# Calculating cyclotron orbits in a lattice

I have a naive question, is it possible to obtain the wavepacket dynamics that give rise to cyclotron orbits from a real space tight-binding Hamiltonian? Consider a square lattice in the presence of a constant magnetic field that amounts to say that the hopping elements acquire a phase. Let $$H = \sum_{} H_{m,n}$$ be the real space tight-binding Hamiltonian describing such a system.

The cyclotron orbits are actually the semi-classical approximation to the above dynamics that $$H$$ captures. These orbits are the paths of the electron $$\dot{\textbf{r}}(t) = \partial H/\partial \textbf{p}$$. How can we obtain this information from the above $$H$$?

Numerically:
We can diagonalize $$H$$, to obtain eigenfunctions $$\psi_{m,n}$$ from this, we can obtain probability densities $$|\psi_{m,n}|^{2}$$.
To obtain wavepacket dynamics $$\Psi(t)$$ , we solve Schrodinger's equation: $$i\partial\Psi(t) = H \Psi$$. This equation can be solved for some initial conditions. The solution tells if we make an excitation in the system at a site $$(m_0,n_0)$$ (initial condition) then how it will propagate in the lattice. This was my attempt, link.
However, this does not give any information about the semi-classical information!

• The state with sufficiently large angular momentum is correlated with the classical orbital movement. Commented Jul 22, 2021 at 11:39
• @AlexTrounev That's right. Does it mean if we consider a state with high angular momentum it will mimic the semi-classical motion? For that, can we try some Gaussian wavepacket? I have no idea how to excite a Gaussian wave in Mathematica. I saw your beautiful solution to my question. Commented Jul 22, 2021 at 12:57
• Are lattice coordinates correspond to space coordinates? Commented Jul 22, 2021 at 18:53
• @AlexTrounev not actually, i think it’s crucial for Gaussian wavepacket? Commented Jul 22, 2021 at 20:26
• What is the physical meaning of this lattice? Commented Jul 22, 2021 at 20:30

The state with sufficiently large angular momentum is correlated with the classical orbital movement. The question is how we can prepare this state? If we use Hamiltonian described on this page and plot all eigenvectors, then we have picture
States from 3 to 10 looks promising, and if we prepare initial state from some mixture of its, then animated peripheral region of lattice looks as follows

It looks like rotation with period of $$T=37$$. I think that for the large lattice we can get something similar to cyclotron orbit.
Update 1. Eigenvalues of Hamiltonian described above are

{2.90519, -2.90519, -2.76278, -2.76278, 2.76278, 2.76278, -2.58201, \
2.58201, -2.58201, 2.58201, 2.52133, -2.52133, -2.39276, 2.39276, \
-2.34139, 2.34139, 2.25356, 2.25356, -2.25356, -2.25356, 2.13097, \
2.13097, -2.13097, -2.13097, -1.95441, 1.95441, -1.95441, 1.95441, \
-1.92521, 1.92521, -1.86377, 1.86377, -1.86377, 1.86377, -1.69775, \
1.69775, -1.61748, 1.61748, -1.58945, 1.58945, -1.58945, 1.58945, \
-1.58691, 1.58691, 1.57744, -1.57744, -1.36333, -1.36333, 1.36333, \
1.36333, 1.29892, -1.29892, -1.29892, 1.29892, 1.27992, -1.27992, \
-1.27992, 1.27992, -1.25195, 1.25195, 1.07599, -1.07599, 1.06731, \
1.06731, -1.06731, -1.06731, 1.04507, -1.04507, -1.04507, 1.04507, \
-1., 1., -1., -1., 1., 1., -0.863067, 0.863067, -0.839226, 0.839226, \
-0.74239, 0.74239, 0.74239, -0.74239, -0.692981, 0.692981, -0.491724, \
-0.491724, 0.491724, 0.491724, -0.35947, 0.35947, -0.229448, \
-0.229448, 0.229448, 0.229448}


We can combine eigenvectors with negative sign of eigenvalues, for example, {2, 3, 4, 7, 9} as initial condition. Then we have picture with illusive rotation

• Thanks a lot, @Alex! Your solution goes to the point. However, if we have to decide the higher angular momentum states, I would think to look for figures where probability amplitudes is mostly localized at the boundary. Like 31-36, someothers. Isn't it? Commented Jul 23, 2021 at 12:13
• Also, I was working for the bigger lattices, is there a way to speed up the code? It looks Manipulate is terribly slow for this case. Commented Jul 23, 2021 at 12:32
• There are many possibilities to get rotation including random combination of all states. What size of lattice do you run? Commented Jul 23, 2021 at 13:05
• For me, it can go upto 2000 lattice sites. Commented Jul 23, 2021 at 17:09