Sometimes ideas come from surprising places. Recently Alexei Kitaev made a splash by pointing out that a problem studied decades ago by Subir Sachdev and Jinwu Ye, who were studying quantum spin glasses, has bearing on the problem of quantum gravity. What is now called the Sachdev-Ye-Kitaev (SYK) model became the subject of intense study as a toy model in which gravity and black hole physics emerges from totally different microphysics: quantum mechanics of many fermions. It has a very simple Hamiltonian:

,

where the coefficients are totally random. Here I’ve written what is called a “complex” SYK model, slightly different from what Kitaev studied, which was written for Majorana fermions. The difference is not crucial for the gravity connection, but it is helpful for us. The key feature is that according to this Hamiltonian electrons move collectively, transitioning from two “orbitals” (kl) to another two (ij) in pairs, never alone. You could visualize it like this:

This is a “toy model” because what emerges (in the limit of a large number of orbitals N) is not the gravity of our world, but a slightly different, and lower-dimensional version. But still it is remarkable and beautiful.

The SYK model also describes the physics of strongly interacting electrons, forming an unconventional metallic state. This metal is what condensed matter physicists refer to as a “non-Fermi liquid”, which behaves differently from typical metals like copper, in which electrons act almost independently. In the SYK metal, you cannot separate motions of single electrons, but instead only see a collective dance of the whole electron fluid. The problem of understanding non-Fermi liquids is a forefront one in the field of quantum materials, but there are very few solvable theoretical models for them. As such, the SYK model is quite valuable.

The drawback of the SYK model is that it is “zero dimensional” — every electron in it interacts with every other one, and so there is no built-in locality to the problem. Because of this we cannot imagine driving a flow from one side of the system to another, and hence cannot discuss conductivity. We should view the SYK model as describing a strongly-interacting “quantum dot”: a little chunk of non-Fermi liquid metal.

A natural thing to do is to try to string together these quantum dots to build up a higher dimensional system. Already a couple of very nice papers along these lines have appeared: this paper, and this one which is a generalization. In both these works, coupling between the SYK dots is chosen to preserve the scaling of the SYK model: electrons must hop in *pairs* between the dots. Consequently the whole system remains a non-Fermi liquid and no new energy scale is introduced by the coupling.

We just posted a paper in which we make a different choice — just ordinary one-electron hopping between the dots:

.

In this model, there is a competition between the SYK interaction U and the hopping t. For t<<U, we have a strongly correlated system. The header image is suppose to show this: electrons zipping between the correlated fluid dots. Fortunately, it remains soluble in the large N limit and one can obtain a rich structure. Briefly, at low temperatures and low energy, the system is a Fermi liquid. We can extract the parameters of the Fermi liquid theory: effective mass, Fermi liquid interactions, quasiparticle residue, etc. However, at higher temperatures it is an incoherent metal, and displays properties that are a mixture of those of the pure SYK model and a regular metal. Together, it is striking how many of the properties are similar to those of experimental correlated metals:

- Small coherence scale E
_{c}= t^{2}/U - Large effective mass (Sommerfeld coefficient) γ ~ m*/m ~ U/t
- Small quasiparticle weight Z ~ t/U
- T
^{2}low-temperature resistivity with Kadowaki-Woods ratio A/γ^{2}= constant *linear*in temperature resistivity at high temperature*linear*in temperature thermal “resistivity” T/κ at high temperature- Fermi liquid Lorenz number L = κ/(T σ) = π
^{2}/3 at low T - non-Fermi liquid Lorenz number L = π
^{2}/8 at high T

Anyway, I’m pretty pleased with how it all works out. Maybe the most remarkable part of this story is that the person who really spearheaded all the calculations, Xue-Yang Song, is still an *undergraduate*. Pretty impressive. You can read the preprint here.