By Martin Greiter

This monograph introduces an actual version for a serious 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 data, the spinons within the normal version brought right here obey non-abelian facts. This manifests itself via topological offerings for the fractional momentum spacings. the overall version is derived by means of mapping distinct versions of quantized corridor states onto spin chains. The booklet starts with pedagogical evaluation of all of the proper types together with the non-abelian records within the Pfaffian corridor kingdom, and is comprehensible to each scholar 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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By choice of gauge, we can implement the phase change of 2π in the superconducting order parameter as a branch cut connecting the vortices to the left boundary of the system, and assume a convention according to which the Majorana fermion in each vortex crossing a branch cut acquires a minus sign, as illustrated in Fig. 6. To obtain the non-Abelian statistics, Ivanov [121] considered permutations of 2n vortices by braiding, which form the braid group B2n [123]. This group is generated by counterclockwise interchanges Ti of particles i and i + 1, which are neighbors with regard to the positions of their branch cuts to the boundary.

5). 34). 1) as the sum of a “kinetic” and a “potential” term, H HS 2π 2 = 2 N N α=β 1 S + S − + Sαz Sβz . 35) We first evaluate the action of the kinetic term on |ψ . ,z M } ψ(ηβ , z 2 , . . , z M )Sβ+ Sz+2 · . . · Sz+M |↓↓ . . ↓ ψ(ηβ , z 2 , . . , z M )Sα+ Sz+2 · . . · Sz+M |↓↓ . . 36) where we have implicitly assumed that each spin configuration in the sum over {z 1 , z 2 , . . 2) appears only once (and not M! times due to permutations of the z i ’s). We write this as Sα+ Sβ− ψ (ηα , z 2 , .

46) that and Stot ψ0HS = 0. , there is no spin current in the ground state. 44) shows that they would have to be singlets and likewise be annihilated by Dα . 45) in Sect. 44) below, there is no need to do so. 44). 2 The Haldane–Shastry Model 39 and recall For S = 1 2 D†α = 1 2 Dα = 1 2 N θαβ i(Sα × Sβ ) − Sβ , β=1 β=α N θαγ i(Sα × Sγ ) + Sγ . 48) and therewith i(Sα × Sβ ) − Sβ · i(Sα × Sγ ) + Sγ = − 3 Sβ Sγ − iSα (Sβ × Sγ ) . 2 This implies N α=1 D†α Dα = − 3 8 N N N θαβ θαγ Sβ Sγ − iSα (Sβ × Sγ ) .

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