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Spin transmutation in (2 + 1) dimensions
~~
NUCLEAR PHYSICSB[FS]
ELSEVIER
Nuclear Physics B 435 [FS] (1995) 690-702
Spin transmutation in (2 + 1) dimensions Wei Chen a,1, Chigak Itoi b,2 a Department of Physics, University of North Carolina, Chapel Hill, NC 27599-3255, USA b Department of Physics and Atomic Energy Research Institute, College of Science and Technology, Nihon University, Kanda Surugadai, Chiyoda-ku, Tokyo 101, Japan
Received 27 April 1994; revised 15 September 1994; accepted 1 December 1994
Abstract
We study a relativistic anyon model with a spin-j matter field minimally coupled to an abelian Chem-Simons gauge field with a statistical parameter t~. A spin and statistics transmutation is shown in terms of the continuum random walk method. An integer or odd-half-integer part of a can be reabsorbed by change of j. We discuss the equivalence of a large class of Chem-Simons matter models for given j and a.
1. I n t r o d u c t i o n
It is well-known now that in ( 2 + 1 ) space-time dimensions there may exist anyons which have arbitrary spin and statistics [ 1 ]. The statistics of particles is changed if an interaction with a statistical gauge potential governed by the Chern-Simons term is introduced [2]. While this issue is settled for non-relativistic matters, there has been debate as far as the correct formalism of relativistic anyons is concerned. In this latter context, the spin degrees of freedom plays an essential role. In the framework of quantum field theory, a Fermi-Bose transmutation has been observed in a ( 2 + 1 ) dimensional system [3]. Namely that a charged scalar field coupled to a Chern-Simons gauge potential at a specific value of Chern-Simons coefficient (or statistical parameter) turns out to be a free fermion field theory. In this and similar examples, one must deal with both statistics and spin transmutations, as these refer to apparently different objects. A statistics transformation affects the Aharonov-Bohm phase of the wavefunction of ] E-mail: [email protected]. 2 E-mail: [email protected].
0550-3213/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 05 50-321 3(94)00566-4
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two identical particles moving around one to the other on a plane, while any spin transmutation should cause corresponding change in the Lorentz group representation of the matter fields. Though there was investigation on anomalous magnetic momentum in a perturbation theory in the literature [4], a clear understanding for spin transmutation seems still lacking. The problem with spin transmutation is discussed in a recent letter [5]. The present paper is to continue the discussion. We see that in a general Chern-Simons matter field theory model, any integer or odd-half-integer part of the Chern-Simons coefficient can be reabsorbed by changing the character of the Lorentz representation describing the matter. This is an observation through a continuous random walk method with certain assumptions with respect to a regularization. In this formalism, the partition function of an anyon system can be formulated in particle path integrals with some phase factor which consists of a spin factor from the matter field and a self-energy and a relative energy from the Chern-Simons gauge field. And then by the topological relations between the spin factor and the relative energy and self-energy we see the spin and statistics transmutations. In these transmutations, due to the quantization of the Chern-Simons relative energy, only the integer or odd-half-integer part of the coupling constant endows the anyons with a change of spin and a "small" part of it ( < ½), if any, has to remain as a residual Chern-Simons interaction. There can be a nice possible application with the spin and statistics transmutation. As we know, the Chern-Simons coefficient is used quite often as a parameter in which a perturbation expansion is conducted. However, when the Chern-Simons coefficient is required to be large, a perturbation expansion is not reliable. Now, with the spin and statistics transmutation, one can trade a large part of the Chern-Simons coefficient for a change of the spin of the matter and keep the remaining Chern-Simons coupling weak. In this case, a perturbation expansion can be a good approximation. In Section 2, we define our models, and introduce SU(2) coherent states to formulate the free energy of spin-j matter fields, treating the Chern-Simons gauge potential as an external field. In Section 3, we analyze the relative and self-energies from the path integral of the Chern-Simons gauge potential, and verify their relations to a topological quantity, the gaussian linking number, and to the spin factor associated with the spin of matter fields. From these the spin transmutation is explicitly observed. We verify the equivalence of a (huge) class of Chern-Simons matter models with a local four-body interaction in Section 4, while concluding remarks are given in Section 5.
2. Path integral of spinning particles The sort of Chern-Simons matter models of interest can be uniformly described by the euclidean lagrangian i , f~(j, a ) = - ~ [ j - l j ~ , ( 3~, + ia~, + iC~,) + M]q~ - 8~ae~uaa~Ouaa
(2.1)
where, to be more general, an external vector field C u, also minimally coupled to the matter field, is introduced.
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Besides the non-vanishing mass parameter M, the system is characterized by two real parameters j, which we call the spin of the matter field q~, and a, the so-called statistical parameter. The spin j is defined by J~Ju = J(J + 1) ,
(2.2)
with the spin matrix ja obeying the SU(2) (Lorentz) algebra:
(2.3)
[j/~,j~,] = ieu~aja . Examples of lower integer and half-integer j spin matrix are
(2.4)
j~ = ½o-~, where o'u are the Pauli matrices, for j = l;
(2.5)
(ju)~a = -ieu,~, , for j = 1; and
1
x/3
2
2
v5
1 25
-x/3
2 -2
v~
1 -1
'
-v5
,15 1 2
'
'
(2.6)
-3 as jl ,j2 and j3, respectively, f o r j = 3 While ( 2 j + l ) x ( 2 j + l ) matrices for integer and odd-half-integer spin j can be similarly constructed, an infinite number of components is required for an arbitrary-j spin matrix [6]. The statistical parameter a represents a change of the statistics of the matter field brought by the Chern-Simons interaction. It represents also a change of the spin of the matter field, as we shall see below. Viewing (2.1) as an interacting field theory, on the other hand, a reflects the strength of the interaction among the matter fields and Chern-Simons gauge potential. To manifest this, one rescales the Chern-Simons field a~, ~ ~ a ~ , so that the Chern-Simons term is normalized to ½, and then the gauge coupling constant is ~ . Now we calculate the partition function of the model (2.1):
Z(j,oO=/Da~,D'~D~exp(-fd3x£(j,a)),
(2.7)
keeping in mind that Z ( j , a ) is a functional of the external gauge field Ca, and pending a gauge fixing concerning the gauge freedom with the Chern-Simons field until the integral over a~, field is to be carried out (A derivation of (2.7) for the spin-1 model
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can be found in Ref. [7] ; with the method used there, derivations can be done for other spin models.) With the minimal couplings, the path integrals over the matter fields and Chern-Simons gauge potential are both of gaussian type. We choose to perform the former first. By a standard method, we obtain
Z(j,a)= f Da~-~.~.((-1)ZJwj)"exP(8-~a f d3xeU"aa~,O,aa) ,
(2.8)
n=0
where the factor ( - 1 ) zj is due to the spin-statistics relation, and the free energy of the spin-j matter field Wj is
Wj = - T r l o g [ - i ( 0 / , + ia/, + iC~,)j~,j-1 + M] .
(2.9)
To render the remaining integral over a u in (2.8) an explicit gaussian, we invoke the path integral representation of spinning particles [8,3] to formulate Wj. To start, we introduce the SU(2) spin coherent states in the spin-j representation [9], which are parametrized by a unit vector e in three dimensions [e) ---exp[-iO(e3
x e).jle3 x e[-l]]0) ,
(2.10)
where e3 = (0,0, 1), 0 is the angle between e3 and e, and I0) is the highest-weight state in the spin-j representation. Note, this definition involves the spin matrix Ju with the magnitude j. The spin coherent state has the following three properties: (i) Unity partition:
f de le)(el = 1
(2.11)
9
where de is the rotationally invariant measure on the two-dimensional unit sphere. (ii) Area law:
(e + 6ele) = exp{ ijA [e, e + 6e, e3] } + O(6e 2) ,
(2.12)
where A [ e, e + 6e, e3 ] is the area of a spherical triangle with the vertices e, e + and e3. (iii) Expectation value:
(eljle) = je .
6e
(2.13)
With these properties, the trace Tr can be represented in terms of path integral over random paths that are closed on a two-dimensional sphere, such as
Tr
[.(0J exp
dtj. S(t)
)]
-
lira Tr ]--[ (1
"
+ Atj. S(ti))
i=1
-- :a D e e x p
ij~[el +j
, 0
(2.14)
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where S(t) is a c-number source, and ~ [ e ] = [ dt dse. [0se x 0tel
(2.15)
,J
D is defined as spin factor, e(t,s) is an extension of e(t) satisfying e ( t , 0 ) = const and e(t, 1) = e(t). Geometrically, the spin factor is the area enclosed by the closed path e(t) (0 <. t <. L) on the two-dimensional unit sphere. Introducing the Schwinger parameter L, we write the free energy of spin-j particles
%as Wj = - T r l o g [ (0~ + ia~ + iC~,)j~j -1 + M] [ --~Tr exp ( - L (i(p + a + C). j j - ! + M ) ) ,
(2.16)
where • is an ultraviolet cutoff. Using the SU(2) coherent states, we have
Wj=
T £
]-[dxidei(xil(eil i=l
x exp (-alL (i(p + a + C). jj-1 + g ) ) lei_l)lXi_l).
(2.17)
The path integrals over xi and ei are subject to periodic boundary conditions x s = x0 and eN = e0. The infinitesimal kernel in terms of the coherent states and the momentum eigenstates is
(xil(ei[ exp ( - A L (i(p + a + C) . j j - i + M)[ ei-l)]Xi-l) = f ~d3pi (eilei - 1) e x p [ - d L ( i ( p i + a i ) . e i + M ) + i p . ( x i - x i _ l ) ] +O(AL 2) .
(2.18)
Therefore, the free energy takes a form of path integrals (X)
wj=fd-~LL fDxDpDe E xexp[-/d,{i(p~-a-~C).eZ-~MZ-ip.,}-~ij,[e]l ,
(2.19)
where the parameter t is rescaled as t ~ Lt, and x = dx(t)/dt. Next, we try to integrate out p, e and L. Let us map the integral over L to the integral over an einbein h, which by definition is an one-form in one-dimensional space. Doing so, we take the advantage of the integral thus being explicitly diffeomorphism invariant in the one-dimensional space. In the Fadeev-Popov procedure, the integral over the Schwinger parameter L is regarded
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as a gauge fixed path integral with respect to the diffeomorphism transformation, with L the zero mode of the einbein. Namely, we have the mapping (30
lim f dL • --.o J L ~
fDh
VDiir
(2.20)
With this replacement, the path integral reads
wj=f DhDx DpDe xexp[-fo {i(p+a+C).eh+Mh-ip.k}+,~i~[e] ] .
(2.21)
Here, p is readily integrated out
Wj=f Dh
~(x
--
eh)
×exp[-idt{i(a+C).eh+Mh}+ij~[e] ] .
(2.22,
In this expression we see h = V ' ~ and
e(t) =k/V~x2 ,
(2.23)
which is the unit tangent vector of the path parametrized by the real variable t. Carrying out the integrals over e and h, finally we obtain
W./:
dttMV~x2+ia.k+iC.kj+ij~ T~[ .
Dxexp-
(2.24)
0
It is worth noticing that the free energies of different spin-j matter fields differ from one another just by a coefficient, the value of spin j, of the spin factor ~ .
3. Spin transmutation via the Chern-Simons term
Wj
With in the form expressed in (2.24), the integral over a u in the partition function (2.8) is readily to carry out, it gives oo
H
.__
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x exp
+ is Zi
[ ~-~.{fdt i=~
(3.1) where
xi(t)
is the position vector of the ith path, and 1
1
l f d t f d s d ~ U dX~(au(xi)av(xk)). ds
~)ik = "~
0
(3.2)
0
As usual, a complete definition of path integral over the Chern-Simons gauge field needs a gauge fixing that involves introducing the ghosts and a gauge function. The former is decoupled and so contributes only to the normalization of the partition function. And for the latter, we choose the covariant gauge function ( 1 / ~ ) (a~a~)2, and further choose the Landau gauge a = 0. Under this gauge choice,
(as,(x )a~(Y) ) =/ Daa au( x)a~(y) exP ( 8-~a / dx "°~ a,r&a,7) x a _ ya = 8~'aeuvaix _ yp .
(3.3)
With the notation of unit vector
e(s, t) = xi(s) - Xk(t) [Xi(S)
--
xk(t)l
E S2 ,
(3.4)
Oi~ takes a compact form 1
1
Oik=/dsfdte.[Ose×Ote]. 0
(3.5)
0
Oik
has been extensively discussed in Refs. [8,3,10], here we present its main features with some comments. First, if i ~ k, i.e. the closed paths denoted by i and k do not coincide, Oik, called the relative energy, is the Gauss linking number and thus is quantized [ 11 ]
Oik E 4zrZ.
(3.6)
From (3.1) and (3.6), we see that to any system that has an integer or odd-half-integer statistical parameter, a E Z/2, the relative energy is irrelevant. In other words, the contribution to the partition function from the relative energy is associated with the "small" portion of statistics a modular integer and odd-half-integer. Secondly, when the paths i and k do coincide, Oil, called the self-energy, is not quantized. Instead it is related to the spin factor ~ (see (2.15)) via
Oii
-
-
2 ~ = 4zr
(mod 8~-) .
(3.7)
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It seems when i = j the definition (3.4) of e(t, s) at the point s = t is ambiguous. To fix this problem, one could use the "ribbon-splitting technique" [ 10], in which one splits the two coinciding paths a little bit, and takes a wisely chosen limit to merge them again later. Here we prefer a resolution to well define the self-energy by introducing a limitation procedure without this splitting [8,3]. Let us change the variables of e(s, t) from (s, t) to (u, t) with u = s - t E [0, 1], and consider the connection conditions at u = 0 and u = 1. At u = 0, one naturally defines a limit
x(t)
e ( u = 0 , t) = lim e ( e , t ) = e ( t ) - - -
,-.+0
Ixl
;
(3.8)
then at u = 1 it must be e(u = 1,t) = lim e(1 - e , t ) = - e ( t ) e---*+0
.
(3.9)
Therefore, e(u, t) satisfies the anti-periodic boundary condition for the variable u e(O,t) = - e ( 1 , t )
= e( t) ,
(3.10)
comparing the periodic boundary condition for the variable t e ( u , t + 1) = e ( u , t )
.
(3.11)
In this way, e and thus the self-energy Oil are well defined on the whole (smooth) path. Moreover, the factor 2 on the left-hand side of (3.7) reflects the fact that the self-energy Oii has two boundaries at u = 0 and u = 1 while the spin factor ~ has only one. Due to the diffeomorphism invariance of Oii - 2 ~ defined on a closed path, one is free to deform the path continuously onto a plane, where (3.7) is readily to obtain. Finally, we should mention that in this approach, one assumes that, when the random paths intersect, a suitable point splitting can always be used to regularize the intersecting point(s). The quantization of the relative energy Oik, (3.6), and the simple relation of the self-energy Oii with the spin factor, (3.7), have very interesting implication that the Chern-Simons coupling therefore can be related to the spin of the matter field coupled. Decreasing a in (3.1) by an integer or odd-half-integer and increasing j at the same time by the same amount (or vise versa), one does not change the partition function at all. Namely, we have for any n E Z Z(j,a)
= Z ( j + ½n, a -
½n) .
(3.12)
(3.12) is our main result here, which explicitly exhibits the spin transmutation. As a special case, when a is an integer or odd-half-integer (so that the relative energy is irrelevant), one may convey the whole amount of Chern-Simons coupling to the spin of matter field by using (3.7), and obtain a "free" theory of anyon with a spin j + a.
4. Equivalence of Chern-Simons matter models Treating anyons as fundamental fields faces great difficulties [ 12]. More practical approach in dealing with anyons is to use integer or odd-half-integer spin-j fields as the
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fundamental fields and couple them to Chern-Simons fields. In this way, traditional and powerful methods to solve interacting field theories can be used directly. In particular, to calculate quantum fluctuations, perturbation expansion over the Chern-Simons coupling is normally used. In many realistic systems, unfortunately, the Chern-Simons interactions are very strong, i.e. ot has to be so large 3 that an expansion in it is not well convergent. With the spin transmutation mechanism seen here, one can now trade a major part of Chern-Simons coupling for higher spin. Then, one may have an effective theory with a remaining small Chern-Simons coupling. For example, it is straightforward now to verify the following three models being equivalent one another, for any given a [5]. The first model has a spin-½ two-component Dirac field 0 as the fundamental field, i £ ( ½ , c 0 = ~[o'~(0~, + ia~, + iC~) + M]~b - 8----~ae~aa~a~a~ ,
(4.1)
where C i, is an external vector field. The second involves a spin-1 complex vector field
B/~, £ ( 1 , a - ½ ) = 1
~ B •~ [ - t e•
~a
• (&, + ~a~ + iG,) + MOt*'~]Ba
i 8zr(ot - ½) e~;ta~O~aA.
(4.2)
Finally, the third contains a spin-~ four-component fermion field £ ( ~ , c ~ - 1) = ~ [ ~ L ~ ( O ~ + i a ~ +iC~) + M]q t
i 8~-(a - 1) 'e~vaa~'O~aa ' (4.3)
where L~, are 4 × 4 matrices satisfying the SU(2) (Lorentz) algebra, as given in (2.6). This claim of equivalence is based on (3.12) which takes a form in the present case Z(1,oO = Z(1,a-
½) = Z ( 3 , a -
1) = . . . .
(4.4)
" . . . " here means obvious extension of the equivalence to (cx~ number) models with "higher" matter spins and "weaker" Chern-Simons interactions. In the view of perturbation expansion over the Chern-Simons couplings, the equivalent models (4.1), (4.2), (4.3), and . . . can be good perturbation theories only around ot = 0, ½, 1, and .... respectively. Moreover, the equivalence can be generalized to some matter field self-interactions. Let us take the four-body interaction as an example. Assume a local four-fermion interaction term ( f f ¢ ) 2 added to the Chern-Simons Dirac fermion model (4.1), the lagrangian now reads
£(½,ot, g) = ~ [ o ' ~ ( O ~ z + i a ~ + i C ~ ) + M ] ~ b + ¼ g ( ~ p ) 2
-
i ~e~vAa~O~,a,~ , (4.5)
3 For example, in a recent successful theory of the half-filled Landau level [ 13], each electron carries about two flux tubes. This corresponds to ot ~ 1 in the Chem-Simons Dime fermion model.
W. Chen, C. ItoilNuclear Physics B 435 [FS] (1995) 690-702
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where g, with dimension one in the unit of mass, is the coupling constant for the four-body interactions. We introduce an auxiliary field b(x) = lig~¢ and turn the four-body interaction into a Yukawa interaction:
Eb(½,0¢,g) = ~[tr#(O# + iajz + iC#) + M + ib]¢ i 8~rae~,,aa~O~,aa + l b2 .
(4.6)
g
The four-body interaction in (4.5) is recovered by performing the gaussian functional integral over b in the partition function
Z(½,ot,g)=/79bZ)a/zD~79~bexp(-ff_.b(½,oL, g)) .
(4.7)
Integrating out the Dirac fields, we have the free energy for the spin-½ field 1
Wl/2=f79xexp[--f
dt((M+ib(x))X/~x2+ia'~z+iC'x}+li~(~xl)].
0
(4.8) Compared to (2.24) (if j = ½ is taken), which involves no self-interaction, the change in (4.8) is only to replace the mass term M with M + b(x). Then, carrying out the integral over a~, as done in the previous section, we obtain
f V ,exp n=l n l J xexp[-~-~(f
d3xb2/
"=
dt((M+b(x))~+iC.xi}
i=1
-½i~(~)
--liolOii)+iotZOik ] .
(4.9)
i
Now, we use (3.6) and (3.7) to shift a by an integer or odd-half-integer. For any n E Z, we obtain z(l,a,g)
= z(l(1 +n),a-
in, g) .
(4.10)
Taking n = 1 as an example, the right-hand side above, Z (1, a - 1, g), is the partition function of the vector Chern-Simons model with four-body interaction I~ZB*~ B ~1~2 £.( 1, oz - l' g) = -ie~,aB~(a~, + ia~,+ iC~,)Ba + MB~BIz + ~;~ i ) e~aa~a~aa . 8Ir ( Ol ½
Now, we have seen the equivalence between the models (4.5) and (4.11).
(4.11 )
700
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It is straightforward to play this game over again for more versions of the ChernSimons matter model for any given a.
5. Concluding remarks In this paper, we have shown explicitly how the spin of particles transmutes when they interact to the Chern-Simons gauge field. An immediate consequence of the equivalence is that the relativistic anyon models become free spinor field with different spin when the coupling constant a is quantized by ½. As the higher-spin matter fields in general carry more degrees of freedom, there seems a problem with a counting of degrees of freedom in equivalent models. We would like to argue that the Chern-Simons interaction affects the number of degrees of freedom of the whole anyon system. As is seen in Sections 3 and 4, an integer or odd-half-integer part of the Chern-Simons coupling a can be reabsorbed by changing the spin of matter field. This change of spin may be accompanied by a change of numbers of degrees of freedom carried by the matter field. There has been debate on whether the only effect of Chern-Simons gauge field is to endow the particle with arbitrary spin or whether residual interactions are present [ 14]. We are now able to shed some lights on this issue. Looking at the partition function (3.1) and the conditions (3.6) and (3.7), it is clear that only the integer or odd-half-integer part of the Chern-Simons coefficient can be absorbed into the spin of the fundamental field. In other words, our analysis here suggests that: only when a is an integer or odd-half-integer, the sole role the Chern-Simons field plays is to change the spin of particle by an amount a; otherwise, a residual Chern-Simons interaction, minimally the part of cr that is less than ½, must remain. The complication is apparently due to the relative energy Oik, which is quantized in a way shown in (3.6). One might be tempted to understand whether the class of models discussed here, under certain conditions, can serve as the critical models that describe the critical behavior or phase transition of anyon systems. For j = ½, i.e. for the Chern-Simons Dirac fermion model, it is indeed the case. In Ref. [15], the authors analyzed a lattice model of anyons in a periodic potential and an external magnetic field which exhibits a secondorder transition from a Mott insulator to a quantum Hall fluid. The continuum limit of this lattice model is shown exactly the Chern-Simons Dirac fermion model (4.1) (with the fermion field doubled and Cu missed), and the transition is characterized by the anyon statistics, a. We would guess that higher odd-half-integer j models are in this nature. On the other hand, however, for j --- 1, i.e. for the Chern-Simons vector boson model (4.2), there appears a kind of discontinuity when the mass parameter M is taken to be zero. In the model (4.2), unlike the Chern-Simons field, the massive vector field carries a local dynamical degree of freedom. When the mass parameter M = 0 is taken, however, the only degree of freedom is missing and the U(1) Chern-Simons (vector) matter theory becomes a topological field theory. To see this, we set a~=eA~,
and
B ~ z_- A 2~ + iA 3 ,
(5.1)
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701
where e = V/4zr(a - ½), and substitute these into (4.2) with Cu = 0. Then we obtain 1•
a
~, = - ~ t e / ~ a ( A / z 3 u A ~
a
1 __abcAa abAc
+ gee
e t / ~ u e t u) ,
(5.2)
upto a total derivative term. (5.2) is the well-known SU(2) Chern-Simons field theory, without reference to a space-time metric. The singular behavior of the spin-1 model at M = 0 raises a question: whether the theory with Chern-Simons coupled to the spin-1 field exists as a continuum limit of a lattice model? This question can be answered only by studying the corresponding lattice model. A research of the quantum diffusion process with the spin factor as a free field theory on the three-dimensional lattice suggests the continuum limit for the spin-1 model exists as well as for the spin-½ model [16]. Meanwhile, this research also shows that M = 0 is indeed a singular point for a model with an arbitrary spin. Since the parity of the two-component Dirac field in three dimensions is specified by the sign of the mass parameter, its massless limit is somehow pathological. On the other hand, the massless limit in the spin-1 model restores topological and non-abelian gauge invariance [17]. Obviously, the equivalence between the spin-1 and spin-½ models cannot be simply extrapolated to the M = 0 case. Finally, the analysis in this work is done in euclidean space, its continuation to Minkowski space for higher spin models is an interesting open problem (for spin-I model, see Ref. [ 18 ] ).
Acknowledgement The authors thank A. Kovner, G. Semenoff, H. Van Dam, and Y.-S. Wu for discussions. The work of W.C. was supported in part by the U.S. DOE under the contract No. DEFGO5-85ER-40219.
Note added After this work was finished, we noticed Ref. [ 19] where a similar issue was discussed in a different way.
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[15] [ 16] [17] [18] [19]
W. Chen, C. Itoi/Nuclear Physics B 435 [FS] (1995) 690-702 S. lso, C. Itoi and H. Mukaida, Nucl. Phys. B 346 (1990) 293; N. Shaju, R. Shankar and M. Sivakumar, Mod. Phys. Lett. A 5 (1990) 593. I. Kogan, Phys. Lett. B 262 (1991) 83; I. Kogan and G. Semenoff, Nucl. Phys. B 368 (1991) 718. W. Chen and C. Itoi, Phys. Rev. Lett. 72 (1994) 2527. R. Jackiw and V.P. Nair, Phys. Rev. D 43 (1991) 1933. W. Chen, Thermodynamics of an anyon system, preprint IFr-494-UNC (1994); R. Amorim and J. Barcelos-Neto, preprint IF-UFRJ-05/94. T. Jaroszewicz and P.S. Knrzepa, Ann. Phys. (N.Y.) 210 (1991) 255; 213 (1992) 135; 216 (1992) 226. J.R. Klauder and B.S. Skagerstam, Coherent states, applications in physics and mathematical physics (World Scientific, Singapore, 1985); A. Perelomov, Generalized coherent states and their applications (Springer, Berlin, 1986); E. Fradkin and M. Stone, Phys. Rev. B 38 (1988) 7215; E. Fradkin, Field theories of condensed matter systems (Addison-Wesley, Reading, MA, 1991). E. Witten, Commun. Math. Phys. 92 (1984) 451; 121 (1989) 351; I. Kogan and A. Yu, Morozov, Sov. Phys. JETP 61 1 (1985). M.S. Plyushchay, Phys. Lett. B 248 (1990) 107; B 273 (1991) 250; B 320 (1993) 91; Int. J. Mod. Phys. A 7 (1992) 7054. B.I. Halperin, P.A. Lee and N. Read, Phys. Rev. B 47 (1993) 7312; V. Kalmeyer and S.°C. Zhang, Phys. Rev. 46 (1992) 9889. J. Hong, Y. Kim and P.Y. Pac, Phys. Rev. Lett. 64 (1990) 2230; R. Jackiw and E. Weinberg, Phys. Rev. Lett. 64 (1990) 2234; R. Jackiw and S.-Y. Fi, Phys. Rev. Lett. 64 (1990) 2969; Phys. Rev. D 42 (1990) 3500; R. Jackiw, K. Lee and E. Weinberg, Phys. Rev. D 42 (1990) 3488. W. Chen, M.A.E Fisher and Y.-S. Wu, Phys. Rev. B 48 (1993) 13749. C. Itoi, Smooth paths on three dimensional lattice, NUP-A-94-9 hep-th/9406123. W. Chen, Phase transition in (2 + 1)d quantum gravity, IFr-490-UNC (1994). S. lso, C. Itoi and H. Mukaida, Phys. Lett. B 236 (1990) 287. S. Forte, Int. J. Mod. Phys. A 7 (1992) 1025; S. Forte and T. Jolicoeur, Nucl. Phys. B 350 (1991) 589.
NUCLEAR PHYSICSB[FS]
ELSEVIER
Nuclear Physics B 435 [FS] (1995) 690-702
Spin transmutation in (2 + 1) dimensions Wei Chen a,1, Chigak Itoi b,2 a Department of Physics, University of North Carolina, Chapel Hill, NC 27599-3255, USA b Department of Physics and Atomic Energy Research Institute, College of Science and Technology, Nihon University, Kanda Surugadai, Chiyoda-ku, Tokyo 101, Japan
Received 27 April 1994; revised 15 September 1994; accepted 1 December 1994
Abstract
We study a relativistic anyon model with a spin-j matter field minimally coupled to an abelian Chem-Simons gauge field with a statistical parameter t~. A spin and statistics transmutation is shown in terms of the continuum random walk method. An integer or odd-half-integer part of a can be reabsorbed by change of j. We discuss the equivalence of a large class of Chem-Simons matter models for given j and a.
1. I n t r o d u c t i o n
It is well-known now that in ( 2 + 1 ) space-time dimensions there may exist anyons which have arbitrary spin and statistics [ 1 ]. The statistics of particles is changed if an interaction with a statistical gauge potential governed by the Chern-Simons term is introduced [2]. While this issue is settled for non-relativistic matters, there has been debate as far as the correct formalism of relativistic anyons is concerned. In this latter context, the spin degrees of freedom plays an essential role. In the framework of quantum field theory, a Fermi-Bose transmutation has been observed in a ( 2 + 1 ) dimensional system [3]. Namely that a charged scalar field coupled to a Chern-Simons gauge potential at a specific value of Chern-Simons coefficient (or statistical parameter) turns out to be a free fermion field theory. In this and similar examples, one must deal with both statistics and spin transmutations, as these refer to apparently different objects. A statistics transformation affects the Aharonov-Bohm phase of the wavefunction of ] E-mail: [email protected]. 2 E-mail: [email protected].
0550-3213/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 05 50-321 3(94)00566-4
w. Chen, C ltoi/Nuclear Physics B 435 [FS] (1995) 690-702
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two identical particles moving around one to the other on a plane, while any spin transmutation should cause corresponding change in the Lorentz group representation of the matter fields. Though there was investigation on anomalous magnetic momentum in a perturbation theory in the literature [4], a clear understanding for spin transmutation seems still lacking. The problem with spin transmutation is discussed in a recent letter [5]. The present paper is to continue the discussion. We see that in a general Chern-Simons matter field theory model, any integer or odd-half-integer part of the Chern-Simons coefficient can be reabsorbed by changing the character of the Lorentz representation describing the matter. This is an observation through a continuous random walk method with certain assumptions with respect to a regularization. In this formalism, the partition function of an anyon system can be formulated in particle path integrals with some phase factor which consists of a spin factor from the matter field and a self-energy and a relative energy from the Chern-Simons gauge field. And then by the topological relations between the spin factor and the relative energy and self-energy we see the spin and statistics transmutations. In these transmutations, due to the quantization of the Chern-Simons relative energy, only the integer or odd-half-integer part of the coupling constant endows the anyons with a change of spin and a "small" part of it ( < ½), if any, has to remain as a residual Chern-Simons interaction. There can be a nice possible application with the spin and statistics transmutation. As we know, the Chern-Simons coefficient is used quite often as a parameter in which a perturbation expansion is conducted. However, when the Chern-Simons coefficient is required to be large, a perturbation expansion is not reliable. Now, with the spin and statistics transmutation, one can trade a large part of the Chern-Simons coefficient for a change of the spin of the matter and keep the remaining Chern-Simons coupling weak. In this case, a perturbation expansion can be a good approximation. In Section 2, we define our models, and introduce SU(2) coherent states to formulate the free energy of spin-j matter fields, treating the Chern-Simons gauge potential as an external field. In Section 3, we analyze the relative and self-energies from the path integral of the Chern-Simons gauge potential, and verify their relations to a topological quantity, the gaussian linking number, and to the spin factor associated with the spin of matter fields. From these the spin transmutation is explicitly observed. We verify the equivalence of a (huge) class of Chern-Simons matter models with a local four-body interaction in Section 4, while concluding remarks are given in Section 5.
2. Path integral of spinning particles The sort of Chern-Simons matter models of interest can be uniformly described by the euclidean lagrangian i , f~(j, a ) = - ~ [ j - l j ~ , ( 3~, + ia~, + iC~,) + M]q~ - 8~ae~uaa~Ouaa
(2.1)
where, to be more general, an external vector field C u, also minimally coupled to the matter field, is introduced.
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Besides the non-vanishing mass parameter M, the system is characterized by two real parameters j, which we call the spin of the matter field q~, and a, the so-called statistical parameter. The spin j is defined by J~Ju = J(J + 1) ,
(2.2)
with the spin matrix ja obeying the SU(2) (Lorentz) algebra:
(2.3)
[j/~,j~,] = ieu~aja . Examples of lower integer and half-integer j spin matrix are
(2.4)
j~ = ½o-~, where o'u are the Pauli matrices, for j = l;
(2.5)
(ju)~a = -ieu,~, , for j = 1; and
1
x/3
2
2
v5
1 25
-x/3
2 -2
v~
1 -1
'
-v5
,15 1 2
'
'
(2.6)
-3 as jl ,j2 and j3, respectively, f o r j = 3 While ( 2 j + l ) x ( 2 j + l ) matrices for integer and odd-half-integer spin j can be similarly constructed, an infinite number of components is required for an arbitrary-j spin matrix [6]. The statistical parameter a represents a change of the statistics of the matter field brought by the Chern-Simons interaction. It represents also a change of the spin of the matter field, as we shall see below. Viewing (2.1) as an interacting field theory, on the other hand, a reflects the strength of the interaction among the matter fields and Chern-Simons gauge potential. To manifest this, one rescales the Chern-Simons field a~, ~ ~ a ~ , so that the Chern-Simons term is normalized to ½, and then the gauge coupling constant is ~ . Now we calculate the partition function of the model (2.1):
Z(j,oO=/Da~,D'~D~exp(-fd3x£(j,a)),
(2.7)
keeping in mind that Z ( j , a ) is a functional of the external gauge field Ca, and pending a gauge fixing concerning the gauge freedom with the Chern-Simons field until the integral over a~, field is to be carried out (A derivation of (2.7) for the spin-1 model
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can be found in Ref. [7] ; with the method used there, derivations can be done for other spin models.) With the minimal couplings, the path integrals over the matter fields and Chern-Simons gauge potential are both of gaussian type. We choose to perform the former first. By a standard method, we obtain
Z(j,a)= f Da~-~.~.((-1)ZJwj)"exP(8-~a f d3xeU"aa~,O,aa) ,
(2.8)
n=0
where the factor ( - 1 ) zj is due to the spin-statistics relation, and the free energy of the spin-j matter field Wj is
Wj = - T r l o g [ - i ( 0 / , + ia/, + iC~,)j~,j-1 + M] .
(2.9)
To render the remaining integral over a u in (2.8) an explicit gaussian, we invoke the path integral representation of spinning particles [8,3] to formulate Wj. To start, we introduce the SU(2) spin coherent states in the spin-j representation [9], which are parametrized by a unit vector e in three dimensions [e) ---exp[-iO(e3
x e).jle3 x e[-l]]0) ,
(2.10)
where e3 = (0,0, 1), 0 is the angle between e3 and e, and I0) is the highest-weight state in the spin-j representation. Note, this definition involves the spin matrix Ju with the magnitude j. The spin coherent state has the following three properties: (i) Unity partition:
f de le)(el = 1
(2.11)
9
where de is the rotationally invariant measure on the two-dimensional unit sphere. (ii) Area law:
(e + 6ele) = exp{ ijA [e, e + 6e, e3] } + O(6e 2) ,
(2.12)
where A [ e, e + 6e, e3 ] is the area of a spherical triangle with the vertices e, e + and e3. (iii) Expectation value:
(eljle) = je .
6e
(2.13)
With these properties, the trace Tr can be represented in terms of path integral over random paths that are closed on a two-dimensional sphere, such as
Tr
[.(0J exp
dtj. S(t)
)]
-
lira Tr ]--[ (1
"
+ Atj. S(ti))
i=1
-- :a D e e x p
ij~[el +j
, 0
(2.14)
W. Chen, C. Itoi/NuclearPhysicsB 435 [FS] (1995) 690-702
694
where S(t) is a c-number source, and ~ [ e ] = [ dt dse. [0se x 0tel
(2.15)
,J
D is defined as spin factor, e(t,s) is an extension of e(t) satisfying e ( t , 0 ) = const and e(t, 1) = e(t). Geometrically, the spin factor is the area enclosed by the closed path e(t) (0 <. t <. L) on the two-dimensional unit sphere. Introducing the Schwinger parameter L, we write the free energy of spin-j particles
%as Wj = - T r l o g [ (0~ + ia~ + iC~,)j~j -1 + M] [ --~Tr exp ( - L (i(p + a + C). j j - ! + M ) ) ,
(2.16)
where • is an ultraviolet cutoff. Using the SU(2) coherent states, we have
Wj=
T £
]-[dxidei(xil(eil i=l
x exp (-alL (i(p + a + C). jj-1 + g ) ) lei_l)lXi_l).
(2.17)
The path integrals over xi and ei are subject to periodic boundary conditions x s = x0 and eN = e0. The infinitesimal kernel in terms of the coherent states and the momentum eigenstates is
(xil(ei[ exp ( - A L (i(p + a + C) . j j - i + M)[ ei-l)]Xi-l) = f ~d3pi (eilei - 1) e x p [ - d L ( i ( p i + a i ) . e i + M ) + i p . ( x i - x i _ l ) ] +O(AL 2) .
(2.18)
Therefore, the free energy takes a form of path integrals (X)
wj=fd-~LL fDxDpDe E xexp[-/d,{i(p~-a-~C).eZ-~MZ-ip.,}-~ij,[e]l ,
(2.19)
where the parameter t is rescaled as t ~ Lt, and x = dx(t)/dt. Next, we try to integrate out p, e and L. Let us map the integral over L to the integral over an einbein h, which by definition is an one-form in one-dimensional space. Doing so, we take the advantage of the integral thus being explicitly diffeomorphism invariant in the one-dimensional space. In the Fadeev-Popov procedure, the integral over the Schwinger parameter L is regarded
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as a gauge fixed path integral with respect to the diffeomorphism transformation, with L the zero mode of the einbein. Namely, we have the mapping (30
lim f dL • --.o J L ~
fDh
VDiir
(2.20)
With this replacement, the path integral reads
wj=f DhDx DpDe xexp[-fo {i(p+a+C).eh+Mh-ip.k}+,~i~[e] ] .
(2.21)
Here, p is readily integrated out
Wj=f Dh
~(x
--
eh)
×exp[-idt{i(a+C).eh+Mh}+ij~[e] ] .
(2.22,
In this expression we see h = V ' ~ and
e(t) =k/V~x2 ,
(2.23)
which is the unit tangent vector of the path parametrized by the real variable t. Carrying out the integrals over e and h, finally we obtain
W./:
dttMV~x2+ia.k+iC.kj+ij~ T~[ .
Dxexp-
(2.24)
0
It is worth noticing that the free energies of different spin-j matter fields differ from one another just by a coefficient, the value of spin j, of the spin factor ~ .
3. Spin transmutation via the Chern-Simons term
Wj
With in the form expressed in (2.24), the integral over a u in the partition function (2.8) is readily to carry out, it gives oo
H
.__
W. Chen, C. ltoi/Nuclear Physics B 435 [FS] (1995) 690-702
696
x exp
+ is Zi
[ ~-~.{fdt i=~
(3.1) where
xi(t)
is the position vector of the ith path, and 1
1
l f d t f d s d ~ U dX~(au(xi)av(xk)). ds
~)ik = "~
0
(3.2)
0
As usual, a complete definition of path integral over the Chern-Simons gauge field needs a gauge fixing that involves introducing the ghosts and a gauge function. The former is decoupled and so contributes only to the normalization of the partition function. And for the latter, we choose the covariant gauge function ( 1 / ~ ) (a~a~)2, and further choose the Landau gauge a = 0. Under this gauge choice,
(as,(x )a~(Y) ) =/ Daa au( x)a~(y) exP ( 8-~a / dx "°~ a,r&a,7) x a _ ya = 8~'aeuvaix _ yp .
(3.3)
With the notation of unit vector
e(s, t) = xi(s) - Xk(t) [Xi(S)
--
xk(t)l
E S2 ,
(3.4)
Oi~ takes a compact form 1
1
Oik=/dsfdte.[Ose×Ote]. 0
(3.5)
0
Oik
has been extensively discussed in Refs. [8,3,10], here we present its main features with some comments. First, if i ~ k, i.e. the closed paths denoted by i and k do not coincide, Oik, called the relative energy, is the Gauss linking number and thus is quantized [ 11 ]
Oik E 4zrZ.
(3.6)
From (3.1) and (3.6), we see that to any system that has an integer or odd-half-integer statistical parameter, a E Z/2, the relative energy is irrelevant. In other words, the contribution to the partition function from the relative energy is associated with the "small" portion of statistics a modular integer and odd-half-integer. Secondly, when the paths i and k do coincide, Oil, called the self-energy, is not quantized. Instead it is related to the spin factor ~ (see (2.15)) via
Oii
-
-
2 ~ = 4zr
(mod 8~-) .
(3.7)
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It seems when i = j the definition (3.4) of e(t, s) at the point s = t is ambiguous. To fix this problem, one could use the "ribbon-splitting technique" [ 10], in which one splits the two coinciding paths a little bit, and takes a wisely chosen limit to merge them again later. Here we prefer a resolution to well define the self-energy by introducing a limitation procedure without this splitting [8,3]. Let us change the variables of e(s, t) from (s, t) to (u, t) with u = s - t E [0, 1], and consider the connection conditions at u = 0 and u = 1. At u = 0, one naturally defines a limit
x(t)
e ( u = 0 , t) = lim e ( e , t ) = e ( t ) - - -
,-.+0
Ixl
;
(3.8)
then at u = 1 it must be e(u = 1,t) = lim e(1 - e , t ) = - e ( t ) e---*+0
.
(3.9)
Therefore, e(u, t) satisfies the anti-periodic boundary condition for the variable u e(O,t) = - e ( 1 , t )
= e( t) ,
(3.10)
comparing the periodic boundary condition for the variable t e ( u , t + 1) = e ( u , t )
.
(3.11)
In this way, e and thus the self-energy Oil are well defined on the whole (smooth) path. Moreover, the factor 2 on the left-hand side of (3.7) reflects the fact that the self-energy Oii has two boundaries at u = 0 and u = 1 while the spin factor ~ has only one. Due to the diffeomorphism invariance of Oii - 2 ~ defined on a closed path, one is free to deform the path continuously onto a plane, where (3.7) is readily to obtain. Finally, we should mention that in this approach, one assumes that, when the random paths intersect, a suitable point splitting can always be used to regularize the intersecting point(s). The quantization of the relative energy Oik, (3.6), and the simple relation of the self-energy Oii with the spin factor, (3.7), have very interesting implication that the Chern-Simons coupling therefore can be related to the spin of the matter field coupled. Decreasing a in (3.1) by an integer or odd-half-integer and increasing j at the same time by the same amount (or vise versa), one does not change the partition function at all. Namely, we have for any n E Z Z(j,a)
= Z ( j + ½n, a -
½n) .
(3.12)
(3.12) is our main result here, which explicitly exhibits the spin transmutation. As a special case, when a is an integer or odd-half-integer (so that the relative energy is irrelevant), one may convey the whole amount of Chern-Simons coupling to the spin of matter field by using (3.7), and obtain a "free" theory of anyon with a spin j + a.
4. Equivalence of Chern-Simons matter models Treating anyons as fundamental fields faces great difficulties [ 12]. More practical approach in dealing with anyons is to use integer or odd-half-integer spin-j fields as the
W. Chen, C. Itoi/Nuclear Physics B 435 [FS] (1995)690-702
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fundamental fields and couple them to Chern-Simons fields. In this way, traditional and powerful methods to solve interacting field theories can be used directly. In particular, to calculate quantum fluctuations, perturbation expansion over the Chern-Simons coupling is normally used. In many realistic systems, unfortunately, the Chern-Simons interactions are very strong, i.e. ot has to be so large 3 that an expansion in it is not well convergent. With the spin transmutation mechanism seen here, one can now trade a major part of Chern-Simons coupling for higher spin. Then, one may have an effective theory with a remaining small Chern-Simons coupling. For example, it is straightforward now to verify the following three models being equivalent one another, for any given a [5]. The first model has a spin-½ two-component Dirac field 0 as the fundamental field, i £ ( ½ , c 0 = ~[o'~(0~, + ia~, + iC~) + M]~b - 8----~ae~aa~a~a~ ,
(4.1)
where C i, is an external vector field. The second involves a spin-1 complex vector field
B/~, £ ( 1 , a - ½ ) = 1
~ B •~ [ - t e•
~a
• (&, + ~a~ + iG,) + MOt*'~]Ba
i 8zr(ot - ½) e~;ta~O~aA.
(4.2)
Finally, the third contains a spin-~ four-component fermion field £ ( ~ , c ~ - 1) = ~ [ ~ L ~ ( O ~ + i a ~ +iC~) + M]q t
i 8~-(a - 1) 'e~vaa~'O~aa ' (4.3)
where L~, are 4 × 4 matrices satisfying the SU(2) (Lorentz) algebra, as given in (2.6). This claim of equivalence is based on (3.12) which takes a form in the present case Z(1,oO = Z(1,a-
½) = Z ( 3 , a -
1) = . . . .
(4.4)
" . . . " here means obvious extension of the equivalence to (cx~ number) models with "higher" matter spins and "weaker" Chern-Simons interactions. In the view of perturbation expansion over the Chern-Simons couplings, the equivalent models (4.1), (4.2), (4.3), and . . . can be good perturbation theories only around ot = 0, ½, 1, and .... respectively. Moreover, the equivalence can be generalized to some matter field self-interactions. Let us take the four-body interaction as an example. Assume a local four-fermion interaction term ( f f ¢ ) 2 added to the Chern-Simons Dirac fermion model (4.1), the lagrangian now reads
£(½,ot, g) = ~ [ o ' ~ ( O ~ z + i a ~ + i C ~ ) + M ] ~ b + ¼ g ( ~ p ) 2
-
i ~e~vAa~O~,a,~ , (4.5)
3 For example, in a recent successful theory of the half-filled Landau level [ 13], each electron carries about two flux tubes. This corresponds to ot ~ 1 in the Chem-Simons Dime fermion model.
W. Chen, C. ItoilNuclear Physics B 435 [FS] (1995) 690-702
699
where g, with dimension one in the unit of mass, is the coupling constant for the four-body interactions. We introduce an auxiliary field b(x) = lig~¢ and turn the four-body interaction into a Yukawa interaction:
Eb(½,0¢,g) = ~[tr#(O# + iajz + iC#) + M + ib]¢ i 8~rae~,,aa~O~,aa + l b2 .
(4.6)
g
The four-body interaction in (4.5) is recovered by performing the gaussian functional integral over b in the partition function
Z(½,ot,g)=/79bZ)a/zD~79~bexp(-ff_.b(½,oL, g)) .
(4.7)
Integrating out the Dirac fields, we have the free energy for the spin-½ field 1
Wl/2=f79xexp[--f
dt((M+ib(x))X/~x2+ia'~z+iC'x}+li~(~xl)].
0
(4.8) Compared to (2.24) (if j = ½ is taken), which involves no self-interaction, the change in (4.8) is only to replace the mass term M with M + b(x). Then, carrying out the integral over a~, as done in the previous section, we obtain
f V ,exp n=l n l J xexp[-~-~(f
d3xb2/
"=
dt((M+b(x))~+iC.xi}
i=1
-½i~(~)
--liolOii)+iotZOik ] .
(4.9)
i
Now, we use (3.6) and (3.7) to shift a by an integer or odd-half-integer. For any n E Z, we obtain z(l,a,g)
= z(l(1 +n),a-
in, g) .
(4.10)
Taking n = 1 as an example, the right-hand side above, Z (1, a - 1, g), is the partition function of the vector Chern-Simons model with four-body interaction I~ZB*~ B ~1~2 £.( 1, oz - l' g) = -ie~,aB~(a~, + ia~,+ iC~,)Ba + MB~BIz + ~;~ i ) e~aa~a~aa . 8Ir ( Ol ½
Now, we have seen the equivalence between the models (4.5) and (4.11).
(4.11 )
700
W. Chen, C. Itoi/Nuclear Physics B 435 [FS] (1995) 690-702
It is straightforward to play this game over again for more versions of the ChernSimons matter model for any given a.
5. Concluding remarks In this paper, we have shown explicitly how the spin of particles transmutes when they interact to the Chern-Simons gauge field. An immediate consequence of the equivalence is that the relativistic anyon models become free spinor field with different spin when the coupling constant a is quantized by ½. As the higher-spin matter fields in general carry more degrees of freedom, there seems a problem with a counting of degrees of freedom in equivalent models. We would like to argue that the Chern-Simons interaction affects the number of degrees of freedom of the whole anyon system. As is seen in Sections 3 and 4, an integer or odd-half-integer part of the Chern-Simons coupling a can be reabsorbed by changing the spin of matter field. This change of spin may be accompanied by a change of numbers of degrees of freedom carried by the matter field. There has been debate on whether the only effect of Chern-Simons gauge field is to endow the particle with arbitrary spin or whether residual interactions are present [ 14]. We are now able to shed some lights on this issue. Looking at the partition function (3.1) and the conditions (3.6) and (3.7), it is clear that only the integer or odd-half-integer part of the Chern-Simons coefficient can be absorbed into the spin of the fundamental field. In other words, our analysis here suggests that: only when a is an integer or odd-half-integer, the sole role the Chern-Simons field plays is to change the spin of particle by an amount a; otherwise, a residual Chern-Simons interaction, minimally the part of cr that is less than ½, must remain. The complication is apparently due to the relative energy Oik, which is quantized in a way shown in (3.6). One might be tempted to understand whether the class of models discussed here, under certain conditions, can serve as the critical models that describe the critical behavior or phase transition of anyon systems. For j = ½, i.e. for the Chern-Simons Dirac fermion model, it is indeed the case. In Ref. [15], the authors analyzed a lattice model of anyons in a periodic potential and an external magnetic field which exhibits a secondorder transition from a Mott insulator to a quantum Hall fluid. The continuum limit of this lattice model is shown exactly the Chern-Simons Dirac fermion model (4.1) (with the fermion field doubled and Cu missed), and the transition is characterized by the anyon statistics, a. We would guess that higher odd-half-integer j models are in this nature. On the other hand, however, for j --- 1, i.e. for the Chern-Simons vector boson model (4.2), there appears a kind of discontinuity when the mass parameter M is taken to be zero. In the model (4.2), unlike the Chern-Simons field, the massive vector field carries a local dynamical degree of freedom. When the mass parameter M = 0 is taken, however, the only degree of freedom is missing and the U(1) Chern-Simons (vector) matter theory becomes a topological field theory. To see this, we set a~=eA~,
and
B ~ z_- A 2~ + iA 3 ,
(5.1)
W. Chen, C. I t o i / N u c l e a r Physics B 435 [FS] (1995) 6 9 0 - 7 0 2
701
where e = V/4zr(a - ½), and substitute these into (4.2) with Cu = 0. Then we obtain 1•
a
~, = - ~ t e / ~ a ( A / z 3 u A ~
a
1 __abcAa abAc
+ gee
e t / ~ u e t u) ,
(5.2)
upto a total derivative term. (5.2) is the well-known SU(2) Chern-Simons field theory, without reference to a space-time metric. The singular behavior of the spin-1 model at M = 0 raises a question: whether the theory with Chern-Simons coupled to the spin-1 field exists as a continuum limit of a lattice model? This question can be answered only by studying the corresponding lattice model. A research of the quantum diffusion process with the spin factor as a free field theory on the three-dimensional lattice suggests the continuum limit for the spin-1 model exists as well as for the spin-½ model [16]. Meanwhile, this research also shows that M = 0 is indeed a singular point for a model with an arbitrary spin. Since the parity of the two-component Dirac field in three dimensions is specified by the sign of the mass parameter, its massless limit is somehow pathological. On the other hand, the massless limit in the spin-1 model restores topological and non-abelian gauge invariance [17]. Obviously, the equivalence between the spin-1 and spin-½ models cannot be simply extrapolated to the M = 0 case. Finally, the analysis in this work is done in euclidean space, its continuation to Minkowski space for higher spin models is an interesting open problem (for spin-I model, see Ref. [ 18 ] ).
Acknowledgement The authors thank A. Kovner, G. Semenoff, H. Van Dam, and Y.-S. Wu for discussions. The work of W.C. was supported in part by the U.S. DOE under the contract No. DEFGO5-85ER-40219.
Note added After this work was finished, we noticed Ref. [ 19] where a similar issue was discussed in a different way.
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