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arXiv:2403.11478 · Physical review. B./Physical review. B

Moore-Read state in half-filled moiré Chern band from three-body pseudopotential

Lu Zhang, Xue-Yang Song

Published Jun 20, 2024• 5 citations
arXivHTMLPDF

Abstract

The moir\'e system provides a tunable platform for exploring exotic phases of materials. This article shows the possible realization of a non-Abelian state characterized by the Moore-Read wavefunction in a half-filled moir\'e Chern band, exemplified by twisted ${\mathrm{MoTe}}_{2}$. This is achieved by introducing short-range repulsive three-body interaction. Exact diagonalization is employed to examine the spectrum in finite size. The incompressibility of the system, the degeneracy of the ground states, and the number of low-energy states provide compelling evidence to identify the ground state as the Moore-Read state. We further interpolate between the three-body interaction and Coulomb interaction to show a phase transition between the composite Fermi-liquid and the Moore-Read state. Finally, we explore the conditions under which the three-body pseudopotential from band mixing effects mimic short-range repulsive three-body interaction. We provide insights toward realizing non-Abelian phases of matter in the moir\'e system.

DOI
10.1103/physrevb.109.245128

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The moir'e system provides a tunable platform for exploring exotic phases of materials. This article shows the possible realization of a non-Abelian state characterized by the Moore-Read wavefunction in a half-filled moir'e Chern band, exemplified by twisted MoTe2{\mathrm{MoTe}}_{2}MoTe2​. This is achieved by introducing short-range repulsive three-body interaction. Exact diagonalization is employed to examine the spectrum in finite size. The incompressibility of the system, the degeneracy of the ground states, and the number of low-energy states provide compelling evidence to identify the ground state as the Moore-Read state. We further interpolate between the three-body interaction and Coulomb interaction to show a phase transition between the composite Fermi-liquid and the Moore-Read state. Finally, we explore the conditions under which the three-body pseudopotential from band mixing effects mimic short-range repulsive three-body interaction. We provide insights toward realizing non-Abelian phases of matter in the moir'e system.
Experimental realizations of Abelian fractional Chern insulators (FCIs) have demonstrated the potentials of moiré systems in synthesizing exotic quantum phases. Remarkably, twisted multilayer graphene system may also host non-Abelian states competing with charge density wave under Coulomb interaction. Here, through larger scale exact diagonalization simulations, we map out the quantum phase diagram for ν=1/2ν=1/2ν=1/2 system with electrons occupying the lowest moirè band of the double twisted bilayer graphene. By increasing the system size, we find the ground state has six-fold near degeneracy and with a finite spectral gap separating the ground states from excited states across a broad range of parameters. Further computation of many-body Chern number establish the topological order of the state, and we rule out possibility of charge density wave orders based on featureless density structure factor. Furthermore, we inspect the particle-cut entanglement spectrum to identify the topological state as a non-Abelian Moore-Read state. Combining all the above evidences we conclude that Moore-Read ground state dominates the quantum phase diagram for the double twisted bilayer graphene system for a broad range of coupling strength with realistic Coulomb interaction.
Breakthrough experiments have recently realized fractional Chern insulators (FCIs) in moiré materials. However, all states observed are Abelian; the possible existence of more exotic non-Abelian FCIs remains controversial both experimentally and theoretically. Here, we investigate the competition between charge density wave (CDW) order, gapless composite fermion liquid (CFL), and non-Abelian Moore-Read states at half filling of a moiré band. Although ground-state (quasi)degeneracies and spectral flow are not sufficient for distinguishing between charge order and Moore-Read states, we find evidence using entanglement spectroscopy that both these states of matter can be realized with Coulomb interactions. By further analyzing the graviton excitations of Moore-Read states, we unveil that the ground states exhibit a mixed behavior of Pfaffian and anti-Pfaffian, despite the weak breaking of particle-hole symmetry. In a double twisted bilayer graphene model, transitions between these phases can be driven by the coupling strength between the layers: at weak coupling there is a CFL phase and at strong coupling a CDW order emerges. Remarkably, however, there is compelling evidence for a non-Abelian Moore-Read FCI phase at intermediate coupling.
Abstract Moiré materials provide a remarkably tunable platform for topological and strongly correlated quantum phases of matter. Very recently, the first Abelian fractional Chern insulators (FCIs) at zero magnetic field have been experimentally demonstrated, and it has been theoretically predicted that non-Abelian states with Majorana fermion excitations may be realized in the nearly dispersionless minibands of these systems. Here, we provide telltale evidence based on many-body exact diagonalization for the even more exotic possibility of moiré-based non-Abelian FCIs exhibiting Fibonacci parafermion excitations. In particular, we obtain low-energy quantum numbers, spectral flow, many-body Chern numbers, and entanglement spectra consistent with the Z3{{\mathbb{Z}}}_{3}Z3​ Z 3 Read–Rezayi parafermion phase in an exemplary moiré system with tunable quantum geometry. Our results hint towards the robustness of moiré-based parafermions and encourage the pursuit in moiré systems of these non-Abelian quasiparticles that are superior candidates for topological quantum computing.
We study fermions on a triangular lattice model that exhibits topological flatbands characterized by nonzero Chern numbers. Our scheme stems from the well-known Hofstadter model but the next-nearest-neighbor hopping is introduced, which is crucial for tuning the lowest band to be nearly flat. Differing from previous proposals with the necessity of multiparticle interactions, we consider the more realistic long-range dipolar interaction combined with two-body short-range attractions between fermions. We show the realization of the non-Abelian $\ensuremath{\nu}=1/2$ Moore-Read fractional Chern insulators, and strong evidence for the existence of the more exotic $\ensuremath{\nu}=3/5$ Read-Rezayi fractional Chern insulators. Our results provide insights for the experimental realization of these exotic states by realistic two-body interactions and thus facilitates the implementation of the universal topological quantum computation.
The recent theoretical discovery of fractional Chern insulators (FCIs) has provided an important new way to realize topologically ordered states in lattice models. In earlier works, on-site and nearest-neighbor Hubbard-like interactions have been used extensively to stabilize Abelian FCIs in systems with nearly flat, topologically nontrivial bands. However, attempts to use two-body interactions to stabilize non-Abelian FCIs, where the ground state in the presence of impurities can be massively degenerate and manipulated through anyon braiding, have proven very difficult in uniform lattice systems. Here, we study the remarkable effect of long-range interactions in a lattice model that possesses an exactly flat lowest band with a unit Chern number. When spinless bosons with two-body long-range interactions partially fill the lowest Chern band, we find convincing evidence of gapped, bosonic Read-Rezayi (RR) phases with non-Abelian anyon statistics. We characterize these states through studying topological degeneracies, the overlap between the ground states of two-body interactions and the exact RR ground states of three- and four-body interactions, and state counting in the particle-cut entanglement spectrum. Moreover, we demonstrate how an approximate lattice form of Haldane's pseudopotentials, analogous to that in the continuum, can be used as an efficient guiding principle in the search for lattice models with stable non-Abelian phases.
Twisted bilayer materials provide a versatile platform for realizing correlated topological states. Motivated by the recent experimental realization of atomic Bose-Einstein condensates in twisted bilayer optical lattices, we investigate the emergence of fractional topological phases in such systems. For experimentally realistic parameters, the single-particle spectrum hosts a pair of quasi-degenerate nearly flat moiré bands. We show using self-consistent Hartree-Fock calculations that atomic interactions spontaneously break time-reversal symmetry, lift the quasi-degeneracy, and generate an isolated nearly flat band with a nonzero Chern number. At fractional filling, exact diagonalization of the Hamiltonian projected to this band reveals a threefold degenerate ground-state manifold and a characteristic particle entanglement spectrum, providing strong evidence for a fractional Chern insulator. The resulting many-body gap is of order 10−3ER10^{-3}E_R10−3ER​, corresponding to a nanokelvin energy scale accessible to state-of-the-art cold-atom experiments. Our scheme requires neither spin-orbit coupling, as in transition metal dichalcogenides, nor a finite magnetic field, as in existing twisted-bilayer-graphene experiments, providing a highly tunable route to strongly correlated topological phases in twisted bilayer optical lattices.