For an exercise question, I'm wondering on how I can determine the dimensionality of a state space spanned by Majorana fermions.

A (Dirac) fermionic excitation ($f , f^\dagger$) has a state space of dimension 2, spanned by the vacuum state $|0\rangle$ and the one-particle-state $f^\dagger |0\rangle$.

For Majorana fermions $\gamma = \gamma^\dagger$, I can't see how to define a meaningful creation / annihilation operator. The answers on this forum seem to suggest that for a single Majorana operator, the notions of creation/destruction/particle number are not meaningful, however given a set of Majorana operators, the corresponding operators can be constructed by essentially mapping to a (Dirac) fermion. From the answer to this question, I would guess that I get a two-dimensional fermionic state space for every pair of Majorana fermion operators I have. So, three of them ($\gamma_x, \gamma_y, \gamma_z$) give me three sets of creation/annihilation operators $c_{xy}, c_{yz}, c_{zx}$ and thus an 8-dimensional state space (for context, see the next paragraph).

For concreteness: I am given a fermion representation of a spin-1/2 lattice model and have to answer the questions whether there is a redundancy when describing the model either via a set of two fermionic creation/annihilation operator pairs or via three Majorana fermion operators. My solution would be to determine whether the state spaces of the spin model (two-dimensional at a single lattice site) and the fermionic representation match in order to see whether there can be a one-to-one correspondence of states. For the Schwinger representation (usual fermionic creation/annihilation), I count 4 fermionic base states ( $|00\rangle , |01\rangle, |10\rangle, |11\rangle$),but for the Majorana representation I'm not sure how to proceed or whether this line of thought is even valid.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.