Bosonization and Duality of Massive Thirring Model

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HEP-TH-9502100 t-curren e Thirring mo del is equiv acult [email protected] In the pro cess of the b osonization e fermion . , Chiba 263, Japan of y hi Kondo ysics, F e-mail: k -dimensional massiv Abstract D t out the imp ortance of the gauge-in ersit ebruary 1995, hep-th/9502100 e Thirring mo del ery massiv Kei-Ic ho ol of Science and T tofPh e p oin ulation of the Thirring mo del as a gauge theory teraction of the curren Massiv e Thirring mo del is mapp ed to the Maxw Bosonization and Dualit Chiba Univ e free scalar eld theory Departmen & Graduate Sc CHIBA-EP-88, F tion to the case of v ulation. Starting from a form y atten dimensional massiv theory and that the (1+1)-dimensional massiv dimensions. Up to the next-to-leading order of 1 to the massiv pa of the Thirring mo del, w form 2) with four-fermion in consider the b osonization of the provided byCERNDocumentServer brought toyouby CORE

1 Intro duction and main results

Recently the Thirring mo del [1, 2 ] was reformulated as a gauge theory and

was identi ed with a gauge- xed version of the corresp onding gauge theory

byintro ducing the Stuc kelb erg eld in addition to the auxiliary vector eld

A which is in that formulation identi ed with the massless gauge eld [3, 4 ].

This is a consequence of the general formalism for the constrained system by

Batalin and Fradkin [5]. This gives the general pro cedure by which the system

with the second class constraintisconverted to that with the rst class one

and the new eld which is necessary to complete this pro cedure is called the

Batalin-Fradkin eld [6]. In the massive gauge theory the Batalin-Fradkin eld

is nothing but the well-known Stuc kelb erg eld as shown in [7]. By maintaining

the manifest gauge symmetry, controversy among the pap ers [8, 9 , 10 , 11 , 12]

will b e resolved as emphasized in [3].

We consider the mapping from quantum eld theory of interacting fermions

onto an equivalent theory of interacting b osons. In this pap er such an equiva-

lent b osonic theory to the original massive Thirring mo del is obtained starting

from the formulation of the D -dimensional Thirring mo del (D 2) as a gauge

theory. This is a kind of b osonization. Esp ecially, in this pap er we consider

the large bare fermion mass limit m 1, in (2+1) and (1+1) dimensional

cases.

In a sp ecial case of (2+1)-dimensions, we show that, up to the next-to-

leading order in the inverse bare fermion mass, 1=m, the (2+1)-dimensional

massive Thirring model is equivalent to the Maxwel l-Chern-Simons theory, the

topological ly massive U (1) gauge theory. This equivalence in three dimensions

has b een shown, to the lowest order in 1=m,byFradkin and Schap osnik [14].

However we think that their treatment is unsatisfactory in the following p oints.

1. The original Thirring mo del has no gauge symmetry. Nevertheless the

equivalent b osonic theory, the Maxwell-Chern-Simons theory, has U (1)

gauge symmetry. Where do es this gauge symmetry come from?

2. The master Lagrangian (which is called the interp olating Lagrangian in

[14]) of Deser and Jackiw [15] was suddenly intro duced without notice

as a device which is needed to show this equivalence. However the origin

of the master Lagrangian was never shown.

3. The master Lagrangian is invariant under the indep endent gauge trans-

formations for the auxiliary vector eld A and another vector eld. In

integrating out another vector eld in the master Lagrangian, they had

to intro duce the gauge- xing term ad ho c.

We resolve these problems by starting from the gauge-invariant or more pre-

cisely the BRS-invariant formulation of the Thirring mo del. Our approachis

more direct than theirs and is able to derive the equivalent Lagrangian to the

interp olating Lagrangian as a natural consequence (in an intermediate step) 2

of b osonization and determines de nitely the pro cedure which is needed to x

the gauge invariance app earing in the master Lagrangian [15 ]. Our metho d

may b e easily extendable to the non-Ab elian case [15, 16 ].

Finally we discuss the (1+1)-dimensional case. The Thirring mo del in

(1+1)-dimensions is rewritten into the equivalent scalar eld theory [1, 2]. It

is well known that the (1+1)-dimensional massless Thirring mo del is exactly

solvable in the sense that the mo del is equivalent to the massless free scalar

theory [1]. In this pap er we show, as a sp ecial case of the ab ove formalism,

the massive Thirring model in (1+1)-dimensions is equivalent to the massive

freescalar eld theory in (1+1)-dimensions, up to the next-to-lowest order in

1=m.

In the previous pap er [4] wehave studied the sp ontaneous breakdown of

the chiral symmetry in the massless Thirring mo del, m =0. In this pap er

we consider another extreme limit m !1. According to the universality

hyp othesis, the critical b ehavior of the mo del will b e characterized by a small

numb er of parameters app earing in the original Lagrangian of the mo del such

as symmetry, range of interaction and dimensionality. Therefore, in the large

bare fermion mass limit, the critical b ehavior of the Thirring mo del will b e

characterized by studying the equivalent b osonic theory according to the ab ove

b osonization.

2 Bosonization

The Lagrangian of the D -dimensional Thirring mo del (D 2) is given by

j j j j j j k k

L = i @ m ( )( ); (1)

where is a Dirac spinor and the indices j; k are summed over from 1 to N ,

and the gamma matrices ( =0; :::; D 1) are de ned so as to satisfy the

Cli ord algebra, f ; g =2g 1=2diag (1; 1; :::; 1).

By intro ducing an auxiliary vector eld A , this Lagrangian is equivalently

rewritten as

g M

j j j j 2

L = i (@ i A ) m + A ; (2)

where wehaveintro duced a parameter M (= 1) with the dimension of mass,

dim[m]= dim[M ] = 1 and put G = . Thanks to the parameter M , all the

elds have the corresp onding canonical dimensions: dim[ ]=dim[ ]=(D

1)=2; dim[A ]= (D2)=2 and then the coupling constant has the dimension:

dim[g ]=(4D)=2; dim[G]=2D.

1 j j

This Lagrangian has O (N ) symmetry.Ifwe adopt the mass term m and put

j =1

m = m for j =1; :::; N k and m = m; :::; m for j = N k +1; :::; N , the Lagrangian

j j

has O (N k ) O (k ) symmetry. See the Vafa-Witten argument [17]. 3

The theory with this Lagrangian is identical with that of the massivevector

eld with which the fermion couples minimally, since a kinetic term for A is

generated through the radiative correction although it is absent originally.As

is known from the study of massivevector b oson theory [7], the Thirring mo del

with the Lagrangian (2) is cast into the form whichisinvariant under the

Becchi-Rouet-Stora (BRS) transformation byintro ducing an additional eld

. The eld is called the Stuc kelb erg eld and identi ed with the Batalin-

Fradkin (BF) eld [6] in the general formalism for the constrained system [5].

Then we start from the Lagrangian with covariant gauge- xing [4]:

g M

j j j j 1 2

00 p

L = i (@ i A ) m + (A NM @ )

A @ B + B + i@ C@ C: (3)

Actually this Lagrangian is invariant under the BRS transformation:

A (x) = @ C (x);

B (x) = 0;

C (x) = 0;

C (x) = iB (x);

(x) = C (x);

j j

(x) = C (x) (x); (4)

where C (x) and C (x) are ghost elds, and B (x) is the Nakanishi-Lautrap

Lagrange multiplier eld.

By intro ducing the scalar eld

i =V

' = N e ; (5)

G, this theory can also b e regarded with the Higgs mo del with V = M=g =1=

(or the gauged non-linear sigma mo del with a lo cal constraint: '(x)' (x)=

) with fermions:

j j j j y

L = i D [A] m +(D [A]') (D [A]')

A @ B + B ; (6)

where D [A] is the covariant derivative: D [A]= @ i A :

First we consider the case of D 3. The two-dimensional case is discussed

separately in the nal part. By intro ducing an auxiliary vector eld f , the

NM @ , "mass term" of the gauge eld is linearized: For K =

Z Z

D 2 2

D exp i d x M (A K )

2 4

Z Z Z

= D D f exp i d x f f + Mf (A K )

Z Z

; (7) f f + Mf A = D f (@ f ) exp i d x

where in the last step wehaveintegrated out the scalar mo de .

Applying the Ho dge decomp osition [18 ] to the 1-form f , f is written as

:::

2 3

f = @ + @ H ; (8)

:::

1 1 1

where wehaveintro duced the antisymmetric tensor eld H of rank

:::

D 2

D 2, which satis es the Bianchi identity. Since f is divergence free, @ f =0,

:::

2 3

we can put f = @ H : Then wehave

:::

1 1

Z Z

2 2 D

M (A K ) Dexp i d x

Z Z

n h

= i D H exp d x

:::

D 2

(1)

::: :::

1 1 2

D 1 D

~ ~

H H + M A @ H ; (9)

::: :::

1 1 2 3

D 1 D

2(D 1)

where wehave de ned

::: ::: :::

1 2 1 3

D 1 D 1 D 1

H = @ H @ H + :::

1 2

D :::

D 2

+(1) @ H : (10)

D 1

This result is a generalization of [31] for D = 3 and coincides with that of Ito

et al. [3].

Then we obtain the equivalent Lagrangian with the mixed term b etween

A and H :

:::

D 2

j j j j

000

L = i D m

(1)

:::

D 1

~ ~

+ H H

:::

D 1

2(D 1)

::: 2

+M A @ H A @ B + B : (11)

:::

1 2 3

Integrating out the fermion eld ; ,wethus obtain the b osonised action

of the Thirring mo del:

S = N ln det[i D [A]+ m]

Let ! be a p-form. Then there is a (p + 1)-form ,a(p1)-form and a harmonic

p-form h (i.e., ob eying h =0=dh) such that

! = + d + h:

We can restrict ourselves to the top ologically trivial space for which there are no harmonic

forms. This is equivalenttosaying that each p-form ! ob eying d! = 0 is of the form ! = d

(Poincare's lemma) and wesay has trivial (co)homology.From nowonwe assume that

the harmonic form is absent: h =0. 5

(1)

D :::

D 1

~ ~

+ d x H H

:::

D 1

2(D 1)

::: 2

+M B : (12) A @ H A @ B +

:::

1 2 3

To see the origin of the eld H ,weintegrate out the gauge eld.

:::

D 2

Then, we obtain the partition function:

Z Z Z Z

Z = D B D D D H

:::

D 2

j j :::

M ( @ H @ B )

:::

2 3

n h

D j j j j

exp i d x i (@ ) m

(1)

::: 2

D 1

~ ~

H H B : (13) + +

:::

D 1

2(D 1) 2

This implies that the dual eld H is a comp osite of the fermion and

:::

D 2

antifermion.

The corresp ondence b etween the original Thirring mo del and the b osonized

theory is generalized to the correlation function. Weintro duce the source b

for the current

j j

J = : (14)

Adding the source term J b to the original Lagrangian Eq. (1), L [b ]

L + J b ,we obtain

g M

j 2 j j j

L [b ] = i (@ i (A ) : (15) A ib ) m +

After intro ducing the BF eld and shifting A ! A b ,we obtain

M g N

j j j j 2

A ) m + L [b ] = i (@ i (A b K )

2 g

A @ B + B + i@ C@ C: (16)

Rep eating the same steps as b efore, we arriveat

:::

000 000

L [b ]= L + M b @ H : (17)

Th Th :::

1 2 3

This leads to the equivalence of the partition function in the presence of the

source b : Z [b ]= Z [b ]. Therefore the connected correlation func-

Th B osonised

tion has the following corresp ondence b etween the Thirring mo del and the

b osonised theory with the action S :

hJ ; :::; J i = h ; :::; i ; (18)

Th B osonised

n n

1 1 6

where

:::

1 1

= @ H : (19)

:::

2 3

G=N

In the following we discuss howtointegrate out the auxiliary eld A to obtain

the b osonic theory which is written in terms of the eld H only.

:::

D 2

3 (2+1)-dimensional case

In the three-dimensional case, D =3,

S = Nln det[i D [A]+ m]

h i

3 2

~ ~

+ d x H H + M A @ H A @ B + B ; (20)

4 2

where

H = @ H @ H : (21)

In what follows we consider the large fermion mass limit, m !1. The

Matthews-Salam determinant in Eq. (20) is calculated with the aid of various

regularization metho ds [19, 20 , 21, 22 , 23 , 24, 25 , 26 , 27, 28 , 29 ]. Appropriate

choice of regularization leads to the regulator indep endent result.

For the 2 2 gamma matrices corresp onding to the two-comp onent fermion,

N ln det[i D [A]+ m] N ln det[i @ + m]

" #

= NT r ln 1+(i @ + m) A

Z Z

2 2 2

ig g @

3 3

= sg n(m) ;(22) d x F A d xF F + O

16 24 jmj m

This should b e compared with the massless limit, m = 0. In this case, it is shown

[20, 30] that up to one-lo op

i g

2 3

A )+ m]= sg n(m) g d x F A + NI [A ]; N ln det [i (@ i

where the parity-conserving term is given by

3=2

1 3 g

3 2

: I [A ]= ( ) d x F

2 N

There are various regularization metho ds: 1) Pauli-Villars [19,20, 21, 22], 2) lattice

[23, 24], 3) analytic [25], 4) dimensional [26, 27], 5) Zavialov [28], 6) parity-invariantPauli-

Villars (variantofchiral gauge invariantPauli-Villars byFrolov and Slavnov) [29], 7) high

covariant derivative [33]. However it should b e remarked that the metho ds 1) and 2) give

regulator dep endent result for the Chern-Simons part. 7

where sg n(m) denotes signature of the bare fermion mass m, sg n(m)= m=jmj.

This is understo o d as follows. Note that

# "

1 g

1 D

= NT r ln 1+(i @ + m) A d xA (x) (@ )A (x)+::::(23)

For D = 3, the vacuum p olarization tensor is given by

@ @

2 2

(@ ;m)+ i @ (@ ; m); (24) (@ )= g

T O

where

g (1 )

2 2

(k ; m)= k d ; (25)

2 2 1=2

2 [m (1 )k ]

and

g m 1

: (26) (k ; m)= d

2 2 1=2

4 [m (1 )k ]

In the large m limit, we obtain Eq. (22).

Thus the Thirring mo del in the large m limit is equivalent to the b osonized

theory with the interp olating Lagrangian:

1 i M

~ ~ ~

L [A ;H ] = H A + F A H H +

4 2 4

2 2

g 1 @

; (27) F F (@ A ) + O

24 jmj 2 m

where

= sg n(m) : (28)

Integrating out the H eld in the interp olating Lagrangian, we obtain the

self-dual Lagrangian:

M i

L [A ] = A A + F A ; (29)

2 4

up to the lowest order of 1=m. This implies that the Thirring mo del is equiv-

alent to the self-dual mo del with the Lagrangian L in the lowest order of

1=m.

P 2 P 2

N N

g g

j j 5

, = sg n(m ) For the mass term m =(N 2k) : Hence, if we

CS j j

j=1 j =1

4 4

take k = N=2 in the rst fo otnote, the theory has O (N=2) O (N=2) symmetry and =0,

although the kinetic term for the eld A is unchanged.

We notice that the interp olating Lagrangian wehave just obtained is es-

sentially equivalent to the master Lagrangian of Deser and Jackiw [15]. The

master Lagrangian for A and H = @ H = H is given by

1 1

L = A A A H + mH~ H : (30)

2 2

Note that the role of the auxiliary eld A and the new eld H is interchanged

in our interp olating Lagrangian compared with that in [14] based on the master

Lagrangian of Deser and Jackiw. Hence the integration over the auxiliary eld

is non-trivial in our interp olating Lagrangian.

On the other hand, for the 4 4 gamma matrices corresp onding to the four-

comp onent fermion, the Chern-Simons term in the determinant disapp ear,

since tr ( )=0. In this case, the Thirring mo del is equivalent to the

b osonised theory with the Lagrangian:

1 g

~ ~

L = H H + M A @ H F F

4 24 jmj

1 @

: (31) (@ A ) + O

2 m

Nowwe pro ceed to the step of integrating out the gauge eld A . By

de ning

J (x) = M @ H (x)= H (x); (32)

and

" #

1 g

K (x; y ) = i @ + @ @ + (@ g @ @ ) (x y );(33)

6 jmj

the interp olating action is written in the form:

~ ~

S = d x H H + A (x)J (x)

Z Z

@ 1

3 3

A (x)K (x; y )A (y )+O : (34) + d x d y

2 m

The A integration in the interp olating action is p erformed by using

Z Z Z Z

3 3 3

D A exp i d x d y A (x)K (x; y )A (y )+ i d xA (x)J (x)

Z Z

3 3 1

= constant exp i d x d y J (x)K (x; y )J (y ) ; (35)

where the constant is indep endent of the eld variable. Here the inverse is

obtained as

1 (1) (2)

K (x; y ) = @ (x; y )+ @ @ (x; y )

1 g

(1)

g @ @ (x; y ) + O ; (36) +

6 jmj m CS

(1) 2 (2) 4 2 (1)

where =1=@ and =1=@ in the sense @ (x; y )=(xy) and

2 2 (2)

(@ ) (x; y )=(xy). By using this formula, A integration is p erformed:

2 2 2

g M M 1

1 2

H @ H + ; (37) J K J = H + O

i 12 jmj m

which is indep endent of the gauge- xing parameter in the original theory.

Thus we arrive at an e ective b osonised Lagrangian for the dual eld H

alone:

2 2

1 g M 2i

~ ~

L = 1+ H H + sg n(m) M H @ H

MCS

4 6 jmj g

+O : (38)

Note that the gauge-parameter dep endence has dropp ed out in the b osonized

theory. In the interp olating Lagrangian L the gauge degree of freedom for

the A eld is xed by the gauge- xing term (@ A ) .However there is an

additional gauge symmetry for the new eld H : the Lagrangian L is invariant

under the gauge trasformation H = @ ! indep endently of A , which leads

to f = 0 in the master Lagrangian. Therefore wemust add a gauge- xing

term for the H eld to the b osonized Lagrangian L .

MCS

Thus, to the leading order of 1=m expansion the Thirring mo del partition

function coincides with that of the Maxwell-Chern-Simons theory. This result

agrees with that in [14 ] obtained up to the leading of 1=m where the less direct

pro cedure is adopted to show this equivalence using the self-dual action byway

of the interp olating action. Up to the next-to-leading order of 1=m,wehave

shown that the equivalence b etween the low energy sector of a theory of 3-

dimensional fermions interacting via a current-current term and gauge b osons

with Maxwell-Chern-Simons term is preserved. The Thirring spin-1/2 fermion

with the Thirring coupling g =N is equal to a spin-1 massive excitation with

2 2

g M

2 2

mass =g 1 + O(1=m ); in 2+1 dimensions. In 2+1 dimensions

6 jmj

there is the following corresp ondence b etween the original Thirring mo del and

the b osonized theory:

j j

$ @ H : (39)

G=N

Esp ecially, for D = 3, the relation Eq (9) shows that the London action

(without a kinetic term F ) for sup erconductivity:

M 1

2 1 2

(@ MA ) = (A M @ ) ; (40) L =

2 2

is equivalent to the mixed Chern-Simons term:

L = H H M A @ H : (41)

4 10

This fact was already p ointed out in [31]. The missing kinetic term for A

is generated through the radiative correction as shown ab ove. The Meissner

e ect in sup erconductivity is nothing but the Higgs phenomenon, the photon

(massless gauge eld) b ecomes massive gauge b oson by absorbing the massless

Nambu-Goldstone b oson (scalar mo de). The mixed Chern-Simons action do es

not break the parity in sharp contrast to the ordinary Chern-Simons term.

Therefore this mo del may b e a candidate for the high-T sup erconductivity

without parity violation, as suggested in [31 ].

Finally we wish to p oint out that, in the large m limit, the Thirring mo del

is also equivalent to the Chern-Simons-Higgs mo del up to the leading order of

1=m and to the Maxwell-Chern-Simons-Higgs mo del up to the next-to-leading

order of 1=m, since

y 3

L = (D ') (D ')+sg n(m) d x F A

CSH

2 2

g @

; (42) d xF F + O

24 jmj m

apart from the gauge- xing term. This result is consistent with the assertion

of Deser and Yang [32].

4 (1+1)-dimensional case

For D = 2, it is easy to show that the b osonized action is given by

S = N ln det[i D [A]+ m]

" #

2 2 2

+ d x (@ H) + M A @ H A @ B + B : (43)

2 2

The determinant is calculated as

det[i D [A]+ m]

N ln

det[i @ + m]

1 @ @

2 2

= d xA (x) g (@ ; m)A (x); (44)

2 @

where

g (1 )

2 2

(k ; m)= k d : (45)

2 2

m (1 )k

In the massless case, m =0,

(k ;m=0)= ; (46)

and hence

g 1 @ @

K = + g @ @ ; (47)

whose inverse is given by

@ @ @ @

g + K = : (48)

2 2 4

g @ @

Putting

J = M @ H; (49)

and integrating out the A eld, we obtain the b osonised theory:

2 2

S = d x 1+ (@ H) : (50)

2 G

The massless Thirring mo del in two dimensions is equivalent to the massless

scalar eld theory, as long as the action is b ounded from b elow, i.e., G :=

2 2

g =M > 0orG<.For more details on the massless case, see reference

[3].

On the other hand, in the large m limit, we nd

" #

2 4 2 4

g 1 k 1 k k

(k ; m)= + + O ( ); (51)

2 4 4

6m 30 m m

and hence

! !

2 2 2 2

g 1 @ @ 1 @ @ @

K = 1 g + ): (52) @ @ + O (

2 2 2 2

6 m 5 m @ m

The inverse is obtained as

! !

2 2 2

6 m 1 @ @ @ @ @ @

K = 1+ g + + O ( ): (53)

2 2 2 2 4 2

g @ 5m @ @ m

Hence we obtain the b osonized theory after integrating out the A eld:

" #

2 4

1 6 3m @

2 2 2

S = d x 1+ (@ H) H ): (54) + O(

2 5G G m

Thus in two dimensions the massive Thirring mo del with large mass m 1

is equivalent to the scalar eld theory with large mass:

6m 6

m = 1+ : (55)

G 5G

In the massive limit, the Thirring mo del is physically sensible for G>0. 12

In two dimensions there is a corresp ondence b etween the Thirring mo del

and the b osonised theory:

j j

$ @ H: (56)

G=N

The ab ove results are reasonable as shown in the following. The massive

Thirring mo del is not exactly soluble even in (1+1)-dimensions. However the

(1+1)-dimensional massive Thirring mo del is equivalent to the sine-Gordon

mo del [2]:

L = @ '@ ' + [cos( ') 1]; (57)

if the following identi cations are made b etween the two theories:

4 G

= ; (58) 1+

= @ '; (59)

cos( '); (60) m =

where a constant in the Lagrangian L is adjusted so that the minimum of

the energy density is zero.

The massless limit m ! 0 of the massive Thirring mo del corresp onds to

the limit ! 0 in the sine-Gordon mo del, i.e., a massless scalar eld theory

in agreement with the ab ove result. On the other hand, the massive limit

m !1 corresp onds to the limit ! 0 (or !1) which inevitably leads to

the limit G !1 as . Hence in this limit the Lagrangian reduces to

2 2 4

L = @ '@ ' ' + O ( ' ): (61)

2 2

Moreover the Eq. (59) recovers the ab ove corresp ondence relation Eq. (56) for

N = 1. Therefore the very massive limit m 1 of the 2-dimensional Thirring

which mo del is equivalent to the massive free scalar eld theory with mass

is identi ed with m given ab ove.

Acknowledgments:

The author would like to thank Atsushi Nakamura, Tadahiko Kimura,

Kenji Ikegami, Toru Ebihara and Koichi Yamawaki for valuable discussions. 13

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