Elastic deformations and Wigner-Weyl formalism in graphene

We discuss the tight-binding models of solid state physics with the Z 2 sublattice symmetry in the presence of elastic deformations in an important particular case—the tight binding model of graphene. In order to describe the dynamics of electronic quasiparticles, the Wigner–Weyl formalism is explored. It allows the calculation of the two-point Green’s function in the presence of two slowly varying external electromagnetic fields and the inhomogeneous modification of the hopping parameters that result from elastic deformations. The developed formalism allows us to consider the influence of elastic deformations and the variations of magnetic field on the quantum Hall effect.


Wigner-Weyl field theory
From the previous talks we've learned that partition function is The Weyl symbols in the above are of the Dirac operator and density matrix Q = iω − H,Ŵ = |Ψ Ψ| The variation of partition function is then We used that Ŵ =Ĝ and employed Peierls substitution for EM potential A, p → p − A(x): δQ = −∂ p k QδA k .
The current density thus It is not an invariant, but the total current is The current density thus It is not an invariant, but the total current is

Conductivity as topological invariant
To obtain the conductivity lets expand the current density in A and its first derivatives.

Conductivity as topological invariant
To obtain the conductivity lets expand the current density in A and its first derivatives.

Conductivity as topological invariant
To obtain the conductivity lets expand the current density in A and its first derivatives.
We have (using Groenewold equation G W * Q W = 1) then in 2 + 1D the average conductivity is given bȳ then in 2 + 1D the average conductivity is given bȳ then in 2 + 1D the average conductivity is given bȳ Using Weyl representation in momenta space, we can also rewrite N = abc 3!4π 2 dl dk dp dq recall G −1 = Q.

Kubo formula
Kubo formula is immediately reconstructed, H, with H |n = E n |n Thus we will have One needed to use that withx understood as i∂p acting on the wave-function in momentum representation Nearest neighbour TB with non-uniform varying hopping parameter where b (j) are the vectors connecting each atom to its nearest M neighbors, j = 1, ..., M . In Fourier representation we get Wigner-Weyl formalism in graphene 26/09 ICNFP 2019 11 / 21 And those with Z 2 symmetry The lattice consist of two Bravais sub-lattices then we introduce a vector wave function and Dirac Hamiltonian becomes

Momentum representation
In momentum representation the Hamiltonian becomes it can be rewritten as if we introduce a shifted Fourier transform

Weyl symbol of Dirac operator
We use the following definition of the Weyl symbol of an operatorÂ: For off-diagonal components of H from above it gives If the hopping parameters are homogeneous, then On the other hand, when the hopping parameters vary, we have

Weyl symbol of Dirac Hamiltonian
Weyl symbol of Dirac Hamiltonian To introduce the EM field we need to use translation operators, i.e.
as a short notation we introduced We obtain thus the Dirac operator symbol We obtain thus the Dirac operator symbol For both t (j) and A that do not vary significantly at the distances of order of lattice spacing we may use above expression for arbitrary values of x.

Weyl symbol of Dirac Operator with EM
We obtain thus the Dirac operator symbol For both t (j) and A that do not vary significantly at the distances of order of lattice spacing we may use above expression for arbitrary values of x.
As before

Elastic deformations
Strained graphene at x 3 = 0 is described by new coordinates y k The displacements have three components u a (x). Induced metric is Elastic deformations change the spatial hopping parameters here β is the Gruneisen parameter. We imply that β|u ij | 1. For arbitrarily varying field u we obtain the following expression for Q W :

Conclusions? Not yet
Thus, we need to obtain Weyl symbol of Dirac operator and of its inverse

Kubo formula revisited
Using Kubo formula for σ = N /2π We decompose the coordinates x 1 , x 2 in relative coordinates ξ i (with bounded values) and center coordinates X i (the unbounded part) x 1 =ξ 1 +X 1 ,x 2 =ξ 2 +X 2 (1) Quite naturally we then come to (in Landau gauge H ≡ H(ξ 1 , ξ 2 )) Assuming that we have two good quantum numbers, |n → |p 2 , m In conventional systems number of occupied levels, N , is counted from the neutrality point. In graphene there are deeply lying levels with large negative Chern numbers, effectively rendering the sum to go from the zero energy. D. Sheng, et al., 2006. Y. Hatsugai, et al., 2006 However, our approximation is probably valid only up to |E F | ∼ t, i.e. in-between the innermost van Hove singularities, and we cannot count deep lying levels. Their contribution is known -it cancels precisely that of N /(2π) at the half filling, σ (0) xy = N (0) /(2π). Finally, where N is counted from the half filling.

Conclusions, finally
Topological invariants in the Wigner-Weyl formalism, applicable to non-uniform Z 2 lattices Total currentJ Both valid in graphene with (slowly) varying external fields and non-uniform mechanical strain.