Statistical mechanics of anyons

Nuclear Physics B251 [FS 13 ] (1985)117-126 © North-Holland Publishing Company

S T A T I S T I C A L M E C H A N I C S OF A N Y O N S

Daniel P. AROVAS

Department of Physics, University of California, Santa Barbara, CA 93106, USA

Robert SCHRIEFFER and Frank WILCZEK

Department of Physics~ University of California and

Institute for Theoretical Physics, Santa Barbara, CA 93106, USA

A. ZEE*

Institute for Advanced Study, Princeton, NJ 08540, USA

Received 31 July 1984

We study the statistical mechanics of a two-dimensional gas of free anyons - particles which interpolate between Bose-Einstein and Fermi-Dirac character. Thermodynamic quantities are discussed in the low-density regime. In particular, the second virial coefficient is evaluated by two different methods and is found to exhibit a simple, periodic, but nonanalytic behavior as a function of the statistics determining parameter.

In two space dimensions, a continuous family of quan tum statistics interpolating

between bosons and fermions is possible [1, 2]. Two examples of particles obeying

exotic statistics have been discussed. The soliton of the (2 + 1)-dimensional 0 (3) nonl inear o-model has a spin which is neither integral nor half-odd integral, and

obeys a statistics which is neither Bose-Einstein nor Fermi ,Dirac [3]. In condensed matter , one finds that the Laughlin quasiparticles [4] in the anomalous quan tum Hall

effect system also possess fractional charge and fractional statistics [5], a result recently derived f rom the adiabatic theorem [6].

We first discuss a method [1] by which the statistics of a two-dimensional system of charged particles can be changed (continuously) via the introduct ion of a fictitious "statist ical gauge field." It is well known that the wave function of a

charged particle interacting with a magnetic flux tube will acquire a phase change due to the mot ion of the particle. If the charge is e* = ve and the flux is

* On leave from the University of Washington 1983-1984.

117

118 D.P. Arovas et al. / Statistical mechanics of anyons

q~ = aeOo = ahc/e, a complete revolution will induce a phase change of eiaY with ay = 2~rav. The basic idea of this method is then to introduce a flux tube at the position of each particle whose magnitude q~ will determine the phase associated with relative particle motion. The gauge field associated with this flux is nondynamical, i.e., its evolution is completely determined by the motion of the particles. It is expected that physical quantities are periodic in the statistics determining parameter O~.

In the symmetric gauge, the vector potential due to a flux tube of magnitude

q, = aq~ 0 takes the form

~.~, ~ _ _ a ¢ o r a ¢ o O = a ¢ O v o . (1) A ' r ' = 2~r2Xr2= 2~r r 27r

The many-particle generalization we seek is therefore

,,-., ",J = , epo Z "v,o,j, (2) A,(ri}=-~--~-zx# ri2j 2~r j

where Oiy = t an - l ( (y j - y , ) / ( x j - x i ) ) is the relative angle between i and j , and the prime on the sum indicates that the term j = i is to be excluded. This leads to the

following many-body hamiltonian:

H(a) = ~_, ~m P i - c *(ri) + V(rl . . . . . rN)' (3) i

where A 0 is the physical vector potential, if present. Suppose one knows an eigenfunction ~o of the bare hamiltonian H 0 = H(0) with energy E o. Then since

[pi,exp ( i aep°2~r 2j ' )] ad~° 2 ;V iOi j=A¢(r i ) / = - ~ J (4)

we see that ~p~ = exp(iaEvairsO, j)q, o is an eigenfunction of H(a) also with energy E 0. The problem is that the function ~k~ will not in general be a single-valued function of

its arguments. For the two-particle problem with no external potential and no interparticle

interactions (free particles), the situation becomes eminently tractable. Recall that the wave function can be decomposed into a product ~Po(r, R) = x(R)~(r) where R is the center-of-mass position and r is the relative coordinate vector. We find x( R ) = e i K ' R , ~( r) = eimOJiml( k r ), E = h2K 2/4M + h2k 2/M (M = particle mass). Imposition of Bose (Fermi) statistics then requires that m be even (odd). The introduction of a statistical gauge field then provides us with new eigenfunctions tk ,( r, R) = ei~°6o( r, R) = eiXRei(m+~)°Jl,,l( kr ). Here, Bose (Fermi) statistics re-

D.P. Arovas et al. / Statistical mechanic's of anyons

quires that m + a = l be even (odd) and we obtain wave functions

119

~ , ( r , R ) = e iX 'Re"°Ji t_ , l ( k r ) . (5)

I f we now in t roduce a circular boundary at some radius R, we find that the allowed

energies are

h2x 2 . / M R 2, J ~ ( x ~ , . ) = 0 (6) ~l,n ~ [ I - a l,

Hence , choosing a to be an odd integer merely shifts the energy spect rum f rom Bose-like to Fermi-like. No te also that the spect rum is periodic in a with per iod zaa = 2. This is in fact true for the N-part icle system, al though explicit (single-valued) wave funct ions are difficult to obta in due to the fact that there are ½ N ( N - 1)

relative angles and only ( N - 1) non -CM angular degrees of freedom. For N = 2, these number s are identical.

This result, eq. (6), can be used to evaluate the second virial coefficient. In two dimensions , it is easily shown that [7]*

(7)

where A is the area of the system X T ~- ( 2 ~ h 2 / M k T ) t /2 is the thermal wavelength, and Z 2 = Tr e -#H~ is the two-particle part i t ion function. The virial expansion is an expans ion of the equat ion of state in the density n: P = n k T [ 1 + Bn + Cn 2 + • • • ].

In pe r fo rming the trace to obta in Z 2, the center-of-mass f reedom is trivially separated, yielding a factor Z 2 = 2AXT2Z2, w h e r e 2 2 is now the single particle par t i t ion funct ion for the relative coordinate problem: 2 2 = Trrele - /~ '~ . This will again be area divergent, and it is therefore convenient to calculate the virial coeff icient B(ct , T ) in terms of a known quantity, i.e., B (2 j , T ) = - 1X2 T or B ( 2 j +

1, T ) = + ¼X 2 , the familiar result for Bose and Fermi systems, respectively ( j ~ Z). Thus, we ob ta in

(8)

W e now appea l to the result (6). Clearly B ( a , T ) must be periodic in a with per iod Act = 2. We will take our original particles to have Bose statistics and expand about even and odd values of a. For a = 2 j + 8, 181 < 1, corresponding to quasi-bosons, the al lowed values of I I - al are 181,2 + 8,4 + 8, etc. For a = 2 j + 1 + 8, 181 < 1, co r responding to quasi-fermions, the allowed values of 1l - a[ are 1 + 8, 3 + 8, etc. Since B mus t be independent of the cutoff k in the limit R ~ oo, and since

* To be precise, we should write B(T) = A[~ - Z2/Z 2 ] with Z 1 = Tre -~ ' , the single particle partition function. With no external fields, we have H 1 =p2/2M and Z 1 ~ AXT 2, which then yields eq. (7)


appears only in the combination M k T k 2 / h 2, it is desirable to rescale k ---, ~ h 2 / M k T h . Expanding about the Fermi point, we find that (8) and (6) give

B ( 2 j + 1 + 3, T) = ¼X 2 + 2X2T

× lim ~ ~ [2e-(X,. , /k)2-e-(X,+*.Jk)2-e-(X,-8, , /k):]. (9) R ~ o o / = 1 n = l

odd

The factor in brackets resembles a second derivative. By expanding in 3, one can then perform the /-sum by means of the celebrated Euler-MacLaurin formula [8]. This leaves

B ( 2 j + 1 + 3, T) = ]X2T -- 2X2T 32 Jim k -2 ~ x,,,, R-~oo n = l

OXl+s,n aS s=O e-(xl""/k)2"

(10)

As n ---, oo, Xl, n ~ O0 and

ax.,. 1 aL(x. .) OP -- J v + l ( X v , n ) alp ' ,

The value of n at which this approximation x~,, - ½v~r + n~r - ¼~r becomes valid is n 0, say, which is certainly independent of R. The sum will then be completely dominated by the terms n o ~< n < oo, the beginning terms being suppressed by the k -2 factor. Making this replacement, and writing E, --* fdXl, n/cr , we obtain

B ( 2 j + 1 + 3, T ) = ¼7~2_ ?~32 lim R - 2 f °° dxxe - (X /k )2= 82G. J~ ---+ oo noq7

(11)

One can check that all other terms in the expansion in 3 and in other approxima- tions employed are formally of order k - 1 a s /~ ~ 00. Thus, we predict

B ( 2 j + 1 + 3, T) = ¼X2T(1 -- 232)p~r, (12)

where the subscript indicates that we are to extend this function for 131 > 1 in a periodic fashion. The complete result has a cusp at Bose values a = 2 j due to the required periodic extension. This in fact follows from the general formula (8). The only difference between the Fermi and Bose expansions is the existence of the

D.P. A rovas et al. / Statistical mechanics of anyons 121

D \ . / . \ . / . .

V V,V - I ~ - ' " - - ' -

Fig. 1. The second virial coefficient B(a , T) as a function of the statistics determining parameter a (T fixed).

I t - al = 181 term:

B ( 2 j + 3,.T) = --¼X2T - ½82X2 r + 2k2T.fim ~_~ [e-(~o../R)~-- e-(~,.,.-/R)~] R ---~ oo n = l

= - ¼G + I~IG- 182G. (13)

This is exactly the form predicted above. Thus, we obtain a picture of B(a, T ) for fixed T as in fig. 1.

This result is also derivable from a path integral approach. The general lagrangian for the many-body system is

L = E ½Mi'i 2 + a E Oij. (14) i pairs

For a system of bosons, the partition function takes the form

1 2 2 Z N = - f f [ fd r l . . . d r N ~_~(rl... rNle-aHqPr~... PrN),

P ( 1 5 )

which may be cast into a path integral form, as done by Feynman [9]. Again, the case N = 2 is considered, and the CM contribution is directly integrated out. This leaves

22 = ½/d2r [(rle-~H~'lr) + (rle-Pnr"ll -- r ) ] ,

Lre I = ¼MP 2 + OtO. ( 1 6 )

The 0 term in L introduces a winding number-dependent phase in the path integral. This problem is in fact equivalent to the Bohm-Aharonov effect [10], the path integral formulation of which was studied extensively by Gerry and Singh [11-14].


The matrix element

g(r, r'; r) = f Dr (t)e ih lf°'dt'L(t')= @le-im/Slr'),

r ( t = 0 ) = r , r ( t + ~') = r ' , fl = i t , (17)

can be decomposed into a sum over contributions of different homotopy sectors, with 0' - 0 -= q~ + 2~rn:

oo

K(r, r'; ~) = Y'~ e2~rinaK--n(r, r'; r), (18) t i e -- ~

F..(r,r'; ¢)= f Dr(t)eih-'Jodi'L(t')'8(O'--O--ep-- 27rn), (19)

M [-M ] 4~rhir exp 4--h--/~ir (r 2 + r '2)

× f= d)teiX(,+2,,),~i~q,, [Mrr'] (20) - ~ " *lxl~2hi z ] ,

where I ,(Z) is the modified Bessel function. In our case, we have Jr' I = IPrl = Irl, and q~ = 0 , ~r:

oo ( M ) ( - M r 2] i.,~ I [Mr 2] K(r,r';¢)=n=~_ ~ ~ exp 2---~-~-~¢ ]e i,_~l~2--~t. 1. (21)

Therefore, we arrive at the result

22=½ ~ fo~dxe-XIl._<(x). (22) even

This is, as expected, formally divergent. A convergence factor e-~x is inserted in the integrand, with e + 0 at the end of the calculation. We use the result [15]

f o ° Fv(a ) - dxe-aX[~(x) =

1 F.(1 + e ) ~ 2-~-e(1 + 2 ~ ) -v

- - ( a + ~ a Z - 1 ) - ' ,

(23)

As before, we appeal to eq. (8). Expanding about Fermi statistics, a = 2 j + 1 + &

D.P. Arovas et al. / Statistical mechanics of anyons 123

and In - a I = 1 + 6, 3 + 8, etc. Thus,

S ( 2 j + 1 + 8, T ) = ¼X~ + 2XZzlim e--*0

[ 1 1 ~_~(1+ 2 ~ ) - " ( 2 - ( 1 + 2 ~ ) 8 - ( 1 + 2 ~ ) - ~ ) ] 2-"~-'-E n = 1

odd

= ¼Xzr- 182X~. (24)

Expanding about Bose statistics introduces a term 181~,~ due to the In - a l - - 181 piece in the sum. Making the required periodic extension recovers the earlier result of eq. (12).

In some sense, the path integral result is more satisfying, because, although one still is presented with the delicacy involved with extracting the (finite) difference of two divergent expressions, there is no necessity to impose a finite volume constraint, which was originally effected in order to perform the mode counting. One might object to our original calculation on the grounds that the virial coefficient might possibly be sensitive to the manner in which we perform the mode counting, since the dominant terms in the sum of eq. (9) are those at the tail end. As we have seen, this fear is unfounded.

A striking result is the nonanalytieity of eq. (12). It would be interesting to know whether cusps also arise in higher-order virial coefficients.

Due to the proliferation of the number of relative angles, such higher-order virial coefficients are exceedingly difficult to evaluate. In the high density limit, one might consider averaging the statistical flux over the entire system, and then consider the effect of a net statistical uniform magnetic field of magnitude B = noteo o, where n is the particle density. It is possible to reproduce the correct form of the free energy to leading order in n in this manner, however, one loses periodicity in a, and only certain values of a actually yield the correct result.

The most significant feature of the statistical interaction is that it is long ranged, hence perturbation expansions in a yield divergences and resummation is necessary, a situation reminiscent of the electron gas. Nevertheless, this statistics transforma- tion process does yield a viable method for interpolating quantum statistics. The representation of a Fermi gas in terms of a Bose gas may be useful in other contexts, such as lattice field theory. Unless the statistical interaction is treated nonperturba- tively, however, divergences may be difficult to handle.

Finally, it is interesting to derive the lagrangian of eq. (14) for the solitons of the nonlinear o-model. Let us briefly recall that the model in question has a unit-vector order parameter na(x) , a = 1, 2, 3 and a conserved topological current

1 I~v~ abc a b c J ~ ' = - ~ e e n Opn Oxn . (25)


The conservation of J " licenses us to manufacture a U(1) "gauge potential" by the curl equation

J~ = e~"XO~Ax. (26)

The crucial point is that we could include a topological term

H= 2~fd~xA~J ~ (27)

in the action, with O a real number (a -= O/rr) , which is analogous to the O-parameter in quantum chromodynamics. H is, in fact, the Hopf invariant describing maps of S 3 to S 2. In a suitable gauge, such as 0A = 0, we can solve for A, and so write H as a nonlocal interaction among the n a fields. The solitons are bosons for O = 0 (a = 0) and fermions for O = ,r (a = 1). In a more general context, any conserved current J~ can be coupled to the vector field A~. If the only other appearance of A, in the lagrangian is the Chern-Simons term [16] e~,pA~a~Ap, then A~ represents a nondynamical field [17] which can be eliminated to give a nonlocal interaction, which will impart anomalous statistics to particles carrying charge associated with the current.

In ref. [3] the statistics of the solitons in this model were determined by invoking the linking number theorem. Here we will determine the statistics directly by interchanging two widely separated solitons, and in the process elucidate the linking number theorem.

For separations large compared to the sizes of the solitons we can approximate the solitons by point particles and the topological current by

J~(x) = Efd¢8¢3~(x- qa(¢)) dq~ (28) d~" '

a

with a = 1,2 and qa( 'r ) describing the trajectories of the two "point solitons." We evaluate H by inserting eq. (28) into eqs. (26) and (27) and keeping only the cross-terms. The divergent self-interaction terms are evidently artifacts of the point approximation. To best understand the situation, we go to euclidean 3-space and think of eq. (26) as one of the time-independent Maxwell's equations V × B = J with the identification of A~ as the magnetic field B. Then H can clearly be interpreted as the work done on a magnetic monopole moving along the trajectory qa(~') by the magnetic field generated by an electric current flowing along the curve q2(~-). With suitable normalization, this is just the number of times curves "1" and "2" wind around each other. We have thus made contact with the explicit form for the linking number between two curves given in mathematical texts [18]. This discussion also defines the linking number between two curves which are not closed.

To evaluate H explicitly, it is easiest to distort one of the curves, say "2," to a straight line q~'(~')= z8 ~°, as we are allowed to do. We find by eq. (26) that

D.P. A rovas et al. / Statistical mechanics of anyons 125

A i = e i j x j / r 2, A 0 = 0, a pure (but topologically nontrivial) gauge. Once again, we could have appealed to (2 + 1)-dimensional electrodynamics, this time interpreting j 0 as B. These remarks make clear that the effect here is essentially the Bohm- Aharonov phenomenon. It is sometimes convenient to transform to a singular gauge wherein A = 0 except along string singularities attached to each particle, across

which A has a jump discontinuity of constant magnitude. In summary, the action describing N of these point particles is just

[ 1 s = f dt L = f dt ½m ~, dt ] -~ -d-t 9~b " (29)

Here x a is a two-dimensional vector locating particle a and 9ab is the angle of particle b relative to particle a, measured from the x-axis, say. The preceding discussion has boiled O H down to the second term in this equation. As we have seen, although this term is a total time derivative and appears as an interaction, it

determines the statistics of the particles. In the original model, the solitons have topological charge Q = f d 3 x J o taking on

all integer values. The I QI > 1 solitons are unstable against breakup. In writing down eq. (29) we have included only Q = + 1 particles. It is easy enough, however, to include Q = - 1 particles as well by noting that the ( + - ) "interaction" has

opposite sign f rom the ( + + ) and ( - - ) "interactions."

DPA would like to thank Stefan Theisen for making the work of Gerry and Singh known to us, and for many useful discussions. This work was supported in part by

the Nat ional Science Foundation under grants DMR82-16285 and PHY77-27084, supplemented by funds from the National Aeronautics and Space Administration. One of us (DPA) is grateful for the support of an A T & T Bell Laboratories

Scholarship.

Note added

This work supersedes the preprint "Interpolating quantum statistics", NSF-ITP-

84-25, by two of the authors (F.W. and A.Z.).

References

[1] F. Wilczek, Phys. Rev. Lett. 49 (1982) 957 [2] Y. Wu, US Dept. of Energy preprint 40048-09P4 (1984) [3] F. Wilczek and A. Zee, Phys. Rev. Lett. 51 (1983) 2250 [4] R.B. Laughlin, Phys. Rev. Lett. 50 (1983) 1395 [5] B.I. Halperin, Phys. Rev. Lett. 52 (1984) 1583 [6] D. Arovas, R. Schrieffer and F. Wilczek, preprint NSF-ITP-84-66, submitted to Phys. Rev. Lett. [7] J.G. Dash, Films on solid surfaces (Academic Press, 1968)

126 D.P. Arovas et aL / Statistical mechanics of anyons

[8] M. Abramowitz and I. Stegun, Handbook of mathematical functions (Dover, 1972) [9] R.P. Feynman, Statistical mechanics (Benjamin, 1972)

[10] Y. Aharonov and D. Bohm, Phys. Rev. 115 (1959) 485 [11] C.C. Gerry and V.A. Singh, Phys. Rev. D20 (1979) 2550 [12] A. Inomata and V.A. Singh, J. Math. Phys. 19 (1978) 2318 [13] C.C. Gerry and V.A. Singh, Nuovo Cim. 73B (1983) 161 [14] S.F. Edwards and Y.V. Gulyaev, Proc. Roy. Soc. London A279 (1964) 229 [15] I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series, and products (Academic Press, 1980) [16] J. Schonfeld, Nucl. Phys. B185 (1981) 157;

S. Deser, R. Jackiw and S. Templeton, Phys. Rev. Lett. 48 (1982) 975; Y.Wu, Washington preprint (1983)

[17] C.R. Hagen, Rochester preprint (1983) [18] H. Flanders, Differential forms (Academic Press, 1963)

Statistical mechanics of anyons

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