Twisted bilayer graphene. V. Exact analytic many-body excitations in Coulomb Hamiltonians: Charge gap, Goldstone modes, and absence of Cooper pairing

We find exact analytic expressions for the energies and wave functions of the charged and neutral excitations above the exact ground states (at rational filling per unit cell) of projected Coulomb Hamiltonians in twisted bilayer graphene. Our exact expressions are valid for any form of the Coulomb interaction and any form of $AA$ and $AB/BA$ tunneling. The single charge excitation energy is a convolution of the Coulomb potential with a quantum geometric tensor of the TBG bands. The neutral excitations are (high-symmetry group) magnons, and their dispersion is analytically calculated in terms of the form factors of the active bands in TBG. The two-charge excitation energy and wave functions are also obtained, and a sufficient condition on the graphene eigenstates for obtaining a Cooper pair from Coulomb interactions is obtained. For the actual TBG bands at the first magic angle, we can analytically show that the Cooper pair binding energy is zero in all such projected Coulomb models, implying that either phonons and/or nonzero kinetic energy are needed for superconductivity. Since Vafek and Kang [Phys. Rev. Lett. 125, 257602 (2020)] showed that the kinetic energy bounds on the superexchange energy are less ${10}^{-3}$ in Coulomb units, the phonon mechanism becomes then very likely. If nonetheless the superconductivity is due to kinetic terms which render the bands nonflat, one prediction of our theory is that the highest $T_c$ would not occur at the highest DOS.

Figure 1

Exact charge $\pm 1$ excitations given by the simplified excitation matrix [Eq. (20)] for three different $w_0/w_1$ at the twist angle $\theta =1.{05}^{\circ }$. Here we change $w_0$ while keeping $w_1=110\mathrm{meV}$ fixed. Other parameters are given in Appendix pp1. These excitations are exact at the charge neutrality point ($\nu =0$) for generic state and are exact at finite integer fillings if the flat metric condition is satisfied. The charge $+$1 and $-1$ excitations are degenerate. The exact charge $\pm 1$ excitations obtained using the full excitation matrix [Eq. (19)] without assuming the flat metric condition are given in Figs. 5 and 6 in Appendix pp3-s4. The charge gap in those cases shrinks considerably.

Figure 2

Exact charge neutral excitations with the flat metric condition being imposed for three different $w_0/w_1$ at the twist angle $\theta =1.{05}^{\circ }$. Here we change $w_0$ while keeping $w_1=110\mathrm{meV}$ fixed. Other parameters are given in Appendix pp1. These excitations are exact at the charge neutrality point ($\nu =0$) for generic states and are exact at finite integer fillings if the flat metric condition is satisfied. The exact charge neutral excitations at different fillings without imposing flat metric condition are given in Figs. 7 and 8 in Appendix pp4-s4. Note the softening of further Goldstone modes from finite to zero $w_0$, reflecting the symmetry enhancement of the first chiral limit. The continuum above the Goldstone modes is fundamentally made of independent particle-hole excitations

Figure 3

The eigenvalues of the mass tensor of the Goldstone mode in the first chiral limit (31). Here $w_1$ is in units of meV.

Figure 4

Exact charge $\pm 2$ excitations with the flat metric condition being imposed for three different $w_0/w_1$ at the twist angle $\theta =1.{05}^{\circ }$. Here we change $w_0$ while keeping $w_1=110\mathrm{meV}$ fixed. Other parameters are defined in Appendix pp1. These excitations are exact at charge neutrality point ($\nu =0$) for generic states and are exact at finite integer fillings if the flat metric condition is satisfied. The charge $+$2 and $-2$ excitations are degenerate. The exact charge $\pm 2$ excitations at different fillings without imposing the flat metric condition are given in Fig. 9 and 10 in Appendix pp5-s5. Note bound states above but not below the two-particle continuum, confirming our analytic proof and showing the lack of Cooper pairing in TBG projected Coulomb Hamiltonians

Figure 5

Exact charge $+1$ (blue) and $-1$ (red) excitations at $\theta =1.{05}^{\circ }$. The flat metric condition is not imposed. In this plot we have used the parameters defined in Appendix pp1: $v_F=5.944\mathrm{eV}\mathring{\mathrm{A}}, \left|K\right|=1.703{\mathring{\mathrm{A}}}^{-1}, w_1=110\mathrm{meV}, U_{\xi }=26\mathrm{meV}$, and $\xi =10\mathrm{nm}$. Note that the excitation gap is largely reduced from the flat-condition limit.

Figure 6

Exact charge $+1$ (blue) and $-1$ (red) excitations at $\theta =1.{05}^{\circ }$. The flat metric condition is not imposed. In this plot we set the screening length as $\xi =20\mathrm{nm}$ and accordingly the interaction strength as $U_{\xi }=13\mathrm{meV}$. The other parameters are same as in Appendix pp1: $v_F=5.944\mathrm{eV}\mathring{\mathrm{A}}, \left|K\right|=1.703{\mathring{\mathrm{A}}}^{-1}$, and $w_1=110\mathrm{meV}$.

Figure 7

Exact charge neutral excitations with $\theta =1.{05}^{\circ }$. The flat metric condition is not imposed. In this plot we have used the parameters defined in Appendix pp1: $v_F=5.944\mathrm{eV}\mathring{\mathrm{A}}, \left|K\right|=1.703{\mathring{\mathrm{A}}}^{-1}, w_1=110\mathrm{meV}, U_{\xi }=26\mathrm{meV}$, and $\xi =10\mathrm{nm}$.

Figure 8

Exact charge neutral excitations with $\theta =1.{05}^{\circ }$. The flat metric condition is not imposed. In this plot, we use the screening length $\xi =20\mathrm{nm}$ and $U_{\xi }=13\mathrm{meV}$ accordingly. The other parameters are same as in Appendix pp1, i.e.,$v_F=5.944\mathrm{eV}\mathring{\mathrm{A}}, \left|K\right|=1.703{\mathring{\mathrm{A}}}^{-1}$, and $w_1=110\mathrm{meV}$.

Figure 9

Exact charge $+2$ (blue) and $-2$ (red) excitations at $\theta =1.{05}^{\circ }$. The flat metric condition is not imposed. In this plot we have used the parameters defined in Appendix pp1: $v_F=5.944\mathrm{eV}\mathring{\mathrm{A}}, \left|K\right|=1.703{\mathring{\mathrm{A}}}^{-1}, w_1=110\mathrm{meV}, U_{\xi }=26\mathrm{meV}$, and $\xi =10\mathrm{nm}$.

Figure 10

Exact charge $+2$ (blue) and $-2$ (red) excitations at $\theta =1.{05}^{\circ }$. The flat metric condition is not imposed. In this plot, we use the screening length $\xi =20\mathrm{nm}$ and the interaction strength $U_{\xi }=13\mathrm{meV}$ accordingly. The other parameters are same as in Appendix pp1: $v_F=5.944\mathrm{eV}\mathring{\mathrm{A}}, \left|K\right|=1.703{\mathring{\mathrm{A}}}^{-1}$, and $w_1=110\mathrm{meV}$.

Figure 11

Approximate charge $+1$ (blue) and $-1$ (red) excitations at $\theta =1.{05}^{\circ }$ at odd fillings in the nonchiral-flat limit. The dashed bands are the excitations in the half-filled valley-spin sectors. The solid blue (red) bands are the $+$1 ($-1$) excitations in the fully empty (occupied) valley-spin sectors. The flat metric condition is not imposed. In this plot we set the screening length as $\xi =10\mathrm{nm}$ and accordingly the interaction strength as $U_{\xi }=13\mathrm{meV}$. The other parameters are same as in Appendix pp1: $v_F=5.944\mathrm{eV}\mathring{\mathrm{A}}, \left|K\right|=1.703{\mathring{\mathrm{A}}}^{-1}$, and $w_1=110\mathrm{meV}$.

Figure 12

Approximate charge neutral excitations at $\theta =1.{05}^{\circ }$ at odd fillings in the nonchiral-flat limit. The flat metric condition is not imposed. Blue, red, black, and green bands are in the empty-half, half-half, empty-filled, and half-filled sectors, respectively. (See the text for the definition of sectors.) In this plot, we set the screening length as $\xi =10\mathrm{nm}$ and accordingly the interaction strength as $U_{\xi }=13\mathrm{meV}$. The other parameters are same as in Appendix pp1: $v_F=5.944\mathrm{eV}\mathring{\mathrm{A}}, \left|K\right|=1.703{\mathring{\mathrm{A}}}^{-1}$, and $w_1=110\mathrm{meV}$.