Superconductivity mediated by quantum critical antiferromagnetic fluctuations: The rise and fall of hot spots

In several unconventional superconductors, the highest superconducting transition temperature $T_c$ is found in a region of the phase diagram where the antiferromagnetic transition temperature extrapolates to zero, signaling a putative quantum critical point. The elucidation of the interplay between these two phenomena—high-$T_c$ superconductivity and magnetic quantum criticality—remains an important piece of the complex puzzle of unconventional superconductivity. In this paper, we combine sign-problem-free quantum Monte Carlo simulations and field-theoretical analytical calculations to unveil the microscopic mechanism responsible for the superconducting instability of a general low-energy model, called the spin-fermion model. In this approach, low-energy electronic states interact with each other via the exchange of quantum critical magnetic fluctuations. We find that even in the regime of moderately strong interactions, both the superconducting transition temperature and the pairing susceptibility are governed not by the properties of the entire Fermi surface, but instead by the properties of small portions of the Fermi surface called hot spots. Moreover, $T_c$ increases with increasing interaction strength, until it starts to saturate at the crossover from hot-spots-dominated to Fermi-surface-dominated pairing. Our work provides not only invaluable insights into the system parameters that most strongly affect $T_c$, but also important benchmarks to assess the origin of superconductivity in both microscopic models and actual materials.

Figure 1

Schematic phase diagram of the spin-fermion model. The antiferromagnetic (AFM) transition temperature is suppressed to zero at $r=r_c$, giving rise to a quantum critical point. According to the results of Ref. [25] for the spin-fermion model, a superconducting (SC) dome then appears, hiding the antiferromagnetic quantum critical point. The maximum $T_c$ is found very close to $r=r_c$.

Figure 2

The two-band spin-fermion model. Fermi surfaces corresponding to the two bands (red and blue curves) in the first Brillouin zone, for different values of $\delta /t$. One of the bands (blue) is displaced by the AFM wave vector $\mathbf{Q}=\left(\pi ,\pi \right)$, which makes both Fermi surfaces appear concentric. In this representation, a pair of hot spots, defined by ${\varepsilon }_{c,{\mathbf{k}}_{\mathrm{hs}}}={\varepsilon }_{d,{\mathbf{k}}_{\mathrm{hs}}+\mathbf{Q}}=0$, correspond to the points at which the two Fermi surfaces overlap. For the system parameters used here, the hot spots are always along the diagonals of the Brillouin zone. By changing the parameter $\delta /t$, the system interpolates between closed nearly nested Fermi surfaces $\left(\delta /t<1/4\right)$ and open Fermi surfaces $\left(\delta /t>1/4\right)$, crossing a Van Hove singularity at $\delta /t=1/4$. The angle ${\theta }_{\mathrm{hs}}$ between the Fermi velocities of a pair of hot spots (red and blue arrows) increases as function of $\delta /t$ (note that one of the Fermi velocities has been multiplied by $-1$ for clarity purposes).

Figure 3

The superconducting transition temperature $T_c$ at the QCP for different band dispersion parameters. (a) The QMC results for $T_c$ and the calculated density of states $N_f$ (calculated directly from the band dispersions) as function of the band dispersion parameter $\delta /t$ (see Fig. 2). We associate a transition temperature $T_c\left(L\right)$ with the temperature at which the BKT condition is met for a system of size $L$, and denote $T_c\left(L_{\max }\right)$ by filled symbols. Analysis of finite-size effects reveals that for most values of $\delta /t,T_c\left(L_{\max }\right)$ is a very good estimate for the thermodynamic-limit value $T_c$. For the systems in which $T_c\left(L\right)$ does not fully converge, namely $\delta /t=0.6$ and $\delta /t=0.8,T_c\left(L_{\max }\right)$ are upper bound values for $T_c$, whereas the stars are lower bound values on $T_c$. Note the enhanced $N_f$ at the Van Hove singularity point $\delta /t=1/4$. (b) The linear relationship between $T_{\mathrm{c}}$ and $sin{\theta }_{\mathrm{hs}}$, where ${\theta }_{\mathrm{hs}}$ is the angle between the two Fermi velocities of a pair of hot spots, calculated directly from the band dispersions.

Figure 4

Universal temperature dependence of the pairing susceptibility ${\chi }_{\mathrm{pair}}$ at the QCP. (a) Temperature dependence of ${\chi }_{\mathrm{pair}}^{-1}$ extracted from QMC simulations for all band dispersion parameters $\delta /t$. The system size is $L=12$. (b) Collapse of the scaled ${\chi }_{\mathrm{pair}}^{-1}\left(T\right)/{\chi }_{\mathrm{pair}}^{-1}\left(3T_c\right)$ as function of $T/T_c$ for all values of $\delta /t$ and all system sizes $L$. For each value of $L$, we used the corresponding $T_c\left(L\right)$. The black dashed curve is the analytical function $f_{\mathrm{pair}}^{\left(\mathrm{hs}\right)}(T/T_c)/f_{\mathrm{pair}}^{\left(\mathrm{hs}\right)}\left(3\right)$ obtained from the hot-spots Eliashberg approximation of the spin-fermion model. (c) The behavior of the QMC-extracted prefactor $A_{\mathrm{pair}}\propto {\chi }_{\mathrm{pair}}\left(3T_c\right)$ of Eq. (2) as function of $\delta /t$.

Figure 5

Dependence of the superconducting transition temperature on the interaction strength. For three values of the band dispersion parameter $\delta /t$, we show the QMC results for $T_c$, in units of the hopping parameter $t$ and normalized by the corresponding value of $sin{\theta }_{\mathrm{hs}}$, as function of the squared coupling constant ${\lambda }^2$ (in units of $8t)$ describing how strong the electrons interact with AFM fluctuations. The system size is $L=12$. The dashed line, which denotes a ${\lambda }^2$ dependence, has the same slope as in Fig. 3, and is expected from the analytical hot-spots Eliashberg solution of the spin-fermion model. The absence of the data point corresponding to $\delta /t=0.8$ and ${\lambda }^2=4.5t$ is because $T_c$ did not converge as a function of the system size for these parameters.

Figure 6

Binder cumulant $\mathcal{B}$ (a) and static spin susceptibility ${\chi }_M$ (b) as a function of $r$ for various inverse temperatures. The set of parameters used here is $\left(\delta /t,L,{\lambda }^2/t\right)=\left(0.6,12,8\right)$. The inverse temperature $\beta$ is in units of $1/t$.

Figure 7

Panel (a) shows the renormalized mass term of the magnetic propagator, $\tilde{r}$, as function of $r_c$. The set of parameters used here is $\left(\delta /t,L,{\lambda }^2/t\right)=\left(0.6,12,8\right)$. The inverse renormalized magnetic propagator ${\tilde{\chi }}^{-1}(\mathbf{q},i{\mathrm{\Omega }}_n)$ at $r=r_c$ is plotted as function of ${\mathrm{\Omega }}_n$ for $\mathbf{q}=0$ (b) and as function of $q$ for ${\mathrm{\Omega }}_n=0$ (c). In (b) and (c), the inverse temperature is $\beta =7/t$.

Figure 8

Static pairing susceptibility ${\chi }_{\mathrm{pair}}^{\left(a\right)}$ in the sign-changing gap channel $\mathrm{[}a=1$; panel (a)] and in the in the sign-preserving gap channel $\mathrm{[}a=2$; panel (b)] as function of the distance to the QCP at $r=r_c$. The inverse temperature $\beta$ is in units of $1/t$ and the susceptibilities are normalized by the noninteracting susceptibility ${\chi }_{\mathrm{pair},0}$ obtained by setting $\lambda =0$. The set of parameters used here is $\left(\delta /t,L,{\lambda }^2/t\right)=\left(0.6,12,8\right)$.

Figure 9

Superfluid density ${\rho }_s(L,T)$ as function of temperature $T$ for the band dispersion $\delta /t=0.6$ and coupling constant ${\lambda }^2=8t$ for various system sizes $L$. The BKT transition temperature for each system size is determined by the condition ${\rho }_s(L,T_{\mathrm{c}})=\frac{2}{\pi }{T}_c$.

Figure 10

(a) The QMC extracted $T_c\left(L\right)$ as function of the inverse system size $1/L$ for all band dispersion parameters $\delta /t$. Interpolated ${\rho }_S\left(T\right)$ curve for $\delta /t=0.4$ (b) and for $\delta /t=0.8$ (c).

Figure 11

(a) ${\chi }_{\mathrm{pair},0}^{(L=12)}$ of the finite-size system $L=12$ plotted as a function of the inverse temperature $\beta$ (in units of $1/t)$ at the Van Hove point $\left(\delta /t=0.25\right)$ and at $\delta /t=0.6$. (b) Comparison between the density of states of the finite-size system, $N_f^{(L=12)}$, and the density of states computed analytically, $N_f$. The results match except very close to the Van Hove singularity, where the divergence is cut off by finite-size effects.

Figure 12

$T_c$ as a function of $sin\left({\theta }_{\mathrm{hs}}\right)$ for ${\lambda }^2/8t=1$, where $T_c$ scales with ${\lambda }^2$, and ${\lambda }^2/8t=4$, where $T_c$ is in the saturation regime (see Fig. 5 of the main text). The linear dependence of $T_c$ on $sin{\theta }_{\mathrm{hs}}$ remains robust for larger values of the interaction parameter $\lambda$. The results in this figure are obtained for $L=12$.