Clifford statistics and the temperature limit in the theory of fractional quantum Hall effect

Using the recently discovered Clifford statistics we propose a simple model for the grand canonical ensemble of the carriers in the theory of fractional quantum Hall effect. The model leads to a temperature limit associated with the permutational degrees of freedom of such an ensemble. We also relate Schur’s theory of projective representations of the permutation groups to physics, and remark on possible extensions of the second quantization procedure. PACS numbers: 73.43.-f, 05.30.-d In a series of papers, building on the work on nonabelions of Read and Moore [1, 2], Nayak and Wilczek [3, 4, 5] (see also [6] on how spinors can describe aggregates) proposed a startling new spinorial statistics for the fractional quantum Hall effect (FQHE) carriers. The prototypical example is furnished by a socalled Pfaffian mode (occuring at filling fraction ν = 1/2), in which 2n quasiholes form an 2-dimensional irreducible multiplet of the corresponding braid group. The new statistics is clearly non-abelian: it represents the permutation group SN on the N individuals by a non-abelian group of operators in the N-body Hilbert space, a projective representation of SN . We have undertaken a systematic study of this statistics elsewhere, aiming primarily at a theory of elementary processes in quantum theory of space-time. We have called the new statistics Clifford, to emphasize its intimate relation to Clifford algebras and projective representations of the permutation groups. The reader is referred to [7, 8, 9] for details. Since the subject is new, many unexpected effects in the systems of particles obeying Clifford statistics may arise in future experiments. One simple effect, which seems especially relevant to the FQHE, might be observed in a grand canonical ensemble of Clifford quasiparticles. In this paper we give its direct derivation first. Following Read and Moore [2] we postulate that only two quasiparticles at a time can be added to (or removed from) the FQHE ensemble. Thus, we start with an N = 2n-quasiparticle effective Hamiltonian whose only relevant to our problem energy level E2n is 2 -fold degenerate. The degeneracy of the ground mode with no quasiparticles present is taken to be g(E0) = 1. Assuming that adding a pair of quasiparticles to the composite increases the total energy by ε, and ignoring all the external degrees of freedom, we can tabulate the resulting many-body energy spectrum as follows: Number of Quasiparticles, N = 2n 0 2 4 6 8 10 12 · · · Degeneracy, g(E2n) = 2 n−1 0 1 2 4 8 16 32 · · · Composite Energy, E2n 0ε 1ε 2ε 3ε 4ε 5ε 6ε · · · (1)