Sachdev-Ye-Kitaev model

The SYK model of $M$ Majoranas with random all-to-all four-body couplings is chaotic, and its level statistics follow random-matrix theory. Which ensemble depends on $M \bmod 8$: within one fermion-parity sector it is GOE for $M \equiv 0$, GUE for $M \equiv 2, 6$ and GSE for $M \equiv 4$, where every level is doubly degenerate.^[1] This page reproduces the three classes from the mean ratio of adjacent level spacings. On the package side it shows how to build a dense random Majorana Hamiltonian symbolically and restrict it to a parity sector, which is where the statistics must be taken.

@majoranas creates operators with $\{\gamma_i, \gamma_j\} = 2\delta_{ij}$, and add!! accumulates the $\binom{M}{4}$ terms in place. The coupling scale is left out, since the spacing ratio does not depend on it.

using FermionicHilbertSpacesimport FermionicHilbertSpaces: add!!using LinearAlgebra, Random, Statistics@majoranas γfunction syk_hamiltonian(M)    ham = 0    for i in 1:M-3, j in i+1:M-2, k in j+1:M-1, l in k+1:M        ham = add!!(ham, randn() * γ[i] * γ[j] * γ[k] * γ[l])    end    return hamendfunction gap_ratio(energies)    s = diff(energies)    return mean(min(a, b) / max(a, b) for (a, b) in zip(s[1:(end - 1)], s[2:end]))end;

The statistics must be taken within one fermion-parity sector: over the full space the two sectors are interleaved, and for $M \equiv 2 \bmod 8$ the odd sector is an exact copy of the even one, so every level is degenerate and the ratio is zero whatever the dynamics. We therefore diagonalize the even sector for a few disorder samples, keeping one level of each degenerate pair for GSE. The table lists the measured mean ratio next to the random-matrix value of the expected class (Atas et al.^[2]); they should agree.

Random.seed!(1)cases = [(M = 16, class = "GOE", ref = 0.5307), (M = 18, class = "GUE", ref = 0.5996), (M = 20, class = "GSE", ref = 0.6744)]results = map(cases) do (; M, class, ref)    Heven = hilbert_space(γ, 1:M, ParityConservation(1))    r = mean(1:20) do _        E = eigvals(Hermitian(representation(syk_hamiltonian(M), Heven, :dense)))        gap_ratio(class == "GSE" ? E[1:2:end] : E)    end    (majoranas = M, expected_class = class, measured_ratio = round(r; digits = 3), random_matrix_ratio = ref)end@assert all(abs(c.measured_ratio - c.random_matrix_ratio) < 0.02 for c in results)results

3-element Vector{@NamedTuple{majoranas::Int64, expected_class::String, measured_ratio::Float64, random_matrix_ratio::Float64}}:
 (majoranas = 16, expected_class = "GOE", measured_ratio = 0.524, random_matrix_ratio = 0.5307)
 (majoranas = 18, expected_class = "GUE", measured_ratio = 0.599, random_matrix_ratio = 0.5996)
 (majoranas = 20, expected_class = "GSE", measured_ratio = 0.678, random_matrix_ratio = 0.6744)

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