We study the quantum phase transition upon varying 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→∞, 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.
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