- Email: [email protected]
The third virial coefficient of free anyons
Physics Letters B 299 (1993) 267-272 North-HoUand
P I-tYSIC S I_ETT ER S B
The third virial coefficient of free anyons
Jan Myrheim 1 a n d
Olaussen
K~re lnstitutt for fysikk, NTH, N-7034 Trondheim, Norway
Received 2 November 1992
We use a path integral representation for the partition function of non-interacting anyons confined in a harmonic oscillator potential in order to prove that the third virial coefficient of free anyons is finite, and to calculate it numerically. Our results together with previously known results are consistent with a rapidly converging Fourier series in the statistics angle.
I. Introduction The two-dimensional "fractional statistics" of anyons [ 1-3 ] is not yet a particle statistics in the same sense as Bose-Einstein or Fermi-Dirac statistics. Not much is known even about the gas of free anyons, beyond the second virial coefficient A2(O) [4,5]. As usual, 0 denotes the angle defining the particle statistics. One known exact result for the third virial coefficient A3 (0) is that it is symmetric under the substitution 0 ~ n - 0 . This follows from the existence of a certain "supersymmetry" transformation [ 6,7 ]. Furthermore, according to perturbation theory there is no variation o f A3(0) to first order in 0 or 101 near the boson point 0 = 0, or in 0 - n or I 0 - n l near the fermion point O=n [ 8 - 1 0 ] , but there is a second order variation at both points [ 11,12 ]. It is not entirely obvious to us that A3 should be a differentiable function o f 0 at 0 = 0 or 0 = n, or indeed anywhere. But if it is analytic, then it ought to be well approximated by only a few terms from the Fourier series, A3 (0) =~ 4(~6 ..[_
+...).
1
sin20+ c4 sin40--t-c6sin60
(1)
This series is consistent with periodicity and "superSupported by the Norwegian Research Council for Scienceand the Humanities, NAVF.
symmetry", and with the second order bosonic and fermionic perturbation expansions. We find it rather remarkable that already the first two terms give a good approximation to our numerical data, with no free parameter. If this lowest order approximation is indeed exact, which is not totally excluded, then there should be hope of calculating it exactly. Following Laidlaw and DeWitt [ 13 ], we consider the path integral representation for the partition function of a system of identical particles in an external potential. This approach emphasizes the connection between topology and particle statistics [ 13,1, 14]. Our path integral formula is reminiscent of (and was inspired by) the cycle expansion for strange attractors [ 15 ]. The partition function of three anyons in a harmonic oscillator potential in the limit of zero frequency gives the third virial coefficient of free anyons [ 9-12,16 ]. The two-anyon harmonic oscillator problem is trivial, it was solved in ref. [ 1 ], and the solution yields the second virial coefficient [9,10]. In the three-anyon problem the complete energy spectrum is not known, but the lowest levels have been calculated numerically [ 17,18], or perturbatively near the boson and fermion points, and an infinite number of exact solutions are known with energies linearly dependent on 0. See e.g. refs. [7,19,20] and references given there. The path integral representation gives the partition function directly without explicit knowledge of the energy levels. One non-trivial result we obtain beyond the numerical values, is a proof that the third virial coefficient of free anyons is always finite. This has so far
0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
267
Volume 299, number 3,4
PHYSICS LETTERS B
been known only to second order in perturbation theory [ 11,12 ]. A more detailed account of our work will be given elsewhere [ 21 ].
2. The partition function By reformulating the trace as a path integral, we get the following formula for the partition function of N anyons in the external potential U,
ZN(~, O) = T r
exp( -
(5)
S)
where Fe, is a probability generating function,
×exp(-iOQ). /~
is
(2)
the
hamiltonian
operator.
R=
( rl , r2 , ..., rN ) E ~7~2Nd e n o t e s t h e N-particle configuration, with r : ~ 2, and R ( r ) denotes an N-particle path
as a function of the imaginary time t = - i z , with 0 <~v <~hfl. ~ (R (v) ) is the path integral measure, and S=j=~ ! d r ( 2
~
+U(rfl)
(3)
is the action in imaginary time, with m the particle mass. The trace is obtained by integration over closed paths. With identical particles, a closed N-particle path may induce a permutation P of the particles. The trace includes a sum over all permutations P~SN, which reduces to a sum over all conjugation classes = SN. Here SN is the symmetric group, and the conjugation class .~ is characterized by a partition of N, i.e. a sequence of non-negative integers Vl, v> ... such that EL vzL=N. Every permutation P E ~ may be factored as a product of commuting cycles with v/_ cycles of length L, and the sign of P is sgn (P) = ( - 1 )N_,, where v = ~ L vL is the number of cycles. In eq. (2), W( ~ ) is the set of N-particle paths inducing an arbitrary, but fixed permutation Pc ~. Q denotes the winding number of the path R(v). Q is always an integer, it is even when the permutation P is even, and is odd when P is odd. The path integral in eq. (2) with 0 = 0 is f ~e(~)
268
This relation holds because a factorization of the permutation Pc ~ into disjoint cycles implies a corresponding factorization of the path integral, and the path integral for one cycle of length L is Z1 (Lfl). In fact, the path integral for L bosons, when the permutation is cyclic, equals the path integral for one particle over L times as long a time interval. Extracting from the path integral a normalization factor given by eq. (4), we write
exp( - Big)
1 = ~ ~t(vL,L~t ) ~ ~(R(z))
Here
28 January 1993
~(R(r"exp(-h)=
S
LI~I[Zt(Lfl)]"z
(4)
F~(fl, 0)=
~
Q= -oo
e~(fl, Q)
exp(-i0Q),
(6)
and P~ (fl, Q) is the probability of the winding number Q, given the partition ~ and the weight exp ( - S/ h) of each path. Since e~,(fl, -Q)=P~,(fl, Q), for the reason that the probability distribution of paths is time reversal invariant, and since Q is even/odd for an even/odd permutation, the following relations hold:
Fg,(fl, O)=F~(fl, -0) =sgn(~)F~(fl, ~-0) .
(7)
In particular, the boson and fermion values are F~(fl, 0 ) = 1 and F~(fl, 7 r ) = s g n ( ~ ) = ( - 1 ) u-~. Thus for N non-interacting bosons or fermions we have that
with the upper sign for bosons and the lower for fermions. This relation is valid in any dimension, as can be proved from the grand canonical partition function, where En is the energy of the nth level in the one-particle system, E,(fl, z ) = 1--[ [1T-zexp(-flE,)] n
~- ~
N=0
~'
zNZN(~).
The proof is by exponentiation of the formula
(9)
V o l u m e 299, n u m b e r 3,4
PHYSICS LETTERS B
( +_z)L exp(-LflE,,)
ln[•(fl, z)] = + Y~ n
t
L=I
oo ( +__z)L[ +_-Zl(Lfl) ] = y
(10)
L
L=I
28 J a n u a r y 1993
the probability generating functions F11, F3 and F21 , which we are only able to calculate numerically, except that the part of Z3 which is odd under the reflection 0 - , ~ - 0 is known from the supersymmetry argument of Sen [6,7 ]. In our notation, Sen's result is that sinh[ ( a - ½)3~] sinh (3~)
3. The harmonic oxcillator
F21(fl, O)=F2(3fl, 0 ) = -
In order to compute the virial coefficients, it is convenient to confine the particles by the harmonic oscillator potential U ( r ) = ½ m t . o 2 1 r l 2 and take the limit co-~0. The well-known one- and two-particle partition functions depend on the dimensionless parameter ~= h~ofl,
The distribution of winding numbers can be found numerically by Monte Carlo generation of paths according to an exact multi-dimensional normal distribution. For this we need the imaginary-time one-particle propagator for the two-dimensional oscillator,
Zl(fl) = [2 sinh(½~) ] - 2 ,
(11)
Z2( fl, 0) = 1 [ZI (fl) 12F11 (fl, O) + ½Zt ( 2fl)Fz( fl, O) cosh[ ( 1 - a ) ~ ] = [2 sinh (½()]2.2 sinh2("
(12)
Here a=a(O) is a "sawtooth" function, o~(0)= IZI/zt when O=x+2nzt, ]XI ~
Fll(fl, 0 ) =
•
Pll(fl, Q) e x p ( - i 0 Q )
(16)
G(s, r; r) = (s[ exp[ - ( r / h ) / 4 ] Ir) mo9 ( m~o - 2~zh sinh(~or) exp - ~ - [tanh(½ogz)ISq-rl 2 + coth ( ½o90 [s-rl 2 ] ) .
(17)
The common starting and ending point r = r ( 0 ) = r(Lhfl) of a random closed path over the interval r = 0 to r = Lhfl has a probability density proportional to G(r, r; Lhfl). This defines a normal distribution of mean zero and standard deviation
even Q
cosh[ (a-½)~] cosh(½~)
ao=
F2(fl, O)= ~ P2(fl, Q) e x p ( - i 0 Q ) odd Q
=-
sinh[ ( a - ½)(]
sinh(½~)
(13)
'
(18)
distribution of mean G=
2~ tanh (½~)
PII(fl, Q ) - ~2+(ztQ): ,
sinh [tO(rb --r) ]ra + sinh [ o g ( r - r~) ]r b sinh[og(rb-- %) ] ,
(14)
•
at =
x / h sinh [ ~ o ( r - r~) ] sinh [cO(rb--r) ] mo9 sinh [to(r b -- ra) ]
The three-anyon partition function is Z3(fl, 0) = ~ (Zl (fl))3F111 (13, O) -]- I Z 1( 3fl)F3(fl, O) + ½ZI (2fl)Zl (fl)F21 (fl, 0)
(19)
and standard deviation
2~coth(½~) ~2+(~Q)2
"
Given three times ra
from which follows that
P2(fl, Q ) -
coth(lL~) ~o~o ~
'
.
( 15)
There are two even partitions, 1 + 1 + 1 = 3 and 3 = 3, and one odd, 2 + 1 = 3. The unknown parts of Zs are
/h(r--%)(rb--r) c°~O~ ~
m(rb--ra)
(20) "
We use these formulae to generate random N-particle paths. Given a value of fl and a partition YL VLL=N, we treat one cycle at a time. A cycle of 269
Volume 299, number 3,4
PHYSICS LETTERS B
length L corresponds to an L-particle path over an imaginary-time interval hfl, or equivalently a closed one-particle path over the interval Lhfl. We generate first the common starting and ending point of this one-particle path, and then a successively finer subdivision of the z-interval. We count the windings by counting sign changes of the relative coordinates.
28 January 1993
We write also
F3(fl, O) =V~°) (0) + O ( ~ ) .
(26)
These expansions imply the non-trivial result that A3(0) is finite for all 0. Explicitly it is
A~(O) = ~ { 1 + ~ (o~- ½)~- ¼( a - ½)~ - F t ~ t (0) - ~F~ °) (0)
+½(a-½)[3F~)(O)+l-4(a-~)2]}.
4. The third virial coefficient
The second and third virial coefficients are, with 2 = h x / ~ f l / m the thermal wavelength,
(27)
Note that all the three functions FI~t,F~ °) and F~ z) are even under the reflection 0 - , n - 0 . Thus we see that A3 is even if and only ifeq. (25) holds.
[z, (/~)]U =~.2[¼- ½( l - a ) 2 ]
,
(21)
4
[z~(p)]: +16 [Z2(fl'O)]2
[Z,(fl)]4
6
Z3(fl, O)'~
[z,(/~)]U"
(22)
In the limit co-,0, eq. ( 18 ) implies that the starting and ending point of an L-particle loop is located inside a region of area inversely proportional to 092. Furthermore, eq. (2) implies that the area covered by the L-particle cycle tends to a non-zero, finite limit as 09~0. The probability that particles belonging to two different cycles wind around each other, is therefore proportional to 092. The probability that three cycles overlap simultaneously, is proportional to 094. These estimates imply for the function F ~ that the windings of the three pairs of particles are uncorrelated up to terms of order e94. Hence,
5. Numerical results an discussion
Due to statistical fluctuations, the Monte Carlo estimates p ~ c (fl, Q) for the probabilities will usually violate the symmetry P ( Q ) = P ( - Q ) , so that the estimated probability generating function F ~c(fl, 0) becomes complex. We may then use the imaginary part as a measure of the statistical uncertainty in the real part. For a given ~= htnfl, F ~c is exact at 0 = 0 and 0= n. One way to check it against some theoretical prediction is therefore the check the derivative with respect to 0. In fig. 1 is plotted n OF~C/O0 versus O/n, based on 105 paths generated at ~= 1. We plot only the interval 0 ~<0 ~<½n, because of the periodicity and symmetry relations. The imaginary part is plotted in or-
]
0
Flll (fl, O) =[F~t(fl, O)]3[l+F~t(O)~4+O(~5)] .
(23)
=F2(fl, 0)[1+F~2)(0)~2+0(~3)].
270
o~2 - -
~3 ~
^
0.4 ^
.
I--
o
n
0.5
~.-~-/~..
I
I -
Im(MCdata)
/ Re(MCdata)
-3(24)
The explicit formula in eq. (16) is of this form, with F~ 2 ) ( 0 ) = - ] + - ~ ( a - ½ ) 2 .
I...A.A
f -
Similarly, for the function F2~ only the two-cycle contributes to the windings up to terms of order coz. Hence, F2, (fl, 0)
o.1 ~./x
(25)
-4 Fig. 1. Real and imaginary parts of nOF~C/O0,at 4= 1, versus 0/n. The smooth curve is the exact result due to Sen.
Volume 299, number 3,4
PHYSICS LETTERS B
der to indicate the statistical uncertainty. The smooth curve is the derivative of eq. (16), it is antisymmetric about 0 = 0 and obviously discontinuous. In addition to statistical fluctuations, there is a systematic deviation of the Monte Carlo generated curve from the exact curve. We interpret this as an example of the Gibbs phenomenon, which is unavoidable when a discontinuous function is approximated by a finite Fourier series. It is characteristic that the finite series "overshoots" close to the discontinuity. Allowing for statistical and systematic errors, we conclude from fig. l that our Monte Carlo simulations agrees with the theoretical result due to Sen, eq. ( 16 ). Thus the comparison serves to verify both. Fig. 2 shows z~OF~C/O0 as a function of 0/~z, together with the straight line 9 ( a - 1 ). 2 × l0 5 paths were generated at ~o= 0, i.e. in the exact free-particle limit. Based on this figure, in comparison with fig. l, we conjecture that the straight line is indeed the exact derivative of F3 in this 0-interval, which would mean that
28 January 1993
0.008-
0.002-
0.1
0
- 1--2-
r.%.
im(/~MCdata) Re(MCdata)~ / ' ~ "
v
0.5
Fig. 3. Real and imaginary parts of ~--( 111 4 ) M c , as computed at ~= 0.25, versus 0/n. The two curves correspond to the two curves in fig. 4. 0.038MCdata
0.036-
0.030.028-
0.0
J.4
o18
o'.8
,'.o
Fig. 4. The third virial coefficient, A3/~. 4 as a function of O/n, compared to eq. (1) with either c4=c6. . . . . 0 or c4=0.00658, c6= -0.00583.
m0 0.3 0.4 _L~_%~L-^~
~
-0.002
0.052-
0.2 __,q - , , , , -
t ~ ~ "" - " Re(MCdata) / / . " Im(MCdata)
,,/,:~/
0.0
This formula is at least a very good approximation, if not exact. The most difficult quantity to extract from Monte Carlo data is the fourth order term F }4~(0). Our result, based on 1.8 X 107 paths generated at ~= 0.25, is plotted in fig. 3. The imaginary part is also plotted and shows that the statistical uncertainty is relatively large. We have verified the (4 dependence and thus
0.1 ,~ _
~"
0.004-
0.034-
1-
/ /
0.006-
(28)
F~ ° ) ( 0 ) = - I + 9 ( a - ½ ) 2 .
N o - p a r a m e t e r fit T w o - p a r a m e t e r fit
0.5 I
- J
-4-5 Fig. 2. Real and imaginary parts of n OF~C/O0,at ~=0, versus 0/~t. The real part is consistent with the straight line 9 ( a - ½).
the extrapolation to ~ = 0 by comparison with Monte Carlo simulations at ~= 0.5 and ( = 0.75. In the figure are also plotted two curves corresponding to the two curves in fig. 4. Fig. 4 shows the numerically calculated third virial coefficient A3 (0), compared to eq. ( 1 ). One theoretical curve represents the first two terms in eq. ( 1 ), with no free parameter, the other curve is a two-parameter least squares fit and has c4=0.00658, c 6 = - 0 . 0 0 5 8 3 . We conclude from this figure and from fig. 3 that the Fourier components c4, c6, ... are small, but we cannot tell with certainty whether or not they vanish exactly. 271
Volume 299, number 3,4
PHYSICS LETTERS B
One further comment is in place. The asymptotic form 1/Q2 of the Lorentz distributions of winding n u m b e r s in eq. ( 1 4 ) is typical o f all w i n d i n g n u m b e r d i s t r i b u t i o n s , thus the p r o b a b i l i t y o f high w i n d i n g n u m b e r s is n o t negligible. In o u r M o n t e C a r l o s i m u lation the high w i n d i n g n u m b e r s o c c u r at e x t r e m e l y small distances, in fact we s i m u l a t e d i s t a n c e d o w n to 10-3o0, which is o f course entirely unrealistic. It w o u l d be m u c h m o r e realistic to i n t r o d u c e s o m e h a r d core r e p u l s i o n b e t w e e n the anyons. T h i s w o u l d cut o f f t h e tails o f t h e w i n d i n g n u m b e r d i s t r i b u t i o n s a n d t h u s g u a r a n t e e t h a t the 0 - d e p e n d e n c e is analytic. See e.g. refs. [ 2 2 , 2 3 ] .
Acknowledgement We t h a n k Per A r n e Slotte for g o o d a d v i c e on rand o m n u m b e r generators.
References [ 1 ] J.M. Leinaas and J. Myrheim, Nuovo Cimento 37 B ( 1977 ) 1. [2] G.A. Goldin, R. Menikoff and D.H. Sharp, J. Math. Phys. 21 (1980) 650;22 (1981) 1664. [3l F. Wilczek, Phys. Rev. Len. 48 (1982) 1144; 49 (1982) 957.
272
28 January 1993
[4] D.P. Arovas, R. Schrieffer, F Wilczek and A. Zee, Nucl. Phys. B 251 (1985) 117. [5] J.S. Dowker, J. Phys. A 18 (1985) 3521. [6] D. Sen, Phys. Rev. Lett. 68 (1992) 2977. [ 7 ] D. Sen, Phys. Rev. D 46 ( 1992 ) 1846. [8] D. Sen, Nucl. Phys. B 360 ( 1991 ) 397. [9] A. Comtet, Y. Georgelin and S. Ouvry, J. Phys. A 22 (1989) 3917; J. McCabe and S. Ouvry, Phys. Lett. B 260 ( 1991 ) 113. [10] A. Comtet, J. McCabe and S. Ouvry, Phys. Lett. B 260 (1991) 372. [ 11 ] A. Dasnibres de Veigy and S. Ouvry, Phys. Lett. B 291 (1992) 130. [ 12] M. Sporre, J.J.M. Verbaarschot and I. Zahed, Nucl. Phys. B 389 (1993) 645. [ 13] M.G.G. Laidlaw and C.M. DeWitt, Phys. Rev. D 3 ( 1971 ) 1375. [ 14] A.P. Balachandran, Intern. J. Mod. Phys. B 5 ( 1991 ) 2585. [15] R. Artuso, E. Aurell and P. Cvitanovi6, Nonlinearity 3 (1990) 325. [16] K. Olaussen, Theor. Phys. Sem. in Trondheim, No. 13 ( 1992 ), submitted to J. Phys. A. [17] M. Sporre, J.J.M. Verbaarschot and I. Zahed, Phys. Rev. Lett. 67 (1991) 1813. [ 18] M.V.N. Murthy, J. Law, M. Brack and R.K. Bhaduri, Phys. Rev. Lett. 67 ( 1991 ) 1817. [ 19] A.P. Polychronakos, Phys. Lett. B 264 ( 1991 ) 362. [20] J.Aa. Ruud and F. Ravndal, Phys. Lett. B 291 (1992) 137. [21 ] J. Myrheim and K. Olaussen, in preparation. [22] A. Suzuki, M.K. Srivastava, R.K. Bhaduri and J. Law, Phys. Rev. B 44 (1991) 10731. [ 23 ] D. Loss and Y. Fu, Phys. Rev. Lett. 67 ( 1991 ) 294.
P I-tYSIC S I_ETT ER S B
The third virial coefficient of free anyons
Jan Myrheim 1 a n d
Olaussen
K~re lnstitutt for fysikk, NTH, N-7034 Trondheim, Norway
Received 2 November 1992
We use a path integral representation for the partition function of non-interacting anyons confined in a harmonic oscillator potential in order to prove that the third virial coefficient of free anyons is finite, and to calculate it numerically. Our results together with previously known results are consistent with a rapidly converging Fourier series in the statistics angle.
I. Introduction The two-dimensional "fractional statistics" of anyons [ 1-3 ] is not yet a particle statistics in the same sense as Bose-Einstein or Fermi-Dirac statistics. Not much is known even about the gas of free anyons, beyond the second virial coefficient A2(O) [4,5]. As usual, 0 denotes the angle defining the particle statistics. One known exact result for the third virial coefficient A3 (0) is that it is symmetric under the substitution 0 ~ n - 0 . This follows from the existence of a certain "supersymmetry" transformation [ 6,7 ]. Furthermore, according to perturbation theory there is no variation o f A3(0) to first order in 0 or 101 near the boson point 0 = 0, or in 0 - n or I 0 - n l near the fermion point O=n [ 8 - 1 0 ] , but there is a second order variation at both points [ 11,12 ]. It is not entirely obvious to us that A3 should be a differentiable function o f 0 at 0 = 0 or 0 = n, or indeed anywhere. But if it is analytic, then it ought to be well approximated by only a few terms from the Fourier series, A3 (0) =~ 4(~6 ..[_
+...).
1
sin20+ c4 sin40--t-c6sin60
(1)
This series is consistent with periodicity and "superSupported by the Norwegian Research Council for Scienceand the Humanities, NAVF.
symmetry", and with the second order bosonic and fermionic perturbation expansions. We find it rather remarkable that already the first two terms give a good approximation to our numerical data, with no free parameter. If this lowest order approximation is indeed exact, which is not totally excluded, then there should be hope of calculating it exactly. Following Laidlaw and DeWitt [ 13 ], we consider the path integral representation for the partition function of a system of identical particles in an external potential. This approach emphasizes the connection between topology and particle statistics [ 13,1, 14]. Our path integral formula is reminiscent of (and was inspired by) the cycle expansion for strange attractors [ 15 ]. The partition function of three anyons in a harmonic oscillator potential in the limit of zero frequency gives the third virial coefficient of free anyons [ 9-12,16 ]. The two-anyon harmonic oscillator problem is trivial, it was solved in ref. [ 1 ], and the solution yields the second virial coefficient [9,10]. In the three-anyon problem the complete energy spectrum is not known, but the lowest levels have been calculated numerically [ 17,18], or perturbatively near the boson and fermion points, and an infinite number of exact solutions are known with energies linearly dependent on 0. See e.g. refs. [7,19,20] and references given there. The path integral representation gives the partition function directly without explicit knowledge of the energy levels. One non-trivial result we obtain beyond the numerical values, is a proof that the third virial coefficient of free anyons is always finite. This has so far
0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
267
Volume 299, number 3,4
PHYSICS LETTERS B
been known only to second order in perturbation theory [ 11,12 ]. A more detailed account of our work will be given elsewhere [ 21 ].
2. The partition function By reformulating the trace as a path integral, we get the following formula for the partition function of N anyons in the external potential U,
ZN(~, O) = T r
exp( -
(5)
S)
where Fe, is a probability generating function,
×exp(-iOQ). /~
is
(2)
the
hamiltonian
operator.
R=
( rl , r2 , ..., rN ) E ~7~2Nd e n o t e s t h e N-particle configuration, with r : ~ 2, and R ( r ) denotes an N-particle path
as a function of the imaginary time t = - i z , with 0 <~v <~hfl. ~ (R (v) ) is the path integral measure, and S=j=~ ! d r ( 2
~
+U(rfl)
(3)
is the action in imaginary time, with m the particle mass. The trace is obtained by integration over closed paths. With identical particles, a closed N-particle path may induce a permutation P of the particles. The trace includes a sum over all permutations P~SN, which reduces to a sum over all conjugation classes = SN. Here SN is the symmetric group, and the conjugation class .~ is characterized by a partition of N, i.e. a sequence of non-negative integers Vl, v> ... such that EL vzL=N. Every permutation P E ~ may be factored as a product of commuting cycles with v/_ cycles of length L, and the sign of P is sgn (P) = ( - 1 )N_,, where v = ~ L vL is the number of cycles. In eq. (2), W( ~ ) is the set of N-particle paths inducing an arbitrary, but fixed permutation Pc ~. Q denotes the winding number of the path R(v). Q is always an integer, it is even when the permutation P is even, and is odd when P is odd. The path integral in eq. (2) with 0 = 0 is f ~e(~)
268
This relation holds because a factorization of the permutation Pc ~ into disjoint cycles implies a corresponding factorization of the path integral, and the path integral for one cycle of length L is Z1 (Lfl). In fact, the path integral for L bosons, when the permutation is cyclic, equals the path integral for one particle over L times as long a time interval. Extracting from the path integral a normalization factor given by eq. (4), we write
exp( - Big)
1 = ~ ~t(vL,L~t ) ~ ~(R(z))
Here
28 January 1993
~(R(r"exp(-h)=
S
LI~I[Zt(Lfl)]"z
(4)
F~(fl, 0)=
~
Q= -oo
e~(fl, Q)
exp(-i0Q),
(6)
and P~ (fl, Q) is the probability of the winding number Q, given the partition ~ and the weight exp ( - S/ h) of each path. Since e~,(fl, -Q)=P~,(fl, Q), for the reason that the probability distribution of paths is time reversal invariant, and since Q is even/odd for an even/odd permutation, the following relations hold:
Fg,(fl, O)=F~(fl, -0) =sgn(~)F~(fl, ~-0) .
(7)
In particular, the boson and fermion values are F~(fl, 0 ) = 1 and F~(fl, 7 r ) = s g n ( ~ ) = ( - 1 ) u-~. Thus for N non-interacting bosons or fermions we have that
with the upper sign for bosons and the lower for fermions. This relation is valid in any dimension, as can be proved from the grand canonical partition function, where En is the energy of the nth level in the one-particle system, E,(fl, z ) = 1--[ [1T-zexp(-flE,)] n
~- ~
N=0
~'
zNZN(~).
The proof is by exponentiation of the formula
(9)
V o l u m e 299, n u m b e r 3,4
PHYSICS LETTERS B
( +_z)L exp(-LflE,,)
ln[•(fl, z)] = + Y~ n
t
L=I
oo ( +__z)L[ +_-Zl(Lfl) ] = y
(10)
L
L=I
28 J a n u a r y 1993
the probability generating functions F11, F3 and F21 , which we are only able to calculate numerically, except that the part of Z3 which is odd under the reflection 0 - , ~ - 0 is known from the supersymmetry argument of Sen [6,7 ]. In our notation, Sen's result is that sinh[ ( a - ½)3~] sinh (3~)
3. The harmonic oxcillator
F21(fl, O)=F2(3fl, 0 ) = -
In order to compute the virial coefficients, it is convenient to confine the particles by the harmonic oscillator potential U ( r ) = ½ m t . o 2 1 r l 2 and take the limit co-~0. The well-known one- and two-particle partition functions depend on the dimensionless parameter ~= h~ofl,
The distribution of winding numbers can be found numerically by Monte Carlo generation of paths according to an exact multi-dimensional normal distribution. For this we need the imaginary-time one-particle propagator for the two-dimensional oscillator,
Zl(fl) = [2 sinh(½~) ] - 2 ,
(11)
Z2( fl, 0) = 1 [ZI (fl) 12F11 (fl, O) + ½Zt ( 2fl)Fz( fl, O) cosh[ ( 1 - a ) ~ ] = [2 sinh (½()]2.2 sinh2("
(12)
Here a=a(O) is a "sawtooth" function, o~(0)= IZI/zt when O=x+2nzt, ]XI ~
Fll(fl, 0 ) =
•
Pll(fl, Q) e x p ( - i 0 Q )
(16)
G(s, r; r) = (s[ exp[ - ( r / h ) / 4 ] Ir) mo9 ( m~o - 2~zh sinh(~or) exp - ~ - [tanh(½ogz)ISq-rl 2 + coth ( ½o90 [s-rl 2 ] ) .
(17)
The common starting and ending point r = r ( 0 ) = r(Lhfl) of a random closed path over the interval r = 0 to r = Lhfl has a probability density proportional to G(r, r; Lhfl). This defines a normal distribution of mean zero and standard deviation
even Q
cosh[ (a-½)~] cosh(½~)
ao=
F2(fl, O)= ~ P2(fl, Q) e x p ( - i 0 Q ) odd Q
=-
sinh[ ( a - ½)(]
sinh(½~)
(13)
'
(18)
distribution of mean G=
2~ tanh (½~)
PII(fl, Q ) - ~2+(ztQ): ,
sinh [tO(rb --r) ]ra + sinh [ o g ( r - r~) ]r b sinh[og(rb-- %) ] ,
(14)
•
at =
x / h sinh [ ~ o ( r - r~) ] sinh [cO(rb--r) ] mo9 sinh [to(r b -- ra) ]
The three-anyon partition function is Z3(fl, 0) = ~ (Zl (fl))3F111 (13, O) -]- I Z 1( 3fl)F3(fl, O) + ½ZI (2fl)Zl (fl)F21 (fl, 0)
(19)
and standard deviation
2~coth(½~) ~2+(~Q)2
"
Given three times ra
from which follows that
P2(fl, Q ) -
coth(lL~) ~o~o ~
'
.
( 15)
There are two even partitions, 1 + 1 + 1 = 3 and 3 = 3, and one odd, 2 + 1 = 3. The unknown parts of Zs are
/h(r--%)(rb--r) c°~O~ ~
m(rb--ra)
(20) "
We use these formulae to generate random N-particle paths. Given a value of fl and a partition YL VLL=N, we treat one cycle at a time. A cycle of 269
Volume 299, number 3,4
PHYSICS LETTERS B
length L corresponds to an L-particle path over an imaginary-time interval hfl, or equivalently a closed one-particle path over the interval Lhfl. We generate first the common starting and ending point of this one-particle path, and then a successively finer subdivision of the z-interval. We count the windings by counting sign changes of the relative coordinates.
28 January 1993
We write also
F3(fl, O) =V~°) (0) + O ( ~ ) .
(26)
These expansions imply the non-trivial result that A3(0) is finite for all 0. Explicitly it is
A~(O) = ~ { 1 + ~ (o~- ½)~- ¼( a - ½)~ - F t ~ t (0) - ~F~ °) (0)
+½(a-½)[3F~)(O)+l-4(a-~)2]}.
4. The third virial coefficient
The second and third virial coefficients are, with 2 = h x / ~ f l / m the thermal wavelength,
(27)
Note that all the three functions FI~t,F~ °) and F~ z) are even under the reflection 0 - , n - 0 . Thus we see that A3 is even if and only ifeq. (25) holds.
[z, (/~)]U =~.2[¼- ½( l - a ) 2 ]
,
(21)
4
[z~(p)]: +16 [Z2(fl'O)]2
[Z,(fl)]4
6
Z3(fl, O)'~
[z,(/~)]U"
(22)
In the limit co-,0, eq. ( 18 ) implies that the starting and ending point of an L-particle loop is located inside a region of area inversely proportional to 092. Furthermore, eq. (2) implies that the area covered by the L-particle cycle tends to a non-zero, finite limit as 09~0. The probability that particles belonging to two different cycles wind around each other, is therefore proportional to 092. The probability that three cycles overlap simultaneously, is proportional to 094. These estimates imply for the function F ~ that the windings of the three pairs of particles are uncorrelated up to terms of order e94. Hence,
5. Numerical results an discussion
Due to statistical fluctuations, the Monte Carlo estimates p ~ c (fl, Q) for the probabilities will usually violate the symmetry P ( Q ) = P ( - Q ) , so that the estimated probability generating function F ~c(fl, 0) becomes complex. We may then use the imaginary part as a measure of the statistical uncertainty in the real part. For a given ~= htnfl, F ~c is exact at 0 = 0 and 0= n. One way to check it against some theoretical prediction is therefore the check the derivative with respect to 0. In fig. 1 is plotted n OF~C/O0 versus O/n, based on 105 paths generated at ~= 1. We plot only the interval 0 ~<0 ~<½n, because of the periodicity and symmetry relations. The imaginary part is plotted in or-
]
0
Flll (fl, O) =[F~t(fl, O)]3[l+F~t(O)~4+O(~5)] .
(23)
=F2(fl, 0)[1+F~2)(0)~2+0(~3)].
270
o~2 - -
~3 ~
^
0.4 ^
.
I--
o
n
0.5
~.-~-/~..
I
I -
Im(MCdata)
/ Re(MCdata)
-3(24)
The explicit formula in eq. (16) is of this form, with F~ 2 ) ( 0 ) = - ] + - ~ ( a - ½ ) 2 .
I...A.A
f -
Similarly, for the function F2~ only the two-cycle contributes to the windings up to terms of order coz. Hence, F2, (fl, 0)
o.1 ~./x
(25)
-4 Fig. 1. Real and imaginary parts of nOF~C/O0,at 4= 1, versus 0/n. The smooth curve is the exact result due to Sen.
Volume 299, number 3,4
PHYSICS LETTERS B
der to indicate the statistical uncertainty. The smooth curve is the derivative of eq. (16), it is antisymmetric about 0 = 0 and obviously discontinuous. In addition to statistical fluctuations, there is a systematic deviation of the Monte Carlo generated curve from the exact curve. We interpret this as an example of the Gibbs phenomenon, which is unavoidable when a discontinuous function is approximated by a finite Fourier series. It is characteristic that the finite series "overshoots" close to the discontinuity. Allowing for statistical and systematic errors, we conclude from fig. l that our Monte Carlo simulations agrees with the theoretical result due to Sen, eq. ( 16 ). Thus the comparison serves to verify both. Fig. 2 shows z~OF~C/O0 as a function of 0/~z, together with the straight line 9 ( a - 1 ). 2 × l0 5 paths were generated at ~o= 0, i.e. in the exact free-particle limit. Based on this figure, in comparison with fig. l, we conjecture that the straight line is indeed the exact derivative of F3 in this 0-interval, which would mean that
28 January 1993
0.008-
0.002-
0.1
0
- 1--2-
r.%.
im(/~MCdata) Re(MCdata)~ / ' ~ "
v
0.5
Fig. 3. Real and imaginary parts of ~--( 111 4 ) M c , as computed at ~= 0.25, versus 0/n. The two curves correspond to the two curves in fig. 4. 0.038MCdata
0.036-
0.030.028-
0.0
J.4
o18
o'.8
,'.o
Fig. 4. The third virial coefficient, A3/~. 4 as a function of O/n, compared to eq. (1) with either c4=c6. . . . . 0 or c4=0.00658, c6= -0.00583.
m0 0.3 0.4 _L~_%~L-^~
~
-0.002
0.052-
0.2 __,q - , , , , -
t ~ ~ "" - " Re(MCdata) / / . " Im(MCdata)
,,/,:~/
0.0
This formula is at least a very good approximation, if not exact. The most difficult quantity to extract from Monte Carlo data is the fourth order term F }4~(0). Our result, based on 1.8 X 107 paths generated at ~= 0.25, is plotted in fig. 3. The imaginary part is also plotted and shows that the statistical uncertainty is relatively large. We have verified the (4 dependence and thus
0.1 ,~ _
~"
0.004-
0.034-
1-
/ /
0.006-
(28)
F~ ° ) ( 0 ) = - I + 9 ( a - ½ ) 2 .
N o - p a r a m e t e r fit T w o - p a r a m e t e r fit
0.5 I
- J
-4-5 Fig. 2. Real and imaginary parts of n OF~C/O0,at ~=0, versus 0/~t. The real part is consistent with the straight line 9 ( a - ½).
the extrapolation to ~ = 0 by comparison with Monte Carlo simulations at ~= 0.5 and ( = 0.75. In the figure are also plotted two curves corresponding to the two curves in fig. 4. Fig. 4 shows the numerically calculated third virial coefficient A3 (0), compared to eq. ( 1 ). One theoretical curve represents the first two terms in eq. ( 1 ), with no free parameter, the other curve is a two-parameter least squares fit and has c4=0.00658, c 6 = - 0 . 0 0 5 8 3 . We conclude from this figure and from fig. 3 that the Fourier components c4, c6, ... are small, but we cannot tell with certainty whether or not they vanish exactly. 271
Volume 299, number 3,4
PHYSICS LETTERS B
One further comment is in place. The asymptotic form 1/Q2 of the Lorentz distributions of winding n u m b e r s in eq. ( 1 4 ) is typical o f all w i n d i n g n u m b e r d i s t r i b u t i o n s , thus the p r o b a b i l i t y o f high w i n d i n g n u m b e r s is n o t negligible. In o u r M o n t e C a r l o s i m u lation the high w i n d i n g n u m b e r s o c c u r at e x t r e m e l y small distances, in fact we s i m u l a t e d i s t a n c e d o w n to 10-3o0, which is o f course entirely unrealistic. It w o u l d be m u c h m o r e realistic to i n t r o d u c e s o m e h a r d core r e p u l s i o n b e t w e e n the anyons. T h i s w o u l d cut o f f t h e tails o f t h e w i n d i n g n u m b e r d i s t r i b u t i o n s a n d t h u s g u a r a n t e e t h a t the 0 - d e p e n d e n c e is analytic. See e.g. refs. [ 2 2 , 2 3 ] .
Acknowledgement We t h a n k Per A r n e Slotte for g o o d a d v i c e on rand o m n u m b e r generators.
References [ 1 ] J.M. Leinaas and J. Myrheim, Nuovo Cimento 37 B ( 1977 ) 1. [2] G.A. Goldin, R. Menikoff and D.H. Sharp, J. Math. Phys. 21 (1980) 650;22 (1981) 1664. [3l F. Wilczek, Phys. Rev. Len. 48 (1982) 1144; 49 (1982) 957.
272
28 January 1993
[4] D.P. Arovas, R. Schrieffer, F Wilczek and A. Zee, Nucl. Phys. B 251 (1985) 117. [5] J.S. Dowker, J. Phys. A 18 (1985) 3521. [6] D. Sen, Phys. Rev. Lett. 68 (1992) 2977. [ 7 ] D. Sen, Phys. Rev. D 46 ( 1992 ) 1846. [8] D. Sen, Nucl. Phys. B 360 ( 1991 ) 397. [9] A. Comtet, Y. Georgelin and S. Ouvry, J. Phys. A 22 (1989) 3917; J. McCabe and S. Ouvry, Phys. Lett. B 260 ( 1991 ) 113. [10] A. Comtet, J. McCabe and S. Ouvry, Phys. Lett. B 260 (1991) 372. [ 11 ] A. Dasnibres de Veigy and S. Ouvry, Phys. Lett. B 291 (1992) 130. [ 12] M. Sporre, J.J.M. Verbaarschot and I. Zahed, Nucl. Phys. B 389 (1993) 645. [ 13] M.G.G. Laidlaw and C.M. DeWitt, Phys. Rev. D 3 ( 1971 ) 1375. [ 14] A.P. Balachandran, Intern. J. Mod. Phys. B 5 ( 1991 ) 2585. [15] R. Artuso, E. Aurell and P. Cvitanovi6, Nonlinearity 3 (1990) 325. [16] K. Olaussen, Theor. Phys. Sem. in Trondheim, No. 13 ( 1992 ), submitted to J. Phys. A. [17] M. Sporre, J.J.M. Verbaarschot and I. Zahed, Phys. Rev. Lett. 67 (1991) 1813. [ 18] M.V.N. Murthy, J. Law, M. Brack and R.K. Bhaduri, Phys. Rev. Lett. 67 ( 1991 ) 1817. [ 19] A.P. Polychronakos, Phys. Lett. B 264 ( 1991 ) 362. [20] J.Aa. Ruud and F. Ravndal, Phys. Lett. B 291 (1992) 137. [21 ] J. Myrheim and K. Olaussen, in preparation. [22] A. Suzuki, M.K. Srivastava, R.K. Bhaduri and J. Law, Phys. Rev. B 44 (1991) 10731. [ 23 ] D. Loss and Y. Fu, Phys. Rev. Lett. 67 ( 1991 ) 294.