Tridiagonal Hamiltonians modeling the density of states of the Double-Scaled SYK model

TL;DR

The paper investigates how the global density of states in the Double-Scaled SYK model determines the mean Lanczos coefficients governing Krylov space dynamics. By constructing finite-dimensional Hamiltonians that replicate the DSSYK DOS and applying tridiagonalization, semi-analytic integral equations, and random-matrix saddle-point methods, it demonstrates that bulk Lanczos coefficients follow a $q$-deformed logarithm across the DOS interpolation between semicircle and normal laws. The results are corroborated by edge analyses via moments and show strong agreement in the bulk, with implications for operator growth and potential holographic interpretations. This work provides a unified framework linking DOS, Lanczos coefficients, and nonperturbative RMT structures in the double-scaled regime.

Abstract

By analyzing the global density of states (DOS) in the Double-Scaled Sachdev-Ye-Kitaev (DSSYK) model, we construct a finite-dimensional Hamiltonian that replicates this DOS. We then tridiagonalize the Hamiltonian to determine the mean Lanczos coefficients within the parameter range. The bulk Lanczos coefficients, especially the Lanczos descent can be analytically expressed as a particular $q$-deformation of the logarithm. Our numerical results are further corroborated by semi-analytical findings, a random matrix potential construction in the bulk, and the analytic results at the edge of the Lanczos spectra using the method of moments.

Figures (5)

• Figure 1: The histogram shows the eigenvalue distribution for the model Hamiltonian $H_m$ at various values of $q$. The red line represents the $q$-normal DOS as defined in \ref{['denB']}. For $q=0$ and $q=1$, it corresponds to the semicircle law and the normal distribution respectively. For intermediate values of $q$, the infinite product is truncated at $k = 100$. The Hamiltonian $H_m$ has a size of $d = 2048$ and the results are averaged over $300$ ensembles.
• Figure 2: The mean value of the Lanczos coefficients for the model Hamiltonian $H_m$ for $q = 0$ (top, left), $q = 0.3$ (top, right), $q = 0.5$ (bottom, left), and $q = 1$ (bottom, right). The red line represents the numerical results, starting from the TFD state and applying the Hessenberg decomposition. The system size is $d = 2048$, and we take averages over $300$ ensembles. The blue line shows the semi-analytic results obtained by solving the integral equation \ref{['masrel']}. Here $\mathsf{b}_n$ with $n \sim o(1)$ and $d-n \sim o(1)$ regions are referred to as the "edge", and "tail" respectively. The "bulk" represents the rest of the spectrum, which is highlighted in blue. Both sets of results exhibit excellent agreement in the bulk of the Lanczos spectrum. Insets highlight the edge results, where the teal-colored points are derived using the moment method from the DOS as per \ref{['momden']}. The analytic expression for $\mathsf{b}_n$ is known and given by \ref{['bnana']} which asymptotes to $1/\sqrt{1-q}$.
• Figure 3: The variation of $f(q)$ and $g(q)$ within the range $q \in [0,1]$. At $q=1$, the function $f(q)$ reaches a value $=1/\sqrt{2} \approx 0.707$, as indicated by the black dashed line. This value is consistent with the Lanczos coefficients for the normal distribution as given in \ref{['eq:lancq1']}.
• Figure 4: The chord diagrams for evaluating $m_4 = \langle \mathrm{Tr}(H^4) \rangle_J$ consist of three diagrams in total. Each diagram features four nodes, representing the four insertions of the Hamiltonian, with chords connecting them in various configurations. The left and middle diagrams have no chord crossings, thus they each evaluate to unity. The right diagram has a single crossing, which evaluates to $q$. Therefore, the total contribution is $m_4 = 1 + 1 + q = 2+q$.
• Figure 5: The mean value of the Lanczos coefficients obtained from the Hamiltonian $H_m$ (red) is compared with those obtained using the saddle-point method in RMT (blue) for $q = 0$ (top, left), $q = 0.1$ (top, right), $q = 0.3$ (bottom, left), and $q = 0.5$ (bottom, right). The parameters for the integrable (red) case are the same as in Fig. \ref{['fig:bnfullq']}, with $d = 2048$ over $300$ ensembles.