Resonating-valence-bond superconductor from small Fermi surface in twisted bilayer graphene

Abstract

Mechanism of superconductivity in twisted bilayer graphene (TBG) remains one of the central problems in correlated moiré materials. The most intriguing question is about the nature of the normal state: is the Cooper pair formed from small Fermi surface or large Fermi surface? In this work we point out the possibility of a symmetric pseudogap metal with small hole pockets, dubbed as second Fermi liquid (sFL). In the sFL phase at $ν=-2-x$, there is a two-component picture: two electrons mainly localize at each AA site and form a paired singlet due to anti-Hund's coupling mediated by the optical phonon, while additional holes doped into the AA sites form small Fermi surfaces. The sFL phase corresponds to an intrinsically strongly interacting fixed point and is topologically distinct to the conventional Fermi liquid. We develop a unified framework to describe both a renormalized FL phase and an sFL phase. We propose that the TBG superconductor emerges from the sFL phase, but is close to the transition toward the FL phase under increasing hole doping. In the superconductor, pairing of local moments is shared to the mobile carriers and a smaller superconducting gap with nodal $p_x$ symmetry is opened on the small hole pockets. This work provides, to our knowledge, the first unified theory that explains both the pseudogap metal above $T_c$ and the two-gap nematic superconductivity below it.

Figures (13)

• Figure 1: (a) Schematic phase diagram illustrating the evolution from the sFL to the FL phase, for example, tuned by decreasing the anti-Hund's coupling $J$. The vertical axis denotes the temperature $T$. The left and right insets show the band structures and Fermi surfaces of the renormalized FL and sFL phases, respectively. $\Delta$ is the pairing of the local moments from the $J$ term. $B$ is the slave boson condensation, which sets a coherence temperature scale $T_{\mathrm{coh}}$. In the phase diagram we also show illustration of tunneling spectrum. (b) Schematic illustration of the relationships among different phases. The FL and sFL phases correspond to two distinct symmetric fixed points. (c) Illustration of the RVB mechanism of superconductivity. We already have local pairing of spinons $\Delta_{\mathrm{loc}}$. Onset of the slave boson condensation $B$ then induces resonance between the local pairing and two mobile carriers, leading to a superconducting pairing $\Delta_{\mathrm{SC}}$ between the mobile carriers, which can be well separated in space.
• Figure 2: (a) The effective mass $m^*/m_e$ as a function of $\gamma/U$ for the sFL phase at $\nu=-2.4$, where effective mass $m^*$ is estimated by the zero energy density of state and $m_e$ is the free electron mass. The horizontal dashed line marks the free electron mass of the $c$ electron in the decoupled limit, with $m^*_0/m_e=0.05$. (b) the quasi-particle weight of $c$, $f$ and $s$ fermions as a function of $\gamma/U$ for $\nu=-2.4$. All the calculations are performed by varying $U$ at fixed $\gamma = -26.184$ meV, with $w_0/w_1=0.8, \theta=1.06^\circ$ and $J=6.0$ meV. The value of $\gamma/U$ used in the main text is indicated by the vertical dashed lines in panels (a) and (b). The slave boson $B$ is artificially set to 0 in the mean-field iteration to get the normal state of sFL.
• Figure 3: Superconducting evolution as a function of the on-site spin interaction $J$, for $\theta=1.06^\circ, w_0/w_1=0.8$, $U=25$ meV, $x=0.4$ and $M=0$. (a) The STM spectrum and (b) the order parameters for different value of $J$. The orange, red and green arrows mark the SC gap, the pseudogap and the inter-band gap. While the pseudogap always increase with increasing $J$, the SC gap first increases and then decreases as a function of $J$. (c) (d) and (e) show three different line cuts of STM at $J=0.5$ meV, $J=3.0$ meV and $J=6.0$ meV, respectively.
• Figure 4: Superconducting evolution as a function of the carrier density $\nu$ for twist angle $\theta=1.06^\circ$, $w_0/w_1=0.8$, $U=25$ meV, $J=6.0$ meV and $M=0$. (a) The STM spectrum as for different charge densities. The orange, red and green arrows mark the SC gap, the pseudogap and the inter-band gap. (b) the order parameters for different value of $\nu$. (c) and (d) show the nematic $d$-wave structure at a specific filling $\nu = -2.5$. (c) The pairing order parameter $|\Delta_{\mathrm{act,1}}(\mathbf{k})|$, projected onto one of the lower Hubbard bands obtained by artificially setting $B_s = 0$. (d) The minimal single particle gap $E_{\mathrm{gap}}$ along different azimuth direction $\phi$. The gap reaches its maximum along $\phi =0,\pi$ directions and its minimum along $\phi \approx \pm \pi/2$ directions.
• Figure 5: Superconducting evolution as a function of spin coupling $J$, with the coefficient $\alpha_B=1.0$ in Eq. \ref{['eqn:self_consistent']}. The parameters are set to $\theta=1.06^\circ, w_0/w_1=0.8$, $U=25$ meV, $M=0$ and $\nu=-2.4$. (a) The STM spectra and (b) the corresponding order parameters for different value of $J$.