# What are the fermions in the SYK model doing?

The Hamiltonian of the SYK model is

$$$$H = \mathcal{N}\sum_{ijkl}^N J^{ijkl} \chi_i \chi_j \chi _k \chi _l$$$$

where $$\mathcal{N}$$ is some normalization to make the energy scale with $$N$$ and $$\chi_i$$ is a Majorana operator. Different reviews on the SYK model call the variables $$i=1,\dots,N$$ sites, others talk about $$\chi$$ as a vector with $$N$$ components like it was a spin $$N$$ particle. In relation to this question, I don't understand whether the number of particles is conserved in this Hamiltonian. In other cases, like the Hubbard model, one gets a term like

$$$$H = \mathcal{N}\sum_{r,r'} J^{r,r'} a^\dagger_r a_{r'}$$$$

where the interpretation is that the Hamiltonian destroys a particle on the site $$r'$$ and creates another one at site $$r$$. On the SYK Hamiltonian, however, since the Majorana fermions are self-adjoint, any operator can work as both creation and annihilation operators. This means that any term in the Hamiltonian could create four particles, destroy four particles or anything in between. So the question is: How should we interpret the SYK Hamiltonian (or any Hamiltonian with Majorana fermions, for that matter)?

It is standard to consider the fermions in the SYK model as all of them being at the same `site' but with an all-to-all coupling. SYK model is considered as a 1-dimensional (or 0-dimensional in condensed matter language) model. While I have seen people refer to them being as different sites, I am not really sure if it matters. There are other papers in which various chains of SYK models are coupled together to construct a higher-dimensional model and in that sense the $$i$$ index should not be confused with spatial sites.

There are a class of models which reproduce the same low-energy physics as that of the SYK model where the fermions are truly components of a vector (these models, known as tensor models, have been studied extensively by Klebanov and collaborators, and Gurau and collaborators. Witten also had a paper on such models in October 2016 showing that their physics is same as that of SYK model). In these cases there are global $$O(N)^3$$ symmetries. However, there is no such symmetry in the SYK model, which only has exchange symmetry in fermions.

Lastly, the fermion number is not conserved in the SYK model as you are correctly pointing out. And if you really want to interpret it as creation and annihilation of particles, then indeed there is transition probability between various states with different particle numbers in this case. Maldacena's paper makes is apparent that eigenstates of the SYK model are in superposition of the particle number basis. You can interpret the SYK model as a system which interacts with some external bath that leads to particle number non-conservation. This bath is a result of disordered average over random couplings $$J_{ijkl}$$.

• Thanks! Great answer. I have one follow up question before marking this as the right answer. Mathematically, what are the $\chi_i$ ? Are they just $NxN$ Matrices? Thanks again – P. C. Spaniel May 6 at 21:10
• You can think of them as $2^{N/2} \times 2^{N/2}$ matrix because the the Hilbert space is $2^{N/2}$ dimensional. The representation using Pauli matrices is discussed in Maldacena's paper that I have linked to in my answer above. – nGlacTOwnS May 6 at 21:25
• I found the matrix representation on page 10 footnoote 5 of the paper, but it seems to have a typo, doesn't it? – P. C. Spaniel May 7 at 21:21
• Yes. It should be $\psi^{2k} = \sigma^2_k \prod_i \sigma_i^3$. – nGlacTOwnS May 13 at 14:20

I'm going to answer your question, and then I'm going to explain why it was the wrong question to be asking.

Perturbatively, you should think of the SYK model as having $$N$$ flavors of fermion. You can use whatever intuition you use when thinking about quarks. The different species interact at four-point vertices. As you said, these can be 1-3, 2-2, or 3-1. They can even be 4-0 or 0-4, at least on the level of virtual particles. The fact that this is in zero spatial dimensions doesn't change any of the intuitions you should be inheriting from more conventional QFTs, at least in a perturbative regime.

But most people don't study the SYK model in a perturbative regime. The coupling constants all have units of energy. So in the low-energy/low-temperature limit, those coupling constants all become huge. Over long timescales/low energies, perturbation theory stops being a good picture at all. Talking about the fundamental fermions doesn't make sense. I won't dive too much into what does emerge, I'll just leave this paper here https://arxiv.org/pdf/1604.07818v1.pdf

• Minor comment to the post (v1): In the future please link to abstract pages rather than pdf files, e.g., arxiv.org/abs/1604.07818 – Qmechanic May 6 at 17:10