Charge fluctuations in lightly hole-doped cuprates: Effect of vertex corrections

Identification of the electronic state that appears upon doping a Mott insulator is important to understand the physics of cuprate high-temperature superconductors. Recent scanning tunneling microscopy of cuprates provides evidence that a charge-ordered state emerges before the superconducting state upon doping the parent compound. We study this phenomenon by computing the charge response function of the Hubbard model including frequency-dependent local vertex corrections that satisfy the compressibility sum rule. We find that upon approaching the Mott phase from the overdoped side, the charge fluctuations at wave vectors connecting hot spots are suppressed much faster than at the other wave vectors. It leads to a momentum dependence of the dressed charge susceptibility that is very different from either the bare susceptibility or from the susceptibility obtained from the random phase approximation. We also find that the paramagnetic lightly hole-doped Mott phase at finite temperature is unstable to charge ordering only at zero wave vector, confirming the results previously obtained from the compressibility. Charge order is driven by the frequency-dependent scattering processes that induce an attractive particle-hole interaction at large interaction strength and small doping.

Figure 1

Compressibility, $\partial n/\partial \mu$, as a function of hole density, $p$, for $U=12t$, $t^{\prime }=-0.3t$, $t^{″}=0.2t$ at various temperatures, either computed directly within the DMFT framework, labeled 1P in the figure and discussed in Sec. 3, or computed from the lattice density-density correlation function, indicated as 2P and discussed in Sec. 5. Inset: The hole density as a function of chemical potential for $T=0.02t$ has a jump for a chemical potential ${\mu }_c\simeq 4.8$ with a discontinuous first-order derivative at that chemical potential, implying the discontinuous compressibility seen in the main panel.

Figure 2

Real (left) and imaginary (right) parts of ${\left[{\mathrm{\Gamma }}_{loc}^{c,irr}({\nu }_n=0)\right]}_{{\omega }_m{\omega }_{m^{\prime }}}$ as a function of two fermionic frequencies ${\omega }_m$ and ${\omega }_{m^{\prime }}$ for hole doping level $p=0.11$ and $T=0.1t$, $U=6t$. The static contribution $U$ is substracted from the real part.

Figure 3

Real part of ${\left[{\mathrm{\Gamma }}_{loc}^{c,irr}({\nu }_n=0)\right]}_{{\omega }_m{\omega }_{m^{\prime }}}$ for hole doping level $p=0.11$ as a function of two fermionic frequencies ${\omega }_m$ and ${\omega }_{m^{\prime }}$ and $T=0.1t$. On the left panel, $U=8t$ while $U=12t$ on the right panel. The static contribution $U$ is subtracted.

Figure 4

Left panel: Dressed susceptibility in the charge channel for $U=12t$ and $T=0.1t$, for several hole doping values. Right panel: Dressed susceptibility in the charge channel for $U=12t$, for several temperatures at hole doping slightly larger than the critical doping.

Figure 5

${\left[{\chi }_{loc}^c({\nu }_n=0)\right]}_{{\omega }_m{\omega }_{m^{\prime }}}$ with ${\omega }_m={\omega }_0=\pi /\beta$ as a function of ${\omega }_{m^{\prime }}$ calculated with $n_b=3$ and $n_b=5$. The doping level is $p=0.06$, $U=12t$.

Figure 6

Spectral weight at zero frequency (left panel) for hole doping levels $p=0.18$ (red), and $p=0.04$ (blue) for $U=12t$, $t^{\prime }=-0.3t$, $t^{″}=0.2t$ at $T=0.1t$ on the square lattice. The dashed line shows the antiferromagnetic Brillouin zone. The wave vectors ${\mathbf{Q}}_{1\cdots 4}$ connect hot spots when $p=0.18$. The corresponding RPA bubble susceptibility is in the right-hand panel.

Figure 7

Compressibility $dn/d\mu$ computed from the density and from the uniform density-density correlation function, labeled by 1P and 2P, respectively, as a function of hole density for $U=8t$ and $T=0.1t$ (top panel) and $U=12t$ and $T=0.4t$ (bottom panel).