Quantum phase transition in the Yukawa-SYK model

We study the quantum phase transition upon varying the fermionic density $\nu$ 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 $\nu$. We show analytically and numerically that the quantum phase transition between these two states is first-order, as for some range of $\nu$ 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.

Figure 1

(Top) The dependence of the spectral asymmetry parameter $\alpha$ on the filling $\nu$. (Bottom) The qualitative (solid line) and numerical (red dots) dependence of the chemical potential $\mu$ on the filling $\nu$. The solid line was obtained by using the low-energy form (5) for the self-energies for all frequencies. The vertical dashed line in the lower panel represents the incompressible phase. We set $N=M$, in which case $x_0=0.16$. The solid line actually has a minute dip at $\nu \approx 0.98$ (see Appendices pp1-s1 and pp2). It is not present in the full numerical solution and likely is an artifact of using Eq. (5) at all energies.

Figure 2

Schematic bosonic and fermionic spectral functions, ${\rho }_F\left(\omega \right)\equiv -ImG(\omega +i\eta )$ and ${\rho }_B\left(\omega \right)\equiv ImD(\omega +i\eta )$, at the first-order transition for $N\sim M$ (a), and at the second-order transition for $N/M\to \infty$ (b). The vertical arrows represent $\delta$-function peaks. The blue and red peaks in (a) come from the coexisting phases: the NFL phase (red) and the incompressible phase (blue).

Figure 3

Same as in Fig. 1, but for $N=60M$. In this case, $x_0=0.45$.

Figure 4

The functions $\alpha \left(\mu \right)$ and $x\left(\mu \right)$ for $M=N$ obtained in Sec. pp1-s1.

Figure 5

The full solution $\tilde{\mathrm{\Sigma }}\left(i\omega \right)$ and $\tilde{\mathrm{\Pi }}\left(i\omega \right)$ for the Schwinger-Dyson equations. We have taken $m_0=3000{\omega }_F$.