Nature of the effective interaction in electron-doped cuprate superconductors: A sign-problem-free quantum Monte Carlo study

Understanding the mechanism of Cooper pairing amounts to determining the effective interaction that operates at low energies. Efforts to achieve such a goal for superconducting materials, especially strongly correlated ones, from both bottom-up and top-down approaches, have been plagued by having to use uncontrolled approximations. Here, we perform large-scale, numerically exact, sign-problem-free zero-temperature quantum Monte Carlo simulations on an effective theory based on “hot spots” plus fluctuating collective modes. Because hot spots are clearly identified by angle-resolved photoemission spectroscopy for electron-doped cuprates, we focus our attention on such materials. Our goal is to determine the minimum effective action that can describe the observed superconductivity and charge-density wave. The results suggest that antiferromagnetic fluctuation alone is not sufficient—the effective action needs to be amended with nematic fluctuations. We believe that our results address the pairing mechanism of high-$T_c$ superconductivity in electron-doped cuprates, and they shed light on the pairing mechanism of hole-doped cuprates.

Figure 1

(a) A prototypical Fermi surface of the cuprates. The dashed lines enclose the AFM Brillouin zone, the hot spots are denoted by green dots, and the purple arrow indicates the AFM ordering wave vector $(\pi ,\pi )$. (b) The Fermi surfaces of the two-band model, where the positions of the hot spots and the Fermi velocity at the hot spots are made to mimic those in the single-band model used to describe the cuprates. The red and blue Fermi surfaces are derived from the bands formed by the $x$ and $y$ orbitals, respectively, described by the action in Eq. (S1). (c) The nematically distorted Fermi surface. The arrows indicate the distortion. (d) AFM order parameter $|〈\overrightarrow{{\phi }_s}〉|$ and $d$-wave SC order parameter $〈{\mathrm{\Delta }}_d〉$ as a function of $r_s$. For each value of $r_s$, we perform a finite-size extrapolation to determine the order parameters in the thermodynamic limit. The blue points represent the maximum $d$-wave bond CDW structure factor, which is reached at the diagonal wave vectors for $L=16$. (f) AFM order parameter $|〈\overrightarrow{{\phi }_s}〉|$ and $d$-wave SC order parameter $〈{\mathrm{\Delta }}_d〉$ as a function of $r_s$ when both AFM and nematic fluctuations are considered. The blue points are the maximum $d$-wave bond CDW structure, which is reached at the horizontal wave vectors for $L=16$. It is remarkable that adding nematic fluctuation can not only enhance $d$-wave superconducting pairing but also induce the CDW wave vectors, consistent with recent experiments [12].

Figure 2

(a) The $s$- and $d$-wave SC pair correlations ${\overline{P}}_{s/d}$ evaluated at maximum separation ${\vec{x}}_{\max }=(L/2,L/2)$ for $L=12,14,16$ and various values of $r_s$. The peak of the enhancement for $d$-wave pairing occurs at $r_s\approx 0.25$, which is close to the AFM quantum critical point. (b) Both $s$- and $d$-wave pair correlations evaluated at ${\vec{x}}_{\max }$ are plotted vs $1/L$ for $L=8,10,12,14,16,18$. By fitting them using $f(1/L)=a+b/L+c/L^2$ and extrapolating to the thermodynamic limit ($L=\infty$), we find that the $d$-wave pairing correlations are extrapolated to a nonzero value, $(3.1\pm 0.8)\times {10}^{-4}$, while the $s$-wave pairing correlations are extrapolated to zero within the error bar. Interestingly, the finite-size dependence of both correlation functions is dominated by $1/L^2$, as shown by the nearly linear dependence of ${\overline{P}}_{d\left(s\right)}$ on $1/L^2$ in the inset of (b). These results clearly indicate that the ground state possesses $d$-wave superconducting long-range order. (c) A comparison of the $d$- and $s$-wave SC pairing correlations at ${\vec{x}}_{\max }$ in the system with $L=14$, without and with nematic fluctuation. In the latter case we set $r_n=0.5$, which places the system on the disordered side of the nematic transition. The results show a considerably enhanced $d$-wave SC correlation by the nematic fluctuations.

Figure 3

(a) $\mathrm{\Delta }{S}_{s/d}\left(\vec{Q}\right)$, the difference between $S_{s/d}\left(\vec{Q}\right)$ with ($r_s=0.5$) and without the AF fluctuations, vs $\vec{Q}$ for $L=16$. $S_{s/d}\left(\vec{Q}\right)$ is the $s$- and $d$-wave bond CDW structure factor vs $\vec{Q}=(k,0)$ (horizontal CDW) or $\vec{Q}=(k,k)$ (diagonal CDW). (b) $\mathrm{\Delta }{S}_{s/d}\left(\vec{Q}\right)$, the difference between $S_{s/d}\left(\vec{Q}\right)$ with ($r_s=0.5,r_n=0.5$) and without the AFM+nematic fluctuations, for $L=16$. (c) $\mathrm{\Delta }{S}_d\left({\vec{Q}}_0\right)$ as a function of $1/L$ for $L=14,16,18$. The red dots represent the enhancement of the peak $d$-wave bond CDW structure factor by both spin ($r_s=0.5$) and nematic ($r_n=0.5$) fluctuations. The black dots are the enhancement by spin fluctuations ($r_s=0.5$) alone. The solid curves are the best fit using $f(1/L)=a+b/L+c/L^2$. In both cases, the structure factors extrapolate to zero within error bars in the thermodynamic limit. This suggests that the bond CDW order induced by AFM and/or nematic fluctuations is short-ranged.

Figure 4

(a) $\mathrm{\Delta }{S}_d\left(\vec{Q}\right)$ for $r_s=0.5$ and $L=16$ is plotted over the first quadrant of the Brillouin zone. The peak is situated around $\vec{Q}=(2\pi /3,2\pi /3)$. (b) $\mathrm{\Delta }{S}_d\left(\vec{Q}\right)$ for $L=16$, driven by both spin ($r_s=0.5$) and nematic ($r_n=0.5$) fluctuations. The strongest peaks are situated around $\vec{Q}=(2\pi /3,0)$ and $(0,2\pi /3)$.

Figure 5

(a) The $d$-wave bond CDW order parameter $\langle B_d\left(i\right)\rangle \equiv \langle {\sum }_{a=\pm \widehat{x}}{\psi }_{i,x}^{†}{\sigma }_0{\psi }_{i+a,x}-{\sum }_{a=\pm \widehat{y}}{\psi }_{i,y}^{†}{\sigma }_0{\psi }_{i+a,y}\rangle$ in a system of size $L=16$ with open boundary conditions. Here ${\sigma }_0$ is the identity matrix in the spin space. The black dots represent the lattice sites. The pattern of bond CDW order parameter modulations obtained numerically is almost perfectly consistent with the expected modulations with a $3a$ period. (b) The concurrence probability $P(B_x,B_y)$ in the QMC simulations is plotted as a function of $B_x$ and $B_y$, where $B_{x/y}$ are the $d$-wave bond CDW order parameters associated with ordering wave vectors $(2\pi /3,0)$ and $(0,2\pi /3)$, respectively. The system size in this computation is $L=15$, and the parameters $(r_s,r_n)$ used are $(0.5,0.5)$.