Volume 252, number 4 PHYSICS LETTERS B 27 December 1990
Fractional statistics and the dynamical gauge symmetry of Yang-Mills-Chern-Simons theory
M a r k Burgess a n d D a v i d J. T o m s Physics Department, University of Newcastle Upon Tyne, Newcastle Upon Tyne NE1 7RU, UK
Received 3 August 1990
We discuss the role of fractional statistics in determining the vacuum of a quantum field theory in 2+ 1 dimensions and show that the 0 parameter for anyons may be determined dynamically, by the presence of a non-zero fermion condensate at finite temperature. By choosing appropriate boundary conditions, in a simplified model, the result suggests that pointlike vortices would have 3n statistics to first order in perturbation theory on the grounds of least energy. The notion of anyons with non-abelian statistics is also considered, and it is shown that the statistics gauge group can be broken by the analogue of the Hosotani mecha- nism, so that there may be domains of statistics around vortices as well as Yang-Mills domains. We speculate on possible observ- able implications of our results.
Extensive studies are in progress to determine properties of systems whose configuration spaces have non-tr ivial topologies, most impor tant ly in connec- t ion with low dimensional models of the quantum Hall effect and high temperature superconductivi ty. It is now well known that, in two dimensions, there is an infinite number of possibil i t ies for the spin sta- tistics of identical particles [ 1-3 ], since angular har- monics on the l -sphere (circle) do not have the def- inite e v e n / o d d pari ty of their higher d imensional analogues, but rather have arbi t rary pari ty due to the infinite connectedness of the configuration space. Particle statistics therefore represent a cont inuous abelian gauge symmetry in 2 + 1 d imensions (0 sta- t is t ics) , with Bose ( 0 = 0 ) and Fermi (0=~r) statis- tics as special cases. For this reason it is somet imes impl ied that the essential difference between fer- mions and bosons, in two dimensions, is removed. This view might be true in quantum mechanics, but it is not necessarily true in a second-quant ized field theory due to the different vacuum results for bosons and fermions.
A necessary requirement for non-s tandard statis- tics is that singular points in the configurat ion space o f a 2 + 1 d imensional system be excluded. Non-stan- dard statistics can then be in t roduced by the addi- tion, and subsequent gauging away, of a C h e r n -
Simons term [4]. Investigations of the fractional quantum Hall effect have shown that the allowed val- ues of the statistical parameter 0 are restricted to cer- tain rat ional fractions [5] , a result which can par- tially be explained in the context of Chern -S imons theory, by the quant izat ion of the Chern -S imons coefficient. However, the range of possibil i t ies for 0 is still infinite and it has been asked whether dynam- ical considerations might lead to a preferred value for 0 in a given system. This issue has been s tudied in connect ion with point vortices (for example in a thin film of superfluid 4He) init ially by Chaio et al. [ 6 ] and more recently by Leinaas and Myrhe im [7,8] amongst others [ 9 ]. These authors show that pairs of such vortices behave either like semions (0 = ½ ~z), i.e. particles whose statistics parameter is half way be- tween that for fermions and bosons, or like anyons with 0= 3~r/2, depending on whether one begins with bosons or fermions respectively. Semions are thought to play an impor tant role in models of high tempera- ture superconductivity [ 10]. The results for the point vortices are based upon a canonical quant izat ion of the hamil tonian equations of mot ion and the result- ing Landau levels of the quasiparticles. The issue of whether they should satisfy ½ ~ statistics or 3~r statis- tics has not, to our knowledge, been considered further.
Volume 252, number 4 PHYSICS LETTERS B 27 December 1990
In this paper, we wish to point out that similar re- suits for the statistics can be obtained, in a simplified model, by considering the effective potential of a sec- ond quantized theory, on the configuration space ~ × S ~. This has the required topology and the bare features o f a vortex-like system. In the special case of two equispaced vortices on a ring, a related calcula- tion selects an statistics uniquely on the grounds o f lowest energy. The implication is that even fields in the vacuum configuration can have fractional statis- tics, at finite temperature, and that these imply a value for the statistics parameter. This dynamical argu- ment can be taken as an ansatz and used to investi- gate anyons of a more general nature, with statistical phases belonging to a non-abelian gauge group G. We show here that non-abelian statistics are indeed al- lowed in this regime, but that symmetry breaking is possible, by a variant o f the Hosotani mechanism, so that the effective gauge group may only be a subgroup of G in practice.
We begin by writing down a general, second-quan- tized gauge theory in 2 + 1 dimensions. We state the lagrangian for our most general case:
_ ! with gauge covariant derivative Du - 0 u +gA~, + qa~, + b u. Fu, is the usual Yang-Mills field strength for A'¢,=.4u +Au. C2(GR) is the quadratic Casimir invariant for the group G in the representation R. M is a mass term which could be taken to contain the extrinsic scalar curvature for the circle and a chemi- cal potential, in the general case. Conventions and techniques are the same as those of our earlier papers [ 11-13 ], the only additional contributions being a second and third Chern-Simons term with gauge fields a~ and b u. Whereas the field A u has been ex- panded around a classical solution (A ~ =Au +Au) the
au and b u fields are always background fields, but not necessarily ones which satisfy the field equations at the classical level. All other fields are to be integrated out in the course of path integration. The classical fields are chosen to have zero field strength and thus their effects are not felt at the tree level, except at the singular points of vortices. The fields a u and b u are not usually thought of as fundamental fields, but more often represent a convenient parametrization o f some scalar field in the superfluid. These are the "ficti- t ious" or "statistical" gauge fields. We shall not re- strict ourselves by specifying the origin of these fields in this paper.
It is possible to perform gauge transformations which eliminate the classical gauge fields from the ac- tion, but which alter the boundary conditions on the quantum field operators. (In the second quantiza- tion this amounts to a twisting of the fields.) The boundary conditions give rise to new statistics for the fields, since in plane polar coordinates (r, ~0) the con- figuration space is non-simply connected when the point r = 0 is excluded. If we presuppose appropriate boundary conditions for the formation of vortices, the configuration space has the topology ~2× S ~ and the problem is directly related to earlier studies o f the vacuum and its abelian limit [ 14-17 ]. On the circle we must specify boundary conditions under rota- tions by 2n (or translations of L = 2nR, the circum- ference). Earlier studies have noted that matter fields may be twisted on the circle, i.e., ~ (~0+2n)= e 2'ria ~(tp) o r ~ ( x + L ) - - - e 2~i6 tit(x) and that the de-
gree o f twisting affects the Casimir energy of the sys- tem. The parameter ~ is constant with respect to x, but its value can be varied to determine the physical value, i.e. the min imum energy configuration [ 14 ].
In anyon models, c~ is simply related to the statis- tics parameter 0 up to a numerical factor, which in turn is related to the abelian gauge field bu.O=2n¢5 and gauge invariance o f the Chern-Simons field [ 18 ] implies that ~= 1/2m where m is an integer other than zero [19]. In the majority of anyon literature the fields A'~, and a u are set to zero; here we allow them for generality.
There is a number of ways in which we may pro- ceed. The simplest, perhaps, is to follow the conven- tions o f our earlier papers [ 12 ]. Ignoring d for a mo- ment, the momentum components (modes) on a circle of length L are of the form 2nn/L . For bosons,
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different n represent harmonics which are symmet- rical between identified points on the circle; for fer- mions, the harmonics are antisymmetric over the in- terval. When 6 is non-zero the eigenvalues are (2n/ L ) ( n + ~). I f we wish to consider the effect of placing N particle-like excitations at equal intervals on the circle then the circle must be effectively partitioned into N equal parts. The field modes of the second- quantized (many-particle) theory must respect the boundary condition implied by this partitioning, so the momentum eigenvalues must be given by (2n/ L) (nN+ ~). This is analogous to induced harmonics on a string. The modes are then either symmetric or antisymmetric over each Nth part of the circle, for bosons and fermions, respectively. ~ does not scale by Nas the fields must still twist by the same amount around the full circle. This gives a result which is con- sistent with the fact that statistics phases add when many particle wave functions are constructed by the multiplication of single particle wavefunctions in or- dinary quantum mechanics. We may now write the modes in the form ( 2 n N / L ) (n + O/N) at the expense of rescaling the potential by an irrelevant constant. It is seen that the effective twist is now fi' =~ /N and O= 2rc~/ N.
It is straightforward to show that the min imum en- ergy value of fi is 0 for bosons and + ~ for fermions. According to this view, then, bosons remain bosons in a many-anyon excitation, whereas fermions ac- quire an extra phase factor 0= + n/N. The total phase on interchange of identical fermions is thus n + zc/N for a 27t/N winding around the circle. For N = 2 this gives the possible values 0= ½ ~, 3 ~z~. In the scheme of our previous calculations these two values are degen- erate with the boson case. This is not surprising how- ever, since we are suggesting nothing more funda- mental than appropriate boundary conditions for the eigenfunctions. A value for the twist can only be de- termined up to a sign, since the vacuum has zero mo- mentum (the sum is over positive and negative n). As remarked in ref. [ 7 ], in a real vortex, the value of n would be bounded from below. A similar calcula- tion to the one in ref. [ 14 ] for the effective potential, with the summation over only positive n leads to the conclusion (after regularization) that ~ = + ½ has lower energy than ~= - ½. Energetically, then, 0=3~r is the preferred value for the statistics parameter and zc+ 7r/Nis the preferred value for the case of N a n y o n
vortices. This is in keeping with the results of ref. [ 6 ] and suggests a possible reason for choosing between their two values.
Fractional statistics can be realized for the vacuum at finite temperature for the following reason. It can be shown from the field equations that the statistical gauge field only gives rise to non-trivial statistics in the presence of particles, or vortex quasiparticles (see, for example, ref. [ 10] ). The phase factor is, in fact, exp(i0p) where p = ( jo ) = ( tpTo~), which repre- sents the number of particles. This is zero in the vac- uum, at the classical level, but in general may be non- zero including one-loop finite-temperature correc- tions, due to the presence of a fermion condensate [ 15 ]. This is basically a correction to the chemical potential (Lop =/t~PT°T). Alternatively one may in- terpret our potential as representing the vacuum po- larization in the space around point sources, in centre of mass coordinates.
It seems natural to ask whether non-abelian statis- tics could occur in nature. They might be realized in dense quark-gluon plasmas, for instance, in which some quasi-laminar structure is produced by a strong magnetic field. The statistical gauge field in such a situation could conceivably be a fundamental colour field. The generalization to non-abelian statistics is straightforward, though the interpretation of ( j o ) is more complicated for mult icomponent fields. Work- ing in periodic euclidean time, and considering only fermions, we set A~ = constant, and bu= constant in ( 1 ) to obtain a finite-temperature result. Consider the vector potential a~=q~H'/2z~r, where i, j= l, ..., rank (G) . H ' are the generators of the Caftan subal- gebra, so the non-abelian field strength for au (f~2) vanishes everywhere except on the singularity at r = 0. This vector field can be gauged to zero everywhere except at r=0 , by the Wilson line phase
A(0) = P e x p ~nkC'H ' = e x p ( i O ) , (2) 0
where c ~ are constants, to be determined and O(~) is now a matrix. 4rtk is required to be an integer for gauge invariance [18]. The fermion multiplet is transformed to A (~) 5 u after a ~ rotation. By consid- ering the field equations to one-loop order, it is found that the Chern-Simons field a u effectively halves the
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acquired phase, as in the abelian case. The equation for ao is
kgf~z = ( j o ) i , (3)
where ( jo ) i is the fermion condensate ( ~P~°Hi~U). It can be written T r ( c ' H ~) for some constant matrix c ' , which is to be determined dynamically, according to our ansatz. I f this matrix is proportional to the iden- tity, and the group G ofau is simple, then the conden- sate is zero due to the tracelessness of the generators. However, this will not be the case if the fermions do not have a min imum at the min imum of the total ef- fective potential. (The effective potential is made up of a number of wavelike contributions which inter- fere [ 17,13 ] to produce a minimum. The minima of the individual contributions need not therefore be lo- cated at the same place as the min imum of the whole potential.) In the absence of one-loop corrections ( j ° ) i = 0 ; however, at finite temperature Ao and bo can conspire to give a non-vanishing expectation value for ( j o ) ~ and thus we have a vortex-like system with the statistics parameter n [ 1 - ( 1 / m ) c i H q ,
where m is a non-zero integer. The matrix O deter- mines the effective gauge group under which ~t rans- forms [ 17 ].
The symmetry group for O can now be determined using the same dynamical ansatz as before [ 15 ]. The behaviour of a closely related model at finite temper- ature has been studied by a number o f authors [20- 22]. Our situation is especially complicated by the fact that we have two non-abelian gauge groups: one for the A~ field and one for the a~ field. Even if the gauge groups G ' and G for A u and a u are the same, the two effects are distinguishable since the statistical gauge field only alters the boundary conditions of gauge invariant matter fields, whereas the finite-tem- perature gauge Au field also affects the boundary con- ditions of its own quantum field component A u. In general the groups might be different. It is known that, in such a system, a high temperature exponentially suppresses the fermion contribution to the vacuum [20], so the position of the min imum is determined predominantly by the gauge field A u. At low temper- ature fermions dominate, shifting the min imum to one which breaks the symmetry o f A u. The situation is complex, owing to the number of parameters which are required to minimize the potential, but it is clear on the grounds of dimensional analysis that a critical
parameter in the minimization is R T , the radius of the vortex times the temperature. At fixed tempera- ture the symmetry of a u may be preserved outside a critical radius and broken inside, giving rise to an abelian core in the vortex.The symmetry of A u fol- lows the same pattern, but the critical radius need not be the same as that for a u. (S ince the quantum con- densate cannot be precisely pointlike there is a length scale associated with it, namely its Compton wave- length. The critical radius R is defined approxi- mately by R T ~ 1.) There can therefore be domains o f symmetry for both A~ and au: the statistics phase has been determined by the appearance of the dy- namical condensate and varies with distance in the space around the vortex system. In view of the pre- cise phase relationships required for specific non- abelian breaking patterns, the most likely occurrence in a generalized non-abelian theory is that the gauge symmetry of the statistics is broken to a maximum number of U ( 1 ) subgroups, giving rise to an effec- tively abelian theory via a peculiar route.
It is interesting to speculate whether vacuum stud- ies could be used to infer more information about 2 + 1 dimensional systems. The interesting result that the effective statistics can depend on both the num- ber of interacting anyons and the temperature prompts the question: does an accelerating observer see different statistics to an inertial observer? It is well known that an accelerating observer does not agree that an inertial observer's vacuum is empty [23,24]. Whether one takes the view that the observer sees ac- tual particles or simply a finite-temperature spec- trum, it is apparent from the foregoing results that the statistics might be affected, if only to a small de- gree. One might ask nevertheless whether an observ- able effect [25 ] might be seen in bremsstrahlung pro- cesses in an anyon system such as a dense two dimen- sional plasma of nuclear matter, particularly for par- ticles passing close to the core of a non-abelian vortex system. In the vicinity of vortices an infinitesimal Faraday effect is expected for radiation owing to the difference in angular momenta L~+ = n + O/2n and L~,_ = - n + O / 2 n , which is non-zero for non-zero 0 [26]. It is plausible that a corresponding polariza- tion effect would be manifest in the radiation emit- ted from rapidly accelerating particles confined by an intense magnetic field.
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Volume 252, number 4 PHYSICS LETTERS B 27 December 1990
We a c k n o w l e d g e A lan M c L a c h l a n for he lp fu l dis-
cus s ion a n d are g ra te fu l to M a r k H i n d m a r s h for
p o i n t i n g ou t ref. [ 2 6 ] . T h e t op i c o f th i s p a p e r was
o r ig ina l ly suggested, in par t , in a t a lk g iven at Os lo
U n i v e r s i t y , Apr i l 1990 by M.B. T h a n k s a re e x t e n d e d
to F. R a v n d a l a n d J .M. L e i n a a s for t h e i r k i n d
i n v i t a t i o n .
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