Pseudogap and Non-Fermi-liquid criticality in double Kondo model for bilayer nickelates

Abstract

Motivated by recent experimental progress on high-temperature superconductivity in bilayer nickelates, we investigate the phase diagram of the normal state in a bilayer Kondo lattice model using single-site dynamical mean-field theory (DMFT). When the interlayer tunneling $t_\perp$ is absent, we identify a non-Fermi-liquid (NFL) critical point tuned by the interlayer spin coupling $J_\perp$ or hole doping $x$, which separates a standard Fermi liquid in the overdoped region from a distinct pseudogap (PG) metal in the underdoped regime. This PG phase, which we term the `second Fermi liquid' (sFL), exhibits small hole pockets and violates the perturbative Luttinger theorem despite the absence of symmetry breaking or fractionalization. The PG metal behaves like a heavy Fermi liquid, with small quasi-particle residue and large effective mass. We also provide an intuitive analytical description of the pseudogap and the ground-state wave function based on an ancilla-fermion framework. Inside the PG phase, we interpret the ancilla fermion as a spin-polaron and demonstrate a Kondo-like resonance peak in the spectral function of this composite fermion directly in DMFT calculation. Extending the analysis to finite $t_\perp$, we apply this framework to the bilayer nickelate $\mathrm{La}_3\mathrm{Ni}_2\mathrm{O}_7$. We propose that current experimental samples ($x \approx 0.5$) reside in the overdoped FL regime, suggesting that the pseudogap phase and the NFL criticality may be accessed via electron doping.

Figures (22)

• Figure 1: (a) Schematic phase diagram of the double Kondo model as a function of temperature $T$ and interlayer coupling $J_\perp$ or doping $1-x$. A quantum critical point is expected at an intermediate value of $J_\perp$, which provides a critical regime with non-Fermi-liquid (NFL) physics. (b) Illustration of how the double Kondo lattice model for bilayer nickelate defiend in Eq. \ref{['eqn:double_kondo']}. Each layer consists of itinerant electron from the $d_{x^2-y^2}$ orbital and a localized spin-1/2 moment from the $d_{z^2}$ orbital, coupled via a ferromagnetic Kondo coupling $J_K=-2J_H<0$, where $J_H$ is the inter-orbital Hund's coupling. The local moments from the two layers are further coupled by an inter-layer super-exchange $J_\perp>0$. (c) and (d) illustrate the distinction between the FL and sFL phases using a $\mathbb{Z}_2$ index $C=Q/2 \,(\mathrm{mod}\,2)$ and their corresponding Fermi-surface volumes. (e) A model wavefunction for the sFL phase using the ancilla framework. The blue layers represent the physical electron from the $d_{x^2-y^2}$ orbital, while the red layers denote hidden ancillary fermions $\psi_{l;\sigma}$. In the final wavefunction, the ancilla Fermions are projected to form rung-singlet and we recover a wavefunction in the physical Hilbert space, but is beyond the Slater determinant framework. The pseudogap arises from the hybridization between the electron and the ancilla fermion, which leads to a small hole-like Fermi pocket around $(\pi,\pi)$.
• Figure 2: Phase diagrams of the double Kondo model in Eq. \ref{['eqn:double_kondo']}, obtained from self-consistent DMFT calculations with $U = 8t$ and $t_\perp=0$, with $t=1$ as the unit. Blue dots and red stars denote even and odd values of the $\mathbb{Z}_2$ charge $C$, characterizing the FL phase (yellow region) and sFL phase (green region), respectively. (a) Phase diagram as a function of $J_\perp$ and doping $x$ at fixed Hund’s coupling $J_K = -12t$ and $t_{\perp;1}=0$. (b) Phase diagram as a function of $J_K$ and $J_\perp$ at fixed doping $x = 0.2$ and $t_{\perp;1}=0$. (c) Fermi surface volume $A_{\mathrm{FS}}$ and quasiparticle weight $Z=m_0/m_{\mathrm{eff}}=(1-\partial_\omega \mathrm{Re}\Sigma(\omega)\vert_{\omega=0})^{-1}$ along the line cut in (a) at fixed $J_\perp=2t$. (d) $-\mathrm{Re}\Sigma(\omega)$ for different doping levels $x$ along the same line cut. Red (blue) curves correspond to the sFL (FL) phase. The red arrow indicates the point where the self-energy develops a divergence. The large peak at energy $\sim 10t$ comes from the Hubbard $U$.
• Figure 3: (a)-(c) Reconstructed momentum-resolved spectral function $A_c(\omega,\mathbf{k})$ along the momentum line cut $(0,0)$–$(\pi,\pi)$ for $J_\perp = 0.2t$, $0.4t$, and $1.0t$, respectively, at fixed $J_K = -10t$, $U = 8t$ and $t_\perp=0$. (e)-(g) Corresponding zero-frequency spectral function $A_c(\omega=0,\mathbf{k})$. (d) Simulated band structure obtained from the ancilla-fermion description Eq. \ref{['eqn:ancilla']}, with $\Phi=0.1t$ and $\mu_\psi=-0.003t$. The red, blue, and black lines represent the bare $c$ band, the $\psi$ band, and the hybridized bands, respectively. (h) Corresponding Fermi surfaces in the ancilla-fermion theory. The Fermi surfaces of the bare $c$ band and the hybridized bands are shown as red and black curves, respectively. (i),(j) Fermi surface and self energies for a finite $t_{\perp;1}=0.2t$ at $J_\perp=0.36t$. (k),(l) Fermi surface and self energies for a finite $t_{\perp;1}=0.2t$ at $J_\perp=0.5t$. A kink structure developed in the self-energy at $\omega\sim -10^{-4}t$ for $J_\perp=0.5t$ in addition to the high energy kinks $\sim \pm 10t$ from Hubbard $U$.
• Figure 4: NFL behavior near the QCP, with parameters chosen as $J_\perp = 0.6t$, $J_K = -10t$, $x = 0.2$, and $U = 8.0t$. (a) The imaginary part of the bosonic correlation function, $-\mathrm{Im}\chi(\omega)/\pi$, which exhibits a frequency-independent (constant) behavior. (b) $-\mathrm{Im}\chi(\omega)/\pi$ showing a linear decay with frequency $\omega$. The shaded blue region marks the NFL regime between $T_{\mathrm{coh}}$ and $T_{\mathrm{NFL}}$. (c) Upper and lower bounds of the NFL regime $T_{\mathrm{NFL}}$ and $T_{\mathrm{coh}}$ along the line cut indicated in Fig. \ref{['fig:phase_diagram']} (a). (d) $T_{\mathrm{NFL}}$ and $T_{\mathrm{coh}}$ as functions of $J_\perp$ for different values of the interlayer hopping $t_{\perp;1}$. The minimum $T_{\mathrm{coh}}$ is of order $\sim 10^{-3}t$ for $t_{\perp;1}=0.2t$. The couplings are fixed at $J_K = -10t$, $U = 8t$, and doping $x = 0.2$. $T_{\mathrm{NFL}}$ depends only weakly on $t_{\perp;1}$ and is shown only for $t_{\perp;1}=0.2t$.
• Figure 5: Spectral functions obtained from the double Hubbard model [Eq. \ref{['eqn:dtJ']}] with $U=8t$ and $V_\perp=0$ for $J_\perp=0.4t$, $1.0t$, and $2.0t$. The case $J_\perp=0.4t$ corresponds to the FL phase, $J_\perp=2.0t$ to the sFL phase, and $J_\perp=1.0t$ lies near the QCP on the sFL side. (a) The spectral functions $A_c(\omega)$ for the physical electron $c_{i;a;\sigma}$. (b) The specturm function $A_{\psi^J}(\omega)$ for the trion operator $\Psi^J_{i;a;\lambda}\sim \sum_\rho c_{i;t;\rho} (\boldsymbol{\sigma}\cdot \boldsymbol{s}_{i;b})_{\rho\lambda}$. The latter vanishes as $\omega \to 0$ in the FL phase, but instead develops a pronounced peak in the sFL phase.