Solution to a Quantum Impurity Model for Moiré Systems: Fermi Liquid, Pairing, and Pseudogap

TL;DR

This work develops and solves a spin-valley Anderson impurity model with valley-anisotropic anti-Hund's interactions to reveal how pairing tendencies and pseudogaps emerge at the single-impurity level. Employing bosonization and refermionization, the authors map to a pair-Kondo problem in the doublet regime, uncovering a Berezinskii-Kosterlitz-Thouless transition from a Fermi liquid to an anisotropic doublet phase and a second-order transition from Fermi liquid to local singlet in the singlet regime. They construct analytic fixed-point solutions and spectral-function ansätze that reproduce pseudogap shoulders observed in spectroscopy and DMFT studies, with NRG confirming the phase diagram and scaling behavior. The results provide a concrete link between single-impurity physics and lattice moiré materials, offering a framework to understand pairing, pseudogap, and potential superconductivity in MATBG/TTG via DMFT and impurity-embedded lattice models.

Abstract

Recent theoretical and experimental studies have revealed the co-existence of heavy and light electrons in magic-angle multilayer graphene, which form a periodic lattice of Anderson impurities hybridizing with Dirac semi-metals. This work demonstrates that nontrivial features -- pairing potential, pseudogap, and continuous quantum phase transitions -- already appear at the single-impurity level, if valley-anisotropic anti-Hund's interactions ($J_S$, $J_D$) are included, favoring either a singlet ($J_S>J_D$) or a valley doublet ($J_D>J_S$) impurity configuration. We derive a complete phase diagram and analytically solve the impurity problem at several fixed points using bosonization and refermionization techniques. When $J_D>J_S,J_D>0$, the valley doublet only couples via pair-hopping processes to the conduction electrons, in sharp contrast to the conventional Kondo scenario. Upon increasing $J_D$, there is a quantum phase transition of the BKT universality class, from a Fermi liquid to an anisotropic doublet phase, the latter exhibiting power-law susceptibilities with non-universal exponents. On the other hand, when $J_S>J_D,J_S>0$, increasing $J_S$ induces a second-order phase transition from Fermi liquid to a local singlet phase, which involves a non-Fermi liquid as an intermediate fixed point. Near the transition towards the anisotropic doublet (local singlet) phase, the renormalized interaction of the Fermi liquid becomes attractive, favoring doublet (singlet) pairing. Based on analytic solutions, we construct ansatz for the impurity spectral function and correlation self-energy, which account for the pseudogap accompanying side peaks, found in recent spectroscopic measurements and a DMFT study. In particular, we obtain a non-analytic V-shaped spectral function with non-universal exponents in the anisotropic doublet phase. All results are further verified by NRG calculations.

Figures (12)

• Figure 1: (a) Energy diagram of the two-electron impurity states in SVAIM. White and black circles indicate valley $l\!=\!\pm$, respectively, and arrows indicate spin $s \!=\! {\uparrow} {\downarrow}$. $J_S$ ($J_D$) is the energy decrease of the singlet $S$ (the valley doublet $D$) compared to the spin triplet $T$; see \ref{['eq:H_AH']}. (b) Schematic phase diagram of SVAIM. The FL phase is separated from the anisotropic doublet (AD) phase by a BKT transition, and separated from the local singlet (LS) phase by a second-order transition. Dashed lines mark the crossover boundaries of the sub-regions in the FL phase, where renormalized interactions turn attractive ("attr.") in certain channels ($S$, $D$, or $T$). AD and LS also exhibit enhanced pairing ("pair.") susceptibilities in the corresponding channels, despite no quasiparticle exists. (c, d) Schematic renormalization group flows for the BKT and second-order transitions, respectively. For NRG results corresponding to panels (b)--(d), see End Matter and \ref{['app:NRG']} in Supplementary Material (SM) supplement.
• Figure 2: (a, b) $A_f(\omega)$ obtained from bosonization. Pseudogap shoulders ($A_f^{(3)}$) correspond to multiplet excitations induced by scattering a bath electron ($\omega\!\le\!-J$) or hole ($\omega\!\ge\!J$), hence are symmetrically pinned around the Fermi energy. For AD, residual longitudinal coupling contributes a constant background ($A^{(1)}_f$, dashed line), while the irrelevant PK coupling contributes a non-analytic kink ($A^{(2)}_f$) above it. (c, d) Lattice spectral function $A(k,\omega)$ in one valley, obtained using the ansätze of $\Sigma_f(\omega)$ derived from single impurity. Insets are contours at $\omega \!=\! 0$, with hexagons denoting the strained moiré Brillouin zone, and white lines indicating the $k$-path of main figures. AD does not have a well-defined Fermi surface, while LS is a Fermi liquid of $c$ electrons. The total density of states $A(\omega) \!=\! \int\frac{\mathrm{d}^2k}{(2\pi)^2} A(k,\omega)$ is also plotted.
• Figure 3: Lattice spectral function in the LS phase, plotted component-wise, $A(k,\omega) \!=\! A_f(k,\omega) \!+\! A_c(k,\omega)$. (a) Same as \ref{['fig:Af']}(d), but with a different $k$-path (inset). Both Hubbard peaks of $f$ lie at high energies $\pm 25$meV, thus the Fermi surface majorly comprises of $c$. (b) Lowering the Hubbard peaks to $\pm 8$meV. Although the Fermi volume is still given by $\frac{\nu-2}{4}$ (inset), hybridization between $c$ bands and the $f$ Hubbard bands has strongly suppressed the Fermi velocity.
• Figure 4: NRG results of the phase diagram, $T_{\rm K}$, and effective parameters of the SVAIM. (a) Phase diagram on $(J_S,J_D)$ plane. The black solid lines sketch the phase boundary, and the color in the FL phase indicates $T_{\rm K}$. The grey dashed line marks $J_S\!=\!J_D$. (b)-(d) The effective interactions in $S,D,T$ channels $\widetilde{E}_{S,D,T}$ compared to $\widetilde{\Delta}_0$ as a function of $J_S,J_D$ in the FL phase. (e)-(g) $\widetilde{J}_S/\pi \widetilde{\Delta}_0, \widetilde{J}_D/\pi \widetilde{\Delta}_0,\widetilde{U}/\pi \widetilde{\Delta}_0$ as functions of: (e) $J_D/J^{(c)}_D$ when $J_S\!=\!0$, (f) $J_S/J^{(c)}_S$ when $J_D\!=\!0.05$, (g) $J_S$ when $J_S\!=\!J_D$. The dashed lines in (f),(g) mark the FL-LS critical point and $J_S\!=\!J_D\!=\!0$, respectively. The numeric labels indicate the regions where the relations in \ref{['tab:eff-int']} hold, while the arrows in (g) show that these relations remain valid upon increasing $|J_S|$ along the line $J_S = J_D$.
• Figure 5: $T_{\rm K}$ near the phase transitions. (a) BKT type, obtained along $J_S\!=\!0$, with critical $J^{(c)}_D \!\approx\! 0.137$. (b) Second-order, obtained along $J_D\!=\!0.05$, with critical $J^{(c)}_S \!\approx\! 0.08026$.