I am studying second quantization for first time. Let we have a a Hamiltonian (corresponding to a chain of particles with nearest neighbor hopping 't', ignore spin, on-site potential and interaction potential between two particles) that is given in first quantization as:
$$H_{1st}=t\sum_{m=1}^N|m><m+1|+h.c.$$
$$H_{1st}=t(|1><2|+|2><3|+|3><4|+|4><1|+h.c.)$$
It can be written in matrix form as following (N=4 and with periodic boundary condition):
$$H_{1st}=t(|1><2|+|2><3|+|3><4|+|4><1|+h.c.)$$
using
$$|1>=\begin{bmatrix}1\\0\\0\\0\end{bmatrix};
|2>=\begin{bmatrix}0\\1\\0\\0\end{bmatrix}...$$ and so on... we can write our matrix as:
$$H_{1st}=t\begin{bmatrix}0&1&0&1\\1&0&1&0\\0&1&0&1\\1&0&1&0\end{bmatrix}$$
In second quantization this Hamiltonian can be written as:
$$H_{2nd}=t\sum_{m=1}^Nc_m^\dagger c_{m+1}+h.c.$$
that can be written as for N=4
$$H_{2nd}=t(c_1^\dagger c_{2}+c_2^\dagger c_{3}+c_3^\dagger c_{4}+c_4^\dagger c_{1}+h.c.)$$
How do we write matrix from this last equation of $H_{2nd}?$
