Cooper pair condensation from entanglement-entropy collapse of many-body states in sheared bilayer graphene

Abstract

It is known that the sheared graphene bilayers can be tuned to have flat low-energy bands for sufficiently large size of their moiré supercell. In this regime, we show by means of a self-consistent Hartree-Fock approximation that the interacting system becomes prone to develop broken-symmetry phases, with valley symmetry breaking as the dominant pattern. We adopt an exact diagonalization approach, on top of the Hartree-Fock approximation, to show how the condensation of Cooper pairs takes place in the strong-coupling limit of a valley-polarized flat band. A key factor of our proposal is the existence of zero entanglement-entropy many-body states, just made of a single Slater determinant, which are immune to the hybridization with the rest of the states under the strong Coulomb interaction. Moreover, we show that single-particle states with reverse sign of valley symmetry breaking have complementary charge distributions in the supercell, leading to a ground state where the Coulomb repulsion is minimized by placing electrons with opposite spin in different valleys. We argue that the collapse of the entanglement entropy causes the formation of many-body ground states with Cooper pairs made of electrons and their respective partners under valley symmetry, unveiling a strong-coupling mechanism of condensation in a flat band with initially no Fermi line.

Figures (8)

• Figure 1: Moiré pattern obtained by applying shear to bilayer graphene, showing the formation of a one-dimensional superlattice with period $L_x \approx 6.4$ nm in the horizontal direction.
• Figure 2: (a) Low-energy bands of a sheared graphene bilayer with period $L_x \approx 56$ nm, obtained from the Hamiltonian of the continuum model in (\ref{['cont']}). The inset is a zoom view showing the two lowest conduction bands close to zero energy. (b) Low-energy bands of the sheared bilayer shown in Fig. \ref{['one']}, modeled by the tight-binding Hamiltonian in Eq. (\ref{['h0']}) with a period $L_y \approx 4.3$ nm of the sinusoidal potential in the $y$ direction. The manifold of eight lowest-energy bands about charge neutrality is printed in solid blue.
• Figure 3: Charge density distributions in the moiré supercell of states with momentum $(\pi/L_x,\pi/L_y)$ (a) and $(\pi/L_x,0)$ (b), from the flat band right below the Fermi level of the phase with valley symmetry breaking shown in Fig. \ref{['five']}(a).
• Figure 4: Phase diagram of the sheared bilayer with noninteracting bands represented in Fig. \ref{['two']}(b), obtained for filling fraction $\nu=1$ with a self-consistent Hartree-Fock approximation and showing different symmetry-breaking order parameters as a function of the strength of the Coulomb potential.
• Figure 5: Low-energy bands (at $k_x=0$) of the sheared bilayer corresponding to Fig. \ref{['two']}(b), obtained with a self-consistent Hartree-Fock approximation for filling fraction $\nu=1$ (a) and $\nu=3$ (b), for interaction strength $e^2/4\pi \epsilon = 1.9$ eV$\times a$ . The red dashed line represents the Fermi level in each case.