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Mapping of Parent Hamiltonians: From Abelian and Non-Abelian by Martin Greiter

By Martin Greiter

This monograph introduces an actual version for a severe spin chain with arbitrary spin S, which include the Haldane--Shastry version because the exact case S=1/2. whereas spinons within the Haldane-Shastry version obey abelian half-fermi facts, the spinons within the common version brought the following obey non-abelian records. This manifests itself via topological offerings for the fractional momentum spacings. the final version is derived via mapping precise types of quantized corridor states onto spin chains. The booklet starts with pedagogical evaluate of the entire appropriate types together with the non-abelian statistics within the Pfaffian corridor nation, and is comprehensible to each pupil with a graduate direction in quantum mechanics.

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Extra resources for Mapping of Parent Hamiltonians: From Abelian and Non-Abelian Quantum Hall States to Exact Models of Critical Spin Chains

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The order parameter 0 (r ) vanishes inside the vortex core. 18) where n labels the modes and are the Bogoliubov quasiparticle operators. 19) where n is an integer and ω0 = 2 /ε F the level spacing. 18) combine creation and annihilation operators of the same spinless (or spin-polarized) fermions. 17) by Hermitian conjugation. We resolve this technical artifact by discarding the negative energy solutions as unphysical. For the n = 0 solution with at E 0 = 0, it implies that we get one fermion solution when we overcount by a factor of two.

As it satisfies the angular momentum algebra, it can be quantized accordingly. Note that L has a component in the er direction: Ler = er L = s0 . 58) If we take the eigenvalue of L 2 to be s(s + 1), this implies s = s0 + n, where n = 0, 1, 2, . . is a non-negative integer (while s and s0 can be integer or half integer, according to number of Dirac flux quanta through the sphere). 53), we obtain 24 2 Three Models and a Ground State = L 2 − s02 . 50) are hence ωc s(s + 1) − s02 2s0 ωc (2n + 1)s0 + n(n + 1) = 2s0 1 n(n + 1) .

The wave function is symmetric (or antisymmetric, depending on the number of electron pairs) under interchange of both groups. The state in the internal Hilbert space spanned by the quasihole affiliations with the two groups will change as we adiabatically interchange two quasiholes belonging to different groups, say ξ3 and ξ4 . Naively, one might think that the dimension of the internal Hilbert space is given by the number of ways to partition the quasiholes at ξ1 , . . , by (2n − 1)!! for 2n quasiholes.

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