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The Many-Body Ground State Manifold of Flat Band Interacting Hamiltonian for Magic Angle Twisted Bilayer Graphene

Kevin D. Stubbs, Michael Ragone, Allan H. MacDonald, Lin Lin

TL;DR

This work provides a rigorous, representation-theoretic characterization of the many-body ground-state manifold for the flat-band interacting Hamiltonian of MATBG in the chiral limit. By exploiting the frustration-free, non-commuting sum structure and a diagonal form factor, the authors show that every ground state lies in the linear span of Slater determinants that are themselves ground states, across three models with increasing internal degrees of freedom. The analysis uses a combination of occupation-vector representations, the highest-weight theorem, and Littlewood–Richardson rules to count and construct the ground-state manifolds, yielding explicit dimension formulas via the hook-length method. The results solidify the ferromagnetic Slater determinant picture in MATBG and illuminate the role of a hidden U(Nocc) × U(Nocc) symmetry in organizing the ground-state degeneracy, with clear pathways for extensions to neutral excitations and possible superconducting contexts.

Abstract

At a magic relative twist angle, magic angle twisted bilayer graphene (MATBG) has an octet of flat bands that can host strong correlation physics when partially filled. A key theoretical discovery in MATBG is the existence of ferromagnetic Slater determinants as exact ground states of the corresponding flat band interacting (FBI) Hamiltonian. The FBI Hamiltonian describes the behavior of electrons that interact with each other in a high-dimensional space, and is constructed from the band structure of the non-interacting Bistritzer--MacDonald model at the chiral limit. A key property of the FBI Hamiltonian for MATBG is that it is frustration free and can be written as a sum of non-commuting terms. In this work, we provide a complete characterization of the ground state manifold of the FBI Hamiltonian, proving that it is precisely the linear span of such ferromagnetic Slater determinants.

The Many-Body Ground State Manifold of Flat Band Interacting Hamiltonian for Magic Angle Twisted Bilayer Graphene

TL;DR

This work provides a rigorous, representation-theoretic characterization of the many-body ground-state manifold for the flat-band interacting Hamiltonian of MATBG in the chiral limit. By exploiting the frustration-free, non-commuting sum structure and a diagonal form factor, the authors show that every ground state lies in the linear span of Slater determinants that are themselves ground states, across three models with increasing internal degrees of freedom. The analysis uses a combination of occupation-vector representations, the highest-weight theorem, and Littlewood–Richardson rules to count and construct the ground-state manifolds, yielding explicit dimension formulas via the hook-length method. The results solidify the ferromagnetic Slater determinant picture in MATBG and illuminate the role of a hidden U(Nocc) × U(Nocc) symmetry in organizing the ground-state degeneracy, with clear pathways for extensions to neutral excitations and possible superconducting contexts.

Abstract

At a magic relative twist angle, magic angle twisted bilayer graphene (MATBG) has an octet of flat bands that can host strong correlation physics when partially filled. A key theoretical discovery in MATBG is the existence of ferromagnetic Slater determinants as exact ground states of the corresponding flat band interacting (FBI) Hamiltonian. The FBI Hamiltonian describes the behavior of electrons that interact with each other in a high-dimensional space, and is constructed from the band structure of the non-interacting Bistritzer--MacDonald model at the chiral limit. A key property of the FBI Hamiltonian for MATBG is that it is frustration free and can be written as a sum of non-commuting terms. In this work, we provide a complete characterization of the ground state manifold of the FBI Hamiltonian, proving that it is precisely the linear span of such ferromagnetic Slater determinants.

Paper Structure

This paper contains 31 sections, 28 theorems, 197 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Contribution
  3. Discussion
  4. Outline
  5. Non-Interacting Model
  6. Chiral, Valley, and Spin Degrees of Freedom
  7. Flat band interacting model
  8. Form Factor of the Spinless, Valleyless Model
  9. Form Factor for with Valley and Spin Degrees of Freedom
  10. Properties of the Many-Body Ground States
  11. An Equivalent Characterization of the Ground States
  12. Proof that \ref{['eq:ferromagnetism']} is Sufficient
  13. On Lattice Terms, $\mathbf{q}' \in \Gamma^*$
  14. Off Lattice Terms, $\mathbf{q}' \not\in \Gamma^*$
  15. Ground State Manifold of the Spinless, Valleyless Model and Occupation Vector Representations
  16. ...and 16 more sections

Key Result

Theorem 1

A many-body state is a ground state of the FBI Hamiltonian for MATBG if and only if it can be written as a linear combination of Slater determinant states, each of which is a ground state of the FBI Hamiltonian.

Figures (1)

  • Figure 1: Occupation vector embedding based on the value of $\lambda_{+}$

Theorems & Definitions (47)

  • Theorem 1: Main Result, informal version of \ref{['thm:single-valley', 'prop:valleyful-case', 'prop:valleyful-spinful-case']}
  • Theorem 2: BeckerEmbreeWittsten2022TarnopolskyKruchkovVishwanath2019WatsonLuskin2021
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Corollary 1
  • proof
  • Proposition 4
  • Lemma 1
  • proof
  • ...and 37 more