Lanczos spectrum for random operator growth

Abstract

Krylov methods have reappeared recently, connecting physically sensible notions of complexity with quantum chaos and quantum gravity. In these developments, the Hamiltonian and the Liouvillian are tridiagonalized so that Schrodinger/Heisenberg time evolution is expressed in the Krylov basis. In the context of Schrodinger evolution, this tridiagonalization has been carried out in Random Matrix Theory. We extend these developments to Heisenberg time evolution, describing how the Liouvillian can be tridiagonalized as well until the end of Krylov space. We numerically verify the analytical formulas both for Gaussian and non-Gaussian matrix models.

Figures (5)

• Figure 1: Comparison of GUE Liouvillian spectrums derived from Convolution of Hamiltonian spectrums versus Simulation. On the left, the simulation is obtained for $N = 60$ over $50$ ensembles, while on the right, the simulation is generated for $N = 600$ over $50$ ensembles, highlighting the degeneracy contribution diminishing by a factor of $1/N$.
• Figure 2: GUE Lanczos coefficients from Liouvillian spectrum.
• Figure 3: Non-Gaussian Liouvillian spectrum from Hamiltonian spectrum for sextic potential $V_s$ (left) and quartic potential $V_q$ (right) at $N=600$ over $50$ ensembles.
• Figure 4: Non-Gaussian Lanczos coefficients from Liouvillian spectrum for sextic potential $V_s$ (left) and quartic potential $V_q$ (right).
• Figure 5: SYK $N=14$ Liouvillian spectrum and Lanczos coefficients over 30 ensembles.