Fermionic Coherent States in .NET framework

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11 Fermionic Coherent States
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the ith level, and the constraint i=1 n i = p expresses the conservation of the total number of fermions These basis states are common eigenstates of the complete set {H i , 1 u i u r} of commuting observables With respect to this basis, the matrix representative E i j of the generator a a j of u(r) has all its entries equal to 0 i with the exception of entry (i, j ), which is equal to 1: a a j E i j i with Ei j
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(1110)
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a notation previously used in (1025) Let us consider the following extremal state for the representation = {1 p , 0r p }:
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p r p
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|extr = |1, 1, , 1, 0, , 0
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(1111)
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It is actually the nonperturbed ground state of a p-fermion Hamiltonian expressed in a mean- eld approximation:
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i, j=1
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(1112)
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The mean- eld approach is necessary when we deal with a many-body system with interactions, since in general there is no way to solve exactly the model, except for extremely simple cases A major drawback (eg, when computing the partition function of the system) is the treatment of combinatorics generated by the interaction terms in the Hamiltonian when summing over all states The goal of mean eld theory (also known as self-consistent- eld theory) is precisely to resolve these combinatorial problems The main idea of mean- eld theory is to replace all interactions with any one body with an average or effective interaction This reduces any multibody problem into an effective one-body problem State (1111) is extremal in the sense that it is annihilated by all elements as a a j |extr = 0 for all i, j, such that 1 u i = j u p or p + 1 u i, j u r Together / i with the H i s, 1 u i u p, these generators span the subalgebra u( p) u(r p) The corresponding group, U ( p) U (r p), is the subgroup of elements in U (r) leaving invariant, up to a phase, the extremal state |extr , that is, it is the stability group of the latter One then de nes a family of coherent states for the group U (r) by
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i j a j a i ) |extr
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(1113)
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These states are in one-to-one correspondence with the points of the coset U (r)/ (U ( p) U (r p)) [153] They represent in a certain sense a quantum version of this compact manifold Note that for a system of bosonic states, the manifold to be considered would be the coset U (r)/(U (1) U (r 1)) Given the set of complex parameters { i j , 1 u j u p , p + 1 u i u r} viewed as the entries of a (r p) ~ p matrix , let us introduce two other (r p) ~ p matrices: Z = sin
+ ,
(1114)
113 Coherent States for Systems of Identical Fermions
= Z (I p Z Z ) 2
(1115)
The matrix is well de ned because I p Z Z is by construction a positive matrix It can be coordinatized by p(r p) complex parameters, denoted , p(r p) being the complex dimension of the coset U (r)/(U ( p) U (r p)) The represent a kind of projective coordinates for this manifold With this notation, if one considers the matrix representation a a j E i j of the u(r) basis elements, we nd that the i displacement operator
exp
1u j u p p+1uiur i j ai a j
i j a j a i ) = | , is represented by the following unitary
or equivalently the coherent state | r ~ r matrix: I r p Z Z Z Z I p Z Z
= (Z ) U (r)
(1116)
The notation is particularly suitable for describing the action of the group U (r) on its coset manifold U (r)/(U ( p) U (r p)) Let us write an element of U (r) in the following block matrix form: