Is k a good quantum number in this infinite wire?
Dear all, I am adding a periodic magnetic field to an infinite wire, where the periods of the lattice and magnetic field are not the same. This would appear to be a violation of the part of Bloch's theorem where $u_k(x)=u_k(x+n.a)$. When I model this in kwant however (minimal working code below), I am able to plot the band structure just fine. I'm not totally sure how kwant is doing this, so I ask what are the implications here? Is k_x a bad quantum number as a result? Have I misunderstood what kwant is doing here? Best wishes, Michael import kwant import kwant.continuum import numpy as np import matplotlib.pyplot hamiltonian = ( "A * (k_x**2 + k_y**2) * kron(sigma_z, sigma_0)" #kinetic terms "- mu * kron(sigma_z, sigma_0)" #chemical potential "+ 0.5 * g * mu_B * M * (kron(sigma_0, sigma_x) * sin(period*x) + kron(sigma_0, sigma_y) * cos(period*x))" #periodic magnetic field "+ delta * kron(sigma_x, sigma_0)" #superconductivity ) ham_template = kwant.continuum.discretize(hamiltonian, coords="xy",grid=20) infinite_wire = kwant.wraparound.wraparound(ham_template).finalized() momenta = np.linspace(-1,1, 100) def spectrum_discrete(**params): kwant.plotter.spectrum( infinite_wire, ('k_x', momenta), params=params, ) spectrum_discrete( period=7.3, A=2.01,#eV/nm^2 g=10, mu_B=57.9e-6,#eV/T M=1,#T mu=0,#eV delta=180e-6,#eV k_y=0, cos=np.cos, sin=np.sin )
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michael.hynes.18@ucl.ac.uk