- #1

Luqman Saleem

- 18

- 3

$$H = -t\sum_i^{L-1} \big[c_i^\dagger c_{i+1} - c_i c_{i+1}^\dagger\big] + V\sum_i^{L-1} n_i n_{i+1}$$

here ##L## is total number of sites, creation (##c^\dagger##) and annihilation (##c##) operators are defined as following

$$

c =

\begin{bmatrix}

0&0\\1&0

\end{bmatrix}

$$

and ##n_i = c_i^\dagger c_i## is number operator.

To exactly diagonalize (for simplicity let's take ##L=4## sites), one can expand ##H## as

$$H = -t\big[

c_1^\dagger \sigma_1^z \otimes c_2\otimes I_3 \otimes I_4 \\

+ I_1 \otimes c_2^\dagger \sigma_2^z \otimes c_3\otimes I_4\\

+ I_1 \otimes I_2 \otimes c_3^\dagger\sigma_3^z \otimes c_4

\big]+h.c.\\

+V\big[

n_1 \otimes n_2 \otimes I_3 \otimes I_4\\

+I_1 \otimes n_2 \otimes n_3 \otimes I_4\\

+I_1 \otimes I_2 \otimes n_3 \otimes n_4

\big]$$

where ##\sigma^z## (Pauli matrix) is just simple matrix multiplication for the sake of anti-commutation relation.

So far so good. (please correct me if I am doing anything wrong)!

__**Question:**__I used the above method and numerically calculated the ground state and found that the above method gives the correct results for ##V=0## but when ##V\ne0## the results are wrong.

Eventually, I get to the point that I am not taking care of the number of particles in the system. How do we numerically diagonalize a Hamiltonian matrix in the subsector of Hilbert space with a chosen number of particles?

__**Note:**__

I want to use this diagonalization in DMRG algorithm (in finding the ground state energy of the Hamiltonian at half-filling)