E D B E H A V I O R O F T H E R E G G E O N F I E L D T H E O R Y . . . 1839

Ref. 11 that the Goldstone boson has observable effects when the ground s ta te i s space and/or t ime dependent; the s t ruc ture of the ground s ta te or iginates f rom local Goldstone-boson condensa- tions in this case.

Written out fully, the in-field expansion of Q H ( 4 i s

Therefore the argument f r o m Eq. (12) to Eq. (19) applies in this c a s e also, and we conclude that spontaneous breakdown in (1 + 1)-dimensional space and in (2 + 1)-dimensional space (T* 0) i s impos- s ible even if a Higgs mechanism i s operating.

We close this note with two comments. F i r s t , the argument presented above must be modified in solid-state physics in accordance with the low- energy spec t rum of the Goldstone boson. If the Goldstone boson sa t i s f ies a wave equation of the f o r m

we find

Therefore , if, a s in the c a s e of the ferromagnet , w(3) -cG2 fo r / jj/ - 0, the resu l t obtained in this paper i s valid fo r T # 0. Superconductivity i s another situation where the conclusions obtained above hold; instead of (30), we have the wave equation

with

w ( ~ ~ ) - c I ~ I , 131-0

A s a second comment, l e t us point out that the argument of this paper can only be applied to s y s t e m s of infinite volume. If the volume i s finite, the allowed momenta a r e d i sc re te and the limit 151 - 0 cannot be taken. In this c a s e we must a l so consider the effect of the surface.

*This work was partially supported by the Graduate Phys. Rev. D 2, 2806 (1974). School of the University of Wisconsin-Milwaukee. 6 ~ . Leplae, F. Mancini, and H . Umezawa, Phys. Rep.

'H. Umezawa, Nuovo Cimento 40, 450 (1965). lOC, 151 (1974). 2 ~ . Matsumoto, N. J. Papastamatiou, and H. Umezawa, 7 ~ x o l e m a n , Commun. Math. Phys. 31, 259 (1973).

Nucl. Phys. E, 45 (1974). 8 ~ . Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 3 ~ . Goldstone, Nuovo Cimento 19, 154 (1961); J. Gold- (1961).

stone, A . Salam, and S. Weinberg, Phys. Rev. 127, 9 ~ . Nakanishi, Prog. Theor. Phys. 50, 1388 (1973). 965 (1962). 'OH. Matsumoto, N. J . Papastamatiou, H. Umezawa, and

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5 ~ . Matsumoto, H . Umezawa, G. Vitiello, and J. Wyly, Nucl. Phys . (to be published).