a r X i v :c o n d -m a t /9404068v 1 21 A p r 1994
Mapping of the Sutherland Hamiltonian to anyons on a ring
Shuxi Li and R.K.Bhaduri
Department of Physics and Astronomy,McMaster University,
Hamilton,Ontario,Canada L8S 4M1
Abstract
It is demonstrated that the Sutherland Hamiltonian is equivalent to particles interacting with the vector statistical interaction on the rim of a circle,to within a nonsingular gauge transformation.
PACS numbers:05.45.+b,03.65.Ge
1
Sutherland[1]obtained a periodic potential by allowing a pair of particles on the circum-ference of a circle to interact any number of times by an inverse square potential.Consider N such particles,each of mass M,described by the Sutherland Hamiltonian H s:
(2M
?x2i
+β(
β
L2 i>j
sinπ(x i?x j)
2M
N
i p i+|e|
A i(r i)=αˉh c
|r i?r j|2.(3)
Note that the strengthαof the interaction is dimensionless.For simplicity,from now on, we putˉh=c=|e|=2M=1.In Eq.(3),r i,r j are radial position vectors in the x-y plane for the i th and j th particles,and?z is a unit vector along the z-direction.The prime on the sum in Eq.(2)implies that j=i.Let the vectors r i,r j make anglesφi,φj with the x?axis, and de?neφij=(φi?φj),r ij=(r i?r j).It is then easy to show that,
r ij=2sin φij
2
+?eφi cos
φij
2α
′ j(?eφi??r i cotφij
?φi +
α
4 i′ j(sinφij
4 i′ j′′ k cotφij2?α2
2cot
φik
3
N(N?1)(N?2),(7)
yielding
H= i(?i?2(N?1))2+α22)?2?α2
Ψ=Ψs exp(?i
α
?φ2i +
α2
2
)?2?α2
12
N(N2?1)
is added to it,and the parameterαabove is related to the coupling strength in Eq.(1)by the relation
α2=β
2?1).(12)
This follows since(x i?x j)=L
24
N(N2?1),(13) with the eigenstate
Ψ0=
√2(N?1) iφi i>j|e iφi?e iφj|β
REFERENCES
[1]Sutherland B1971Phys.Rev.A42019
[2]Sutherland B1972Phys.Rev.A51372
[3]Calogero F1969J.Math.Phys.102197
[4]Simons B D,Lee P A and Altshuler B L1993Phys.Rev.Lett.704122
[5]Simons B D,Lee P A and Altshuler B L1994Phys.Rev.Lett.7264
[6]Sriram Shastry B1993Proc.of the16th Taniguchi Int.Sym.on the Theory of Condensed
Matter Shima,Japan(Eds.Kawakami N and Okiji A,Springer Verlag,New York1994)
[7]Sutherland B1971J.Math.Phys.12246
[8]Leinaas J M and Myrheim J1977Nuovo Cimento37B1
[9]Wilczek F1982Phys.Rev.Lett.481144;ibid49957.See also Wilczek F1990Fractional
Statistics and Anyon Superconductivity(World Scienti?c,Singapore)
[10]Murthy M V N,Law J,Bhaduri R K and Date G1992J.Phys.A256163
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