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Quantum Phase Transition in the Yukawa-SYK Model

We study the quantum phase transition upon variation of the fermionic density $ν$ in a solvable model with random Yukawa interactions between $N$ bosons and $M$ fermions, dubbed the Yukawa-SYK model. We show that there are two distinct phases in the model: an incompressible state with gapped excitations and an exotic quantum-critical, non-Fermi liquid state with exponents varying with $ν$. We show analytically and numerically that the quantum phase transition between these two states is first-order, as for some range of $ν$ the NFL state has a negative compressibility. In the limit $N/M\to \infty$ the first-order transition gets weaker and asymptotically becomes second-order, with an exotic quantum-critical behavior. We show that fermions and bosons display highly unconventional spectral behavior in the transition region.