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Anyon statistics with single-valued wave functions
Volume 180, number 4
PHYSICS LETTERS B
20 November 1986
ANYON STATISTICS WITH SINGLE-VALUED WAVE FUNCTIONS R a m a n S U N D R U M 1 a n d L.J. T A S S I E
Department of Theoretical Physics, Research School of Physical Sciences, Australian National University, GPO Box 4, Canberra, ACT2601, Australia Received 23 August 1985; revised manuscript received 1 August 1986
The fractional statistics of the anyons proposed by Wilczek are demonstrated in a simple manner using single-valued wave functions. Taking the magnetic flux tube and charge comprising each anyon to be bosons, the wave function for two identical anyons is symmetrical with respect to the interchange, but for e~ = ~r, where e is the charge and ¢, the magnetic flux in each anyon, the anyons behave as fermions, and for other values of eO, the anyons obey intermediate statistics.
Wilczek [ 1] has proposed a simple model in two dimensions of particles obeying fractional statistics and has called such particles anyons. An anyon is a composite of a charge e attached to a tube of magnetic flux qb. Although the properties of anyons have been extensively discussed [ 1 - 3 ] , the discussion involves such complications as singular gauge transformations, multi-valued wave functions, the distinction between the use of canonical and mechanical angular m o m e n t u m as the rotation generator to interchange anyons, or arguments involving the spin of an anyon. The purpose of this letter is to give a simple demonstration of the statistics of anyons without such complications. We find the solutions to the wave equation using a hamiltonian for two identical anyons. The flux tube and charge comprising each anyon are taken to be bosons so that the solution to the wave equation for two identical anyons must be symmetric under interchange of the anyons. Nevertheless it is shown that for the product of charge and flux, e ~ = 7r, the solutions describe fermionic behaviour, and that fractional statistic occurs for other values of e ~ . Units in which ~/= c = 1 are used throughout. We consider the hamiltonian for two identical anyons moving in two dimensions and each carrying charge e and magnetic flux ~ . Wu [3] gives this hamiltonian when the electrostatic repulsion between anyons is neglected. When the repulsion potential is included the full hamiltonian becomes H = ([_iV1 _ 2eA(Xl _ x 2 ) ] 2 + [_iV 2 _ 2eA(x2 _ X l ) ] 2}/2m + e2/lXl _x21 '
(1)
where
a(r,O) ~/2~r
(2)
=
in polar coordinates. Using centre-of-mass and relative coordinates R =(x 1 +x2)/2,
r = x I - x 2,
(3)
we have
H = H R +Hr,
(4)
where i Present address: Department of Physics, Yale University, New Haven, CT 06520, USA. 381
Volume 180, number 4
H R = (-iVR)2/4m,
PHYSICS LETTERS B
20 November 1986
H r = ( - i V r - 2eA(r))2/m + e2/Irl.
(5,6)
We denote Hr, for @ = 0, by H O. Then H R + H 0 is just the hamiltonian for a pair of particles with charge e, interacting purely through electrostatic repulsion. Since the hamiltonian (4) separates, we have eigenfunctions of H of the form
~b(R,r) = [2(R )x(r),
(7)
where ~2(R) and x(r) are eigenfunctions o f H R and H r respectively. In polar coordinates, r = (r, 0),
1 ~[~r~2+la+ 1 (a_~r -~ -~
Hr = - m
ia
)2 1
e2 + --'r
(8)
where a = ecb/~r. Let QXs(r) denote the eigenfunctions of 1 [~ 2
1 a ar
- m ~ r 2 +r
s2~+d r 2] r
s>0,
'
with eigenvalues Ex, normalised so that
21r fQ*X'(r)QsX(r)r dr = 8(X - X'). Then the single-valued eigenfunctions o f H r are of the form x(r,0) =Ql~_al(r) exp(i/0),
IE z.
(9)
Since we assume the components of the anyons are bosons, we require symmetry under interchange o f x 1 and x2, so the physical eigenfunctions o f H r are restricted to those of the form
x(r, 0) = Q~2l-~ i(r) exp(i210),
l E Z.
(10)
Thus the allowed eigenfunctions of H are of the form ff(R,r) = ffZ(R)a~t_~l(r) exp(i210),
IEZ,
(11)
where ~(R) is an arbitrary eigenfunction o f H R . We note that the generator of relative rotations for wave functions is the canonical angular m o m e n t u m - i a / a o since the wave functions are single-valued,
~(R,r,O + A0) = ~2(R)a~2l_al(r) exp[i2l(O + A0)] = exp[iA0(--i 0/a0)] ~b(R,r,O).
(12)
But we shall see that these anyons display intermediate statistics. If gp = 0, a = 0, then the relative eigenfunctions Q~2!l(r) exp(i210) are eigenfunctions o f H O, and so the composites are just bosons interacting purely under electrostatic repulsion. We see that for edp = rr, ~ = 1, the relative eigenfunction x(r) = Qi2l_ll(r) x exp(i210) = exp(i0) (Q~21_ll(r) e x p [ i ( 2 l - 1)0]}
(13)
is just a phase exp(i0) multiplied by an antisymmetric eigenfunction o f H O, i.e. an eigenfunction corresponding to identical fermions interacting purely by electrostatic repulsion. However, the factor exp(i0) does not affect probabilities, and so for et = 1 the anyons behave precisely as charged fermions. For 0 < ~ < I, an intermediate effect occurs corresponding to neither charged bosons nor charged fermions. This can be considered to be intermediate statistics. The variation of the statistics with ct can be illustrated by examining a scattering experiment. Consider two 382
Volume 180, number 4
PHYSICS LETTERS B
20 November 1986
identical anyons known to be at (r0/2, 0) and (r0/2,7r) in the centre-of-mass frame, so that the symmetric wave function describing them is q~(r) = (l/x/2) ([6(r - r012 ) + 6(r + r0/2)] }
*x x + G fOj~ll_~l(ro)Qit_~l(r) exp[U(0 + ~)1 dX) • IEZ
At a time t later, the wave function is
X(r) X/7(/~Z fQI l-~l(rO)Qll-~l( ) exp(i/O) e x p ( - i E x t ) dk +
*h h ) ~, fQMt_o~l(rolQit_,~i(r) expli/(0 + ¢r)l e x p ( - i E x t ) d~, ,
(14)
I~Z
which can be written as x(r) = (1/'v/21 exp(ia0)
[fc~(r,O) + exp(ilroOfa(r,O + n)],
(15)
where
f,~(r, o) = .~,_fQ~(,.o) Q~-~(r) exp [ i ( / -
cOO] exp(-iExt ) dh.
(16)
Note that in generalf a is not a 2n-periodic function of 0, though the physical wave function X clearly is by (14). The physically observable quantity is from (15) Ix(r) l2 = ½Ifa(r,O ) + exp(irca)fe~(r,O
+ lr) l2.
(17)
Consider the case q~ = 0, a = 0. Then from (16), fo(r, O) can be seen to be just the single-valued relative wave function for two distinguishable particles interacting only through their electrostatic repulsion, while fo(r, 0 + re) is the corresponding exchange amplitude. For a = 0, these two amplitudes are added in (17) and thus the composites are ordinary identical, charged bosons. From (16)we see that
fl(r,O) =fo(r,O),
(18)
and so for ~ = 1, [x(r)[2 =
½[fo(r,O) _fo(r, 0 + ~)12.
Thus for a = 1, the anyons behave like two ordinary identical, charged fermions. To further illustrate the result, consider the amplitude for two distinguishable charged particles to be at -r/2,
xd(r) =f0(r).
(19)
r/2 and (20)
It is clear from (17) and (20) that 2[Xa=0(0)[ 2 > 2 Ixct(0) l2 > 2 IXa=l(0)l 2 = 0.
(21) 383
Volume 180, number 4
PHYSICS LETTERS B
20 November 1986
By the continuity of wave functions, we have that, for r sufficiently near the origin, 2 I×~=0(r)l 2 > IXd(r)l 2 + I×d(-r)l 2 > 21×5 = l(r)l 2.
(22)
So the probability for two particles to be near each other is enhanced for a = 0 anyons and diminished for a = 1 anyons, relative to that for distinct charges, which is just the well-known result for identical bosons and identical fermions respectively. Also, from (16),
fo(r, n/2) =fo(r, 37r/2),
(23)
so
Ix~= l(r' ./2)12 = 0,
(24)
another phenomenon characteristic of fermion-fermion scattering. For 0 < a < 1, it can be seen from (17) that the composites show behaviour intermediate between that of bosons and that o f fermions. This behaviour cannot be simply related to the wave function for distinct charged particles,fo(r,O), as was done for a = 0 or 1, since in generalf a is not simply related to f0" It might be thought, since Bose and Fermi statistics are obtained when ~ = 0 and t~ = 1 respectively, that for some intermediate value o f a the anyons will behave as purely charged distinguishable particles (corresponding to Boltzmann statistics). This is easily seen not to be the case. By just considering the spectrum of relative angular momentum (canonical or mechanical) we see that the anyon system has allowed angular momenta separated by 2(h) because the wave functions are symmetric, while the angular momentum spacing for distinguishable particles is of course l(/i). So anyons are physically quite distinct from distinguishable charges. Thus Boltzmann statistics does not lie on the anyon interpolation between Fermi and Bose statistics.
References [1] F. Wilczek, Phys. Rev. Lett. 48 (1982) 1144;49 (1982) 957. [2] A.S. Goldhaber, Phys. Rev. Lett. 49 (1982) 905; D.H. Kobe, Phys. Rev. Lett. 49 (1982) 1592; H.J. Lipkin and M. Peshkin, Phys. Lett. B 118 (1982) 385; R. Jackiw and A.N. Redlich, Phys. Rev. Lett. 50 (1983) 555; G.A. Goldin and D.H. Sharp, Phys. Rev. D 28 (1983) 830. [3] Y. Wu, Phys. Rev. Lett. 53 (1984) 111.
384
PHYSICS LETTERS B
20 November 1986
ANYON STATISTICS WITH SINGLE-VALUED WAVE FUNCTIONS R a m a n S U N D R U M 1 a n d L.J. T A S S I E
Department of Theoretical Physics, Research School of Physical Sciences, Australian National University, GPO Box 4, Canberra, ACT2601, Australia Received 23 August 1985; revised manuscript received 1 August 1986
The fractional statistics of the anyons proposed by Wilczek are demonstrated in a simple manner using single-valued wave functions. Taking the magnetic flux tube and charge comprising each anyon to be bosons, the wave function for two identical anyons is symmetrical with respect to the interchange, but for e~ = ~r, where e is the charge and ¢, the magnetic flux in each anyon, the anyons behave as fermions, and for other values of eO, the anyons obey intermediate statistics.
Wilczek [ 1] has proposed a simple model in two dimensions of particles obeying fractional statistics and has called such particles anyons. An anyon is a composite of a charge e attached to a tube of magnetic flux qb. Although the properties of anyons have been extensively discussed [ 1 - 3 ] , the discussion involves such complications as singular gauge transformations, multi-valued wave functions, the distinction between the use of canonical and mechanical angular m o m e n t u m as the rotation generator to interchange anyons, or arguments involving the spin of an anyon. The purpose of this letter is to give a simple demonstration of the statistics of anyons without such complications. We find the solutions to the wave equation using a hamiltonian for two identical anyons. The flux tube and charge comprising each anyon are taken to be bosons so that the solution to the wave equation for two identical anyons must be symmetric under interchange of the anyons. Nevertheless it is shown that for the product of charge and flux, e ~ = 7r, the solutions describe fermionic behaviour, and that fractional statistic occurs for other values of e ~ . Units in which ~/= c = 1 are used throughout. We consider the hamiltonian for two identical anyons moving in two dimensions and each carrying charge e and magnetic flux ~ . Wu [3] gives this hamiltonian when the electrostatic repulsion between anyons is neglected. When the repulsion potential is included the full hamiltonian becomes H = ([_iV1 _ 2eA(Xl _ x 2 ) ] 2 + [_iV 2 _ 2eA(x2 _ X l ) ] 2}/2m + e2/lXl _x21 '
(1)
where
a(r,O) ~/2~r
(2)
=
in polar coordinates. Using centre-of-mass and relative coordinates R =(x 1 +x2)/2,
r = x I - x 2,
(3)
we have
H = H R +Hr,
(4)
where i Present address: Department of Physics, Yale University, New Haven, CT 06520, USA. 381
Volume 180, number 4
H R = (-iVR)2/4m,
PHYSICS LETTERS B
20 November 1986
H r = ( - i V r - 2eA(r))2/m + e2/Irl.
(5,6)
We denote Hr, for @ = 0, by H O. Then H R + H 0 is just the hamiltonian for a pair of particles with charge e, interacting purely through electrostatic repulsion. Since the hamiltonian (4) separates, we have eigenfunctions of H of the form
~b(R,r) = [2(R )x(r),
(7)
where ~2(R) and x(r) are eigenfunctions o f H R and H r respectively. In polar coordinates, r = (r, 0),
1 ~[~r~2+la+ 1 (a_~r -~ -~
Hr = - m
ia
)2 1
e2 + --'r
(8)
where a = ecb/~r. Let QXs(r) denote the eigenfunctions of 1 [~ 2
1 a ar
- m ~ r 2 +r
s2~+d r 2] r
s>0,
'
with eigenvalues Ex, normalised so that
21r fQ*X'(r)QsX(r)r dr = 8(X - X'). Then the single-valued eigenfunctions o f H r are of the form x(r,0) =Ql~_al(r) exp(i/0),
IE z.
(9)
Since we assume the components of the anyons are bosons, we require symmetry under interchange o f x 1 and x2, so the physical eigenfunctions o f H r are restricted to those of the form
x(r, 0) = Q~2l-~ i(r) exp(i210),
l E Z.
(10)
Thus the allowed eigenfunctions of H are of the form ff(R,r) = ffZ(R)a~t_~l(r) exp(i210),
IEZ,
(11)
where ~(R) is an arbitrary eigenfunction o f H R . We note that the generator of relative rotations for wave functions is the canonical angular m o m e n t u m - i a / a o since the wave functions are single-valued,
~(R,r,O + A0) = ~2(R)a~2l_al(r) exp[i2l(O + A0)] = exp[iA0(--i 0/a0)] ~b(R,r,O).
(12)
But we shall see that these anyons display intermediate statistics. If gp = 0, a = 0, then the relative eigenfunctions Q~2!l(r) exp(i210) are eigenfunctions o f H O, and so the composites are just bosons interacting purely under electrostatic repulsion. We see that for edp = rr, ~ = 1, the relative eigenfunction x(r) = Qi2l_ll(r) x exp(i210) = exp(i0) (Q~21_ll(r) e x p [ i ( 2 l - 1)0]}
(13)
is just a phase exp(i0) multiplied by an antisymmetric eigenfunction o f H O, i.e. an eigenfunction corresponding to identical fermions interacting purely by electrostatic repulsion. However, the factor exp(i0) does not affect probabilities, and so for et = 1 the anyons behave precisely as charged fermions. For 0 < ~ < I, an intermediate effect occurs corresponding to neither charged bosons nor charged fermions. This can be considered to be intermediate statistics. The variation of the statistics with ct can be illustrated by examining a scattering experiment. Consider two 382
Volume 180, number 4
PHYSICS LETTERS B
20 November 1986
identical anyons known to be at (r0/2, 0) and (r0/2,7r) in the centre-of-mass frame, so that the symmetric wave function describing them is q~(r) = (l/x/2) ([6(r - r012 ) + 6(r + r0/2)] }
*x x + G fOj~ll_~l(ro)Qit_~l(r) exp[U(0 + ~)1 dX) • IEZ
At a time t later, the wave function is
X(r) X/7(/~Z fQI l-~l(rO)Qll-~l( ) exp(i/O) e x p ( - i E x t ) dk +
*h h ) ~, fQMt_o~l(rolQit_,~i(r) expli/(0 + ¢r)l e x p ( - i E x t ) d~, ,
(14)
I~Z
which can be written as x(r) = (1/'v/21 exp(ia0)
[fc~(r,O) + exp(ilroOfa(r,O + n)],
(15)
where
f,~(r, o) = .~,_fQ~(,.o) Q~-~(r) exp [ i ( / -
cOO] exp(-iExt ) dh.
(16)
Note that in generalf a is not a 2n-periodic function of 0, though the physical wave function X clearly is by (14). The physically observable quantity is from (15) Ix(r) l2 = ½Ifa(r,O ) + exp(irca)fe~(r,O
+ lr) l2.
(17)
Consider the case q~ = 0, a = 0. Then from (16), fo(r, O) can be seen to be just the single-valued relative wave function for two distinguishable particles interacting only through their electrostatic repulsion, while fo(r, 0 + re) is the corresponding exchange amplitude. For a = 0, these two amplitudes are added in (17) and thus the composites are ordinary identical, charged bosons. From (16)we see that
fl(r,O) =fo(r,O),
(18)
and so for ~ = 1, [x(r)[2 =
½[fo(r,O) _fo(r, 0 + ~)12.
Thus for a = 1, the anyons behave like two ordinary identical, charged fermions. To further illustrate the result, consider the amplitude for two distinguishable charged particles to be at -r/2,
xd(r) =f0(r).
(19)
r/2 and (20)
It is clear from (17) and (20) that 2[Xa=0(0)[ 2 > 2 Ixct(0) l2 > 2 IXa=l(0)l 2 = 0.
(21) 383
Volume 180, number 4
PHYSICS LETTERS B
20 November 1986
By the continuity of wave functions, we have that, for r sufficiently near the origin, 2 I×~=0(r)l 2 > IXd(r)l 2 + I×d(-r)l 2 > 21×5 = l(r)l 2.
(22)
So the probability for two particles to be near each other is enhanced for a = 0 anyons and diminished for a = 1 anyons, relative to that for distinct charges, which is just the well-known result for identical bosons and identical fermions respectively. Also, from (16),
fo(r, n/2) =fo(r, 37r/2),
(23)
so
Ix~= l(r' ./2)12 = 0,
(24)
another phenomenon characteristic of fermion-fermion scattering. For 0 < a < 1, it can be seen from (17) that the composites show behaviour intermediate between that of bosons and that o f fermions. This behaviour cannot be simply related to the wave function for distinct charged particles,fo(r,O), as was done for a = 0 or 1, since in generalf a is not simply related to f0" It might be thought, since Bose and Fermi statistics are obtained when ~ = 0 and t~ = 1 respectively, that for some intermediate value o f a the anyons will behave as purely charged distinguishable particles (corresponding to Boltzmann statistics). This is easily seen not to be the case. By just considering the spectrum of relative angular momentum (canonical or mechanical) we see that the anyon system has allowed angular momenta separated by 2(h) because the wave functions are symmetric, while the angular momentum spacing for distinguishable particles is of course l(/i). So anyons are physically quite distinct from distinguishable charges. Thus Boltzmann statistics does not lie on the anyon interpolation between Fermi and Bose statistics.
References [1] F. Wilczek, Phys. Rev. Lett. 48 (1982) 1144;49 (1982) 957. [2] A.S. Goldhaber, Phys. Rev. Lett. 49 (1982) 905; D.H. Kobe, Phys. Rev. Lett. 49 (1982) 1592; H.J. Lipkin and M. Peshkin, Phys. Lett. B 118 (1982) 385; R. Jackiw and A.N. Redlich, Phys. Rev. Lett. 50 (1983) 555; G.A. Goldin and D.H. Sharp, Phys. Rev. D 28 (1983) 830. [3] Y. Wu, Phys. Rev. Lett. 53 (1984) 111.
384