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Macroscopic approximation of tight-binding models near spectral degeneracies and validity for wave packet propagation

Guillaume Bal, Paul Cazeaux, Daniel Massatt, Solomon Quinn

TL;DR

The paper develops a general semiclassical framework to derive macroscopic PDEs from tight-binding models near spectral degeneracies, with rigorous error controls for wave packets under slow spatial variations. By representing TB and continuum models as Weyl pseudo-differential operators, it derives macroscopic Hamiltonians that capture Dirac-type dynamics and higher-order corrections, and extends the theory to position-dependent degeneracies and moiré/strain scenarios in graphene. The approach is validated on Haldane and graphene-based systems, including twisted bilayer graphene, with numerical simulations confirming long-time accuracy for edge and bulk states. It also reveals that higher-order macroscopic descriptions can alter spectral topology unless regularized, highlighting subtle connections between semiclassical approximations and topological invariants. Overall, the work provides a systematic pathway from microscopic TB models to accurate macroscopic descriptions with quantified errors, relevant for transport, strain engineering, and moiré materials.

Abstract

This paper concerns the derivation and validity of macroscopic descriptions of wave packets supported in the vicinity of degenerate points $(K,E)$ in the dispersion relation of tight-binding models accounting for macroscopic variations. We show that such wave packets are well approximated over long times by macroscopic models with varying orders of accuracy. Our main applications are in the analysis of single- and multilayer graphene tight-binding Hamiltonians modeling macroscopic variations such as those generated by shear or twist. Numerical simulations illustrate the theoretical findings.

Macroscopic approximation of tight-binding models near spectral degeneracies and validity for wave packet propagation

TL;DR

The paper develops a general semiclassical framework to derive macroscopic PDEs from tight-binding models near spectral degeneracies, with rigorous error controls for wave packets under slow spatial variations. By representing TB and continuum models as Weyl pseudo-differential operators, it derives macroscopic Hamiltonians that capture Dirac-type dynamics and higher-order corrections, and extends the theory to position-dependent degeneracies and moiré/strain scenarios in graphene. The approach is validated on Haldane and graphene-based systems, including twisted bilayer graphene, with numerical simulations confirming long-time accuracy for edge and bulk states. It also reveals that higher-order macroscopic descriptions can alter spectral topology unless regularized, highlighting subtle connections between semiclassical approximations and topological invariants. Overall, the work provides a systematic pathway from microscopic TB models to accurate macroscopic descriptions with quantified errors, relevant for transport, strain engineering, and moiré materials.

Abstract

This paper concerns the derivation and validity of macroscopic descriptions of wave packets supported in the vicinity of degenerate points in the dispersion relation of tight-binding models accounting for macroscopic variations. We show that such wave packets are well approximated over long times by macroscopic models with varying orders of accuracy. Our main applications are in the analysis of single- and multilayer graphene tight-binding Hamiltonians modeling macroscopic variations such as those generated by shear or twist. Numerical simulations illustrate the theoretical findings.
Paper Structure (45 sections, 24 theorems, 327 equations, 13 figures, 1 table)

This paper contains 45 sections, 24 theorems, 327 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Degenerate Tight Binding Hamiltonians and Main results
  3. Tight binding models and pseudo-differential representation.
  4. Degenerate points of TB Hamiltonian.
  5. Taylor expansion near degenerate points.
  6. Macrosopic model, localized wave packets, and approximation error.
  7. Position-dependent degenerate points.
  8. Higher-order approximations.
  9. Applications
  10. Haldane model
  11. Microscopic description.
  12. Macroscopic description and Dirac model.
  13. Strain and twist effects in graphene
  14. Strain effect.
  15. Twist effect.
  16. ...and 30 more sections

Key Result

Proposition 2.4

If $H_\delta = {\rm Op}^w a_\delta$ satisfies eq:a_delta-eq:reg_a with $\nu_0 = \nu_1 = \nu_2 = 0$, then $H_\delta$ satisfies Assumption assumption:commutator for all $m \in \mathbb{N}_0$.

Figures (13)

  • Figure 1: Geometry of graphene bipartite lattice. Left: spatial lattice. Right: dual lattice.
  • Figure 2: Numerical band structure for a straight edge profile, computed using the continuum Hamiltonian of order 1, $H_1$. Highlighted respectively in green and red is the support for the two localized edge modes studied in section \ref{['sec:edgestatenum']}.
  • Figure 3: Error plot as a function of time for both the first and second order continuum models compared to tight-binding for the fast edge mode supported by the green momentum-energy region in Figure \ref{['fig:edgebands']}.
  • Figure 4: Profile of the gating parameter $\omega(x_1,x_2) = \omega m \left ( \frac{\delta}{\omega} {\bf x} \right )$ used for the edge mode propagation simulations in section \ref{['sec:edgestatenum']}.
  • Figure 5: Error plot as a function of time for both the first and second order continuum models compared to tight-binding for the slow edge mode supported by the red momentum-energy region in Figure \ref{['fig:edgebands']}.
  • ...and 8 more figures

Theorems & Definitions (54)

  • Remark 2.2
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • Proposition 2.6
  • Theorem 2.7
  • proof : Proof
  • Definition 2.8
  • Remark 2.10
  • Theorem 2.12
  • ...and 44 more