The parastatistics of braided Majorana fermions

This paper presents the parastatistics of braided Majorana fermions obtained in the framework of a graded Hopf algebra endowed with a braided tensor product. The braiding property is encoded in a $t$-dependent $4\times 4$ braiding matrix $B_t$ related to the Alexander-Conway polynomial. The nonvanishing complex parameter t defines the braided parastatistics. At $t = 1$ ordinary fermions are recovered. The values of $t$ at roots of unity are organized into levels which specify the maximal number of braided Majorana fermions in a multiparticle sector. Generic values of $t$ and the $t =-1$ root of unity mimick the behaviour of ordinary bosons.


Introduction
Braided Majorana fermions have been intensively investigated since the [1] Kitaev's proposal that they can be used to encode the logical operations of a topological quantum computer which offers protection from decoherence (see also [2][3][4]).In this talk I present consequences and open questions about the parastatistics of 2 -graded braided Majorana qubits derived from the results of [5]; this paper applied to 2 -graded qubits the [6] framework of a graded Hopf algebra endowed with a braided tensor product.A nonvanishing complex braiding parameter t controls the spectra of multiparticle Majorana fermions.Inequivalent physics is derived for the set of t roots of unity which are organized into different levels (L 2 , L 3 , . . ., L ∞ ).The levels interpolate between ordinary fermions (L 2 for t = 1) and the spectrum of bosons ("L ∞ " recovered at t = −1).The intermediate levels L k for k = 3, 4, 5, . . .implement a special type of parafermionic statistics (see [7][8][9]) which allows at most k − 1 braided Majorana excited states in any given multiparticle sector.
The paper is structured as follows.In Section 2 the braiding of 2 -graded qubits is illustrated.In Section 3 the truncations of the spectra at roots of unity are discussed.The consequences for the parastatistics are presented in Section 4.

Braiding -graded qubits
We present the main ingredients of the construction.A single Majorana fermion can be described as a 2 -graded qubit which defines a bosonic vacuum state |0〉 and a fermionic excited state |1〉: The operators acting on the 2 -graded qubit close the gl(1|1) superalgebra.In a convenient presentation they can be defined as Their (anti)commutators are The diagonal operators α, β are even, while β, γ are odd, with γ the fermionic creation operator.
The excited state is a Majorana since it is a fermion which coincides with its own antiparticle.This is a consequence of the fact that the (2) matrices span the Clifford algebra C l(2, 1) which, see [10,11], is of real type (implying that the charge conjugation operator is the identity).
The braiding of the graded qubits is realized by introducing a braided tensor product ⊗ br such that, for the operators a, b ( is the identity) one can write where the right hand side operator Ψ(a, b) satisfies braided compatibility conditions.For the purpose of braiding 2 -graded qubits it is only necessary to specify the braiding property of the creation operator γ: ( A consistent choice for the right hand side is to set where B t is a 4 × 4 constant matrix which depends on the complex parameter t = 0.The dot in the right hand side denotes the standard matrix multiplication.The braiding compatibility condition is guaranteed by assuming B t to be given by since B t satisfies The matrix B t is the R-matrix of the Alexander-Conway polynomial in the linear crystal rep on exterior algebra [12] and is related, see [13], to the Burau representation of the braid group.

Truncations at roots of unity
The requirement that for some n = 2, 3, ... finds solution for the n − 1 roots of the polynomial b n (t).This set of polynomials is defined as The set of b k (t) polynomials enters the construction of multiparticle states.The n-particle vacuum |0〉 n is given by the tensor product of n single-particle vacua: The fermionic excited states are created by applying powers of tensor products involving the single-particle creation operator γ.For n = 2, 3 one has, e.g., that the first excited state is created by By taking into account the braided tensor product one obtains, for the second and third excited states, This construction works in general.The b k (t) = 0 roots of the polynomials produce truncations at the higher order excited states and the corresponding spectrum of the theory.

The levels and the associated parastatistics
The single-particle Hamiltonian H can be identified with the operator δ in (2).It follows that the single-particle excited state has energy level E = 1.This is also true (due to the property of the Hopf algebra coproduct) for the first excited state in the multiparticle sector.Each creation operator produces a quantum of energy.In the n-particle sector the energy spectrum of the theory depends on whether t produces a truncated or untruncated spectrum.The notion of truncation level acquires importance.
A "level-k" root of unity, for k = 2, 3, 4, . .., is a a solution The physical significance of a level-k root of unity is that the corresponding braided multiparticle Hilbert space can accommodate at most k − 1 Majorana spinors.
The special point t = 1, being the solution of the b 2 (t) ≡ 1 − t = 0 equation, is a level-2 root of unity.It gives the ordinary total antisymmetrization of the fermionic wavefunctions.The t = 1 level-2 root of unity encodes the Pauli exclusion principle of ordinary fermions.
With an abuse of language, the t = −1 root of unity, which does not solve any b k (t) = 0 equation, can be called a root of unity of ∞ level.
The physics does not depend on the specific value of t, but only on the root of unity level.A generic t which does not coincide with a root of unity produces the same untruncated spectrum of the t = −1 "L ∞ " level.
The following energy spectra are derived.
Case a, truncated L k level: the n-particle energy eigenvalues E are a plateau is reached for the maximal energy level k − 1; this is the maximal number of braided Majorana fermions that can be accommodated in a multiparticle Hilbert space; Case b, untruncated (t = −1) L ∞ level: the n-particle energy eigenvalues E are E = 0, 1, . . ., n for any n; there is no plateau in this case.The energy eigenvalues grow linearly with N .

Conclusion
The [5] braided multiparticle quantization of Majorana fermions produces truncations of the spectra at certain values of t roots of unity.This feature points towards a relation between the braided tensor product framework here discussed and the representations of quantum groups at roots of unity where similar truncations, see [14,15], are observed.The precise connection of the two approaches is on the other hand not yet known and still an open question.The representations of the quantum group U q (gl(1|1) at roots of unity have been classified and presented in [16] (see also [17]).A possibility to investigate the connection seems to be offered by the scheme of [18] which shows how a quasitriangular Hopf algebra can be converted into a braided group.
On a separate issue it should be mentioned that a forthcoming paper will present, with the help of intertwining operators, the construction of the braided tensor product ⊗ br in terms of an ordinary tensor product ⊗.This construction relates the observed parastatistics of Majorana fermions to the "mixed brackets" (which interpolate ordinary commutators and anticommutators) that have been introduced in [19] in defining the Volichenko algebras.

Figure 1 :
Figure 1: Roots of unity up to level 8