Symmetric Kondo Lattice States in Doped Strained Twisted Bilayer Graphene

We use the topological heavy fermion (THF) model and its Kondo lattice (KL) formulation to study the possibility of a symmetric Kondo (SK) state in twisted bilayer graphene. Via a large-$N$ approximation, we find a SK state in the KL model at fillings $\nu =0,\pm 1,\pm 2$ where a KL model can be constructed. In the SK state, all symmetries are preserved and the local moments are Kondo screened by the conduction electrons. At the mean-field level of the THF model at $\nu =0,\pm 1,\pm 2,\pm 3$ we also find a similar symmetric state that is adiabatically connected to the symmetric Kondo state. We study the stability of the symmetric state by comparing its energy with the ordered (symmetry-breaking) states found in [H. Hu et al., Phys. Rev. Lett. 131, 026502 (2023)., Z.-D. Song and B. A. Bernevig, Phys. Rev. Lett. 129, 047601 (2022).] and find the ordered states to have lower energy at $\nu =0,\pm 1,\pm 2$. However, moving away from integer fillings by doping the light bands, our mean-field calculations find the energy difference between the ordered state and the symmetric state to be reduced, which suggests the loss of ordering and a tendency toward Kondo screening. In order to include many-body effects beyond the mean-field approximation, we also performed dynamical mean-field theory calculations on the THF model in the nonordered phase. The spin susceptibility follows a Curie behavior at $\nu =0,\pm 1,\pm 2$ down to $\sim 2\mathrm{K}$ where the onset of screening of the local moment becomes visible. This hints to very low Kondo temperatures at these fillings, in agreement with the outcome of our mean-field calculations. At noninteger filling $\nu =\pm 0.5,\pm 0.8,\pm 1.2$ dynamical mean-field theory shows deviations from a $1/T$ susceptibility at much higher temperatures, suggesting a more effective screening of local moments with doping. Finally, we study the effect of a $C_{3z}$-rotational-symmetry-breaking strain via mean-field approaches and find that a symmetric phase (that only breaks $C_{3z}$ symmetry) can be stabilized at sufficiently large strain at $\nu =0,\pm 1,\pm 2$. Our results suggest that a symmetric Kondo phase is strongly suppressed at integer fillings, but could be stabilized either at noninteger fillings or by applying strain.

Figure 1

(a) Band structure of the noninteracting THF model at $\nu =0$. (b),(c),(d) Band structure of the SK phase at $\nu =0,-1,-2$ respectively.

Figure 2

Doping dependence of the ground state energy difference $\mathrm{\Delta }E=E_{\mathrm{sym}}-E_{\text{order}}$ near integer fillings $\nu =0,-1,-2,-3$ in THF model. The ordered states we considered are KIVC ($\nu =0$), $\mathrm{KIVC}+\mathrm{VP}$ ($\nu =-1$), KIVC ($\nu =-2$), and VP ($\nu =-3$).

Figure 3

DMFT solution of the THF model. (a) Doping $\nu$ dependent low-energy spectral function at the Fermi level [$A(\omega =0)$] for the full system $A_{\mathrm{tot}}$, the $c$ ($A_c$) and the $f$ electrons ($A_f$) at 11.6 K. Also shown is the scattering rate ${\mathrm{\Gamma }}_f$ as extracted from the local $f$ electron self-energy. (b) Effective local moment $T\times {\chi }_{\mathrm{spin}}^{\mathrm{loc}}(\omega =0)$ as a function of temperature $T$ for different doping levels $\nu$. The full set of values of the screened local moments ${\mu }_{\mathrm{eff}}$ and screened temperature $T_{\odot }$ is provided in SM, Sec. VI [133].

Figure 4

Energy difference $\mathrm{\Delta }{E}^{\text{strain}}=E_{\mathrm{sym}}^{\text{strain}}-E_{\text{order}}^{\text{strain}}$ between the symmetric state that only breaks $C_{3z}$ symmetry ($E_{\mathrm{sym}}^{\text{strain}}$) and the ordered state ($E_{\text{order}}^{\text{strain}}$) as a function of $\alpha$—a parameter characterizing the strain amplitude. Inside: $\mathrm{\Delta }{E}^{\text{strain}}$ at $\nu =-2$ over an extended parameter region $0\mathrm{meV}\le \alpha \le 45\mathrm{meV}$. We note that even at zero strain $\alpha =0$, a symmetric state that only breaks $C_{3z}$ symmetry has lower energy than the fully symmetric state. We also note that there are small kinks due to the transition between two ordered phases at $\alpha \sim 3\mathrm{meV}$ for $\nu =-1$, at $\alpha \sim 5\mathrm{meV}$ for $\nu =-2$, and at $\alpha \sim 3\mathrm{meV}$ for $\nu =-3$ (Sec. V, SM [133]).