Energy Spectrum of lattice Model

Question

Here the author has done numerical calculation and has plotted energy spectrum and wave function for Kiteav Chain whose hamiltonian is given by $$H=-\mu\sum_n c_n^\dagger c_n-t\sum_n (c_{n+1}^\dagger c_n+\textrm{h.c.}) + \Delta\sum_n (c_n c_{n+1}+\textrm{h.c.})$$ I was wondering how can we also do the same using mathematica or any other tools.

Question So, for given hamiltonian how can we calculate the energy spectrum and wave function numerically?

Note: The author has used $n=35$ as lattice size.

If we write $ d_m := \sum_n U_{m,n} c_n $, what kind of matrix can $U$ be if we want to preserve the fermionic commutation relations? How does this transform the Hamiltonian? How would the Hamiltonian look if we want it to be diagonal in these new $d_m$ variables? By going through this logic carefully, you will see how you can diagonalize the Hamiltonian numerically for thousands of sites in a matter of seconds. — Ruben Verresen, Jun 25 '20 at 05:01
@RubenVerresen Do you mean diagonalising by doing Fourier transform followed by Bogolibouv transformation? — Kartik Chhajed, Jun 26 '20 at 01:53

score 1 · Answer 1 · answered Jun 25 '20 at 05:02

1

Choose a basis.
Write the Hamiltonian matrix in that basis (e.g. like it's done here.)
Diagonalise the Hamiltonian. That will give you the eigenenergies and energy wavefunctions. On Mathematica, for instance, you'd use Eigensystem.

answered Jun 25 '20 at 05:02

SuperCiocia

24,596

In principle matrix for $\hat a$ or $\hat a^\dagger$ is infinite-dimensional. But since I say: I like to work with 35 sites on lattice! Then my matrix dimension of $\hat a$ or $\hat a^\dagger$ is 35x35? – Kartik Chhajed Jun 26 '20 at 02:02
Well, you'd need a different $a$ and $a^\dagger$ for each site – SuperCiocia Jun 26 '20 at 02:03
Since we are dealing with fermions $\hat a$ will be 2x2 matrix? – Kartik Chhajed Jun 26 '20 at 02:06
I think so. Provided you find a matrix representation such that $a^\dagger$$^2 = 0$. – SuperCiocia Jun 26 '20 at 02:08
For fermions it might be more complicated though: look here https://physics.stackexchange.com/questions/62604/matrix-representation-for-fermionic-annihilation-operator – SuperCiocia Jun 26 '20 at 06:00

Energy Spectrum of lattice Model

1 Answers1