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Contact interactions of anyons
Physics Letters B 268 ( 1991 ) 222-226 North-Holland
P H YSIC S L ETT ER S B
Contact interactions of anyons Cristina Manuel and Rolf Tarrach 1 Departament Estructura i Constituents de la Matkria, Facultat de Fisica, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain
Received 16 July 1991
We prove that anyons, in spite of the centrifugal barrier which characterizes them, can contact interact in the broadest sense, and usually do so breaking scale invariance. This is seen by trading the contact interaction for a boundary condition, or by regularizing the contact interaction, or finally by identifying s-wave anyons with s-wavebosons in higher dimensions.
Anyons are a gift o f two-dimensional space [ 1,2 ]. Two-dimensional systems exist in condensed matter physics, where anyons play an important role in the fractional q u a n t u m Hall effect [ 3 ], and may be relevant to high temperature superconductivity [4]. They have their q u a n t u m field theoretical setting in the Chern-Simons theories [ 5 ] and no doubt will be an important subject o f theoretical and experimental research in the future. Several reviews on anyon physics have been written [6,7 ]. Here we will address an issue of basic anyon quantum mechanics: contact interactions of anyons. Since this issue is related to a certain n u m b e r of recent publications, we will first shortly recall what are anyons and what is understood under contact interaction, and then proceed with the discussion of the literature. First, one anyon: this is a concept related to the multiple connectedness o f the two-dimensional rotation group and, o f course, prior to statistics. When the wavefunction o f a particle in two-dimensional quantum mechanics acquires under a 2re rotation a phase different from _+ 1 we call the particle an anyon [ 2 ]. Now, two identical anyons: this is a concept related to the multiple connectedness of the configuration space o f two identical particles in two dimensions [ 1 ]. When the wavefunction & t h e two identical particles in two-dimensional quantum mechanics acquires under the exchange o f the particles a phase different from _+ 1 we call the particles anyons. A genl Bitnetaddress: ROLF@EBUBECMI. 222
eralized spin-statistics theorem [8] relates both definitions: the fact that due to the abelian character of SO (2) spin is not quantized and thus can take a n y value has its counterpart in t h e fact that under exchange of identical particles the phase can take a n y value too, hence anyons [ 2 ]. Let us remind here too that for any spin the representations of the rotation group are one-dimensional: neither fermions nor anyons have components in two-dimensional quantum mechanics. For fermions this is also seen by recalling the two-component character of the two-dimensional Dirac equation, where the two components correspond to the particle-antiparticle degrees of freedom, but not to spin. Contact interactions are interactions represented by a Dirac delta distribution i n the quantum-mechanical equation of motion, be it the Schrrdinger or the Dirac equation. In two dimensions we know that Dirac deltas are too singular to allow for the standard analysis of quantum mechanics. There are basically two ways of treating them: regularizing them or substituting them by boundary conditions which ensure the self-adjointness of the hamiltonian [ 9,10 ]. We also know that for the Schrrdinger equation they are only nontrivial if attractive and they generally break scale invariance in spite of being classically scale invariant [ 1 1 ], a usual feature of q u a n t u m field theory but unusual in q u a n t u m mechanics. Finally they are known not to exist for Dirac particles [ ! 2 ]. It is thus enough to study the contact interactions o f anyons which satisfy a SchrSdinger equation. Before doing
0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
Volume 268, number 2
PHYSICS LETTERS B
so, let us shortly mention related work, most of which studies fermions in an Aharonov-Bohm field [ 13 ]. As just mentioned no contact interactions are allowed here but, as shown in ref. [ 14], the self-adjointness of the hamiltonian requires a boundary condition which relates both components of the spinor. On the other hand, two SchriSdinger equations with contact interaction are obtained by iteration from the original Dirac equation [15 ]. These specific contact interactions have a strength determined by the Aharonov-Bohm magnetic flux and they are due to the singularity of the gauge field in the original Dirac equation. Their study in refs. [ 15,16 ] reveals that the most outstanding feature of contact interactions in two dimensions, namely scale invariance breaking, is absent. It Will be seen from our work why this is so. Some other related work is found in ref. [17]. We will now prove that anyons can contact interact in the broadest sense, except in the limit in which they become fermions. The simplest wording for the proof is: in s-wave, anyons interpolate between s-wave bosons in two-dimensional and s-wave bosons in fourdimensional space, and these are known to contact interact below four space dimensions [ 18 ]. Consider a particle in a central potential in two space dimensions. The hamiltonian is
H=-
h2 (0 2 1 0 l~+V(r). ~-~\Or 2 + r o t + r o~ /
(1)
Because of the multiple connectedness of SO(2) a rotation of 27~ leads to an arbitrary phase in the wave function:
g~o(r, ~o+2zr) = e x p ( i 0 ) g/o(r, ~o) ,
(2)
where 0~ [0, n] (values 0~ (n, 2n) correspond to complex conjugate wavefunctions). The phase 0 counts windings around the origin. A wavefunction whose phase counts windings is well known from the Aharonov-Bohm effect [ 13 ]. We can thus trade the multivaluedness of the wavefunction for a magnetic flux ~ located at the origin and an electric charge q located at the particle, such that q ~ / h c = - O. This amounts to performing a gauge transformation
7t(r, ~o)=exp(--iO~o/2n) ~o(r, ~o)=~v(r, ~0+2n) under which the hamiltonian becomes
(3)
10 October 1991
h2
Ho=--~
02
10_+
~r2 + rOr
1
0 +i~_~n
-fi -~
+V(r) (4)
Since this is a separable operator and from (3) ~ ( r , {0) ~--~eim~ 6 ( r ) ,
m~Z,
(5)
so that the SchfiSdinger equation becomes
I
h2 [d2 - ~dfir
1d 2 + r dr
v2)+V(r)] 7
~(r)
=E0(r),
(6)
with
(7)
v- m+ O .
Then, as long as v ¢ 0 and as long the centrifugal barrier dominates over the potential at short distances, the wavefunction vanishes at the origin,
0(0) = ~,o(0, ~o)=0.
(8)
Consider two identical anyons. Now r and ~0are the relative coordinates. A rotation of zc already leads to a physically indistinguishable state, so that
~vo(r, q~+ re) = e x p ( i 0 ) ~to(r, (o)
(9)
and 0 determines the statistics. For bosons 0 = 0 and the wavefunction is symmetric, for fermions 0= ~rand the wavefunction is antisymmetric and for other values of 0 we speak about anyons. One can now go through the same steps as before, with qq~/hc=- 0 (two factors 2 cancel: both particles carry flux and charge, but we rotate only zr), q ~ 0 so that unwanted Coulomb interactions are avoided: The gauge transformation is q/(r, ~o)=exp(-i0~o/zr)
~to(r, ~o)=q/(r, ~o+zr)
(10)
and the transformed hamiltonian
( °l]
ro2 1 0 1 0 + i ~ /~o=- ~ - ~ L ~ + 7 ~ + 7 U~
+V(r) (11)
M being half the anyon mass. Eq. (5) becomes ~(r, ~) -~-e2imp' 0 ( r ) ,
mET],
(12)
SO that (6) still holds but with 223
Volume 268, number 2
PHYSICS LETTERS B
10 October 1991
measure dr z", precisely requires ( 16 ), ( 17 ) or v = 2 m + O1_
(13) du(r)
We are now ready to prove that anyons, in spite of the centrifugal barrier which usually makes their wavefunction vanish at the origin, can contact interact in the broadest sense, and usually do so breaking scale invariance, exactly as bosons do. Recall from the introduction that there are basically two ways o f dealing with Dirac deltas localized at the origin. One o f them is based on the fact that all that they can do is changing the regularity condition o f the wavefunction at the origin. By modifying this condition to one which involves both the wavefunction and its derivative one is taking into account the whole effect o f the Dirac delta interaction. Specifically, the general solution of (6) for V ( r ) - - 0 when r > 0, and for positive energy is
O(r)=AJ~(kr)+BY.(kr),
k=
h
'
(14)
where J . and Y. are the first and second kind Bessel functions, respectively. One then introduces the regular function
u(r)=-r"¢(r) ,
(15)
so that the standard regularity condition u(0)=0
u ( 0 ) + , t R o,~2. ~d u ( r ) .=o = 0 ,
(17)
where Ro is an arbitrary length. To see where this condition comes from notice that any matrix element of the hamiltonian can be rewritten with the help of (15) as co
f
( d2
rdrC~(r) ~
ld
+ r dr
v2)
;7 01(r)
0
]2 = 2 v i dr2"u~(r) (\ d~--~d2, , / ul(r).
(18)
0
It is now very easy to check that self-adjointness o f (d/drZ") 2 in LZ(~ +, dr2"), i.e. in the Hilbert space of square integrable functions on the hairline with 224
(19)
As we will see in more detail later, (16) and (19) are limiting cases o f the general scale invariance breaking condition (17), to which we turn now. F r o m the definition B tan ~ ( k ) = - ~ ,
(20)
one finds the phase shift tan ~ ( k ) sin 7rv = cos zcv-y (2/Rok)Z"F(1 + v ) / F ( 1 - v) '
(21)
where we limit ourselves to s-waves (m = 0), i.e. v ~ 1. Notice, that there is no contact scattering for two identical fermions, v = 1. For higher waves (m > 0) v > 1 and normalizability only holds for ( 16 ). For negative energy states, i.e. bound states, which certainly only exist for attractive Dirac deltas,
O(r)=CK.(x/-2EoM/h
2 r),
(22)
where K. is the modified Bessel function, which is normalizable for v < 1, and (17) with the upper sign leads to a single bound state o f energy
(16)
selects the free solution B = 0. With interaction at r = 0 one imposes the boundary condition
=0.
dr2V ,=o
Eo =
2h 2 ( F ( l + v ) ) MR 2 \F--~--~
1/" (23)
Again, there is no bound state for u= 1. From (21) with the upper sign and (23) one can write tan ft.(E) =
sin 7cv cos roy- ( - E o / E ) ~ "
(24)
The case (19) corresponds to R 2~ ~ and leads from (21 ) to d, (k) = g v; it is intermediate between the two situations considered above and still corresponds to an attractive interaction, with strength such that just the bound state merges into the continuum. This case corresponds to the one o f ref. [ 15 ], and indeed no scale is introduced because it tends to infinity. The case (16) corresponds to R2"-~0 in the lower sign expression (17). The interaction vanishes. The limit Ro2"-~0 in the upper sign expression (17) leads to collapse: the hamiltonian is unbounded from below. There are therefore contact interactions of anyons
Volume 268, number 2
PHYSICS LETTERS B
in the boundary condition formalism. They are always attractive. Only two identical fermions do not contact interact. Let us now reproduce these results in the formalism in which the Dirac delta is regularized so that the argument by which the wavefunction has a zero at the origin still goes through. It is well known that only attractive potentials o f this type has lead to interaction in two space dimensions [ 18 ]. Consider thus the circular well potential V(r)=-2MR~
c2 1+4#
O(R-r),
(25)
where c~_, is the first positive zero o f Ju-,, J u - l (cu_ ~) = 0 and # = 0/n. This potential can be solved immediately, and it provides a regularization of the Dirac delta potential, - 2 d ( r ) , because its range vanishes for R ~ 0 and because its depth diverges for R ~ 0 in such a way that 2-= ( 1 / 2 M ) zch2c2_ 1. It is of course its subdominant term which leads to breaking o f scale invariance and it is such that one finds for the upper sign o f (25) a unique s-wave bound state o f finite energy and a finite phase shift when R ~ 0 . One can work out these limits easily and the results are precisely (21 ) and (23) with v-- #. But of course now (16) holds all the time, as (25) is not singular at r = 0 and the interior solution ( r < R), J,, vanishes at r = 0 . Thus anyons interact at a range smaller than any positive number. For higher waves the very same potential (25) is trivial as R~O, that is it does not support b o u n d states nor scatters. As is characteristic of contact interactions, only S-waves notice them (this explains why the same interaction corresponds to different boundary conditions for different waves). Let us now consider three specific values o f u, v = 0 (identical bosons), ~,_!_2 (identical semions), and u = 1 (identical fermions). It is immediate to see that for these values (6) coincides with the s-wave hamiltonian for two identical bosons in two, three and four space dimensions and for the wavefunction ~t(r)=_e)(r), r~/Zq)(r) and rO(r) respectively. The standard results [ 9 ] o f Dirac delta potentials in two space dimensions, tan do(k) =
n log(E/-Eo)
three space dimensions,
'
(26)
tan do(k) = - ~
10 October 1991
,
(27)
and four space dimensions, tan do(k) = E o = 0 ,
(28)
are reobtained. The last result is just the triviality o f Dirac deltas in four space dimensions, where even bosons do not contact interact. It corresponds, according to our results, to identical fermion contact interaction which, indeed, is impossible, even in two space dimensions. Anyons, however, can contact interact, as bosons can. Their s-wave behaviour is equivalent to the bosonic s-wave behaviour in d = 2 ( 1 + t,) space dimensions, and Friedman's theorem states that below d = 4 bosons do allow for contact interactions. We thank J. Soto, P. Roy and J.I. Latorre for helpful comments. Financial support under contract AEN90-0033 is acknowledged. C.M. acknowledges the Ministerio de Educaci6n y Ciencia for an FPI grant.
References [ 1] J.M. Leinaas and Myrheim, Nuovo Cimento 37 B (1977) 1. [2]F. Wilczek, Phys. Rev. Lett. 48 (1982) 1144; 49 (1982) 957. [3] S.C. Zhang, T.H. Hansen and S. Kivelson, Phys. Rev. Lett. 62 (1989) 82. [ 4 ] A. Fetter, C. Hanna and R. Laughlin,Phys. Revl B 39 (1989) 9679; P.B. Wiegmann, Phys. Rev. Lett. 60 (1988) 821; A.M. Polyakov, Mod. Phys. Lett. A 3 (1988 ) 325. [5] S. Deser, R. Jackiw and R: Templeton, Phys. Rev Len. 48 (1982) 975; Ann. Phys. (NY) 140 (1982) 372. [6] R. Mackenzie and F. Wilczek, Intern. J. Mod. Phys. A 3 (1988) 2877. [7] R.B. Laughlin, Science 242 (1988) 525; F. Wilezek, States of anyon matter, Institute for Advanced Study, Princeton preprint IASSNS-HEP-90/29 (1990); M. Stone, Essentials ofanyons, University of Illinois preprint ILL-TH-89/4 ( 1989); G.S. Canright and S.M. Girvin, Science 247 (1990) 1197. [8] A.P. Balachandran, A. Daughton, Z.C. Gu, G. Marrno, R.D. Sorkin and A.M. Srivastava, Mod. Phys. Lett. A 5 (1990) 1575. [ 9 ] P. Gosdzinsky and R. Tarrach, Am. J. Phys. 50 ( 1991 ) 70. 225
Volume 268, number 2
PHYSICS LETTERS B
[ 10] R. Jackiw, Delta-function potentials in two- and threedimensional quantum mechanics, MIT preprint CTP 1937 (January 1991 ). [ 11 ] C. Thorn, Phys. Rev. D 19 (1979) 639. [ 12 ] E.C. Svendsen, J. Math. Anal. Appl. 80 ( 1981 ) 551; F.A.B. Coutinho and Y. Nogami, Phys. Rev. A 42 (1990) 5716. [ 13] Y. Aharonov and D. Bohm, Phys. Rev. 115 (1959) 485. [14]P. Gerbert and R. Jackiw, Commun. Math. Phys. 124 (1989) 229;
226
10 October 1991
Pi Gerbert, Phys. Rev. D 40 (1989) 1346; M.G. Alford and F. Wilczek, Phys. Rev. Lett. 62 (1989) 1071. [ 15] C.R. Hagen, Phys. Rev. Lett. 64 (1990) 503. [ 16 ] T. Blum, C.R. Hagen and R. Ramaswany, Phys. Rev. Lett. 64 (1990) 709. [ 17 ] R. Jackiw and S.-Y. Pi, Phys. Rev. Lett. 64 (1990) 2969; S.M. Girvin et al., Phys. Rev. Lett. 65 (1990) 1671. [ 18] C. Friedman, J. Funct. Anal. 10 (1972) 346.
P H YSIC S L ETT ER S B
Contact interactions of anyons Cristina Manuel and Rolf Tarrach 1 Departament Estructura i Constituents de la Matkria, Facultat de Fisica, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain
Received 16 July 1991
We prove that anyons, in spite of the centrifugal barrier which characterizes them, can contact interact in the broadest sense, and usually do so breaking scale invariance. This is seen by trading the contact interaction for a boundary condition, or by regularizing the contact interaction, or finally by identifying s-wave anyons with s-wavebosons in higher dimensions.
Anyons are a gift o f two-dimensional space [ 1,2 ]. Two-dimensional systems exist in condensed matter physics, where anyons play an important role in the fractional q u a n t u m Hall effect [ 3 ], and may be relevant to high temperature superconductivity [4]. They have their q u a n t u m field theoretical setting in the Chern-Simons theories [ 5 ] and no doubt will be an important subject o f theoretical and experimental research in the future. Several reviews on anyon physics have been written [6,7 ]. Here we will address an issue of basic anyon quantum mechanics: contact interactions of anyons. Since this issue is related to a certain n u m b e r of recent publications, we will first shortly recall what are anyons and what is understood under contact interaction, and then proceed with the discussion of the literature. First, one anyon: this is a concept related to the multiple connectedness o f the two-dimensional rotation group and, o f course, prior to statistics. When the wavefunction o f a particle in two-dimensional quantum mechanics acquires under a 2re rotation a phase different from _+ 1 we call the particle an anyon [ 2 ]. Now, two identical anyons: this is a concept related to the multiple connectedness of the configuration space o f two identical particles in two dimensions [ 1 ]. When the wavefunction & t h e two identical particles in two-dimensional quantum mechanics acquires under the exchange o f the particles a phase different from _+ 1 we call the particles anyons. A genl Bitnetaddress: ROLF@EBUBECMI. 222
eralized spin-statistics theorem [8] relates both definitions: the fact that due to the abelian character of SO (2) spin is not quantized and thus can take a n y value has its counterpart in t h e fact that under exchange of identical particles the phase can take a n y value too, hence anyons [ 2 ]. Let us remind here too that for any spin the representations of the rotation group are one-dimensional: neither fermions nor anyons have components in two-dimensional quantum mechanics. For fermions this is also seen by recalling the two-component character of the two-dimensional Dirac equation, where the two components correspond to the particle-antiparticle degrees of freedom, but not to spin. Contact interactions are interactions represented by a Dirac delta distribution i n the quantum-mechanical equation of motion, be it the Schrrdinger or the Dirac equation. In two dimensions we know that Dirac deltas are too singular to allow for the standard analysis of quantum mechanics. There are basically two ways of treating them: regularizing them or substituting them by boundary conditions which ensure the self-adjointness of the hamiltonian [ 9,10 ]. We also know that for the Schrrdinger equation they are only nontrivial if attractive and they generally break scale invariance in spite of being classically scale invariant [ 1 1 ], a usual feature of q u a n t u m field theory but unusual in q u a n t u m mechanics. Finally they are known not to exist for Dirac particles [ ! 2 ]. It is thus enough to study the contact interactions o f anyons which satisfy a SchrSdinger equation. Before doing
0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
Volume 268, number 2
PHYSICS LETTERS B
so, let us shortly mention related work, most of which studies fermions in an Aharonov-Bohm field [ 13 ]. As just mentioned no contact interactions are allowed here but, as shown in ref. [ 14], the self-adjointness of the hamiltonian requires a boundary condition which relates both components of the spinor. On the other hand, two SchriSdinger equations with contact interaction are obtained by iteration from the original Dirac equation [15 ]. These specific contact interactions have a strength determined by the Aharonov-Bohm magnetic flux and they are due to the singularity of the gauge field in the original Dirac equation. Their study in refs. [ 15,16 ] reveals that the most outstanding feature of contact interactions in two dimensions, namely scale invariance breaking, is absent. It Will be seen from our work why this is so. Some other related work is found in ref. [17]. We will now prove that anyons can contact interact in the broadest sense, except in the limit in which they become fermions. The simplest wording for the proof is: in s-wave, anyons interpolate between s-wave bosons in two-dimensional and s-wave bosons in fourdimensional space, and these are known to contact interact below four space dimensions [ 18 ]. Consider a particle in a central potential in two space dimensions. The hamiltonian is
H=-
h2 (0 2 1 0 l~+V(r). ~-~\Or 2 + r o t + r o~ /
(1)
Because of the multiple connectedness of SO(2) a rotation of 27~ leads to an arbitrary phase in the wave function:
g~o(r, ~o+2zr) = e x p ( i 0 ) g/o(r, ~o) ,
(2)
where 0~ [0, n] (values 0~ (n, 2n) correspond to complex conjugate wavefunctions). The phase 0 counts windings around the origin. A wavefunction whose phase counts windings is well known from the Aharonov-Bohm effect [ 13 ]. We can thus trade the multivaluedness of the wavefunction for a magnetic flux ~ located at the origin and an electric charge q located at the particle, such that q ~ / h c = - O. This amounts to performing a gauge transformation
7t(r, ~o)=exp(--iO~o/2n) ~o(r, ~o)=~v(r, ~0+2n) under which the hamiltonian becomes
(3)
10 October 1991
h2
Ho=--~
02
10_+
~r2 + rOr
1
0 +i~_~n
-fi -~
+V(r) (4)
Since this is a separable operator and from (3) ~ ( r , {0) ~--~eim~ 6 ( r ) ,
m~Z,
(5)
so that the SchfiSdinger equation becomes
I
h2 [d2 - ~dfir
1d 2 + r dr
v2)+V(r)] 7
~(r)
=E0(r),
(6)
with
(7)
v- m+ O .
Then, as long as v ¢ 0 and as long the centrifugal barrier dominates over the potential at short distances, the wavefunction vanishes at the origin,
0(0) = ~,o(0, ~o)=0.
(8)
Consider two identical anyons. Now r and ~0are the relative coordinates. A rotation of zc already leads to a physically indistinguishable state, so that
~vo(r, q~+ re) = e x p ( i 0 ) ~to(r, (o)
(9)
and 0 determines the statistics. For bosons 0 = 0 and the wavefunction is symmetric, for fermions 0= ~rand the wavefunction is antisymmetric and for other values of 0 we speak about anyons. One can now go through the same steps as before, with qq~/hc=- 0 (two factors 2 cancel: both particles carry flux and charge, but we rotate only zr), q ~ 0 so that unwanted Coulomb interactions are avoided: The gauge transformation is q/(r, ~o)=exp(-i0~o/zr)
~to(r, ~o)=q/(r, ~o+zr)
(10)
and the transformed hamiltonian
( °l]
ro2 1 0 1 0 + i ~ /~o=- ~ - ~ L ~ + 7 ~ + 7 U~
+V(r) (11)
M being half the anyon mass. Eq. (5) becomes ~(r, ~) -~-e2imp' 0 ( r ) ,
mET],
(12)
SO that (6) still holds but with 223
Volume 268, number 2
PHYSICS LETTERS B
10 October 1991
measure dr z", precisely requires ( 16 ), ( 17 ) or v = 2 m + O1_
(13) du(r)
We are now ready to prove that anyons, in spite of the centrifugal barrier which usually makes their wavefunction vanish at the origin, can contact interact in the broadest sense, and usually do so breaking scale invariance, exactly as bosons do. Recall from the introduction that there are basically two ways o f dealing with Dirac deltas localized at the origin. One o f them is based on the fact that all that they can do is changing the regularity condition o f the wavefunction at the origin. By modifying this condition to one which involves both the wavefunction and its derivative one is taking into account the whole effect o f the Dirac delta interaction. Specifically, the general solution of (6) for V ( r ) - - 0 when r > 0, and for positive energy is
O(r)=AJ~(kr)+BY.(kr),
k=
h
'
(14)
where J . and Y. are the first and second kind Bessel functions, respectively. One then introduces the regular function
u(r)=-r"¢(r) ,
(15)
so that the standard regularity condition u(0)=0
u ( 0 ) + , t R o,~2. ~d u ( r ) .=o = 0 ,
(17)
where Ro is an arbitrary length. To see where this condition comes from notice that any matrix element of the hamiltonian can be rewritten with the help of (15) as co
f
( d2
rdrC~(r) ~
ld
+ r dr
v2)
;7 01(r)
0
]2 = 2 v i dr2"u~(r) (\ d~--~d2, , / ul(r).
(18)
0
It is now very easy to check that self-adjointness o f (d/drZ") 2 in LZ(~ +, dr2"), i.e. in the Hilbert space of square integrable functions on the hairline with 224
(19)
As we will see in more detail later, (16) and (19) are limiting cases o f the general scale invariance breaking condition (17), to which we turn now. F r o m the definition B tan ~ ( k ) = - ~ ,
(20)
one finds the phase shift tan ~ ( k ) sin 7rv = cos zcv-y (2/Rok)Z"F(1 + v ) / F ( 1 - v) '
(21)
where we limit ourselves to s-waves (m = 0), i.e. v ~ 1. Notice, that there is no contact scattering for two identical fermions, v = 1. For higher waves (m > 0) v > 1 and normalizability only holds for ( 16 ). For negative energy states, i.e. bound states, which certainly only exist for attractive Dirac deltas,
O(r)=CK.(x/-2EoM/h
2 r),
(22)
where K. is the modified Bessel function, which is normalizable for v < 1, and (17) with the upper sign leads to a single bound state o f energy
(16)
selects the free solution B = 0. With interaction at r = 0 one imposes the boundary condition
=0.
dr2V ,=o
Eo =
2h 2 ( F ( l + v ) ) MR 2 \F--~--~
1/" (23)
Again, there is no bound state for u= 1. From (21) with the upper sign and (23) one can write tan ft.(E) =
sin 7cv cos roy- ( - E o / E ) ~ "
(24)
The case (19) corresponds to R 2~ ~ and leads from (21 ) to d, (k) = g v; it is intermediate between the two situations considered above and still corresponds to an attractive interaction, with strength such that just the bound state merges into the continuum. This case corresponds to the one o f ref. [ 15 ], and indeed no scale is introduced because it tends to infinity. The case (16) corresponds to R2"-~0 in the lower sign expression (17). The interaction vanishes. The limit Ro2"-~0 in the upper sign expression (17) leads to collapse: the hamiltonian is unbounded from below. There are therefore contact interactions of anyons
Volume 268, number 2
PHYSICS LETTERS B
in the boundary condition formalism. They are always attractive. Only two identical fermions do not contact interact. Let us now reproduce these results in the formalism in which the Dirac delta is regularized so that the argument by which the wavefunction has a zero at the origin still goes through. It is well known that only attractive potentials o f this type has lead to interaction in two space dimensions [ 18 ]. Consider thus the circular well potential V(r)=-2MR~
c2 1+4#
O(R-r),
(25)
where c~_, is the first positive zero o f Ju-,, J u - l (cu_ ~) = 0 and # = 0/n. This potential can be solved immediately, and it provides a regularization of the Dirac delta potential, - 2 d ( r ) , because its range vanishes for R ~ 0 and because its depth diverges for R ~ 0 in such a way that 2-= ( 1 / 2 M ) zch2c2_ 1. It is of course its subdominant term which leads to breaking o f scale invariance and it is such that one finds for the upper sign o f (25) a unique s-wave bound state o f finite energy and a finite phase shift when R ~ 0 . One can work out these limits easily and the results are precisely (21 ) and (23) with v-- #. But of course now (16) holds all the time, as (25) is not singular at r = 0 and the interior solution ( r < R), J,, vanishes at r = 0 . Thus anyons interact at a range smaller than any positive number. For higher waves the very same potential (25) is trivial as R~O, that is it does not support b o u n d states nor scatters. As is characteristic of contact interactions, only S-waves notice them (this explains why the same interaction corresponds to different boundary conditions for different waves). Let us now consider three specific values o f u, v = 0 (identical bosons), ~,_!_2 (identical semions), and u = 1 (identical fermions). It is immediate to see that for these values (6) coincides with the s-wave hamiltonian for two identical bosons in two, three and four space dimensions and for the wavefunction ~t(r)=_e)(r), r~/Zq)(r) and rO(r) respectively. The standard results [ 9 ] o f Dirac delta potentials in two space dimensions, tan do(k) =
n log(E/-Eo)
three space dimensions,
'
(26)
tan do(k) = - ~
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,
(27)
and four space dimensions, tan do(k) = E o = 0 ,
(28)
are reobtained. The last result is just the triviality o f Dirac deltas in four space dimensions, where even bosons do not contact interact. It corresponds, according to our results, to identical fermion contact interaction which, indeed, is impossible, even in two space dimensions. Anyons, however, can contact interact, as bosons can. Their s-wave behaviour is equivalent to the bosonic s-wave behaviour in d = 2 ( 1 + t,) space dimensions, and Friedman's theorem states that below d = 4 bosons do allow for contact interactions. We thank J. Soto, P. Roy and J.I. Latorre for helpful comments. Financial support under contract AEN90-0033 is acknowledged. C.M. acknowledges the Ministerio de Educaci6n y Ciencia for an FPI grant.
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