Supersymmetric anyon quantum mechanics

Physics Letters B 274 (1992) 59-64 North-HoUand

PHYSICS LETTERS B

Supersymmetric anyon quantum mechanics Pinaki Roy l and Rolf Tarrach

2

Departament Estructura i Constituyentes de la Mat~ria, Facultat de Fisica, Universitat de Barcelona, Diagonal 647, E-8028 Barcelona, Spain

Received 11 October 1991

We develop a framework for anyons within supersymmetric quantum mechanics. We prove that supersymmetry can be realized spectrally but necessarily requires a contact interaction which modifies in a specific way the usual regularity condition of the wavefunction at the origin.

It is now well known [ 1-3 ] that particles in two space dimensions can have fractional statistics which correspond to neither bosons nor fermions. These particles are called anyons [ 4 ]. A feature characterizing anyons is that under exchange of positions or rotations by 2n the anyon wavefunction acquires a phase 0. This statistical phase or continuous spin can be understood in terms of a singular gauge interaction closely related to the A h a r o n o v - B o h m effect [ 5 ] as well as the A h a r o n o v - C a s h e r effect [ 6 ]. Anyons have been studied from various viewpoints. In particular the symmetry aspect of free anyons have been considered [ 7 ]. In the present paper we study how anyons can be realized within the framework of supersymmetry and our investigation will be within the context of supersymmetric quantum mechanics [ 8,9 ]. It may be pointed out here that very recently anyons in supersymmetric q u a n t u m field theory or statistical mechanics have been studied [ 10-12 ]. We also note that, although in quantum mechanics the relation of anyons to supersymmetry has to our knowledge not been discussed before, some cases related to the present one have been considered previously [ 13,14 ]. Here we shall analyze general supersymmetric anyon quantum mechanics. In particular we shall closely analyze which of the various boundary conditions allowing for anyons are compatible with supersymmetry. As we will see, supersymmetric anyons always contact interact nontrivially.-Also the question o f supersymmetry breaking will be considered. Let us start recalling the form o f the hamiltonian o f a supersymmetric particle in two dimensions,

l

H=~(Qx+a~+i[Qx,

Qy]az)=

0)

H_

'

where Q = (Qx, Qy) are the supercharges [ 8 ]. R e m e m b e r that the multivaluedness o f the anyon wavefunction can always be traded for a pointlike magnetic field o f finite flux of A h a r o n o v - B o h m type by a suitable gauge transformation. Thus for "free" (i.e. with only the defining ingredients) supersymmetric anyons one would have Q=P-eA,

(2)

with the singular gauge field given by Permanent address: Indian Statistical Institute, Calcutta, India. 2 Bitnet address: ROLF@EBUBECM1. 0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

59

Volume 274, number 1

q~ ( - y , x) p2

A = 2n

PHYSICS LETTERS B

2 January 1992

q~ 2~V~°,

(3)

where p 2= x z+ y 2 and ~0= arctan (y/x). This gauge field indeed leads to a magnetic field

B= V×A=EaOiAJ = _ i [Qx, Qy] = O S ( r ) ,

(4)

e

which is pointlike and of flux O, related to the phase 0 through 0 = - e q ) . We learn from (4) and (1) that supersymmetric anyons necessarily have zero range or contact interactions. Contact interactions in two dimensions cannot be studied directly by working with a Dirac delta; instead one has to regulate the interaction or trade it for a boundary condition at the interaction point [ 15,16 ]. By the use of either of these procedures they have been studied recently for anyons [ 17 ], and they exist, in a variety of forms. Thus the two hamiltonians of ( 1 ) have solutions, but this is not enough if one wants supersymmetry to hold spectrally, which is what matters for physics. This furthermore requires that the solutions of one hamiltonian be the supersymmetric partners of the solutions of the other hamiltonian, except, at most, the zero energy ground state, if supersymmetry is not broken spontaneously. This is then the subject of this study: whether contact interactions ofanyons provide enough freedom to make the spectra of both supersymmetric sectors match (except the zero energy eigenvalue, at most). Notice that defining eq~> 0, the zero range interaction of H+ is attractive and exactly of the same strength as the one of H_, which is repulsive. We choose to work in the boundary condition formalism. In order to do so let us introduce the supercharge

Q=Qx+iQy,

(5)

which in polar coordinates reads Q=exp(i~o)

(-

10

~+p~+R(p,~o)+i

.;,.)

a_,

(6)

with adjoint Q+=

(.olo

- p ~ p p - p~-~ +R(p,~o) - i I ( ~ o )

a+ exp(-i~0) ,

(7)

and where for "free" supersymmetric anyons, for which (3) holds, R(p, ~o)=0 and I(p, ~ 0 ) = - e O / 2 n . The departure of I and R from these values reflects additional supersymmetric interactions, beyond the defining interaction of"free" supersymmetric anyons. The hamiltonian ( 1 ) can be written as

H=I{Q+,Q},

(8)

and it reads from (6) and (7)

1 { 02

10

1 02

0

OR(p,~o)

+ i ~pp + i - - 0p

H = - ~ k 0 - ~ + p ~ + - - - - + Z° i) R p 2( p0 ,q~) 2

R (p' ~°) P

RE(p, ~0)

(9) For a rotational invariant system R and I only depend on p and writing the wavefunction as ( exp(im~o) ~+(p) ] ~(r) ----\ e x p [ i ( m + 1 )~0] q~_( p ) ] '

60

meZ

(10)

Volume274, number 1

PHYSICSLETTERSB

2 January 1992

one obtains the radial hamilton~an

m2+rn+~ +2iR(p)_~_(2m+l,I(p___))_R2(p)_Ia(P)+iR,(p)+i~.~)

l(d 2 1 d H=-2\dp2+pc ~

p2

l(['(p) m+½

+ 2\ p

__

_ p2

_

p - )T

I(p)~a p2 "] z"

192

(11)

The dependence in R (p) as well as the terms linear in derivatives can be gauged away by the transformation #

(0+(P)~

~peXp(-i

f o

\¢-(P)]

'

"~/~+(P)'~

(12)

which leads to the transformed hamiltonian

H=-

l(I'(p)m+½+/(p)~ 1 ( m + 2~ t ( p ) )2 2 dp2 -I- 2 +2\ p p2 ] crz,

1 d__~_ ~

(13)

acting on the Hilbert space ~_+(p)e L 2(R +, dp). We will limit ourselves to additional interactions which are regular, i.e. I(p) + e~/2n~p a for small p. Notice that H in (13) is supersymmetric in •+ with supercharge Q = ( - i ~ cd -iW(x))a_

(14)

and

W ( x ) = - °t + V(X)

a = m + ½ - ~ - eq~ ~

(15)

with V(x) ~x. The contact interaction (4) is not seen as such in (13) but instead is reflected in the boundary condition which has to be satisfied at x = 0 to ensure self-adjointness of (13) on ~+. For regular potentials these boundary conditions are well known [ 1 ]. They are (i)

~(0)=0,

~,'(0)~0,

(ii) ~u'(0)=0,

~u(0)#0,

(iii)~u(0)#0,

~u'(0)#0,

~'(0)=c~(0),

ce~.

(16)

Notice that because of the superposition principle c has to be independent of 9'. The boundary conditions (16) are the only ones for which a consistent quantum mechanics can be defined on the hairline for a regular potential. Let us recall that (i) is the standard regularity condition while (ii) and (iii) imply an interaction at the origin, that is, a Dirac delta. The converse is not true as Dirac deltas are often trivial, in particular any repulsive ones are, i.e. correspond to (i) and have thus no physical consequences whatsoever [ 15-17 ]. But ( 13 ) reads 1

d2

1 or(a-T- 1 )

H+_= - 2dx~2 + 2

x2

+V±(x)

(17)

where the singular part is explicitly shown. With v_+= [ot T- 11 the hamiltonians in (17) are of the type 1 d2

V2-1

H=-~dx--7+~+V(x),

v>0.

(18)

Let us now obtain for them the boundary conditions analogous to ( 16 ) which ensure that ( 18 ) is self-adjoint. Notice 61

Volume274, number 1

PHYSICSLETTERSB

dxqJ*~(x) -~dx---2+--~xz-+V(x)

xdx¢~(x)

~u,(x)=

o

2 January 1992

2dx 2

2xdx + ~ x ~ + V(x) 0, (x)

0 oo

0

=

dxZ"u~(x)

-u

+

jul(x),

(19)

0

where

~'(x) -xl/ZO(x) --x'/2-~u(x)

(20)

and ~(x)eLZ(~+,dx),

q~(x)~L2(~+,xdx),

u(x)~L2(R+,x~-2"dx).

(21)

One now sees from the last two equalities of (19) and by analogy with (16) the self-adjointness of (18) requires any of the following boundary conditions to hold: (i)

u(O)=O,

du(x)[ dx2,

SO, xmO

(ii)

du(x)] dx2~

=0,

u(0)~0,

S0,

u(0)¢0

x=0

du(x)] (iii) dx2~ X=0

du(x)[ ~ '

dx

=cu(x) Ix=o

ceR '

(22) "

As an example consider V(x) = 0. Then the general solution of (18) is, for positive energy and 0 < u < 1,

u(x)=x"[aJ~(kx)+bY,(kx) ] , k=x/~-E,

(23)

which admits all three choices of (22). For u i> 1 normalizability implies necessarily b= 0 and only condition (i) is allowed. This just reflects the fact that contact interactions are not seen if the centrifugal barrier is strong enough. Recall finally that a single hound state exists for 0 < u< 1 for the condition (iii) when c < 0 which corresponds to an attractive enough Dirac delta [ 17 ]. Let us come back to supersymmetry, i.e. to H+ and H_ of (17). According to the previous analysis they have to be supplemented with boundary conditions of the type (22) to ensure self-adj ointness. These boundary conditions cannot be independent of H+ and H_ if one wants supersymmetry to hold spectrally, which is what matters for physics. On R + we have (8), i.e. the algebraic realization of supersymmetry is not enough to imply the spectral degeneracy characteristic of supersymmetry. This requires furthermore that an eigenfunction of H+, which has to satisfy the boundary conditions characteristic of the domain of self-adjointness of H+, transforms under supersymmetry into an eigenfunction of H_, which satisfies the boundary conditions characteristic of the domain of self-adjointness of H_. Otherwise boundary conditions will explicitly break supersymmetry. Indeed, if ~t+ (x) are eigenstates of H e which are supersymmetric partners, one has ix/~u-(x)=i or

62

( - dx d + xa - v ( x ) ) ~ , + ( x ) ,

(24)

Volume 274, number 1

u_ (x) = - x . . . . . .

PHYSICS LETTERS B

(

d

1 ( ½_ v+ - a ) u+ (x) + 2 ~'+ x 2~+ dx2~l---~U+(X) +xV(x)

2 January 1992

)

u+ (x) .

(25)

Then the following cases hold: (1) ol>_.-~ (m>_. 1 ) = ~ , + 1> 1. F r o m the discussion following (23) we know that normalizability requires u + (0) = 0. Is this compatible with supersymmetry, i.e. with (25 )? it can be seen that it is, that is u+ (0) = 0 =~u_(0) = 0 .

(26)

(2) 3 > a > ½ (rn = 0) =~ 1 > v + > 0, v_ > 1. Normalizability implies u_ (0) = 0. Supersymmetry implies from (25) u _ ( 0 ) = 0 =~u+(0) = 0 .

(27)

(3) ½> ot > - ½ (m = - 1 ) =~ 1 > v_+> 0. Thus all three boundary conditions (22) are allowed by normalizability. H o w does supersymmetry relate them? Specifying (25) to this case gives x/2Eu_(x)=-

2v+

dx2~+ +x . . . . . V(x)u+(x)

(28)

and

x / ~ ddux 2_V _( x )

( --

d )u+(x) E-½V2(x)-v+V(x)x2~+-ldx2~+ v_

(29)

where the Schrtidinger equation has been used for u+ (x). Then one obtains (i)

u+(0)=0=~

(ii)

du+(x) I a~2,+

du_(x) I dx2.U_

x=0

--.~-0,

=0 ~u_(0)=0,

X~0

.... d u + ( x ) 111) ~ x = O

d u - (x)l =CU+(0) *

dx2V -

E u _ (0)

Ix=o-- 2cv_v+

(30) "

The first two cases lead to allowed boundary conditions for u_ (x), but not the last one, because o f the energy dependence which appears. For or< - ½ the analysis is analogous to ( 1 ) and (2). The cases o~= + ½ lead to v_+= 0 so that our analysis o f ( 18 ) does not apply. But we do not need the corresponding analysis, as for anyons v > 0. The case (3) requires some comments. The unavoidable presence o f a boundary condition of type (22) (ii) reflects the Dirac delta of (4), which only has consequences on the wavefunctions o f the sector where both superpartners have an effective angular m o m e n t u m v smaller than 1. With our conventions e ~ > 0 actually only ( 30 ) (ii) applies, since only attractive contact interactions can modify the wavefunction at the origin. N o w we come to the question o f supersymmetry breaking. To this end let us note that the zero energy ground states are

~,.s.(x):N+_exp[+idx'(~,-V(x'))],

(31)

so that normalizability requires V(x) ~/3/x% y~< 1 for large x, as one can easily verify. Indeed, for any value of ot one finds values offl for which one of the solutions (31 ) is normalizable. Since in many cases the value o f p for which this happens does not depend on o~, and a varies with the orbital angular m o m e n t u m m, one can have 63

Volume 274, number 1

PHYSICS LETTERS B

2 January 1992

o n e g r o u n d state per wave, i.e. i n f i n i t e l y m a n y g r o u n d states. T h i s is n o t surprising i f o n e recalls a result b y A h a r o n o v a n d C a s h e r o n g r o u n d states i n t w o - d i m e n s i o n a l m a g n e t i c fields [ 18 ]: a n i n f i n i t e flux i m p l i e s a n i n f i n i t e n u m b e r o f zero energy states. It m i g h t be worth n o t i n g that the fact that the allowed b o u n d a r y c o n d i t i o n s d e p e n d o n the orbital a n g u l a r m o m e n t u m m (a feature characteristic o f c o n t a c t i n t e r a c t i o n s ) is n o t i n c o n t r a d i c t i o n with the fact that the b o u n d a r y c o n d i t i o n has to be the same for all w a v e f u n c t i o n s for which the s u p e r p o s i t i o n p r i n c i p l e applies: w a v e f u n c t i o n s with different value o f rn can o n l y be a d d e d i n E2 since their h a m i l t o n i a n s in R + differ, a n d E2 does n o t have a b o u n d a r y at the origin. In c o n c l u s i o n : a n y o n s c a n be s u p e r s y m m e t r i c b o t h algebraically a n d spectrally b u t only by h a v i n g b o u n d a r y c o n d i t i o n s o f the type ( 2 2 ) (i) for all waves except for o n e for which a b o u n d a r y c o n d i t i o n o f type ( 2 2 ) (ii) is necessary in one o f the s u p e r s y m m e t r i c sectors. T h i s reflects a zero range interaction. T h i s f i n d i n g fits the result that in s u p e r s y m m e t r i c C h e r n - S i m o n s theories a four-field i n t e r a c t i o n is necessary [ 12 ] a n d reproduces the c o n n e c t i o n b e t w e e n h a r d core i n t e r a c t i o n s a n d s u p e r s y m m e t r y o f ref. [ 11 ]. O n e can o f course give u p spectral s u p e r s y m m e t r y [ 19,20 ] b u t here the issue was the opposite: if o n e could m a i n t a i n it for a n y o n s . N o t i c e finally that b o u n d a r y c o n d i t i o n s o f type ( 2 2 ) (iii) w h i c h i n t r o d u c e a scale a n d which are allowed for a n y o n s [ 17 ] are n e v e r c o m p a t i b l e with s u p e r s y m m e t r y .

Acknowledgement We t h a n k C. M a n u e l a n d J. Soto for helpful c o m m e n t s . F i n a n c i a l s u p p o r t u n d e r c o n t r a c t A E N 9 0 - 0 0 3 3 is acknowledged. P.R. t h a n k s the M i n i s t e r i o de E d u c a c i 6 n y C i e n c i a for a postdoctoral grant.

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