Itinerant quantum critical point with frustration and a non-Fermi liquid

Employing the self-learning quantum Monte Carlo algorithm, we investigate the frustrated transverse-field triangle-lattice Ising model coupled to a Fermi surface. Without fermions, the spin degrees of freedom undergo a second-order quantum phase transition between paramagnetic and clock-ordered phases. This quantum critical point (QCP) has an emergent U(1) symmetry and thus belongs to the (2+1)D XY universality class. In the presence of fermions, spin fluctuations introduce effective interactions among fermions and distort the bare Fermi surface towards an interacting one with hot spots and Fermi pockets. Near the QCP, non-Fermi-liquid behaviors are observed at the hot spots, and the QCP is rendered into a different universality with Hertz-Millis–type exponents. The detailed properties of this QCP and possibly related experimental systems are also discussed.

Figure 1

(a) Illustration of our model. Fermions reside on two of the layers $\left(\lambda =1,2\right)$ with intralayer nearest-neighbor hopping $t$. The middle layer is composed of Ising spins $s_i^z$, subject to nearest-neighbor antiferromagnetic Ising coupling $J$ and a transverse magnetic field $h$. Between the layers, an onsite Ising coupling is introduced between fermion and Ising spins $\left(\xi \right)$. (b) The bare FS of the $H_{\text{fermion}}$ (yellow circle) and the folded FS (orange circles), coming from translating the bare FS by momentum $\mathbf{Q}$ (blue arrow). The folded FS contains Fermi pockets and hot spots. (c) Semiquantitative phase diagram. The dashed lines mark the phase boundaries of the naked bosonic model $H_{\text{spin}}$, with a QCP (open magenta dot) at $h_c=1.63\left(1\right)$ [40, 41]. The filled areas are the phases with fermions. The orange area is the clock phase with long-range order of Ising spins and Fermi pockets and hot spots, the pink area is the BKT phase with power-law correlation functions, the blue area is the quantum critical region, and the white area is the disordered paramagnetic phase. The QCP (solid blue dot) is shifted to a higher value $h=1.83\left(1\right)$ in comparison to the naked bosonic one, where non-Fermi-liquid behavior emerges near the hot spots.

Figure 2

Quasiparticle weight $Z\left(T\right)$ [(a) and (b)] and fermion self-energy $\mathrm{\Sigma }(\mathbf{k},{\omega }_n)$ [(c) and (d)]. (a, c) The ordered phase $h=1.0<h_c$, (c, d) at the QCP $h\approx h_c$. In (a) and (c), $Z\left(T\right)$ is suppressed on the hot spot due to the gap opening and correspondingly $\text{Im}\mathrm{\Sigma }(\mathbf{k},{\omega }_n)$ diverges, whereas in (b) and (d), $Z\left(T\right)$ is suppressed at the hot spot due to the emergence of non-Fermi-liquid behavior and correspondingly, the $\text{Im}\mathrm{\Sigma }(\mathbf{k},{\omega }_n)$ saturates at low frequency.

Figure 3

Spin susceptibility $\chi (T,h_c,\mathbf{q},\omega )$. To determine the $q$ and $\omega$ dependence, we plot $\frac{{\chi }^{-1}\left(\mathbf{q}\right)-{\chi }^{-1}\left(0\right)}{\left|\mathbf{q}\right|}$ and $\frac{{\chi }^{-1}\left(\omega \right)-{\chi }^{-1}\left(0\right)}{\omega }$ in (a) and (b). Although linear behaviors are observed in both figures, the difference in intersections [(a) zero and (b) finite] is crucial and indicates a clean quadratic momentum dependence $c_q{\left|\mathbf{q}\right|}^2$ and a crossover $c_{\omega }\omega +c_{\omega }^{\prime }{\omega }^2$ frequency dependence. (c) The $T$ dependence. At high $T$, the linear relation in the inset $({\chi }^{-1}$ vs $T^2)$ indicates a $T^2$ dependence. However, the low-$T$ part deviates strongly from $T^2$ and fits well to linear $T$. The constant $g$ here is the finite-size gap, which scales to zero in the thermodynamic limit. (d) The equal-time correlation function. Here we compared numerical results (dots) and the analytic theory in Eq. (8) (red solid line). The theoretical curve contains no adjustable parameters, where all values are determined from (b) and the $L=18$ results in (c). Dashed lines (yellow and green) show the asymptotic behaviors at low and high $T$, respectively.

Figure 4

We set $h=h_c,{\omega }_n=0$, and plot $ln[{\chi }^{-1}(h_c,T,\left|\mathbf{q}\right|,0)-{\chi }^{-1}(h_c,T,0,0)]$ as a function of $ln\left(\right|\mathbf{q}\left|\right)$ to obtain the power-law behavior $c_q{\left|\mathbf{q}\right|}^{a_q}|$ in the momentum dependence of bosonic susceptibility.

Figure 5

$T$ dependence of spin susceptibility $\chi (T,h_c,\mathbf{q},\omega )$ near the quantum critical point. (a), (b), and (c) are for system sizes $L=18$, 24, and 30, respectively. The obtained fitting parameters are summarized in Table 1.

Figure 6

(a) Correlation ratio vs transverse field with $\beta =L$ scaling relation and (b) binder ratio vs transverse field with $\beta =L$ scaling relation. The crossing point of the correlation ratio $R_c$ and the binder ratio $R_b$ converge to $h_c=1.83\left(1\right)$, which is the quantum critical point of our model.

Figure 7

(a) $\mathbf{q}$ dependance of the 2+1D XY dynamic bosonic susceptibility $\frac{{\chi }^{-1}\left(\mathbf{q}\right)-{\chi }^{-1}\left(0\right)}{\left|\mathbf{q}\right|}$. (b) $\omega$ dependance of the 2+1D XY dynamic bosonic susceptibility $\frac{{\chi }^{-1}\left(\omega \right)-{\chi }^{-1}\left(0\right)}{\omega }$. (c) $T$ dependence of the 2+1D XY dynamic bosonic susceptibility ${\chi }^{-1}(T,h_c,\left|\mathbf{q}\right|=0,\omega =0)$. The fits are performed according to Eq. (C1).

Figure 8

$K_{yy}(q_x=0,q_y)$ for different temperatures at $h=h_c$ for a $L=24$ and $L=30$ system. When $q_y$ approaches zero, the superfluid density approaches a negative number, indicating a paramagnetic response, and hence there are no strong superconductivity fluctuations.