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arXiv:2009.02354 · Physical review. B./Physical review. B

Efficient simulation of moiré materials using the density matrix renormalization group

Tomohiro Soejima, Daniel E. Parker, Nick Bultinck, Johannes Hauschild, Michael P. Zaletel

Published Nov 11, 2020• 118 citations
arXivHTMLPDF

Abstract

Despite considerable interest, it has been difficult to conclusively determine the ground state of twisted bilayer graphene (tBLG), due to the long-range nature of the Coulomb interaction and the fragile topology of the tBLG flat bands. This paper puts forward a method for studying tBLG more efficiently: mixed-$x\phantom{\rule{0}{0ex}}k$ space density-matrix renormalization group (DMRG) with matrix product operator (MPO) compression. The method enables the authors to calculate accurately the ground state of single-valley single-spin tBLG, and it puts DMRG with more degrees of freedom, such as valley, within reach.

DOI
10.1103/physrevb.102.205111

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Despite considerable interest, it has been difficult to conclusively determine the ground state of twisted bilayer graphene (tBLG), due to the long-range nature of the Coulomb interaction and the fragile topology of the tBLG flat bands. This paper puts forward a method for studying tBLG more efficiently: mixed-x\phantom{\rule{0}{0ex}}k space density-matrix renormalization group (DMRG) with matrix product operator (MPO) compression. The method enables the authors to calculate accurately the ground state of single-valley single-spin tBLG, and it puts DMRG with more degrees of freedom, such as valley, within reach.
We report an implementation of the momentum space quantum Monte Carlo (QMC) method on the interaction model for the twisted bilayer graphene (TBG). The long-range Coulomb repulsion is treated exactly with the flat bands, spin and valley degrees of freedom of electrons taking into account. We prove the absence of the minus sign problem for QMC simulation when either the two valleys or the two spin degrees of freedom are considered. By taking the realistic parameters of the twist angle and interlayer tunnelings into the simulation, we benchmark the QMC data with the exact band gap obtained at the chiral limit, to reveal the insulating ground states at the charge neutrality point (CNP). Then, with the exact Green’s functions from QMC, we perform stochastic analytic continuation to obtain the first set of single-particle spectral function for the TBG model at CNP. Our momentum space QMC scheme therefore offers the controlled computation pathway for systematic investigation of the electronic states in realistic TBG model at various electron fillings.
Abstract. Recent experiments have shown that magic angle twisted bilayer graphene (MATBG) can exhibit correlated insulator behavior at half-filling. Seminal theoretical results towards understanding this phase in MATBG have shown that Hartree–Fock ground states (with a positive single-particle excitation gap) can be exact many-body ground states of an idealized flat band interacting (FBI) Hamiltonian. We prove that in the absence of spin and valley degrees of freedom, the only Hartree–Fock ground states of the FBI Hamiltonian for MATBG are two ferromagnetic Slater determinants. Incorporating spin and valley degrees of freedom, we provide a complete characterization of the Hartree–Fock ground state manifold, which is generated by a [Formula: see text] hidden symmetry group acting on five elements. We also introduce new tools for ruling out translation symmetry breaking in the Hartree–Fock ground state manifold, which may be of independent interest.
Abstract Magic-angle twisted bilayer graphene has recently become a thriving material platform realizing correlated electron phenomena taking place within its topological flat bands. Several numerical and analytical methods have been applied to understand the correlated phases therein, revealing some similarity with the quantum Hall physics. In this work, we provide a Mott-Hubbard perspective for the TBG system. Employing the large-scale density matrix renormalization group on the lattice model containing the projected Coulomb interactions only, we identify a first-order quantum phase transition between the insulating stripe phase and the quantum anomalous Hall state with the Chern number of ±1. Our results not only shed light on the mechanism of the quantum anomalous Hall state discovered at three-quarters filling, but also provide an example of the topological Mott insulator, i.e., the quantum anomalous Hall state in the strong coupling limit.
Magic-angle twisted bilayer graphene has recently become a thriving material platform realizing correlated electron phenomena taking place within its topological flat bands. Several numerical and analytical methods have been applied to understand the correlated phases therein, revealing some similarity with the quantum Hall physics. In this work, we provide a Mott-Hubbard perspective for the TBG system. Employing the large-scale density matrix renormalization group on the lattice model containing the projected Coulomb interactions only, we identify a first-order quantum phase transition between the insulating stripe phase and the quantum anomalous Hall state with the Chern number of ±1. Our results not only shed light on the mechanism of the quantum anomalous Hall state discovered at three-quarters filling, but also provide an example of the topological Mott insulator, i.e., the quantum anomalous Hall state in the strong coupling limit.
Abstract Twisted bilayber graphene (TBG) has revealed fascinating quantum many-body states, and unstrained chargeneutral TBG at 1.08° is understood, both experimentally and theoretically , as a correlated insulator due to the interplay of long-range Coulomb interactions and the quantum metrics of flat bands. However, the fate of the ground state as one continuously tunes the twisted angle is still largely unknown. Here, theoretically, by employing a newly developed momentum-space continuous-field quantum Monte Carlo method that can fully take into account the long-range Coulomb interactions and quantum metrics of flat bands with system sizes that were not possible before , we show that charge-neutral TBG realizes an angle-tuned quantum phase transition with Gross-Neveu criticality at Θc ~ 1.20(1)°, from the analyses of single-particle spectra, free energy and the order parameter of inter-valley coherence. This shows the exciting possibility that by tuning away from 1.08°, pristine TBG can mimic a fundamental phase transition of Dirac fermions.
Single-layer and Bilayer of graphene are new classes of two-dimensional electron systems with unconventional band structures and valley degrees of freedom. The ground states and excitations in the integer and fractional quantum Hall regimes are investigated on torus and spherical geometries with the use of the density matrix renormalization group (DMRG) method. At nonzero Landau level indices, the ground states at effective filling factors 1, 1/3, 2/3 and 2/5 are valley polarized both in single-layer and bilayer graphenes. We examine the elementary charge excitations which could couple with the valley degrees of freedom (so called valley skyrmions). The excitation gaps are calculated and extrapolated to the thermodynamic limit. The largest excitation gap at effective filling 1/3 is obtained in bilayer graphene, which is a good candidate for experimental observation of fractional quantum Hall effect.