Quantum chaos in supersymmetric Yang-Mills-like model: equation of state, entanglement, and spectral form-factors

Abstract

We analyze in detail a sharp transition between the low-energy, low-dimensional eigenstates and the high-energy chaotic bulk of the spectrum for a simple supersymmetric quantum-mechanical model with Hamiltonian ${\hat{H}}_S=\left({\hat{p}}_1^2+{\hat{p}}_2^2+{\hat{x}}_1^2{\hat{x}}_2^2\right)\otimes I+{\hat{x}}_1\otimes {\sigma }_1+{\hat{x}}_2\otimes {\sigma }_3$, which mimics the structure of the Banks-Fischler-Susskind-Stanford (BFSS) matrix model, the spatially compactified $\mathcal{N}=1$ super-Yang-Mills theory. We conjecture that this transition might be similar to the transition between the $D0$-brane and $M$-theory regimes in the BFSS model, and find that it does not lead to irregularities in the thermodynamic equation of state. We demonstrate that real-time spectral form-factor for our supersymmetric model exhibits the ``ramp'' behavior typical for quantum chaos. We also analyze the entanglement entropy and the spectrum of the reduced density matrix of the eigenstates of ${\hat{H}}_S$, considering one of the bosonic degrees of freedom as a subsystem. The entanglement entropy of low-energy eigenstates appears to be practically energy-independent. Exactly at the onset of random-matrix-type level spacing fluctuations, this behavior rapidly changes into a steady growth of entanglement with energy. We demonstrate that the spectrum of the reduced density matrix also exhibits universal level-spacing fluctuations towards its higher end, even for the ground state of the supersymmetric model. Thus even the regularly spaced, non-chaotic eigenstates contain some information about semi-classical chaotic dynamics at high energies.

Comments: Proceedings of the 39th International Symposium on Lattice Field Theory (Lattice 2022), 8-12 August 2022, Bonn, Germany. v2: updated references, published version. 17 pages, 6 figures, PoS style