The finite-difference parquet method: Enhanced electron-paramagnon scattering opens a pseudogap

Abstract

We present the finite-difference parquet method that greatly improves the applicability and accuracy of two-particle correlation approaches to interacting electron systems. This method incorporates the nonperturbative local physics from a reference solution and builds all parquet diagrams while circumventing potentially divergent irreducible vertices. Its unbiased treatment of different fluctuations is crucial for reproducing the strong-coupling pseudogap in the underdoped Hubbard model, consistent with diagrammatic Monte Carlo calculations. We reveal a strong-coupling spin-fluctuation mechanism of the pseudogap with decisive vertex corrections that encode the enhanced, energy-dependent scattering amplitude between electrons and antiferromagnetic spin fluctuations.

Figures (20)

• Figure 1: (a) Comparison of the two-particle propagator $\Pi$ and its fd counterpart $\tilde{\Pi}$ for an exemplary AIM fd-PA calculation. Both quantities are ($L^{\infty}$-)normalized to reveal the rapid decay of $|\tilde{\Pi}(\omega, \nu)|$ in $\nu$. (b) Charge susceptibility $\chi_{\mathrm{D}}(T)$ in the AIM from different methods. Our fd-D$\Gamma$A(HA) employs the Hubbard atom as reference and is soluble irrespective of the 2PI vertex divergence at $T \approx 1.58$. Inset: Spin susceptibility.
• Figure 2: HM at PHS and $U_0= 8.356$, $T = 2$, $t' = 0$. (a) DMFT 2PI vertex in the density channel just below and above the first 2PI vertex divergence ($U_\pm = U_0 \pm 0.002$) and (b) its fd counterpart at $\mathbf{q}=\Gamma$ obtained from (fd-)pD$\Gamma$A. (c) Self-energy and susceptibilities, where the tilde denotes the difference from DMFT, at $U_0$ from (fd-)pD$\Gamma$A, (self-consistent) $\ell$D$\Gamma$A, PA, DQMC. The vertical line is the DQMC error bar. $\tilde{\chi}_\mathrm{{D}}$ in PA is outside the plotted range ($\sim 25 \times 10^{-3}$).
• Figure 3: Spectral functions from fd-pD$\Gamma$A for the HM at $U \!=\! 5.6$, $T \!=\! 0.2$, $t' \!=\! -0.3$, and 4% hole doping. (a) Matsubara proxy for the spectral function at the Fermi level, $A_\mathbf{k}^{\mathrm{imag}} \!=\! -\frac{1}{\pi} \operatorname{Im} \frac{1}{\mu - \varepsilon_\mathbf{k} - \Sigma_\mathbf{k}(\pi T)}$. The gray dashed line denotes the Fermi surface determined by $\mu - \varepsilon_\mathbf{k} - \operatorname{Re} \Sigma_{\mathbf{k}}(\pi T) \!=\! 0$, which almost coincides with the maximum of $A_\mathbf{k}^{\mathrm{imag}}$ (cf. Ref. Simkovic2024). (b) Real-frequency spectral function from maximum-entropy analytic continuation Kaufmann2023anacont of the self-energy. The gray dashed line shows the bare dispersion.
• Figure 4: HM at $U = 5.6$, $T = 0.2$, $t' = -0.3$, and 4% hole doping. (a) Matsubara frequency dependence of $\operatorname{Im} \Sigma_\mathbf{k}(\nu)$ at the node $\mathbf{k}_\mathrm{{N}} = (1.47, 1.47)$, hot spot $\mathbf{k}_\mathrm{{HS}} = (2.26, 0.88)$, and antinode $\mathbf{k}_\mathrm{{AN}} = (3.04, 0.49)$ in pD$\Gamma$A using Eq. \ref{['eq:local_corr']}, compared to DiagMC (data from Ref. Wu2017). (b) Momentum dependence of $\Delta \Sigma_\mathbf{k} = \operatorname{Im} \Sigma_\mathbf{k}(\pi T) - \operatorname{Im} \Sigma_\mathbf{k}(3\pi T)$ in pD$\Gamma$A. The dashed line indicates the Fermi surface obtained from $\varepsilon_\mathbf{k} + \operatorname{Re}\Sigma_\mathbf{k}(\pi T) = \mu$. (c,d) Same as (a,b), but for $\ell$D$\Gamma$A.
• Figure 5: Fluctuation diagnostics (parquet decomposition) Gunnarsson2016 for $\Sigma$ from Fig. \ref{['fig:Wu_self_energy']}. (a) Result of the SDE at $\mathbf{k}_\mathrm{{N}}$ and $\mathbf{k}_\mathrm{{AN}}$ from the full vertex $F$, the fd reducible vertices in the magnetic (M), density (D), and singlet (S) channels, and the DMFT impurity vertex ($f$). (b) Interaction-reducible part of the $\tilde{\Phi}_\mathrm{{M}}$ contribution at $\mathbf{k}_\mathrm{{AN}}$. Lines distinguish different combinations of the Hedin vertex and susceptibility from pD$\Gamma$A and $\ell$D$\Gamma$A calculations. Inset: Magnetic Hedin vertex at bosonic frequency $\omega=0$ and wavevector $\mathbf{q} = (\pi, \pi)$ from pD$\Gamma$A and $\ell$D$\Gamma$A, showing enhancement and reduction, respectively.