 01454 Detlef Lehmann
 Exact Diagonalization of the Fractional Quantum Hall
ManyBody Hamiltonian in the Lowest Landau Level
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Dec 6, 01

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Abstract. For a gaussian interaction V(x,y)=\lambda e^{(x^2+y^2)/r^2} with long
range r>>l_B, l_B the magnetic length, we rigorously prove that the
eigenvalues of the finite volume Hamiltonian H_{N,LL}=P_{LL} H_N P_{LL},
H_N=\sum_{i=1}^N [i\hbar \nabla_{x_i}eA(x_i)]^2+\sum_{i,j; i\ne j} V(x_ix_j),
\rotA=(0,0,B), and P_{LL} the projection onto the lowest Landau level,
are given by the following set: Let M be the number of flux quanta
flowing through the sample such that \nu=N/M is the filling factor.
Then each eigenvalue is given by
E=E(n_1,...,n_N)=\sum_{i,j=1;i\ne j}^N W(n_in_j).
Here n_i\in {1,2,...,M}, n_1<...<n_N and the function W is given by
W(n)=\lambda \sum_{j\in Z} e^{{1/r^2}(L{n/M}jL)^2} if the system is
kept in a volume [0,L]^2. The eigenstates are also explicitely given.
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