Abstract
Strong interactions between electrons occupying bands of opposite (or like) topological quantum numbers (Chern=\pm1=±1), and with flat dispersion, are studied by using lowest Landau level (LLL) wavefunctions. More precisely, we determine the ground states for two scenarios at half-filling: (i) LLL’s with opposite sign of magnetic field, and therefore opposite Chern number; and (ii) LLL’s with the same magnetic field. In the first scenario – which we argue to be a toy model inspired by the chirally symmetric continuum model for twisted bilayer graphene – the opposite Chern LLL’s are Kramer pairs, and thus there exists time-reversal symmetry (\mathbb{Z}_2ℤ2). Turning on repulsive interactions drives the system to spontaneously break time-reversal symmetry – a quantum anomalous Hall state described by one particle per LLL orbital, either all positive Chern |{++\cdots+}\rangle|++⋯+⟩ or all negative |{--\cdots-}\rangle|−−⋯−⟩. If instead, interactions are taken between electrons of like-Chern number, the ground state is an SU(2)SU(2) ferromagnet, with total spin pointing along an arbitrary direction, as with the \nu=1ν=1 spin-\frac{1}{2}12 quantum Hall ferromagnet. The ground states and some of their excitations for both of these scenarios are argued analytically, and further complimented by density matrix renormalization group (DMRG) and exact diagonalization.
Highlights
The ground states and some of their excitations for both of these scenarios are argued analytically, and further complimented by density matrix renormalization group (DMRG) and exact diagonalization
When a uniform magnetic field B is applied to a sample of electrons, the sample can be understood to be “discretized" in the sense that the non-interacting single-particle states, i.e Landau levels take on an integer degeneracy N which is equal to the division of the sample area A by the area of the quantized cyclotron orbit 2πlB2
We provide a simple model, constructed out of continuum lowest Landau levels, which captures the essential physics of two strongly interacting flat bands when the bands have identical
Summary
The unprecedented nature of the phenomena exhibited by twisted bilayer graphene (TBG) devices at various electron densities about charge neutrality (CN), including correlated insulated states [1,2,3] and superconductivity [4, 5], has opened the door to many experimental [1,2,3,4,5,6,7] and theoretical [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26] studies since their initial discovery by Y. The larger degeneracy in the latter brings with it more integer fillings and possible phases; for the purposes of this study, we focus on Scenario (2) and assume valley polarization, such that there is only a single intra-valley Kramer pair In this limit, Scenario (1) and (2) have the same degeneracy structure, both describing an interacting system of fermions occupying opposite-Chern bands. III we construct pseudopotentials for both scenarios by placing the problem onto a cylinder, which makes one direction compact and reduces the 2D continuum problem to an effective 1D discrete chain; following this, in Sec. IV, we use DMRG to solve for both the ground state manifolds of the constructed pseudopotentials and the system size scaling of the energy gap.
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