Spectrum of vector and fermionic matrix models

Abstract

We study fermionic vector and matrix models, and we are interested in their connection to random matrices. We expect them to share many of the properties of the SYK model, but without disorder. We compute the spectrum for some of these vector models, like SO(N)×SO(2)×SO(2). We then use representation theory to determine the eigenstates and their degeneracies, as this allows us to obtain energy states in a very efficient way relative to exact diagonalization. A normal computer can reach up to N ≈120 and higher relying on this method.To reach a higher number N, we use an approximation that allows us to look at larger N vector models, till N = 10000. . We observe for large N and small energies that ρ(E) ≈√E−E0, which is expected to be an universal behavior for random matrix models. For larger energies, we observe a behavior like ρ(E) ≈e (E−E0)Nlog(N) 2J , for a section of the spectrum between the ground state energy E0 and E = 0. Even at smaller N ≈100, we note that the density of the states has the form ρ(E) ≈eβEα, where α is a constant throughout the spectrum. For larger N, we expect α →1 till very close to the E = 0 energy. We repeat a similar calculation for matrix models. The qualitative behaviour of the density of energies spectra for these matrix models is different from the behavior of vector models. For matrix models, the spectrum is much smoother, and we do not see the cusp at E = 0, as seen in vector models. Also, if for the matrix model we plot the degeneracies for the odd E1,E3,E5,..., and the even energies E0,E2,E4,.. separately, we see that the spectrum is very smooth and regular for both of these plots. For matrix models, the density around the E = 0 energy has a form ρ ≈e−AE2, which corresponds to Wigner’s semi-circle law for Gaussian matrices,whileforthevectormodelwehave ρ ≈e−A|E| aroundthe0energyofthespectrum. For the density of states of matrix models, we suspect that the density might behave like ρ(E) ≈ (E−E0)2 near the ground state, but we need to calculate the spectrum for higher N.