Nonergodic Extended States in the Sachdev-Ye-Kitaev Model

We analytically study spectral correlations and many body wave functions of a Sachdev-Ye-Kitaev model deformed by a random Hamiltonian diagonal in Fock space. Our main result is the identification of a wide range of intermediate coupling strengths where the spectral statistics is of Wigner-Dyson type, while wave functions are nonuniformly distributed over Fock space. The structure of the theory suggests that such manifestations of nonergodic extendedness may be a prevalent phenomenon in many body chaotic quantum systems.

Figure 1

Left: Cartoon of Fock space sites $n,m,l,\dots$ (indicated by dots) connected by hopping operator $\mathcal{P}$ (solid lines). For $\mathrm{\Delta }\gg 1$ exceeding the bandwidth of the unperturbed model, one may approach the problem perturbatively, i.e., taking the isolated eigenstates of levels $v_n$, $v_m$, $v_l$ as a starting point. The hybridization leads to level broadening $\kappa$ of resonant neighbors (indicated by hatched link) which have both energies $v_n$, $v_m\lesssim 1$ within the SYK band. Right side: Typical energy distributions of Fock-space neighbors connected by $\mathcal{P}$. The hybridization does, in general, not generate overlap between neighboring sites. For $\mathrm{\Delta }<N^2$ wave functions are thus extended ($\to$ Wigner-Dyson statistics) yet confined to only a fraction $\sim 1/{\mathrm{\Delta }}^2$ of the total Fock space.

Figure 2

The scattering of wave function amplitudes in Fock space. Variables $Y_{n{n}^{\prime }}^{s{s}^{\prime },\sigma {\sigma }^{\prime }}$ describe the correlated propagation of resolvents (solid lines) labeled by a conserved index $(s,\sigma )$. Scattering processes (indicated by dots) can be distinguished into those dressing propagators by “self energies” (dashed lines connecting same resolvent) and vertex contributions (dashed lines connecting different resolvents). Hatched regions summarize repeated, ladder-diagrams of vertex contributions and define the slow modes in the system. Inset: Self consistency equation for self-energy Eq. (4).

Figure 3

Inverse participation ratio as a function of $\mathrm{\Delta }$, normalized by $I_2(\mathrm{\Delta }=0)$, from exact diagonalization $N=13$; the analytical prediction Eq. (7) is indicated by the solid line. Left inset: Relative entropy (Kullback-Leibler) between numerical and Wigner Dyson (dashed), respectively, Poisson (solid) distributions. Right inset: Inverse participation ratio as a function of $N$, normalized by $I_2(N=7)$, from exact diagonalization at $\mathrm{\Delta }=10$; solid line is the analytical prediction from Eq. (7) [36].