Nuclear Physics B352 (1991) 897-921. North-Holland. USA. SEL -. E ... Uni~~ersity of California,. Santa Barbara, CA 93106, USA. Received 7 December 1989.
Nuclear Physics B352 (1991) 897-921 North-Holland
SEL -
E Soo-Jong REY* and A. ZEE
Department of Physics and Institute for Theoretical Physics, Uni~~ersity of California, Santa Barbara, CA 93106, USA Received 7 December 1989 (Revised 27 August 1990) We describe a precise self-duality relation between the electric charge and magneüc flux vortex sectors in the (2 + 1)-dimensional 71 N gauge Higgs model with Chern-Simons term. The phase structure of this model is deduced using the self-duality . In particular, we find that the Chern-Simons term suppresses the confinement phase . Compact U(1) pure Chern-Simons theory corresponds to an N ~ x, infinite bare gauge coupling and zero bare mass limit. The self-duality of Chern-Simons theory is used to explain the hierarchy of odd-denominator filling fractions in the fractional quantum Hall effect and to seggest similar hierarchical ground-state structures in theories of the anyon and high-temperature superconductivity . Possible extensions to the nonabelian gauge groups and connection to the two-dimensional conformal field theories are also discussed .
1. Introduction In four-dimensional 71N gauge theory whose continuum limit is related to the compact U(1) gauge theory, one has a beautiful self-duality relation interpolating between the electric charge sector and the magnetic monopole sector (an extensive review from a modern and generalized point of view is lucidly explained in ref. [1]). This is of considerable interest since nonabelian SU( N )gauge theory, according to 't Hooft [2], may be described, in some domain of the gauge coupling constant 1/g 2, by the compact U N -' (1) subgroup of SU(N ). Inclusion of the CP-violating 8-term leads to a new possibility of electric and magnetic charge composite excitations realizing 't Hooft's oblique confinements. Its underlying mathematical structure has been identified with the invariance under the modular transformation SL(2, 71) involving the coupling constants 1/g 2 and 8/2-rr [3-5]. The same self-duality is revealed in the compactifications of strings viewed as two-dimensional field theory . The space-time background is indistinguishable under generalized Kramers-Wannier-type transformations [5, 6]. * Present address: Institute for Fundamental Theory, University of Florida, Gainesville, FL 3L611, USA. 0550-3213/91/$03 .50©1991 - Elsevier Science Publishers B.V. (North-Holland)
S9S
S.-J. Rey, A . Zee / Chers-Simons theory
I~o we have self-duality structure in odd-dimensional field theories? If so, are there any relations between even- and odd-dimensional field theories that admit self-dualities? In this paper, we make some progress towards the answers to these questions . Specifically, we consider the 7L N Higgs model with Chern-Simons term in 2 + 1 dimensions and prove its self-duality between the electric charge and the magnetic flux sectors. The limit N -~ ~ corresponds to the abelian gauge group. The self-duality of pure Chern-Simons theory is deduced from this by considering the strong bare coupling and zero bare mass limit. In particular, we identify the duality transformation of S-generator of the SL(2, 7~) modular group. Nothing was known about the phase structure of the 7 N Higgs model when the Chern-Simons term is included . This is an interesting and important problem by itself and for other applications we mention below . Using the proven self-duality, we deduce the phase structure of this model. The Chern-Simons term makes the confinement phase shrink to a finite domain in the three-dimensional space of coupling constants . ®ur study of this model is not just a theoretical curiosity. We have in mind physical implications of self-duality to various field theories in which the Chern-Simons term is incorporated . In the fractional quantum Hall effect [7, 8], the nature of broken parity and time reversal symmetries plays a crucial role. The mean-field theory includes Chern-Simons term and induces fractional charge and statistics to quasi-particle excitations . The hierarchical structure of the quasiparticle spectrum in this theory resembles strikingly the phase structure deduced from the self-duality of Chern-Simons theory coupled to a conserved current . For example, as Shapere and Wilczek [5] pointed out, filling fraction v transformations in the Laughlin-Jastrow wave function, v~l-v,
v -i -~v - ~+2
are transcribable [5,11] into those of Chers-Simons term coefficient 8/2Tr in the effective lagrangian of statistical gauge fielu a~
e
4-rr'
E~~ .aa~c~~,a~ +J~a~
as S
® -~ 2~ - ~ e
e
-.
A similar hierarchical structure is also expected in some proposed mean-field theories of high-temperature superconductivity [9] and in anyon gas superconductivity [10]. Kalmeyer and Laughlin [11] constructed an approximate mapping of the frustrated Heisenberg antiferromagnet system in two dimensions, which, when
S.-J. Rey, A. Zee / Chern-Simons theory
g99
properly doped, ~s thought to capture the essential physics of high-temperature superconductivity, into tht fractiona', quantum Hall system of bosons on a triangular lattice . The bosonic fractional quantum Hall effect corresponds to the eL~en denominator in the filling f action v. We see that eq. (l .l) also gives rise to a mapping among even denominators and, hence, we expect a similar hierarchical structure of the ground states in the frustrated Heisenberg antiferromagnet upon doping. The present paper is organized as follows. In sect . 2 we recapitulate Cardy's work [4] on the partition function of the 7L N gauge model with 8-term . In particular, we derive the self-duality relation by decomposing the partition function into self-dual and anti-self-dual blocks, which is a direct analog of the structure of partition functions in two-dimensional rational conformal field theories. In sect . 3, we show self-duality of the (2 + 1)-dimensional abelian Higgs model with Chern-Simons term included . By taking a strong coupling limit, we derive the self-duality and the modular transformation for pure Chern-Simons theory . Using this self-duality, we deduce the structure of phase domains in three-dimensional coupling constant space. In sect. 4, we discuss two condensed matter realizations in which the self-duality of Chern-Simons theory is relevant: the fractional quantum Hall effect, and the anyon and high-temperature superconductivity . We stress the importance of self-duality to the underlying phase structure and quasiparticle excitation spectrums. In sect. 5, we summarize the present paper and discuss a possible connection to the c = 1 chiral conformal field theory . 2. Self-duality of 7LN gauge theory with 8-te First, we review the classic results obtained by Cardy [4] on the self-duality of the compact U(1) gauge theory including ®-term in four dimensions. Our approach is somewhat different from Cardy's. We extend some of his results and view them in a new light. We consider a discrete version of the compact U(1) gauge theory, namely ~ N gauge theory on a four-dimensional hypercube lattice with periodic boundary condition . Our objective is to derive a manifestly self-dual form of the partition function under a modular transformation of the complex coupling constant 8 ? .. Tri ~ - 2-rr + Ng2
~
The 71 N gauge theory is defined by restricting A~ to take values 2~rk/N, where k is an integer from 0 to N - 1 (we will denote this restrictiu~i by A~ E (2Tr/N)7~N). The partition function of 71 N gauge theory is written as
v+
1 ~F ~2 ~~ g, el = i exp - 4g2 M, A p ~
iN8
32?T2
~ E~vaßF P®P~`
~Faß
.
.2)
S.-J. Rey, A. Zee / Chern-Simon: theory
900
®ur notation is as follows. The plaquettes and links are denoted as P and L, respectively, and their duals as P* and L* . The normalization of each term in the action follows the Einstein convention of summing over independent tensor indices . The subscript P, P*, etc. in the sum indicate on which lattice (original or dual) each term is defined. E~v«p interpolatThe tensor density E~,°«ß (P - P*) is a lattice version of E ~v«ß = ing between plaquette P and its dual plaquette P* . It is defined by E~v"Rf( R P - R P~ ), where R P and 1~ P* are the center of the plaquette P and the dual plaquette P*, respectively, and f(x) is a sharp function which satisfies E~v«ß for J dx f(x) = 1 . In what follows, we will, for the sake of simplicity, take every E~""ß. Since the plaquette operator of compact gauge rneory is defined by Up expii(®~A v - wA~)], an expansion up to quadratic order gives rise to the first term in eq. (2.2) in which the gauge field is defined by
I-Iere M~~, is an integer-valued describing magnetic monopoles . The magnetic monopole current J~ _ --',EU~v«ß®vM~aR~ is conserved: ~~`J~ = 0. Thus magnetic monopole current loops are closed . The difference operator in the ~u,th direction is denoted by ®~. Fourier transforming the ~~-valued gauge field into an angular variable defined over Au ~ ~ - ~rr, ~r], we get
M,JE71
-~r
p
g
~v
32?r
Lt.v
P®P*
cY
L
( 2 .4)
The electric current J~ is a Fourier transform conjugate variable to the gauge field A~ and must be conserved, ®~J~ = 0, to maintain the gauge invariance. Let us decompose the gauge field configurations into self-dual and anti-self-dual ones by introducing new variables, F~v = 2 [ Ffcv + ? E N.va/3 Fa R ]
~
F~.~ - 2 [ F~.t.v
-
2
E~t.va~i FaR , ~
(2 .5)
9 (2 .6)
[Strictly speaking, there is no notion of self-duality and anti-self-duality on a lattice, since F~ and E~v«ßF"ß are defined on different lattices. A precise definition of our field decomposition is possible with E~~"~ replaced by
S.-J. as
Rey, A. Zee / Chern-Simons theory
901
It is straightforward algebra to show that the partition function can be rewritten
M,A
where the self-dual and anti-self-dual blocks a N~
~ ] = exp + A ~~ 4 2~r P®P*
(2 .7)
~M,A~~~ ~M,ALS J ~ ~M, A [ ~]
( F~.v )
~[rl
and 2LM, A [ ~] are denoted by
-
i N~
exp
4 27r
P®P *
(F~v)2
'
respectively . Here, the complex coupling constant ~ is defined as in eq. (2.1) and F~v are self-dual and anti-self-dual field strengths according to eqs. (2.5) and (2.6). Complex conjugation is denoted by an overbar . We can now rewrite eq. (2.4) into
M,JE71
~A~ ~M,A[~~ ~M,A[S ]eXp IN~J~A~
_~r
L
(2 .9)
We now make a gaussian transformation to the field strength and introduce continuous rank-two conjugate variables ~fl~v E [ - oo, oo] to find ~M, A ~ S ~ -
1
2
i 2 ~r
./ Y'
~
P®P*
o0
i P®P*
and similarly for 2~M.A [i']. The prefactor .~t~° denotes a normalization constant . We obtain the partition function ~°[~, ~] :
_
~ f~~~v
M, JE7L
x exp -
v 1~
l 27T
~A~ exp(iA~[ NJ~ - 2 vv(~~v - ~~,v) - âErwaßw(~ ß + ~~ß)~ )
~
4 ~N P®P*
- 2?Tl
~
l (~~v)2 +
2?T
4 ~N P®P*
~ ( 2 (~,w - ~~, v)
P®P* ~u,
We thus find that the partition function .,~`[~, ~] is self-dual since
is a symmetry of the partition function . Note that we have dropped the nondynamical, overall constant factor .,-1 ~ in the gaussian transformation in eq. (2.10). There is another symmetry of the partition function associated with the 8parameter, It is easier to see this from the expansion of the Last term in eq. (2.2). We have
Thus, upon 8 -~ 8 + 2-rr71, the first term induces the electric current shift J~ -~ J~ +J~ while the second term shifts by an integer multiple of 2~r. In fact, the shift in the second term is cancelled by the induced electric charge-magnetic monopole interaction [ 13]. This should be so since in three spatial dimensions, the spin-statistics theorem allows only fermions or bosons, and all potentially B-dependent pieces must cancel out. In any case, we find that each term is periodic with respect to 8 E [0, 2~]. Of course, this periodicity due to monopoles reflects the periodic 8-vacua structure of the underlying compact gauge theories [ 14]. Therefore we find that there is an additional symmetry given in eq . (2.19b). These two transformations of eq. (2.19) generate the modular group SL(2, 7~) in Siegel's upper half-plane [ 15] of the complex coupling constant ~. Thus the structure of four-dimensional 71 N gauge theory is very similar to what one would have in the one-loop partition function of the conformal field theories (modulo an infinite tower of Hilbert space spectrum for the former) . Indeed, the two theories have many similarities at least on a formal level . Namely, the holomorphic and antiholomorphic decomposition in the partition function of rational conformal field theory was possible due to the complex structure in two dimensions . Its four-dimensional counterpart in gauge theory is precisely the self-dual and antiself-dual decompositions of field strengths as we gave in eqs . (2.5) and (2.6). The modular parameter is now played by the coupling constant, 16~r`
1~®P~~
.
so that it resembles strikingly the structure of the one-loop partition function of
S.-J. Rey, A. Zee / Chern- Situons
904
~hPOry
rational conformai field theory, (2.21) The trace is taken over the Hilbert space ~ of 1 N gauge theory . Putting it in a different way, one may sum over the gauge field A~ in eq. (2.4). The result is 2 2~r
J,JE71
L®L
*( ~.
~)
Im~
'
v
~
( 2 .22)
where denotes the Green function with GIl v(x
- Y)
= SIl z,
x ®2 1 Y
~lt v(x - Y) =
~
x
E~v« ßR
® ß
R«®«®«2
Y
.
(2 .23)
An arbitrary gauge-fixing vector R~` is introduced in eq. (2.23). The partition function written as current-current interaction in eq. (2.22) displays manifestly the SL(2, 7) modular invariance . Finally, for the sake of completeness, we indicate how we can recover the U(1) gauge theory by taking the N ~ ~ limit with g 2N fixed. In eq. (2.2), we let g' = e 2/N and with the Planck constant ~i (set equal to unity in eq. (2 .2)) replaced by ~z = hN- ' . So, in terms of the new coupling constants (setting fi = 1), ~[e, ®] _
Au
exp -
1
4e2
P
Fv +
i8
32Tr2
P®P
*
E ~v«ßF~ vF«R
~
(2 .24)
where A~ E [0, 2~rr] does not involve topologically nontrivial configurations . The gauge transformation A~ ~ À~ + a~ ~ is restricted to the range [0, 2-rr] . Now, the sum over 111 may be included into the A~-definition, to get 4e2
P
~
32Tr2
P®P*
~`
ß
~`
ß
Finally, we get the continuum limit by taking the lattice spacing a to zero. Thus, with ßu 4 --- j d 4x, we have
the abelian gauge theory with ®-term in four-dimensional space-time .
S.-J. Rey, A . Zee / Chern-Simons theory
905
3. Self-duality of 71N iggs model with Chern-Simons te We now study a discrete gauge group version of the abelian Higgs model [16,17] with a C hern-Simons term on a three-dimensional hypercube lattice with periodic boundary condition . This model is known to possess vortices and magnetic monopoles whose masses are inversely proportional to the gauge coupling constant . To describe their excitations, we take a compact gauge potential A~ E (2~r/N)7 N and a compact scalar field ~ E (2~r/N)7L N . Thus the partition function of the 7 N Higgs model including Chern-Simons term is 1 A, M, ~, V
g
m
2
~(A~+~~~-2~rV~)2
2
P
L
+ iNe ~ ~~ vA( A~ + ~~~ - 2 TrV~)( Cv A A - ®aAv - 2~rM ,~) ~ (3 .1) ôTl P®L*
The first term is as in sect. 2 with M~~ taking integer values . The second term comes from the quadratic expansion of cos[~L(A~ + ~~~)] so that V~ takes integer values. The third term is the lattice v,;rsion of the Chern-Simons term. We use an obvious definition of the Chern-Simons term, with the expectation that the long-distance limit of the version in eq. (3.1) approaches the continuum limit in a correct manner. In fact, a more rigorous definition of the lattice Chern-Simons term is already available [18]. As in sect. 2, we use the Einstein summation convention with a suitable normalization for each term in the action. Again P, L, S denote plaquettes, links anû sites of the three-dimensional hypercube lattice, respectively, while ~ signifies the dual lattice . The tensor function E~v~ is expected to approach E~" A In the continuum limit . Again we change the summation over discrete A~ and ~ into integrals by using the Poisson summation formula, ~ .~A~ ~~ exp - 1 2 ~ ( p~Av-pvA~-2?TM~v)2 J 4g p J, IC, M, V -~r +
~Ne
, ~ E~`va(A~+~~~-2~v~,)(v,,A~-p~AV-2TrM A) ôTl` L* ®P
m 2
~ ( Ak +~ ~-2-rrV )z L
+iN~J~A~+iN~K~ . L
S
(3 .2)
Let us first recall that there exist two types of topological excitations in this model: "magnetic monopole" instantons and magnetic flux vortices. In the absence of the Chern-Simons term, the phase structure of the model is well studied
S.-J. Rey, A. Zee / Chern-Si~nons theory
906
[16,17]. The pure compact 71 N gauge theory is known to have a confining phase for all values of the coupling constant . Once the Higgs field is introduced, the confinement phase competes against the Higgs phase and results in the Coulomb phase as well as the Y'~gs phase . An important question we address in the present paper is to understand the modified phase structures once the Chern-Simons term is included . We will see that the modification is a three-dimensional analog of the oblique confinement [2] . The variables M~~, E 7 and V~ E 7 describe magnetic monopole and vortex topological excitations, respectively. Magnetic monopole density and magnetic flux current are p m = ; E~ ~~®~ M,,~ and J~ _ ; E~,,a~ M ~, + ~,V~ - ® V ), respectively . They are related to each other as ~~J~ = p m . The Chern-Simons term induces a mixing of electric charge and magnetic flux by modifying the current couplings of J~ and K to the gauge potential A ~, J~ -~ J~ +
e
e
K -~ K + 2 ~ p m
2-rr J~ '
(3 .3)
.
The sum over gauge field and scalar field as well as the sum over magnetic monopoles and magnetic flux vortices M~,, and V~ are highly redundant . We could fix the gauge, for example, by taking ~ = U. We find it more convenient not to fix this gauge redundancy . This will generate an infinite multiplicative factor to the gauge-fixed partition function, which does not affect our final conclusions. We decompose the gauge field configurations into self-dual and anti self-dualpieces, G~ = A( A~ + ®~~ - 27rV~ ) +
G- = ~t(A r~ a
1 2~
E~~,~(
~,,Aa
- ®~A,, - 27rM,,~ ) ,
1 + ®~ ~ - 27rV~ ) - 2~ E~.~ , ~ ( ®,~ A a - ® A a - 27rM, ) .
Here, A~ = mg. The partition function can be rewritten as
A
v n~
2 47r P®L~
~`
Here, the complex coupling constant ~ is defined as 8
27rî
2 Tr
Ng/na
2 47T P®L
W
(3 .4)
S.-J.
Rey, A . Zee / Chern-Simons theory
907
After converting the gauge field into a continuous variable, we obtain [~~ ~] _
V, M, JET1
, .l -~.~dA~ ~v,A[~] ~v,A[~]exp iN~A~J~ L
where
iN~
2
iN~
_
P®L*
P®LY (3 .6)
We now gaussian transform the partition function to get i 4Tr
x x
~ P®1*
P®L*
and a similar expression for `lw. A[~]. Thus we obtain
~~n~ß~)
-Nr ~
l
2~rri ~ Jl(~l+ - ~- )V + 1 (,~+ The summation over Vim , M~f, requires that ~l~ satisfy ~1(~fl+ -~1 - )
E
7L,
1 (~+ +~ -)
E
71 .
(3 .9)
We now integrate over the gauge field A~ and change to discrete values of ~~ to
S.-J. Rey, A. Zee / Chern-Simons theory
908
get ex - -
- i
4Tr
z
~+
+ i
4Tr
(~- )z
J, ,fl
+ + ®",f2 ) - NJ~ . (3 .10 x ® ~ ~+ - ~l® + 1 E "ß (®",fl ~ ~~ ~ a~ ~ 2~ ( ~ ~)
The I~ronecker delta function constraint is seen to be solved by N 4Tr
_ 1 + 2~
"Bß ® ®~ B" E~"~( ®
® 2TrE"ß ) + J~(B~ + ®~X - 2TrW~) ,
such that 2 E ~t,vAL
Eva + 2Vv ~~j~ ]
.
Since ,~~ should satisfy eq. (3.9), we have B~, X (2-rr/N )71N and E~.,,, W~ 1 . Again the dual gauge field and scalar field are summed over without gauge fixing . VVe thus get ,f~~
N
_ 1
2~rE~~ß ) +1~(B~, + ®~X - 2~rW~) 4~r + 2~ ~"~( ®" B~ ® ®~ B" ® E
n = 2rr~~ (3 .12)
H
~ have the same form as G~ . Note that The partition function now reads i 4 Tr B,W
~ P* ®L
_Tr
z l'* ®L
1
W,JE71
i4~
z
~
_
1
~
L*
By comparing with eq. (3.5) we obtain one of our main results, (3 .14)
VVe thus find that the duality transformation exchanges two complex conjugate coupling constants,
S.-J. Rey, A. Zee / Chern-Simons theory
909
Also there is another approximate symmetry associated with the coefficient ® of the Chern-Simons term,
To see this, we expand the Chern-Simons term in eq. (3 .2) to get ~Ne L* ®P
4~,2
E~.v~A,~® AA +
iN
e
2~r
(A~J~ + pm~)
+ iN 8 (iE~vw~MvA) .
(3 .16)
The first term gives rise to the topological mass for the gauge potential. In a weak coupling and large bare mass limit we can ignore this contribution compared to the first two terms in eq. (3.2). The second term is the interaction of induced electric charge couplings from monopoles and vortices. As discussed in eq. (3.3), we find that ® -~ 8 + 2-rr shifts J~ ~ J~, + J~ and K ~ K + p m . Since we sum over integervalued J~, and K as well as J~, and p m , ® --> ® + 2~r leaves the contribution of the second term to the partition function invariant . The third term is the interaction of integer-valued magnetic flux currents . Again, we find that 8 -> B + 2~r is an invariance of this term . The above two transformations in eq. (3.15) constitute the complete self-duality of the (2 + 1)-dimensional 7~ N Higgs model with ChernSimons term. The transformations (3.15) here differ from the modular transformation SL(2, 71) of four-dimensional TA N gauge theory with A-term we studied in sect . 2. This may be not at all unexpected . There is no generalization of complex structure in odd dimensions and we should not have anticipated that the self-dual and anti-self-dual structures that we started with originally remain preserved (cf. eq. (3 .4)). The mapping S* is just the generator S of SL(2, ~) followed by a time reversal transformation ~ --, -~.
(3 .17)
Thus S *, T together with .W constitute SL(2, 71). To get a pure abelian 71 N Chern-Simons theory, we may take formally the g a, m 2 -> ~ 1 imit of the above 7~ N Higgs model while keeping ~ fixed. In this limit, the gauge field acquires an infinite mass gap, since the contribution of the Chern-Simons term to the gauge boson mass is gN8/2 ~r. Thus, topological gauge field configurations constitute the only nontrivial information left in the theory. Thus the duality transformation of a pure abelian Chern-Simons theory is seen to be S.
e
2~
2 ~r -> ® ,
(3 .18)
S. -J. Rey, A . Zee / Chern -Situons theory
91 U
identified as the S-generator of the modulator group SL(2, 7). This is one of the main results of the present paper . The above self-duality relation (3.18) could have been derived directly from a pure Chern-Simons theory . The reason we have taken a long route with g 2, m - 2 first kept finite is twofold. First, we tried to make clear the origin of the physical spectrum left over in the strong coupling limit, i.e. magnetic flux vortices with induced electric charges. Second, the structure of self-duality of the three-dimensional abelian Higgs model with Chern-Simons term is an interesting topic by itself that we will encounter in sect. 5. Once again, we can go over to a U(1) gauge theory by letting N --~ ~ while holding e 2 = g 2N and M 2 = m2/N fixed. Restoring the lattice spacing, we find that the gauge field quadratic term takes the form 1 4g2
~a4 (F~. v)2 =
a
4g2
~
~a~(F ,,)2 .
Similarly, the Higgs field quadratic term is 2mz
2 L_
(A~ +
D~~)2
_~a
1 m2 2 + a ~a~(A~ 1_
D~.~)2 .
Thus we can take the continuum limit by letting a --~ 0 while keeping g a /a and m2/a fixed. In this limit, ßl 2 = gm scales as a . The meaning of B/2~rr is clear from eq. (3.2). From eq. (3 .3), a unit vortex current J~ induces 8/2Tr units of electric current J~ . In particular, the electric charge and magnetic flux are related by Q
e
= 2~r ~ .
(3 .19)
Thus ®/2~rr measures the ratio of electric charge to the magnetic flux carried by the vortex. This the (2 + 1)-dimensional version of the Witten effect [14). On the lattice the flux is given by
where the summation is over all spatial plaquettes . We have to exercise some care in going over to the continuum limit. As explained earlier, the procedure is to let 2Tr(M;~ + D;V) - a 2F;~. Thus, in the
S.-J. Rey, A . Zee ,l Chern-Situons theory
911
continuum limit, eq. (3.20) correspond to ~- ~
d2x
47r E,JF''
27r ,~
d2xF~z
= 2 ~r ~cont ~
(3 .21)
Therefore, Q = (8 /4~ 2 )~cont~ , Let us recall some established results from the continuum theory . Consider the lagrangian density
where the lagrangian ~~~ admits a conserved number current Jam . Solving the equation of motion for A~, we find that a localized particle with charge Q j d`xJ~~ = 1 (this is unity by definition) carries a flux given by ~c~nt = Q/2a. When two such particles are interchanged, their wave function acquires a phase e'" with
~ - 2Q~cont -
z Q 4C1,
(3 .23)
The statistical angle O is '-, of what one might expect from a naive application of the Aharonov-Bohm effect . This may be understood in terms of a polarization effect which implies that the effective charge Qeft~ _ '-, Q [13]. If we now put together eqs. (3 .19), (3.21) and (3.23), we find 2~r 2 ThusS ~ vortex with unit charge Q = 1 is characterized by a statistics, given by O
2 ~r
(3 .25)
~- e
It is sometimes convenient to rewrite this, using eq. (3.19), as O ~ _
Q
.
(3 .26)
We must keep in mind that eqs. (3.25) and (3 .26) only hold when Q = 1 . According to eq. (3.25), the duality transformation S : 8/2Tr -~ 2Tr/8 in eq. (3.18) corresponds to (for Q = 1) ( 3 .27)
91 2
S.-J. Rey, A . Zee / Chern-Situons theory
Finally, we study the phase structure of the 7 N Higgs model with Chern-Simons term using the self-duality property . The phase diagram is defined in a threedimensional space of the coupling constants g 2, m 2 and 8/2Tr . It is well known that the 7 N Higgs model without Chern-Simons term (two-dimensional subspace of the phase diagram with 8/2~ = 0) has a confinement phase for strong coupling and smaü ûare r~~ass [17] and a Higgs phase for ~:~eak coupling and large bare mass. This is for N < 4. If N is larger than 4, there appears a Coulomb phase for weak coupling and small bare mass. The Higgs phase and the confinement phase are continuously connected if the scalar field carries a unit electric charge (which is the case we are looking at) while they may be separated by a phase boundary for a higher-charge scalar field . Naively, the limit N -~ ~ makes all topological excitations completely suppressed and approaches a noncompact abelian gauge theory, since the Coulomb phase expands while the other phases are shrinking in this limit. However, even a tiny amount of topological excitations is enough to destroy the Coulomb interactions . Thus, the N -~ ~ limit should be taken with care. The duality transformation ß .15a) interchanges the electric charges and the magnetic flux vortices. The self-duality dictates that the phase diagram must be symmetric under the inversion of m2/g 2. Indeed, this is the case, since the reflection line m2 = gz in the g 2 - m 2 phase diagram bisects symmetrically the Coulomb phase and also between the Higgs phase and the confinement phase. We expect that these phase domains extend smoothly to 8/2Tr ~ 0. Thus, ICI = 1, which is a fixed surface ("surface" since it involves all three coupling constants g 2, m2 and ®/2-rr) of the S-mapping in eq. ß.15a), should be the inversion symmetric surface of the phase diagram . As ®/2Tr is increased, we see that the domain of the confinement phase is diminishing, while the Coulomb phase is expanding. Thus, the effect of the Chern-Simons term is to destroy both the electric and magnetic confinements . Since ®/2-rr = 1, g 2 = m 2 = 0 is a fixed point of the duality transformation, we find that the Coulomb phase extends for all values of g 2 at m2 = 0 and 8/2Tr = l . ®n the other hand, the consequences of the approximate periodicity of eq. ß .15b) are less clear. We have argued before that the periodicity is expected to be present in the weak coupling and large bare mass regime . This corresponds to the Higgs phase. Thus, if we take the periodicity ß .15b) seriously, we should find that the Higgs phase repeats itself as we traverse along 8/2-rr. Thus, as in the four-dimensional 71 N gauge theory we discussed in sect . 2, the possible condensations in each phase domain are determined by the value ®/2Tr. If it is a rational number, 8/2T = p/q, where p, q are relative prime, namely the electric charge and the magnetic flux of the condensate take values (Q, ~) _ ( p, q), we have a finite number ( p) of phases in the strong coupling limit characterized by different integer-valued electric charge and magnetic flux quantum numbers of condensate order parameters. We view this as the three-dimensional version of oblique confinement à la 't Hooft [2]. All the other excitations with different
S.-J. (Q,
Rey, A . Zee / Chern-Simons theory
913
~) quantum numbers cannot be Coulomb screened, leading to confinements .
On the other hand, if 8/2 ~rr becomes irrational, we should expect an infinite number of different phases and generic confinement . This picture may have many implications for condensed matter realizations we discuss in sect. 4. Vd z recall that a complete Higgs model has also radial field excitations . Our model in eq. (3 .2) amounts to freezing this radial excitations to zero. Once we include the radial field excitations, the model is no longer self-dual . We do not expect that there exists an inversion-symmetry surface. Indeed, analytic and Monte Carlo simulation studies [ 19] show that there appears a new phase between the Coulomb and the confinement phase once the radial field excitations are taken into account. Due to the absence of the self-duality, it is not easy to deduce the phase structure. This problem of including the Chern-Simons terms deserves a deeper investigation, though, since it has direct relevance to the mean-field theory of the fractional quantum Hall effect [20, 21 ] and to the Hubbard model relevant to high-temperature superconductivity [22, 23] we will discuss in sect. 4. Still, we expect that the exact self-duality of the Higgs model with Chern-Simons term without radial field excitations may provide an important guide in this model too. 4.
ierarchical structures in the condensed matter systems
In this section, we apply the results of sect. 3 to some condensed matter realizations exhibiting hierarchical structure of quasiparticle excitations . There are interesting condensed matter systems that have properties related to the theories we discussed . One is the fractional quantum Hall effect and the other is the recently discovered high-temperature superconductivity. Common features of these system are as follows : (1) To a very good approximation, the physics is 2 + 1 dimensional . (2) These systems are not invariant under parity and time reversal transformations through either explicit or spontaneous breaking . Thus an effective mean-field theory lagrangian describing these would involve the Chern-Simons term, which is odd under time reversal or parity transformations. We first discuss implications of modular invariance to the hierarchical structures of the fractional quantum Hall effect . Some aspects of the following discussion have also been given by Shapere and Wilczek [5], which was a motivation of the present section . An exact ground-state wave function of N electrons in the fractional quantum Hall system with filling fraction v is given by the Laughlin-Jastrow wave function [24] (see also sect. 7 of ref. [9])
