# Reducible Fermi Surface for Multi-layer Quantum Graphs Including Stacked Graphene

@article{Fisher2020ReducibleFS, title={Reducible Fermi Surface for Multi-layer Quantum Graphs Including Stacked Graphene}, author={L. R. Fisher and Wei Li and Stephen P. Shipman}, journal={Communications in Mathematical Physics}, year={2020}, volume={385}, pages={1499 - 1534} }

We construct two types of multi-layer quantum graphs (Schrödinger operators on metric graphs) for which the dispersion function of wave vector and energy is proved to be a polynomial in the dispersion function of the single layer. This leads to the reducibility of the algebraic Fermi surface, at any energy, into several components. Each component contributes a set of bands to the spectrum of the graph operator. When the layers are graphene, AA-, AB-, and ABC-stacking are allowed within the same…

## 16 Citations

### Irreducibility of the Fermi surface for planar periodic graph operators

- MathematicsLetters in Mathematical Physics
- 2020

We prove that the Fermi surface of a connected doubly periodic self-adjoint discrete graph operator is irreducible at all but finitely many energies provided that the graph (1) can be drawn in the…

### On the Spectra of Periodic Elastic Beam Lattices: Single-Layer Graph

- Physics, Mathematics
- 2021

We present full description of spectra for a Hamiltonian defined on periodic hexagonal elastic lattices. These continua are constructed out of Euler-Bernoulli beams, each governed by a scalar-valued…

### Tight-binding reduction and topological equivalence in strong magnetic fields

- MathematicsAdvances in Mathematics
- 2022

### Stable defect states in the continuous spectrum of bilayer graphene with magnetic field

- Physics
- 2022

. In a tight-binding model of AA-stacked bilayer graphene, it is demonstrated that a bound defect state within the region of continuous spectrum can exist stably with respect to variations in the…

### Fermi isospectrality for discrete periodic Schrodinger operators

- Mathematics
- 2021

. Let Γ = q 1 Z ⊕ q 2 Z ⊕· · ·⊕ q d Z , where q l ∈ Z + , l = 1 , 2 , · · · , d . Let ∆+ V be the discrete Schr¨odinger operator, where ∆ is the discrete Laplacian on Z d and the potential V : Z d →…

### Fermi Isospectrality of Discrete Periodic Schrödinger Operators with Separable Potentials on $$\mathbb {Z}^2$$

- MathematicsCommunications in Mathematical Physics
- 2022

. Let Γ = q 1 Z ⊕ q 2 Z with q 1 ∈ Z + and q 2 ∈ Z + . Let ∆+ X be the discrete periodic Schr¨odinger operator on Z 2 , where ∆ is the discrete Laplacian and X : Z 2 → C is Γ-periodic. In this paper,…

### Irreducibility of the Bloch Variety for Finite-Range Schrödinger Operators

- MathematicsJournal of Functional Analysis
- 2022

### Irreducibility of the Fermi variety for discrete periodic Schrödinger operators and embedded eigenvalues

- MathematicsGeometric and Functional Analysis
- 2022

Let $$H_0$$ H 0 be a discrete periodic Schrödinger operator on $$\ell ^2(\mathbb {Z}^d)$$ ℓ 2 ( Z d ) : $$\begin{aligned} H_0=-\Delta +V, \end{aligned}$$ H 0 = - Δ + V , where $$\Delta $$ Δ is the…

### Topics on Fermi varieties of discrete periodic Schrödinger operators

- MathematicsJournal of Mathematical Physics
- 2022

This is a survey of recent progress on the irreducibility of Fermi varieties, rigidity results and embedded eigenvalue problems of discrete periodic Schrödinger operators.

### Spectra of Regular Quantum Trees: Rogue Eigenvalues and Dependence on Vertex Condition

- Materials ScienceAnnales Henri Poincaré
- 2021

We investigate the spectrum of Schrödinger operators on finite regular metric trees through a relation to orthogonal polynomials that provides a graphical perspective. As the Robin vertex parameter…

## References

SHOWING 1-10 OF 65 REFERENCES

### Reducible Fermi surfaces for non-symmetric bilayer quantum-graph operators

- MathematicsJournal of Spectral Theory
- 2019

This work constructs a class of non-symmetric periodic Schrodinger operators on metric graphs (quantum graphs) whose Fermi, or Floquet, surface is reducible. The Floquet surface at an energy level is…

### Dirac cones for bi- and trilayer Bernal-stacked graphene in a quantum graph model

- PhysicsJournal of Physics A: Mathematical and Theoretical
- 2020

A quantum graph model for a single sheet of graphene is extended to bilayer and trilayer Bernal-stacked graphene; the spectra are characterized and the dispersion relations explicitly obtained; Dirac…

### Dirac cones for graph models of multilayer AA-stacked graphene sheets

- Physics
- 2021

Abstract We propose an extension, of a quantum graph model for a single sheet of graphene, to multilayer AA-stacked graphene and also to a model of the bulk graphite. Spectra and Dirac cones are…

### The low energy electronic band structure of bilayer graphene

- Physics
- 2007

Abstract.We employ the tight binding model to describe the electronic band structure of bilayer graphene and we explain how the
optical absorption coefficient of a bilayer is influenced by the…

### Properties of graphene: a theoretical perspective

- Physics
- 2010

The electronic properties of graphene, a two-dimensional crystal of carbon atoms, are exceptionally novel. For instance, the low-energy quasiparticles in graphene behave as massless chiral Dirac…

### From graphene to graphite : Electronic structure around the K point

- Physics
- 2006

Within a tight-binding approach we investigate how the electronic structure evolves from a single graphene layer into bulk graphite by computing the band structure of one, two, and three layers of…

### Landau Level Splittings, Phase Transitions, and Nonuniform Charge Distribution in Trilayer Graphene.

- PhysicsPhysical review letters
- 2016

The results indicate the importance of a previously unaccounted band structure parameter which, together with a more accurate estimate of the other tight-binding parameters, results in a significantly improved determination of the electronic and Landau level structure of TLG.

### Electronic Energy Spectrum of Two-Dimensional Solids and a Chain of C Atoms from a Quantum Network Model

- Physics
- 2004

The aim of this study is to explain how a quantum network can be used as simple model to calculate complex band structures. The paper contains an introduction, a mathematical exposure of the method,…

### Zigzag periodic nanotube in magnetic field

- Mathematics
- 2006

We consider the magnetic Schr\"odinger operator on the so-called zigzag periodic metric graph (a quasi 1D continuous model of zigzag nanotubes) with a periodic potential. The magnetic field (with the…

### Cantor spectrum of graphene in magnetic fields

- PhysicsInventiones mathematicae
- 2019

We consider a quantum graph as a model of graphene in magnetic fields and give a complete analysis of the spectrum, for all constant fluxes. In particular, we show that if the reduced magnetic flux…