Gaussian approximation of the Gross-Neveu model in the functional Schrödinger picture

PHYSICAL REVIEW

0

VOLUME 40, NUMBER 8 15OCTOBER 1989

Gaussian approximation

of

the Gross-Neveu

model

in

the functional

Schrodinger

picture

S.

K.

Kim and

J.

Yang

Department

_of

Physics, Ehwa Women's University, Seoul 120-750, Korea

K.

S.

Soh

Department

of

Physics Education, Seoul National University, Seoul 151742, K-orea

J.

H.

Yee

Department ofPhysics, Yonsei University, Seoul 120-749, Korea (Received 9November 1988;revised manuscript received 2June 1989)

The Gross-Neveu model is analyzed by the Gaussian approximation in the functional Schrodinger picture. It is shown that in the large-N limit the Gaussian approximation exactly reproduces the Gross-Neveu results, but for finite Nit contains more information than the large-N approximation. There are two nontrivial phases of the theory depending upon the sign of the

infinitesimal bare coupling constant. Dynamical symmetry breaking occurs in one ofthe phases. We also apply our analysis tothe chiral Gross-Neveu model.

I.

INTRODUCTION

@(x)

=

1

—

u

(x)+

5

2 5u

t(x)

iP (x)

=

u

(x)+

2 5u

x

to satisfy the equal-time anticom mutation relation

Ig;(x,

t),

QJ(x',

t)I

=5;.

5(x

—

x'),

where u and

ut

are an-ticommuting Grassmann variables. The time-indepen-dent functional Schrodinger equation is then given by

H

u

+,

u+

5

~e) =E~%')

5u ' gg~

(1.

2)

and we can take the trial wave functional, for the varia-tional approximation, in the Gaussian form

~%)

—

+~G)

=,

exp

I

ut(x)G(x,

y)u(y)

(detG

)'

(1.

3a) and its dual

It

is believed that the Schrodinger-picture Gaussian variational method is quite promising for the study

of

quantum structures

of

field theories. ' This method can be easily applied for boson field theories. In the case

of

fermion field theories, on the other hand, it isnot easy to take the wave functional in Gaussian form.

Recently, Floreanim and Jackiw proposed that the fermion field operator

f

and its conjugate momentum i

_g

be realized as

(4/~(G/=

exp

I

u

_(x)G(x,

_y)u(y)

(detG)'"

(1.

3b) where

G=(G

)

The purpose

of

this paper is to use the realization

(1.

1)

to establish the Gaussian variational method for fermion

theories. We believe that the Cxross-Neveu (GN) model is most suitable for this purpose since the model is solv-able in the large-X limit. We note that a variational analysis

of

the GN model already has been carried out by

Latorre and

Soto.

Our work is different from theirs in two features. Their wave functional is adelta functional, while ours is a Gaussian one. They introduced constant

background spinor fields and expressed the effective

po-tential (V,s.) in terms

of

a mass parameter which is con-structed from constant spinor fields. Following Gross

and Neveu who used the vacuum expectation value

of

the compositive operator

(gpss):—

—

o for V,

s,

we.expressed

Vffwith an effective scalar field u which is analogous to

the 0. variable so that our results can be compared with the GN model. Despite the differences in the choice

of

the trial wave functionals and the variational parameters the results turn out to be equivalent. Still another work by Lou and Ni, who take acoherent-state wave function-al, give similar results. The equivalence

of

the results suggest that the Gaussian variational approximations

of

fermion fields share the same essential core albeit ap-parently different formulations.

As Latorre and Soto pointed out there is a strong parallel between A,_P in four-dimensional spacetime and

the GN model. In scalar theory there exist two phases:

the precarious and the autonomous phases. In the

pre-carious phase the bare coupling constant is negative infinitesimal, while in the autonomous phase it is positive infinitesimal. In our approach to the GN model there are

S.

K.

KIM,

J.

YANG,

K.

S.SOH, AND

J.

H.YEE

indeed two phases exactly analogous to scalar theory. In Sec.

II

we introduce our notation and the overall schemes.

It

is well known in the scalar theory that the Gaussian eff'ective potential contains the leading

1/X

re-sult as its formal

X—

+

~

limit. We will show explicit-ly in Sec.

III

that the large-X limit is a special approxi-mation to the Gaussian method. In this section we will also present how the variational equation can be solved to

obtain the wave functional, which demonstrates essentials

of

mathematical techniques without getting involved in the complexity

of

the full equation. In

Sec.

IVwe obtain the effective potential

of

the GN model and the proper-ties

of

two phases are investigated. In the last section we summarize our results and brieAy present the Gaussian analysis

of

the chiral Gross-Neveu model. '

'"

II.

SCHRODINGER PICTURE GAUSSIAN APPROXIMATION

The GN model isdefined by the Lagrangian density

X~=/'(i8)g'+ ,

'g

—(f'g'),

a

=1,

. . .

,

N,

(2.1)

in two-dimensional space-time. Note that we take the plus sign for _g as Gross and Neveu did. _{The y} algebra

of

the theory is de6ned by

1

0

[Yp&}vj gpv& gpv

₀

Byemploying the Gaussian trial wave functional ~G&

and its dual defined in Eqs.

(1.

3a) and

(1.

3b),we calculate the expectation value

of

the Hamiltonian

of

the system in

the form

&GiHiG &

—

=

&H&

2

=

—,'

f

dx

dy[tr[

—

iy y'B(x,

y)Q(y,

x)]

_j

—

f

dx_t

tr[yaQ(x, x

)]tr[y0Q(x, x

)]

+

tr[2Q(x, x }5(x,x)

—

_y

Q(x,

x)yDQ(x,

x

)]

j,

(2.2)

where

S—

:

(G

+

G) and

I

denotes the identity matrix. The divergent function

5(x,

x)

can be defined as, in momentum space,

5(x,

x)=

f

dp

2m (2.4)

If

we are interested only in obtaining the ground-state en-ergy

of

the system, we take variations on &H& directly

with respect to

6

and

6:

where tr denotes the trace taken over Dirac spinor and

color indices. The matrix

Q(x,

y)is defined as

Q(x,

y)

=

2&GI

P(x

)Pt(y)lG&

(2.3)

We are, however, interested in obtaining the effective potential to see

if

dynamical symmetry breaking occurs in the system.

For

this purpose, we de6ne the effective Hamiltonian

H,

z=H

+

f

dx

a

u+

—

tr[y

Q(x,

x)]

(2.8) where

H

is obtained by replacing —,

'g tr[y

Q(x, x

)]

~

—

_{o in &H}

& by

Eq.

(2.2). More explicitly, we write it

as

H

=

_,

'

f

dx d—y

tr[

iy

y'i3(—

x,

y)Q(y,

x)]

,

'

f

dx

—

o—

2

f

dx

tr[2Q(x,

x)5(x;x)

5,

,

-&H

&

=0

.

(2.5)

—

_yDQ(x,

x)y

Q(x,

x)]

.

(2.₉₎

These conditions yield the equations

(I —G)hn(I+G)=0,

(I+

G)h

n(I

—

G)

=0,

(2.6a)

(2.6b)

The n 6eld introduced in

H,

&is the Lagrange multiplier

auxiliary 6eld.

If

we take a variation on

H,

z with respect

to u we obtain

where

hn(x,

y)

—

=

i

y

y'B(x,

y—)

g2

_I(x,

_{y)[y tr[y}

_Q(x,

_x)]

cr

= —

—

_tr[y

_{Q(x, x}

_)]

= —

_g&

f(x)g(x)

&,

&H

&=H.

=H„.

Other variational equations are

(2. 10)

+I(x,

x}

—

tr[y

Q(x,

x)y

]j

.

(2.7) Equation (2.6a) can be solved exactly for G,

to

obtain the ground-state energy

of

the system. The result will be presented later. One can show that the condition (2.6b)is equivalent

to Eq.

(2.6a).

5

H,

a=0,

5gH,

~=0,

56H,

~=O

.

(2.11a) (2.11b) (2.11c) In obtaining the extremum value

of

H,

&, Eqs. (2.10)

—

(2.11)are equivalent to Eqs. (2.6a) and (2.6b). We now explain our strategy in detail.

We first solve Eqs. (2.1la)

—

(2.

11c)

to determine G,

6,

and o.as functions

of

a:

i.

e.

,

G=G(a), G=G(a), cr=o(a)

.

It

turns out that the condition

(3.

3c)is equivalent

to

the condition (3.3b).

Multiplying

Eq. (3.

3b) by hN from the left yields Using these results we determine

H,

ffand thus V,ff as a

function

of

cx:

hN

K—

N+

f hN, KN

]

0—

, where

(3.

4)

Heff H~ff(a)

—

f

dx Vtff(a) (2.12)

E~

—

—

h~G

.

H,

ff(a)

=0

.

da

(2.13)

Note that we never use

Eq.

(2.10) in computing V,fffrom

Eq.

(2.12). The information

of Eq.

(2.10}is determined, in this case, by

Eq.

(2.

13).

In the next section we solve

Eqs. (2.11a)

—

(2.

1lc)

in the large-N limit and obtain the

e6'ective potential V,ff.

The extremum value

of

H,

ffis then determined from the

condition

0=I,

I

1=i/1, I

2=/0/1

(3.

5}

I

"s

are taken tobe Hermitian and satisfy the commuta-tion relacommuta-tions

One trivial solution

of Eq.

(3.4) isKN

=

khN, which leads

to

H

ff with vanishing quantum corrections.

To

obtain

nontrivial solutions

of Eq.

(3.4), we decompose the

2X2

matrix KN(x,

y)

in terms

of

four

Dirac

matrices. We denote them collectively by

I':

III.

EFFECTIVEPOTENTIAL INTHELARGE-N LIMIT In the large-X limit,

H,

ffcan be written in the form

II',

I

JJ

=25,

._{, i}

=1,

_2,

3,

[r',

r

J]

=2ie,

_,

„r",

[rP,

I']

=0

.

(3.

6) HN

=

—

f

dx dy tr'[hN(x, y)Q(y,

x)]

+

f

dx(acr

—

—,'cr

},

where

(3.

1)

KN(x,

y)

is then decomposed in the form

K

(xy)=

_y

r'f

d

"e-'~'"

»KN-.

_(p).

a=0 (3.7)

hN(x,

y)

=

iy y—

'B(x,

y)+ga5(x

—

y)y

(3.

2)

cr

=a,

(I

6)hN(I

+

G—₎

=0,

(I+6)hN(I

6)

=0

.

—

(3.3a) (3.3b)

(3.

3c) 1

and the tr' denotes the trace taken over Dirac spinor in-dices only. The variation

_{of H,}

_ff with respect to o.,

6,

and G yields, respectively, hN(x,

y)=

(

—

pI

+gaI')e

2'

hN(x,

y)

—

=

f

d»N(x,

~)hN(r,y) P

_(P2+g2a2)e

—lp(»—y)

2'

(3.

8)

(3.9}

Other functions relevant

to Eq.

(3.4) are written in the form

KN(x,

y)=

f

dp

[hN

KN](»y) =21

f

_2'

[I'[

—

PKN3(P)

gaKN2(p}]+f'gaK—

Nl(P)+I'PKN1(p)

Ie

"'

"

3 3

KNp(p}+

g

KN

(p}

+2

g

I

'KNp(p)KN (p)

(3.

10)

(3.11}

Substituting Eqs. (3.

7)-(3.

11)into

(3.

4) yields

3

KNp+

g

KN;=p

+g a

KNPN1 (PKN3+g KN2}

KN& N2 1_{g aKN1}

KN+N3

=

lpKN1

From the equations we find

(3.

12a)

(3.

12b)

(3.

12c) (3.12d)

K

(x

y)=%I

+p

+g a

e

2'

GN(x,

y)

=

(hN 1KN

)(x,y)

P d (

—

pI +gaI

)e

Qp

2+g

2a2 (3.14)

We can thus determine the matrices KN(x,

y)

and GN(x,

y),

the solution

of

Eq.

(3.3b), uniquely except for the sign

2650 S.

K.

KIM,

J.

YANG,

K.

S.SOH, AND

J.

H.YEE

Gx=G+=G~

—1

[h~,

K]=[h~,

G~]=0

.

Using Eqs. (2.3) and (3.15)we find

Q=Q~=(I+Ging

)S~ '(I+Ging)

=(I+6')

.

Wecan thus determine

H,

z in the large-N limit:

Hiv,ff=

—

f

dx

tr'K(x,

x)+

,

'

f

—dxcr2(x)

fdx

i

2+~f

d

PQ

2+

2 2 2

_2~

(3.15)

H,

ff=

f

dx(aa

—

—,

'o

)

+

—

f

dx dy

tr'[h (x,

_y)Q(y,

x

)

_]

f

dx

tr'[y

Q(x, x)y Q(x,

x)],

g2N w~ere h

_(x,

_y) iy—

'y

'a(x,

_y} 2

+y'&(x

—

y)

ag+

[y'Q(x,

x)y'

—

5(x,

x)]

(4.1)

f

dx

_jeff

' (3.17)

We take the minus sign from

Eq.

(3.14) in.order to have

the correct zero-point energy

E0..

(4.2) Taking variations on

_H,

_ff with respect to o._{, G, and G} yields the same equation as Eqs. (3.3a)—

(3.

3c)with hz re-placed by h:

i.e.

,

dp

Eo

=

—

g

_,

' fico

=

N

_—f

—

+p

(3.18) (4.3a) in the free fermion field theory limit.

If

we take the plus

sign from

Eq.

(3.14), the system will be unstable.

The effective potential is then given by

V

=

—

'a

—

Xf

d

+p+ga

(I

—

G)h(I+G)=0,

(I

+

G )h

(I

G)

=

0

.

—

(4.3b)

(4.3c)

lt

again turns out that

Eq.

(4.3c) is equivalent

to

Eq.

(4.3b). Multiplying

Eq.

(4.3b) by h froin the left yields h

—

K

+[h,

K]=0,

where 2 2

=

—

'a

—

_X

I

₊

_I(M)

1 2 0 g2(g2 _lng2+2

—

₁

4~

(3.19) ho(p)h;(p)

ill

~

(4»

gh

(p)

g

h;(p)

where

K

=hG.

Nontrivial solution

of Eq.

(4.4) for

K

is uniquely determined in exactly the same way as the solu-tion

of Eq.

(3.4)for

Kz.

The results are

3 1/2

I,

=f"

lpl,

Io(M)

=

2ir

_+p

_+M

(3.

20) (3.21) h;(p) 3 h; (p)

i=1

-1r2

~"

Gi~'

(4.6)

Here

M

is an arbitrary constant. The effective potential

(3.

19) is the same as that

of

Gross and Neveu. After re-normalization we obtain the renormalized efFective

poten-tial f

2 2

V(a)=

—

'a

₊

_a

4m ln

_a0

—

3

(3.

22)

IV. EFFECTIVEPOTENTIAL BEYOND THK LARGE-N LIMIT

We first write

H,

ff defined in

Eq.

(2.8)in the form,

tak-en trace over color indices,

It

is straightforward to show that this potential exhibits spontaneous symmetry breaking as is done in

Ref.

3.

Ac-cordingly the Gaussian approximation in the Schrodinger

picture is equivalent to the Gross-Neveu approach in the large-N limit. where 3 h

=ho++

h,

l'.

K=K~,

h

=h

[h,

K]=[h,

G]=0,

Q=I+G

.

(4.7b) (4.7c) (4.7d) Using the results (4.6) and (4.7d), we can write

h(x,

y) in

the form

We again choose the minus sign in Eqs. (4.5) and (4.6)to

be consistent with the zero-point energy

of

the system as shown in

Eq. (3.

18). We note the

l"

component

of

Gis

zero. The results given in Eqs. (3.15) and

(3.

16) are also reproduced

if

we drop the index N,

i.

e.

,

a(x,

y)=

f

P

—

g

G,

(O)r'

—

g

G,

(O)+p

r'

2~

2

'

2 2

+

g _G

_(O)+ag

r'

_e 2 3 g

Io(M)

=

2 or

—

2,

for which the effective potential can be made finite. We

consider each

of

these two cases separately.

(i)The case with _g

ID=2/(2N

—

1):

The renormaliza-tion condirenormaliza-tions are

where

G;(0)'s

are defined in momentum space:

G;(0)

=

f

G;(p)

.

(4.8) (4.9) 1 1

(N

—

—

—,'

)Io(M),

g (4.14a) (4.14b)

From Eqs. (4.6), (4.8), and (4.9)we obtain the consistency

condition

r

G&(0)=

—

G3(0)+ag

f

p

+

G3(0)

forwhich the consistency condition becomes

2% 2N

—

1

and the effective potential becomes

(4.15) 2 —1/2 V

g=

—aR₂

—

N 77l g 2 ₂ Nm m

4~

"M

2 m

=ag+

_G3(0),

2

the condition becomes

(4. 11) (4.10) and

G,

(0)=G2(0)=0.

Defining the effective mass m as

r 1 2N

—

1 2 2N 2 2

aR+

ln

—

₃ '2

aR

ap 2N

—

1 2N

ag

=F71 g 771 2 1 1 Pl 2m

M

'2

—

_Io(M)

_(4.12)

where

Io(M)

is given by (3.

21).

We are now ready to evaluate _{V,z as} a function

of

a

and m.

It

is

(4.16) where we have set

M=[(2N

—

1) 2/N]g~ aeo. In this

case the bare coupling constant becomes positive infinitesimal as the cutoff goes to infinity. The behavior

of

_{V,z is qualitatively} similar

to

the GN result, and in the large-N limit it reproduces the GN result.

It

has a minimum at 2 V,s

=

—,

'a

—

N

G3(0)

+

—

tr'X(x,

x

)

_az

_=aoexp

1—

2 _(4.17)

=

—,

a

2

—

N m 2 2 Nm 2 1 m ln

—

1 2m'

—

_Io(M)

_(4.13)

where we have neglected irrelevant infinite constants.

We notice that the condition (4.12) may be understood as

8V/Bm

=0,

which reminds us

of

the large-N approxima-tion

of

O(N)-symmetric scalar theory. The ground state

of

the theory is determined by the stationary points

of

V,

i.

e.

,by the equations BV/Bm

=0

and

BV/Ba

=0.

Since the GN model is renormalizable the divergences in (4.12) and (4.13) can be absorbed by a suitable renor-malization

of

coupling constant and wave function. We can do this by defining (d V/dm )~

=

I/gii or

equivalently by adding counterterms to the original

La-grangian.

If

we define

"'Vea

de

2

_~

o gR2

and require that the renormalized coupling constant gR be finite, we find two cases,

+

—,

'Io(M),

g

(4.17a)

(4.17b)

Although the renormalization conditions (4.17)can make the effective potential (4.13) finite and this phase seems similar to a phase in the case

of

the scale A,_P

theoryone,

has to question the existence

of

this phase since the

right-hand side

of

(4. 17b) contains an infinite term due to

Eq.

(2.10). We think that this fact may be related

to

the point raised by the authors

of Ref.

11and warrants

fur-ther investigation.

and therefore the symmetry breaking occurs for any finite

N. The behavior

of

the effective potential resembles that

of

the so-called autonomous phase

of

A,_{P theory} and our

result is qualitatively equivalent to the result

of

Lou and Ni who used a coherent wave functional which is some-what different from ours.

(ii) The case with _g

Io=

—

2:

In this case the

2652 S.

K.

KIM,

J.

YANG,

K.

S.SOH, AND

J.

H.YEE V. DISCUSSIONS

For

the Gaussian effective potential evaluation

of

the

GN model, so far there have been developed three different formulations for the wave functionals

of

fermion fields: namely, the delta wave functional, the coherent

wave functional, and Floreanini-Jackiw wave furictional which was used in our work. We have found that these apparently different approaches lead to qualitatively equivalent results, and furthermore the results are quite similar to the A,_{P theory} case: there are two phases,

i.

e.

_,

the precarious phase in which the bare coupling constant g is negative infinitesimal and the autonomous phase where _g is positive infinitesimal, and symmetry breaking

I

occurs forany finite

Ã

in the latter phase.

Both in A,_{P theory and in the} GN model the Gaussian

variational approach contains the leading

1/X

result asN

goes to infinity. ' ' _{However, in the chiral Gross-Neveu} (CGN) model this does not hold. We will _{briefly present}

our analysis on this model showing that the Gaussian ap-proximation isequivalent

to

the

1/N

approximation.

The CGN model isdefined by the Lagrangian density L

=f'&'8f'+

_,

'g

[(—fg)

(Py

—

g)

],

a=1,

2,

. .

.,N .

(5.1)

The expectation value

of

the Hamiltonian with respect to a trial wave functional is

(H)

=

_,

'

f

d—x_dy

_tr[

_iy

—

y'B(x,

_y)Q(y,

x)]

2

+

_f

dx(trI[y

Q(x,

x)]

—

[y

y

Q(x,

x)]

I

—

[try Q(x,

x)]

+[try

y

Q(x,

x)]

—

4tr[5(x, x)Q(x,

x)])

.

(5.2)

Weintroduce auxiliary fields o. and m as

o

=

—

—

(gpss)

=

_g

_tr[y

_{Q(x, x}

_)],

2

in=

—

(g.

—

_1Tystt

₎

=

—,

'g tr[y

y

Q(x,

x)]

.

In order tostudy the effective potential in terms

of o

and m we use an effective Hamiltonian

(5.3a)

(5.3b)

H,

z=H

+

f

dx

a

a+

—

tr[y

Q(x,

x)] +P

n

—

—

'

tr[y

_y

Q(x,

x)]

(5.4)

where

a

and

_P

are Lagrange multiplier fields and

H

= ,

'

f

_dx

—dy

tr[

iy

y'B—

(x,

y)Q(y,

x)]

—

—,'

f

dx(cr

+n

)

2

f

dx

trI2Q(x,

x)5(x,x)

—

[y

Q(x,

x)]

+[y

y

Q(x,

x)]

I

.

(5.5)

After performing similar calculations to the previous sections we obtain the effective potential V

=

'(a

+P

) N

f

—d [p

+—

g

(a

+P

)]'

—

I(0)I(0)

2

eff

2'

₂

—

1

—

_Io(M)

. , (5.6)

where _P

=a +P,

and irrelevant infinite constants are neglected. This is indeed

of

the same form as the

1/X

efFective potential (3.19), and after renormalization one obtains the finite effective potential

of

the CGN model

A2

V

=

—

'P2+

g P2

ln~

—

3 (5.7)

Thus the Gaussian variational approximation is equiva-lent tothe large-N approximation in the case

of

the CGN

model, while the former contains the latter as a special limit in the A,_P scalar theory and the GN model.

The effective potential (5.7) has a local maximum at

P

=0,

and the absolute minimum at P

=foe

~ g. This implies that the chiral symmetry is spontaneously broken and the fermions acquire masses. But in two space-time

I

dimensions, itiswell known that spontaneous breaking

of

acontinuous symmetry is not possible. As shown in the

case

of

the large-X approximation by Witten, ' therefore,

the appearance

of

the nontrivial minimum in the Gauss-ian effective potential (5.7) must be interpreted not as an indication

of

the symmetry breaking

of

the CGN model, but as an indication

of

aphase transition which gives rise

tomass generation

of

the physical fermions. ACKNOWLEDGMENTS

We thank

Dr.

W.

Namgung for many useful discus-sions. This research was supported in part by the Korea Science and Engineering Foundation and the Ministry

of

Education through the Research Institute

of

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