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Canonical quantization of the CPN model with a θ-term
Volume 224, number 3
PHYSICS LETTERS B
29 June 1989
CANONICAL QUANTIZATION OF THE CP N MODEL WITH A 0-TERM A. K O V N E R School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv, Israel Received 30 March 1989
We quantize canonically the (2 + 1)-dimensional CP~' model with a 0-term. It is found that the rotational anomaly is present for any N. As a consequence, solitonic excitations acquire spin (0/2n+integer). At 0= n the lowest excitation is twice degenerate for N= 1 and nondegenerate for N> 1.
During the last year we have been witnessing a renewed interest in the (2 + 1 )-dimensional O (3) sigma-model and in related models. The push was given by Polyakov's rediscovery [ 1 ] of the fact (initially noted by Wilczek and Zee [2 ] ) that when the H o p f invariant is added to the action of the model, solitonic solutions behave q u a n t u m mechanically as particles with fractional spin and statistics. Since the model describes the continuum limit of the Heisenberg antiferromagnet, and antiferromagnetism is characteristic for the recently discovered high Tc superconductors, it is speculated that this statistics changing effect might be important in understanding the nature of high Tc superconductivity [3 ]. It was also recently suggested that similar phenomena may be connected with the fractional quantum Hall effect [ 4 ]. On the purely theoretical side the interest is enhanced by the hope that a better understanding of the effect may give a clue for the generalization of BoseFermi mapping (bosonization) to more than two spacetime dimensions. Therefore, it would be interesting to find more examples of models which exhibit the same phenomenon. In this letter, we consider the family o f C P u nonlinear sigma models in 2 + 1 dimensions governed by the lagrangian 2 . . L = ~f ( OuzO~z*- zc~ OuZ,~Zp~uz ~)
0 ~,~Z(z, a,,z, -c.c.) (a~z~0zz~).
8zcz
z,~ is an ( N + 1 )-component complex scalar field constrained by
Cl = z ' z -
1=0.
(2)
The lagrangian is invariant (up to a total derivative) under the U ( 1 ) gauge transformation
z(x)~exp[ia(x)
] z(x).
For N = 1 this is just the O (3) sigma-model in the CP ~ notation, and the 0-term is the H o p f invariant [ 5 ]. In this case the 0-term can be locally represented as a total derivative (the corresponding form on the phase space is closed) and it does not influence classical equations o f motion. However, q u a n t u m mechanically it changes the spectrum of the solitonic excitations and affects their spin. For N > 1 the 0-term is no longer a total derivative, so it has influence already on the classical level. We shall quantize canonically the theory in order to see whether the statistics changing effect survives for any N. The canonical quantization of the O (3) model was performed in ref. [6], and for the, related, CW theory with a C h e r n Simmons term in ref. [ 7 ]. In both cases it was found that this effect exhibits itself via an anomalous term in the angular m o m e n t u m operator. The conjugate m o m e n t a are defined as 0L 0 H,~= ~)z,~ - 8~rz EiJ(z*Oiz*OJz+2z*O'zOJz*)"
( 1)
0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
(3)
(4)
The conservation in time of the constraint (2) and 299
Volume 224, number 3
PHYSICSLETTERSB
the gauge invariance (3) lead to two more constraints on the phase space variables:
29 June 1989
[ [//~ ( x ) , / 7 ~ ( y ) ] = - i
~lT~(x)D,z~(y)~)~'G(x-y)
\
C2=Hz+ z*IP=O,
(5)
+ ~0 (½zp(y) [D,z,~(x)]*c'J~V#(x-y)
C3= Hz- z*II*- 405 ~iJOiz*ajz=O.
(6)
+ {alz/j(y) [Diz~ (x) ] * - z~(y)Al(y)a,z* (x) }
One must also supply a gauge fixing condition in order to eliminate all the unphysical degrees of freedom. We choose to work in the Coulomb gauge: C4 =z*OiO,z-c.c.=0.
(7)
x a ; ~ ' G ( x - y ) ) ) - [ (~, x)--, (~, y) ]*, ( 10 cont'd) where Diz~(x) = aiz~ (x)
+iA,(x)z~(x),
Now we have a theory with four constraints of the second kind, (2), ( 5 ) - (7). The standard prescription to quantize such a system is [ 8 ]
At(x) = ½i(z*a,z-c.c. ),
{Ct, b}D--~--i[A, B],
G(x-y)
(8)
where a and b denote classical variables, A and B the corresponding quantum operators, square brackets denote the quantum commutator and curly brackets the classical Dirac brackets. The Dirac brackets are expressed via canonical Poisson brackets as follows: {A, B}D ={A, B)p -{A, Ci}pC;f I {Cj, g}p,
C,j = {C,,
C,}P,
(9)
where C~ are all the constraints of the second kind. Applying this prescription to our model, we find the following nonvanishing commutators:
[c~p-z~,(x)z*~(y) ] O ( x - y ) -z,~(x) [D~z/~(y) ]*~-~G(x-y)}, [z~ ( x ) , / 7 ~ ( y ) ] =iz~(x)D~zp(y)6~'G(x-y),
1
= ~ l n ( x - y ) 2.
( 11 )
The angular momentum operator is calculated from the lagrangian ( 1 ):
j=(ij
J d2xxiToj,
(12)
where Toj are components of the energy-momentum tensor. Since the 0-term does not depend on the metric, it does not contribute to T~. We therefore find
Toj-- ~2
[ (Doz~)*aiz~ +h.c. ] =/70~z+h.c.,
j=¢0j d2xxi(HOjz+h.c.).
(13)
[z~ ( x ) , / / p ( y ) ] =i{
[/7~(x),//~(y) ] = i +/-/~ (x) [D~z~(y) 0 + 4z~2 [~z~(x)
There does not seem to be any anomalous term in the angular momentum operator. However, since the algebra of Ha and z~ is not the usual Heisenberg algebra, we have to calculate the commutator of J with the fields to know what transformation it induces. After some algebra we find
(ll=z*p~(x-y) ]*O)'G(x-y)
[J,z~(x)] =i((Jx, ajz,~+ i~-~z,~(x)f d2x e~J~,Aj)
[D,z~(y)]*¢°~fi(x-y)
+ { O~z~(x) [Dlzp(y)]*+z*(x)Ai(x)~tz*~(y)}
×O)'-O~'G(x-y)]) -
300
[ (a, x ) ~ (//, y) ],
=i (10)
(JxiOjz~(x)+ ---o-Qz~(x) ,
(14)
where (2 is the global U ( 1 ) charge Q=ifd2x (Hz - h . c . ) , and the last equality follows by eq. (6). So the anomalous contribution is present in J although it is not as apparent as in the 0 ( 3 ) formulation of the CP 1model [ 6 ]. It is also clear in this formulation
Volume 224, number 3
PHYSICSLETTERSB
that the anomaly is the same for any N. We should now like to consider more closely the one-soliton sector of the model for two main reasons. First, we want to see how this anomaly exhibits itself on the soliton excitations. Second, if one wanted to fermionize this theory (that is to write down the local fermionic lagrangian having the same spectrum, if it exists), the fermion operators would presumably be the ones that create the lowest one-soliton states. So it is interesting to know the degeneracy of the lowest soliton states in order to see what number of fermion operators must be involved. We shall perform the semiclassical zero mode quantization of the soliton sector. We start with the static solution to the classical equations of motion:
2
4(2-,u) nc+c*nt=0.
Because of (21) the second term in (20) can be dropped. We next rewrite the hamiltonian in terms of 2N real variables 0, and their conjugate momenta, defined by Ca=Oa+iOa+N,
(15)
Since the 0-term is linear in the time derivatives, it does not change the static solutions. So the function g(r) for any N is the same as for N = 1. It satisfies the boundary conditions g ( 0 ) = 0 , g(oe)=re. This classical solution is degenerate with the set of solutions a = 1, ..., N,
(16)
where U is an N × N unitary matrix. The ZN+ ~cannot be changed since that would change the boundary conditions on the gauge invariant quantities. For a zero mode quantization, we take U,a(t) time dependent and promote its components to quantum variables. After some algebra we find the effective lagrangian: L = 2 dct dc -dt - - - dt -
P ctdC 2 dt
-i
O
dc
Ct--dt,
(17)
1 S~bSat,+ /¢=~2 42(2-/t) a
47~ f
u= 7 ]
cos4(~gr) dr.
(18)
ctc= 1.
(23)
Sab=Pa~b --PhO~ , S=i(nc-cbrt)=
N
~ S a'a+N.
(24)
a=l
The angular momentum operator in this sector takes form J=-S+
0 ~.
(25)
The operator S, as follows from its definition, is quantized and therefore the angular momentum has the spectrum (0/2re+integer). This is the manifestation of the rotational anomaly discussed above. This property is true for any N. There is, however, a certain difference between N = 1 and N > 1. For N = 1 the hamiltonian (23) becomes
(S_O) 2.
(26)
For the value 0=z~ when the solitons become fermions, the ground state is twice degenerate: S = 0 , 1.
The variables are
c,=U,N,
/z 2re]
where S a~'are the generators of the SO(2N) group:
H= U 4 (~a--- S cos2(½gr) dr,
(22) 2
1
where
I
7~a=½(Pa--ipa+u),
(4n)2/z '
Z°+1 = e ~° sin [ ½g(r) ].
z,=U,~pz°~,
(21)
02
Z°u=COS[tg(r)],
2= -7 J
1
H = ~1 ~r,~(~,~ctc_c,~cta)~z,p + -4 ( 2-- / 2 ) (~zc+ctrct) 2
a
z°=0,
29 June 1989
(19)
Performing a Legendre transformation, we obtain the hamiltonian and an additional constraint:
Indeed the whole spectrum is twice degenerate in this case. This is not so for N > 1. The simplest example is N = 2 . The SO(4) group is the direct product SU (2) X SU (2). The generators are 301
Volume 224, number 3 S[ = ~(V,-Ai),
PHYSICS LETTERS B
S~ = ½ ( ~ + A , ) ,
(27)
Vi=
($23 , 831, S12),
Ai =
($41 , $24, S43).
The generators of the S U ( 2 ) symmetry group J , = i 0 t a i c - h . c . ) coincide with S~. The U ( 1 ) generator S = - 2S~. The spectrum (for 0 = ~) is
E= ~2 [ n ( n + l ) + m ( m + l ) ] /t + 42(2-/0
(_2/_
2__2_)2_ 1 2/zJ 4/.t'
(28)
where 2n and 2m are arbitrary positive integers and l = - n , ..., n. Clearly the ground state is a singlet of SU (2)R, m = 0. This is true for any N: the ground state is a singlet of the unbroken S U ( N ) symmetry of the theory. It is easy to see that the ground state is nondegenerate and corresponds to n=0,
(30)
Indeed, it can be checked that the transformation (30) interchanges the two degenerate lowest lying solitonic excitations and is responsible also for the degeneracy o f all the levels. In the case N > 1 the situation is entirely different. Since the 0-term is not a closed form any more, it is not confined to taking integer values. Therefore, the theory is not parity invariant for any nonzero value of 0. To conclude we summarize our results. The canonical quantization of the model reveals that the rotational anomaly is present for any N. The lowest lying soliton excitation has a spin 0/2~r. For 0 = 7r there is a double degeneracy in the soliton spectrum for N = 1 connected to parity invariance. For N > 1 the solitons are nondegenerate for any value of 0.
/=0.
So we conclude that for N = 1 the ground state is twice degenerate while for N > 1 it is nondegenerate at 0 = zc. This difference can be traced back to the symmetries of the lagrangian (1). As mentioned earlier, the 0term for N = 1 is just the H o p f invariant for maps $ 3 ~ S 2 and takes only integer values. Due to this at 0 = ~r the theory possesses an additional discrete parity symmetry
z(x, y) ~z* ( - x , y)a2.
(29)
Under this transformation the Hopfinvariant changes sign, the action is changed by 8 S = 20n and the generating functional Z = fdz exp(iS) is invariant at 0 = 7r. This symmetry is broken by the vacuum configuration z , = (1, 0), but the following symmetry
302
(parity plus isospin rotation) is unbroken:
z(x, y) ~z* ( - x , Y)a3.
where
29 June 1989
References
[ 1] A.M. Polyakov, Mod. Phys. Len. A 3 (1988) 325. [2] F. Wilczek and A. Zee, Phys. Rev. Lett. 51 (1983) 2250. [3] I. Dzialoshinsky, A. Polyakov and P. Wiegmann, Phys. Len. A 127 (1988) 112; P. Wiegmann, Phys. Rev. Lett. 60 (1988) 821. [4 ] V. Kalmeyer and R.B. Laughlin, Phys. Rev. Lett. 59 (1987 ) 2035; R.B. Laughlin, Phys. Rev. Lett. 60 (1989) 2677. [5] Y. Wu and A. Zee, Phys. Lett. B 147 (1984) 325. [6] M. Bowick, D. Karabali and L.C.R. Wijewardhana, Nucl. Phys. B 271 (1986) 417. [7] P.K. Panigrahi, S. Roy and W. Scherer, Phys. Rev. Lett. 61 (1988) 2827. [8] P.A.M. Dirac, Lectures in quantum mechanics, Belfer Graduate School of Science, Yeshiva University (New York, 1964).
PHYSICS LETTERS B
29 June 1989
CANONICAL QUANTIZATION OF THE CP N MODEL WITH A 0-TERM A. K O V N E R School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv, Israel Received 30 March 1989
We quantize canonically the (2 + 1)-dimensional CP~' model with a 0-term. It is found that the rotational anomaly is present for any N. As a consequence, solitonic excitations acquire spin (0/2n+integer). At 0= n the lowest excitation is twice degenerate for N= 1 and nondegenerate for N> 1.
During the last year we have been witnessing a renewed interest in the (2 + 1 )-dimensional O (3) sigma-model and in related models. The push was given by Polyakov's rediscovery [ 1 ] of the fact (initially noted by Wilczek and Zee [2 ] ) that when the H o p f invariant is added to the action of the model, solitonic solutions behave q u a n t u m mechanically as particles with fractional spin and statistics. Since the model describes the continuum limit of the Heisenberg antiferromagnet, and antiferromagnetism is characteristic for the recently discovered high Tc superconductors, it is speculated that this statistics changing effect might be important in understanding the nature of high Tc superconductivity [3 ]. It was also recently suggested that similar phenomena may be connected with the fractional quantum Hall effect [ 4 ]. On the purely theoretical side the interest is enhanced by the hope that a better understanding of the effect may give a clue for the generalization of BoseFermi mapping (bosonization) to more than two spacetime dimensions. Therefore, it would be interesting to find more examples of models which exhibit the same phenomenon. In this letter, we consider the family o f C P u nonlinear sigma models in 2 + 1 dimensions governed by the lagrangian 2 . . L = ~f ( OuzO~z*- zc~ OuZ,~Zp~uz ~)
0 ~,~Z(z, a,,z, -c.c.) (a~z~0zz~).
8zcz
z,~ is an ( N + 1 )-component complex scalar field constrained by
Cl = z ' z -
1=0.
(2)
The lagrangian is invariant (up to a total derivative) under the U ( 1 ) gauge transformation
z(x)~exp[ia(x)
] z(x).
For N = 1 this is just the O (3) sigma-model in the CP ~ notation, and the 0-term is the H o p f invariant [ 5 ]. In this case the 0-term can be locally represented as a total derivative (the corresponding form on the phase space is closed) and it does not influence classical equations o f motion. However, q u a n t u m mechanically it changes the spectrum of the solitonic excitations and affects their spin. For N > 1 the 0-term is no longer a total derivative, so it has influence already on the classical level. We shall quantize canonically the theory in order to see whether the statistics changing effect survives for any N. The canonical quantization of the O (3) model was performed in ref. [6], and for the, related, CW theory with a C h e r n Simmons term in ref. [ 7 ]. In both cases it was found that this effect exhibits itself via an anomalous term in the angular m o m e n t u m operator. The conjugate m o m e n t a are defined as 0L 0 H,~= ~)z,~ - 8~rz EiJ(z*Oiz*OJz+2z*O'zOJz*)"
( 1)
0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
(3)
(4)
The conservation in time of the constraint (2) and 299
Volume 224, number 3
PHYSICSLETTERSB
the gauge invariance (3) lead to two more constraints on the phase space variables:
29 June 1989
[ [//~ ( x ) , / 7 ~ ( y ) ] = - i
~lT~(x)D,z~(y)~)~'G(x-y)
\
C2=Hz+ z*IP=O,
(5)
+ ~0 (½zp(y) [D,z,~(x)]*c'J~V#(x-y)
C3= Hz- z*II*- 405 ~iJOiz*ajz=O.
(6)
+ {alz/j(y) [Diz~ (x) ] * - z~(y)Al(y)a,z* (x) }
One must also supply a gauge fixing condition in order to eliminate all the unphysical degrees of freedom. We choose to work in the Coulomb gauge: C4 =z*OiO,z-c.c.=0.
(7)
x a ; ~ ' G ( x - y ) ) ) - [ (~, x)--, (~, y) ]*, ( 10 cont'd) where Diz~(x) = aiz~ (x)
+iA,(x)z~(x),
Now we have a theory with four constraints of the second kind, (2), ( 5 ) - (7). The standard prescription to quantize such a system is [ 8 ]
At(x) = ½i(z*a,z-c.c. ),
{Ct, b}D--~--i[A, B],
G(x-y)
(8)
where a and b denote classical variables, A and B the corresponding quantum operators, square brackets denote the quantum commutator and curly brackets the classical Dirac brackets. The Dirac brackets are expressed via canonical Poisson brackets as follows: {A, B}D ={A, B)p -{A, Ci}pC;f I {Cj, g}p,
C,j = {C,,
C,}P,
(9)
where C~ are all the constraints of the second kind. Applying this prescription to our model, we find the following nonvanishing commutators:
[c~p-z~,(x)z*~(y) ] O ( x - y ) -z,~(x) [D~z/~(y) ]*~-~G(x-y)}, [z~ ( x ) , / 7 ~ ( y ) ] =iz~(x)D~zp(y)6~'G(x-y),
1
= ~ l n ( x - y ) 2.
( 11 )
The angular momentum operator is calculated from the lagrangian ( 1 ):
j=(ij
J d2xxiToj,
(12)
where Toj are components of the energy-momentum tensor. Since the 0-term does not depend on the metric, it does not contribute to T~. We therefore find
Toj-- ~2
[ (Doz~)*aiz~ +h.c. ] =/70~z+h.c.,
j=¢0j d2xxi(HOjz+h.c.).
(13)
[z~ ( x ) , / / p ( y ) ] =i{
[/7~(x),//~(y) ] = i +/-/~ (x) [D~z~(y) 0 + 4z~2 [~z~(x)
There does not seem to be any anomalous term in the angular momentum operator. However, since the algebra of Ha and z~ is not the usual Heisenberg algebra, we have to calculate the commutator of J with the fields to know what transformation it induces. After some algebra we find
(ll=z*p~(x-y) ]*O)'G(x-y)
[J,z~(x)] =i((Jx, ajz,~+ i~-~z,~(x)f d2x e~J~,Aj)
[D,z~(y)]*¢°~fi(x-y)
+ { O~z~(x) [Dlzp(y)]*+z*(x)Ai(x)~tz*~(y)}
×O)'-O~'G(x-y)]) -
300
[ (a, x ) ~ (//, y) ],
=i (10)
(JxiOjz~(x)+ ---o-Qz~(x) ,
(14)
where (2 is the global U ( 1 ) charge Q=ifd2x (Hz - h . c . ) , and the last equality follows by eq. (6). So the anomalous contribution is present in J although it is not as apparent as in the 0 ( 3 ) formulation of the CP 1model [ 6 ]. It is also clear in this formulation
Volume 224, number 3
PHYSICSLETTERSB
that the anomaly is the same for any N. We should now like to consider more closely the one-soliton sector of the model for two main reasons. First, we want to see how this anomaly exhibits itself on the soliton excitations. Second, if one wanted to fermionize this theory (that is to write down the local fermionic lagrangian having the same spectrum, if it exists), the fermion operators would presumably be the ones that create the lowest one-soliton states. So it is interesting to know the degeneracy of the lowest soliton states in order to see what number of fermion operators must be involved. We shall perform the semiclassical zero mode quantization of the soliton sector. We start with the static solution to the classical equations of motion:
2
4(2-,u) nc+c*nt=0.
Because of (21) the second term in (20) can be dropped. We next rewrite the hamiltonian in terms of 2N real variables 0, and their conjugate momenta, defined by Ca=Oa+iOa+N,
(15)
Since the 0-term is linear in the time derivatives, it does not change the static solutions. So the function g(r) for any N is the same as for N = 1. It satisfies the boundary conditions g ( 0 ) = 0 , g(oe)=re. This classical solution is degenerate with the set of solutions a = 1, ..., N,
(16)
where U is an N × N unitary matrix. The ZN+ ~cannot be changed since that would change the boundary conditions on the gauge invariant quantities. For a zero mode quantization, we take U,a(t) time dependent and promote its components to quantum variables. After some algebra we find the effective lagrangian: L = 2 dct dc -dt - - - dt -
P ctdC 2 dt
-i
O
dc
Ct--dt,
(17)
1 S~bSat,+ /¢=~2 42(2-/t) a
47~ f
u= 7 ]
cos4(~gr) dr.
(18)
ctc= 1.
(23)
Sab=Pa~b --PhO~ , S=i(nc-cbrt)=
N
~ S a'a+N.
(24)
a=l
The angular momentum operator in this sector takes form J=-S+
0 ~.
(25)
The operator S, as follows from its definition, is quantized and therefore the angular momentum has the spectrum (0/2re+integer). This is the manifestation of the rotational anomaly discussed above. This property is true for any N. There is, however, a certain difference between N = 1 and N > 1. For N = 1 the hamiltonian (23) becomes
(S_O) 2.
(26)
For the value 0=z~ when the solitons become fermions, the ground state is twice degenerate: S = 0 , 1.
The variables are
c,=U,N,
/z 2re]
where S a~'are the generators of the SO(2N) group:
H= U 4 (~a--- S cos2(½gr) dr,
(22) 2
1
where
I
7~a=½(Pa--ipa+u),
(4n)2/z '
Z°+1 = e ~° sin [ ½g(r) ].
z,=U,~pz°~,
(21)
02
Z°u=COS[tg(r)],
2= -7 J
1
H = ~1 ~r,~(~,~ctc_c,~cta)~z,p + -4 ( 2-- / 2 ) (~zc+ctrct) 2
a
z°=0,
29 June 1989
(19)
Performing a Legendre transformation, we obtain the hamiltonian and an additional constraint:
Indeed the whole spectrum is twice degenerate in this case. This is not so for N > 1. The simplest example is N = 2 . The SO(4) group is the direct product SU (2) X SU (2). The generators are 301
Volume 224, number 3 S[ = ~(V,-Ai),
PHYSICS LETTERS B
S~ = ½ ( ~ + A , ) ,
(27)
Vi=
($23 , 831, S12),
Ai =
($41 , $24, S43).
The generators of the S U ( 2 ) symmetry group J , = i 0 t a i c - h . c . ) coincide with S~. The U ( 1 ) generator S = - 2S~. The spectrum (for 0 = ~) is
E= ~2 [ n ( n + l ) + m ( m + l ) ] /t + 42(2-/0
(_2/_
2__2_)2_ 1 2/zJ 4/.t'
(28)
where 2n and 2m are arbitrary positive integers and l = - n , ..., n. Clearly the ground state is a singlet of SU (2)R, m = 0. This is true for any N: the ground state is a singlet of the unbroken S U ( N ) symmetry of the theory. It is easy to see that the ground state is nondegenerate and corresponds to n=0,
(30)
Indeed, it can be checked that the transformation (30) interchanges the two degenerate lowest lying solitonic excitations and is responsible also for the degeneracy o f all the levels. In the case N > 1 the situation is entirely different. Since the 0-term is not a closed form any more, it is not confined to taking integer values. Therefore, the theory is not parity invariant for any nonzero value of 0. To conclude we summarize our results. The canonical quantization of the model reveals that the rotational anomaly is present for any N. The lowest lying soliton excitation has a spin 0/2~r. For 0 = 7r there is a double degeneracy in the soliton spectrum for N = 1 connected to parity invariance. For N > 1 the solitons are nondegenerate for any value of 0.
/=0.
So we conclude that for N = 1 the ground state is twice degenerate while for N > 1 it is nondegenerate at 0 = zc. This difference can be traced back to the symmetries of the lagrangian (1). As mentioned earlier, the 0term for N = 1 is just the H o p f invariant for maps $ 3 ~ S 2 and takes only integer values. Due to this at 0 = ~r the theory possesses an additional discrete parity symmetry
z(x, y) ~z* ( - x , y)a2.
(29)
Under this transformation the Hopfinvariant changes sign, the action is changed by 8 S = 20n and the generating functional Z = fdz exp(iS) is invariant at 0 = 7r. This symmetry is broken by the vacuum configuration z , = (1, 0), but the following symmetry
302
(parity plus isospin rotation) is unbroken:
z(x, y) ~z* ( - x , Y)a3.
where
29 June 1989
References
[ 1] A.M. Polyakov, Mod. Phys. Len. A 3 (1988) 325. [2] F. Wilczek and A. Zee, Phys. Rev. Lett. 51 (1983) 2250. [3] I. Dzialoshinsky, A. Polyakov and P. Wiegmann, Phys. Len. A 127 (1988) 112; P. Wiegmann, Phys. Rev. Lett. 60 (1988) 821. [4 ] V. Kalmeyer and R.B. Laughlin, Phys. Rev. Lett. 59 (1987 ) 2035; R.B. Laughlin, Phys. Rev. Lett. 60 (1989) 2677. [5] Y. Wu and A. Zee, Phys. Lett. B 147 (1984) 325. [6] M. Bowick, D. Karabali and L.C.R. Wijewardhana, Nucl. Phys. B 271 (1986) 417. [7] P.K. Panigrahi, S. Roy and W. Scherer, Phys. Rev. Lett. 61 (1988) 2827. [8] P.A.M. Dirac, Lectures in quantum mechanics, Belfer Graduate School of Science, Yeshiva University (New York, 1964).