Phase diagram of the spin-$\frac{1}{2}$ Yukawa–Sachdev-Ye-Kitaev model: Non-Fermi liquid, insulator, and superconductor

We analyze the phase diagram of the spin-$\frac{1}{2}$ Yukawa–Sachdev-Ye-Kitaev model, which describes complex spin-$\frac{1}{2}$ fermions randomly interacting with real bosons via a Yukawa coupling, at finite temperatures and varying fermion density. A recent work [Wang and Chubukov, Phys. Rev. Res. 2, 033084 (2020).] showed that, upon varying the filling or chemical potential, a first-order quantum phase transition exists between a non-Fermi-liquid (NFL) phase and an insulating phase. Here we show that in such a model with time-reversal symmetry this quantum phase transition is preempted by a pairing phase that develops as a low-temperature instability. As a remnant of the would-be NFL-insulator transition, the superconducting critical temperature rapidly decreases beyond a certain chemical potential. On the other hand, depending on the parameters, the first-order quantum phase transition extends to finite temperatures and terminates at a thermal critical point, beyond which the NFL and the insulator become the same phase, similar to that of the liquid-gas and metal-insulator transition in real materials. We determine the pairing phase boundary and the location of the thermal critical point via combined analytic and quantum Monte Carlo numeric efforts. Our results provide a model realization of the transition of NFLs towards superconductivity and insulating states and therefore offer a controlled platform for future investigations of the generic phase diagram that hosts a NFL, insulator, and superconductor and their phase transitions.

Figure 1

(a) Phase diagram of the Yukawa-SYK model at $N=4M,{\omega }_0=1,m_0=2$. From the large-$N$ calculation, one sees the NFL become a superconductor at low temperature in a wide range of the chemical potential and the first-order hysteresis region denoted by the red points and lines, obtained in the absence of pairing instability within large $N$. In the presence of superconductivity the position of this region is renormalized, hence the dashed line. The thermal critical point that terminates the first-order transition locates at $({\mu }_c=0.194,T_c=0.015)$. The blue triangles are the transition points from a NFL to a superconductor obtained from QMC at finite $N,M$ (see Fig. 4), which are consistent with the results obtained from large-$N$ calculations (black squares). (b) Phase diagram of the Yukawa-SYK model at $N=M,{\omega }_0=1,m_0=2$ from the large-$N$ calculation. The first-order hysteresis region denoted by the red points and lines is obtained in the absence of pairing instability within large $N$. The dashed-line portion of this boundary is renormalized by superconductivity phase. The black dashed line denotes the boundary of the superconducting phase within the hysteresis region and is depicted only qualitatively. The QMC $n-\mu$ curves in Fig. 2 are along blue dashed paths. The thermal critical point at $({\mu }_c=0.3825,T_c=0.07)$ is denoted by the black star.

Figure 2

Filling $n$ versus $\mu$ for selected $T$ in the vicinity of the thermal critical point in Fig. 1 (b) for $N=M,{\omega }_0=1,m_0=2$. At higher $T=0.125$, one sees the $n\left(\mu \right)$ curves from both large $N$ and QMC are smooth. $T=0.125,\mu =0.375$ (marked by the star) is a thermal critical point. At temperature $T=0.083$ close to the thermal critical point, there is a sharp turn in $n\left(\mu \right)$ signifying the divergence of the compressibility $dn/d\mu$. At lower $T$, there is a range of $\mu$—the hysteresis region—where the filling is double valued. The lower branch represents the NFL behavior, and the upper branch represents the insulating behavior; a first-order transition connects the two at a chemical potential given by a Maxwell construction. The dashed line delimits the region where solutions are unstable. Note that the QMC results at $T=0.050$ (the blue triangles) further differ from the first-order transition behavior (the blue curves). At finite $M,N$ and at finite temperatures, the system is finite and does not have phase transitions. Instead, the system undergoes crossovers, which become phase transitions only in the thermal dynamical limit. We have verified that, numerically, as one increases $M,N$, the numerical data indeed tend to approach the large-$N$ curve.

Figure 3

(a) The QMC fermionic Green's functions and (b) the bosonic Green's functions with different $\mu$. $M=4, N=16, \beta =32, {\omega }_0=1, m_0=2$, plotted with semilog axes. For convenience both $G_f$ and $G_b$ have been normalized to 1 at $\tau =\beta$. The system becomes gapped with the increase of the chemical potential. The sharp downturn of the large-$N$ result in (a) is an artifact of keeping finite-frequency points.

Figure 4

Pair susceptibility $P_s$ measured at different chemical potentials $\mu$. The obtained $T_c$ (${\beta }_c$) are denoted by the blue triangles in Fig. 1. From the temperature dependence of $P_s$ with different system sizes $(M,N)$ we perform the data collapse using mean-field exponents ${\gamma }_0=1, {\nu }_0=2$, and the transition temperatures $T_c$ (${\beta }_c$) are obtained accordingly. The parameters are $N=4M, {\omega }_0=1, m_0=2$. $\mu =0$ in (a) and (b), $\mu =0.05$ in (c) and (d), $\mu =0.125$ in (e) and (f), $\mu =0.2$ in (g) and (h), $\mu =0.25$ in (i), and $\mu =0.35$ in (j). The superconducting transition temperature decreases as $\mu$ increases. For $\mu =0.25$ in (i) and $\mu =0.35$ in (j) the pairing susceptibilities are not divergent at larger $N$, and the system enters a gapped insulator phase.

Figure 5

Spin-$\frac{1}{2}$ Yukawa-SYK model. There are $i,j=1,\cdots ,M$ quantum dots; each dot acquires $\alpha ,\beta =1,\cdots ,N$ flavors. Fermions are Yukawa coupled via the random hopping $t_{ij}$ and antisymmetric bosonic field ${\varphi }_{\alpha \beta }$. The system goes through a phase transition from non-Fermi liquid to a pairing state when the temperature is below the superconducting critical temperature $T_{sc}$.

Figure 6

Pair susceptibility for different $\beta$ and data collapse, ${\omega }_0=1, m_0=1, \mu =0$. The superconducting transition temperature is around $T_{sc}\sim 0.1 ({\beta }_c\sim 10)$, which is higher than the superconducting transition temperature around $T_{sc}\sim 0.067({\beta }_{sc}\sim 15)$ at ${\omega }_0=1,m_0=2$, and $\mu =0$.