In reading a recent preprint [1] contrasting bosonic models with local (tensor product) Hilbert spaces with SYK-like models of fermions, I realized I was confused about something. While I have a vague understanding that fermions are fundamentally non-local degrees of freedom (because of the non-trivial exchange statistics), I was surprised to read that the Hilbert space of the SYK model is non-local — that is, not a tensor product of local Hilbert spaces.

Naively, I would expect one could write down the Hilbert space as a tensor product of Majorana Hilbert spaces, just as in a spin model one uses a tensor product of spins. I guess this is complicated for Majorana fermions because each Majorana is somehow "half of a degree of freedom" and does not exist independently. Actually, I am not even sure how to write down the Hilbert space of a single Majorana fermion (or if that is even a notion that makes sense). I wonder if this complication could be removed by using Dirac fermions instead of Majoranas.

My specific questions are:

Q1) Can someone provide an explicit description of the non-local Hilbert space of a collection of Majorana fermions, as in the SYK model?

Q2) Is the non-locality (lack of tensor product structure) a consequence of using Majorana fermions, or would it also appear in the case of Dirac fermions?

[1] C. L. Baldwin, B. Swingle, Quenched vs Annealed: Glassiness from SK to SYK, https://arxiv.org/abs/1911.11865

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    $\begingroup$ Not to make the situation worse for you, but there is an extra complication in the case of the SYK model due to the fact that there is disorder averaging. In models with disorder averaging the notion of Hilbert space is complicated by the fact that there isn't a one-to-one map between states and their weights. But perhaps your question is meant to focus first on the structure of fermionic Hilbert spaces independent of any particular Hamiltonian. EDIT: I should say there are versions of the SYK model which don't have disorder averaging (e.g. Witten's approach). $\endgroup$ – miggle Dec 7 '19 at 5:38
  • $\begingroup$ @miggle I was trying to think about the Hilbert space without worrying about the hamiltonian, but would you mind elaborating on your comment? In particular I'm not sure I understand what you mean by the "weight" of a state. $\endgroup$ – d_b Dec 11 '19 at 22:25
  • $\begingroup$ Sure - when computing a partition function one sums over all microstates of the system weighted by some number. In the canonical ensemble this is just the exponential of the Hamiltonian, for example. My comment is just that when you disorder average, each instance of the disorder will assign a different weight to the state because you change the Hamiltonian. $\endgroup$ – miggle Dec 12 '19 at 20:38

I think I have an answer, which is mostly borrowed from the page Braiding of Majoranas. As with most things it is almost obvious in retrospect. We can write down the Hilbert space of $2N$ Majorana fermions as a tensor product of $N$ Dirac fermion Hilbert spaces. The Dirac fermions $c_j^{\dagger}$ are non-local linear combinations of the Majoranas $\gamma_j$, namely \begin{align} c_j^{\dagger} = \frac{1}{2}\left(\gamma_{2j-1} + i \gamma_{2j}\right), \end{align} and the Hilbert space is spanned by Dirac fermion occupation number states $|n_1, n_2, \ldots, n_N\rangle$. It seems the non-locality of the Majorana Hilbert space disappears if we consider Dirac (complex) fermions.


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