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On the Hartree-Fock Ground State Manifold in Magic Angle Twisted Graphene Systems

Kevin D. Stubbs, Simon Becker, Lin Lin

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 has shown that Hartree-Fock ground states (with a positive charge 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 ${\rm U}(4) \times {\rm U}(4)$ 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.

On the Hartree-Fock Ground State Manifold in Magic Angle Twisted Graphene Systems

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 has shown that Hartree-Fock ground states (with a positive charge 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 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.
Paper Structure (16 sections, 12 theorems, 82 equations)

This paper contains 16 sections, 12 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Discussion and open question
  4. Organization
  5. Notation
  6. The Flat-Band Interacting Hamiltonian for Twisted Graphene
  7. Hartree-Fock Theory for Flat-Band Interacting Hamiltonians
  8. A Review of Hartree-Fock Theory without Translation Symmetry
  9. Hartree-Fock Energy of Flat-Band Interacting Hamiltonians
  10. Generalized Ferromagnetism in Flat-Band Interacting Hamiltonians
  11. The Sylvester Equation and Translation Breaking
  12. Application to Spinless, Valleyless TBG
  13. Extension to Valleyful and Spinful Models
  14. An Algorithm for Determining Translation Symmetry Breaking
  15. Calculation of the Hartree-Fock Energy
  16. ...and 1 more sections

Key Result

Lemma 2.1

Any many-body ground state $\ket{\Phi}$ of a frustration-free FBI Hamiltonian is half-filled. That is, it satisfies where $\hat{N}$ is the number operator $\hat{N} = \sum_{\mathbf{k} \in \mathcal{K}} \sum_{m \in \mathcal{N}} \hat{f}_{m\mathbf{k}}^\dagger \hat{f}_{m\mathbf{k}}$.

Theorems & Definitions (21)

  • Conjecture : Many-Body Ground States of Frustration-Free FBI Hamiltonians
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • Proposition 1: Generalized Ferromagnetism
  • Corollary 1
  • proof : Proof of \ref{['coro:uniform-filling']}
  • Lemma 4.1: Lemma 7.1 BeckerLinStubbs2023
  • proof : Proof of \ref{['prop:pseudo-ferromagnetism']}
  • Lemma 5.1
  • ...and 11 more