DMRG study of strongly interacting $\mathbb{Z}_2$ flatbands: a toy model inspired by twisted bilayer graphene

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.

Twisted-bilayer FeSe and the Fe-based superlattices

“…Within a spin-sector, each copy is composed of two degenerate bands of opposite Chern number -a 2 pair. By introducing a finite λ, the degeneracy is lifted, similar to the action of the sublattice-symmetry breaking in tBG [85], but without violating any symmetries here.…”

“…Within a fixed spin sector, the previously neglected spin-orbit λ enters the problem in a similar fashion to the sublattice splitting in hexaboron nitride-aligned tBG [85], but without violating any space-group symmetries. Its effect is to split the degenerate flatbands at zero into C = ±1 pairs of flat bands, separated in energy by 2λ.…”

“…Since the Hamiltonian is everywhere real valued in the absence of spin-orbit, turning on non-zero λ, and therefore gapping the node at Γ , guarantees a Chern number 1 (or −1 depending on the sign of λ) for the upper superlattice band. Because λ only reduces the SU(2) spin-symmetry down to U(1), opposite spin sectors decouple within degenerate bands, which are necessarily degenerate and have opposite Chern number due to time-reversal [85].…”

Symmetric Kondo Lattice States in Doped Strained Twisted Bilayer Graphene

“…Alternatively, an approach based on a momentum space model has been considered [92][93][94][95][96][97][98][99][100], in which correlated insulators [101][102][103][104][105][106][107][108], superconductivity [109][110][111][112][113][114], and other correlated quantum phases [115][116][117][118][119] have been identified and studied. Besides, various numerical calculations [120][121][122][123][124][125][126][127] have also been performed to investigate the correlated nature of the phenomena. However, the active phase diagram including the states at non-integer fillings is not well understood.…”

Phase transition in bilayer quantum Hall system with opposite magnetic field

“…[52,54] Then those FQHL states can be viewed as an analogy of the FQH states and then be realized in a particular BQHS. [55,56] Motivated by the above considerations we prefer to use the BQHS as a powerful ideal model for realizing ideal proposals. We focus on the FQHL system with strong interaction.…”