Hierarchy of energy scales in an O(3) symmetric antiferromagnetic quantum critical metal: A Monte Carlo study

We present numerically exact results from sign-problem free quantum Monte Carlo simulations for a spin-fermion model near an O(3) symmetric antiferromagnetic (AFM) quantum critical point. We find a hierarchy of energy scales that emerges near the quantum critical point. At high-energy scales, there is a broad regime characterized by Landau-damped order parameter dynamics with dynamical critical exponent $z=2$, while the fermionic excitations remain coherent. The quantum critical magnetic fluctuations are well described by Hertz-Millis theory, except for a $T^{-2}$ divergence of the static AFM susceptibility. This regime persists down to a lower-energy scale, where the fermions become overdamped and, concomitantly, a transition into a $d$-wave superconducting state occurs. These findings resemble earlier results for a spin-fermion model with easy-plane AFM fluctuations of an O(2) spin density wave (SDW) order parameter, despite noticeable differences in the perturbative structure of the two theories. In the O(3) case, perturbative corrections to the spin-fermion vertex are expected to dominate at an additional energy scale, below which the $z=2$ behavior breaks down, leading to a novel $z=1$ fixed point with emergent local nesting at the hot spots [Schlief et al., Phys. Rev. X 7, 021010 (2017)]. Motivated by this prediction, we also consider a variant of the model where the hot spots are nearly locally nested. Within the available temperature range in our study ($T\ge E_F/200$), we find substantial deviations from the $z=2$ Hertz-Millis behavior, but no evidence for the predicted $z=1$ criticality.

Figure 1

Fermi surface with hot spots across the Brillouin zone. The black lines correspond to the Fermi surfaces of the two fermion flavors, ${\psi }_x$ (solid) and ${\psi }_y$ (dashed). One band has been shifted by $Q=(\pi ,\pi )$ such that hot-spot pairs (red points) occur at crossing points. The energy of the ${\psi }_x$ band is shown as background (color shading).

Figure 2

Electron hopping and local interactions on the square lattice. (a) Illustration of nearest (green), next-nearest (blue), and next-next-nearest neighbor (red) hopping processes and (b) local Yukawa interaction (dashed lines) of the two fermion flavors ${\psi }_x,{\psi }_y$ and the magnetic order parameter $\varphi$ present in model (1).

Figure 3

Scaling regimes suggested by perturbation theory. There is a parametrically large (in the number of hot-spot pairs $N$) Landau-damped regime. For the spin-fermion model (1) with a Fermi surface as in Fig. 1, $N=4$.

Figure 4

Effect of nesting on the vertex energy scale ${\mathrm{\Omega }}_{\lambda }$ given in Eq. (8). Here, $\theta$ is the relative angle between the Fermi velocities (as indicated in the insets). Perfect local nesting corresponds to $\theta =0$ and antinesting to $\theta =\pi$.

Figure 5

Phase diagram of the spin-fermion model, given by Eq. (1), for (a) $\lambda =2$ and (b) $\lambda =1$. The correlation length ${\xi }_{\mathrm{AFM}}$ is shown for a $L=12$ system (color coding, parameter grid indicated by gray points). In accordance with the Mermin-Wagner theorem, there is no magnetic phase transition at finite temperatures, but only a crossover originating from the QCP at $r=r_c$. For large tuning parameter values $r\gg r_c$, the system is in an ordinary Fermi-liquid phase. In the opposite limit, $r\ll r_c$, the SDW correlation length diverges as $T\to 0$. For $\lambda =2$, an extended dome-shaped $d$-wave superconducting phase is masking the QCP (blue). The maximal $T_c$ is of the order of $E_F/20$. For $\lambda =1$, the system stays nonsuperconducting down to the lowest considered temperature, $T=0.025$. Depending on the coupling strength, the quantum critical point, marking the onset of SDW order at $T=0$, lies at $r_c\approx 3.85$ ($\lambda =2$) or $r_c\approx -1.89$ ($\lambda =1$), respectively.

Figure 6

Nature of the superconducting state. Momentum-resolved equal-time pairing correlations across the first Brillouin zone with $d$-wave (left) and $s$-wave (right) symmetry close to the quantum critical point for $\lambda =2$, $\beta =20$, and $L=12$. While the $s$-wave correlations are featureless, there is a distinct peak in the $d$-wave channel.

Figure 7

Frequency dependence of the magnetic susceptibility $\chi$ close to the quantum critical point ($r=-1.74$) at inverse temperature $\beta =40$ and $\lambda =1.0$. The different colors indicate different momenta. The solid lines are linear low-frequency fits.

Figure 8

Momentum dependence of the magnetic susceptibility $\chi$ close to the quantum critical point ($r=-1.74$) at inverse temperature $\beta =40$ and $\lambda =1.0$. The different colors indicate the lowest few Matsubara frequencies. The solid lines are linear fits.

Figure 9

Tuning parameter dependence of the inverse magnetic susceptibility, given by Eq. (12), close to the quantum critical point at inverse temperature $\beta =40$ and $\lambda =1.0$. We show a linear fit of the data (light green line) with root $r_c\approx -1.89$, an estimate for the location of the QCP.

Figure 10

Temperature dependence of the magnetic susceptibility $\chi$ at $r\approx -1.86$, close to the quantum critical point, for $\lambda =1.0$. The black line is a quadratic fit, $f\left(x\right)=a_2{x}^2+a_{1}x+a_0$, with $a_2\approx 0.366$, $a_1\approx -0.005$, and $a_0\approx 0.009$, for system sizes $L>8$.

Figure 11

Imaginary part of the fermionic self-energy as a function of Matsubara frequency for (a) $r=-1.95<r_c$, (b) $r=-1.85\approx r_c$, and (c) $r=-1.55>r_c$. Shown are results for $\lambda =1$ and $L=12$. The dotted line indicates the non-Fermi-liquid crossover scale, $-\mathrm{Im}\mathrm{\Sigma }\left({\omega }_n\right)={\omega }_n$. The solid lines represent the self-energy at the hot spots, $\mathbf{k}={\mathbf{k}}_{\mathrm{hs}}$, and dashed lines represent the self-energy at the Fermi point at $k_y=0$, namely, $\mathrm{\Sigma }[\mathbf{k}={\mathbf{k}}_{\mathbf{x}}=(k_f,0)]$. The data points are averaged over different twisted boundary conditions, while the width of the line is the standard deviation of $\mathrm{Im}\mathrm{\Sigma }$ between the different boundary conditions, thus representing the uncertainty due to finite-size effects; see text.

Figure 12

Fermi surface close to local nesting near the hot spots across the Brillouin zone. The black lines correspond to the two fermion flavors, ${\psi }_x$ (solid) and ${\psi }_y$ (dashed). One band has been shifted by $Q=(\pi ,\pi )$ such that hot-spot pairs (red) occur at crossing points.

Figure 13

Frequency dependence of the magnetic susceptibility $\chi$ close to the quantum critical point ($r=1.625$) at ultralow temperature $T=1/100$ and $\lambda =1.0$ for the almost locally nested Fermi surface in Fig. 12. The solid line is the noninteracting fermionic susceptibility ${\mathrm{\Pi }}_0$.

Figure 14

Momentum dependence of the magnetic susceptibility $\chi$ close to the quantum critical point ($r=1.625$) at ultralow temperature $T=1/100$ ($T=1/40$ in the inset) for the almost locally nested Fermi surface in Fig. 12 with $\lambda =1.0$. The solid lines are linear fits.

Figure 15

First-order transition occurring at low temperatures for $u=0$. Shown is the tuning parameter dependence of the inverse magnetic susceptibility ${\chi }^{-1}$ close to the quantum critical point for various temperatures.

Figure 16

Continuous tuning parameter dependence of model (1) for the locally nested Fermi surface in Fig. 12 with vanishing quartic coupling, $u=0$, at inverse temperature $\beta =40$.

Figure 17

Charge density correlations across the phase diagram for $\lambda =2$ (top panel) and $\lambda =1$ (bottom panel). Shown is the temperature dependence of the maximum (excluding $\mathbf{q}=0$) of the correlations defined in Eq. (C1) for different tuning parameter values $r\ll r_c$ (purple), $r\gtrsim r_c$ (green), and $r\gg r_c$ (pink). For $\lambda =2$, the dashed gray line indicates the temperature at which superconductivity sets in (for $r=4.3\gtrsim r_c$).

Figure 18

Temperature dependence of the superfluid density of model (1) for $\lambda =2$. The colors indicate three different tuning parameter values: $r=4.2\approx r_{T_c^{max}}$ (blue), $r=5.1>r_{T_c^{max}}$ (orange), and $r=6.0\gg r_{T_c^{max}}$ (green). The dashed line indicates the critical BKT value $\mathrm{\Delta }{\rho }_s=2T/\pi$.