gistfile1.txt

using StaticArrays
using PyPlot

const gap = 0.0 # gap at the Dirac points. 0 for graphene
const n = 80 # number of k point per direction
const t2 = 0.0

#http://www-personal.umich.edu/~sunkai/teaching/Fall_2014/Chapter6.pdf

# K = 1/3,-1/3
# K' = -1/3,1/3

# near K/K', H(k) = (Jk)_1 σ1 + (Jk)_2 σ2, with
# J = pi (√3 -√3)
#        (1    1)
# for K, and columns exchanged for K'

const x = (0:1/(n):(n-1)/n)-0.5

# see e.g. https://courses.cit.cornell.edu/mse5470/handout11.pdf
const δ = @SMatrix [1/sqrt(3)    -1/(2sqrt(3))   -1/(2sqrt(3));
                    0            1/2             -1/2]
# lattice vectors
const A = @SMatrix [sqrt(3)/2   sqrt(3)/2;
                    1/2         -1/2     ]
# recp lattice vectors
const B = inv(A')
#H(k+K) = tau^|K|*H(k)*(tau^|K|)'
const tau = @SMatrix [1 0;0 exp(im*2π/3)]

f(k1,k2) = sum(cis.(2π.*δ'*(B*(@SVector [k1,k2]))))
function H(k1,k2)
    fk = f(k1,k2)
    M = gap + 2t2*(sin(2π*k1) + sin(2π*k2) + sin(2π*(k1-k2)))
    @SMatrix [M    fk';
              fk    -M]
end

function plot_gaps()
    figure()
    gapk(k1,k2) = eig(Hermitian(H(k1,k2)))[1][2] - eig(Hermitian(H(k1,k2)))[1][1]
    gaps = [gapk(k1,k2) for k1 in x, k2 in x]
    pcolor(x,x,gaps)
    colorbar()
end