1. Berry phase in graphene: a semi-classical perspective
Denis Ullmo & Pierre Carmier
(LPTMS, Université Paris-Sud)
Discussion with: folks from the Orsay graphene journal club (Mark Goerbig, Jean Noel Fuchs, Gilles Montambaux , etc..)
Reference : Phys. Rev. B 77, (2008)
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2. Graphene (a short introduction)B A
Sp2 hybridization of carbon → hexagonal lattice

3. Tight binding model (nearest neighbor approx)
- Low energy (continuum) limit

4. How hard is it to make graphene ?
Mechanical cleavage (a.k. scotch tape method)
[Novoselov, et al., science 2004]
Epitaxial graphite [Berger et al. J. Chem Phys B 2004]
graphite

5. Quantum Hall effect in graphene
Landau levels
- graphene
- GaAs/AlGaAs
[Novoselov et al. Nature (2005), Zhang et al., Nature (2005)]
(cf also [Sadowski et al. (2006)] (spectroscopy) )

6. Magnetic field dependence
Mean (Weyl) density of states independent of the magnetic field
Landau level degeneracy

7. And thus

8. Zero energy state r Maslov indices Need of an additional phase
Focal points
Need of an additional phase

9. Berry phase [Berry, Proc. R. Soc. Lond. (1984)]
Parameter space
Context
time dependent Hamiltonian :
( eigenstates , eigenvalues )
Adiabatic evolution from
Berry phase
Dynamical phase factor

10. Circuit encircling a degeneracy
“Thus, we obtain the pleasant result […] that the geometrical phase associated with C is
where Ω(C) is the solid angle that C subtends at the degeneracy point” [Berry 1984]
Here : C and degeneracy (Dirac) point in
the same plane
Exactly compensate the Maslov phase C

11. However
Direct relation between the Landau levels and the Berry phase not straightforward (after all, this is not a time dependent problem).
Case of a (possibly small) mass term (asymmetry between A and B sub-lattices)
(no degeneracy)
Nevertheless (Haldane 1988) :
→ require also exact cancelation of Maslov phase π

12. Semiclassical Green function [Bolte and Keppeler, Ann. Phys. (1999)]
Graphene Hamiltonian
(α = ±1 → K or K’ , ∏ = -i ħ∂ + e A(r) )
Green function
2 * 2 matrices

13. Far from the source r”: WKB approximation
Assume G of the form O(ħ⁰) equation: (Hamilton Jacobi) O(ħ) equation: (transport)
2 * 2 matrices

14. Matching with exact solution near r”
Exact “free” Green function
Asymptotic expressions:

15. Semiclassical Green function
classical action, stability determinant,
→ classical Hamiltonian
: eigenvector

16. semi-classical vs adiabatic (Berry) phase
Semiclassical phase
Adiabatic (Berry) phase

17. Specify to the evolution of V⁺ and switch to
bra/ket notations:
First order perturbations:
Inserting in def of
Thus:

18. Particular case : m=0, U(r) arbitrary
Eigenstates :
In the absence of a mass term:
i) semiclassical phase and Berry phase are identical
ii) both are simply related to the rotation angle of ∏

19. Landau level with of a mass term
Constant mass term → Berry-like phase: no dependence on the mass

20. Conclusion
Semiclassical formalism is the proper setup to understand the “extra phase” responsible for zero energy states in graphene Landau Level.
Two cases are particularly simple: i) m=0 ii) cte masse and electric potential. If m=0, the semiclassical phase is identical to the Berry phase, and simply related to the winding of the momentum.
In general however (cf. Littlejohn and Flynn 1991) both differ as they correspond to different concept.

21. NB: things one can compute easily with the semiclassical formalism
Landau levels in monolayer and bilayer graphene.
Corrections due to nonlinear term in the graphene dispersion relation, or to next nearest neighbor hopping.
Landau levels in monolayer graphene with a finite mass
(cf Phys. Rev. B 77, (2008))
n-p junction in graphene with strong magnetic field (in progress)