A Quantum Breakdown Model: from Many-body Localization to Chaos with Scars

We propose a quantum model of fermions simulating the electrical breakdown process of dielectrics. The model consists of $M$ sites with $N$ fermion modes per site, and has a conserved charge $Q$. It has an on-site chemical potential $\mu$ with disorder $W$, and an interaction of strength $J$ restricting each fermion to excite two more fermions when moving forward by one site. We show the $N=3$ model with disorder $W=0$ show a Hilbert space fragmentation and is exactly solvable except for very few Krylov subspaces. The analytical solution shows that the $N=3$ model exhibits many-body localization (MBL) as $M\rightarrow\infty$, which is stable against $W>0$ as our exact diagonalization (ED) shows. At $N>3$, our ED suggests a MBL to quantum chaos crossover at small $W$ as $M/N$ decreases across $1$, and persistent MBL at large $W$. At $W=0$, an exactly solvable many-body scar flat band exists in many charge $Q$ sectors, which has a nonzero measure in the thermodynamic limit. We further calculate the time evolution of a fermion added to the particle vacuum, which shows a breakdown (dielectric) phase when $\mu/J<1/2$ ($\mu/J>1/2$) if $W\ll J$, and no breakdown if $W\gg J$.