Entanglement Hamiltonian and correlation matrix of a system

Question

I am new to quantum entanglement and condensed matter physics. I encounter a difficulty in understanding the relation between entanglement Hamiltonian and correlation function. Currently, I am reading a paper written by Ingo Peschel, regarding the reduced density matrix of a subsystem. The thing that we want to compute is the correlation function of the subsystem since it directly relates to the entanglement Hamiltonian of the subsystem. Following the notation of Peschel's paper, the index $n,m$ will be used for the whole system and the index $i,j$ denotes the subsystem. My central question is that can we simply treat the correlation function of the subsystem $C_{ij} = \langle c^{\dagger}_{i} c_{j} \rangle$ as a part of the correlation function of the whole system $\hat{C}_{nm}$? \begin{equation} C_{ij} = \hat{C}_{nm} ~~,~~ \text{where} 1 \leq n,m \leq M \end{equation} where $M$ is the number of site of the subsystem.

Answer 1

Yes, the correlation matrix of the subsystem is the submatrix of the correlation matrix for the entire system.

This is since the correlation matrix is defined as $$ C_{ij} = a \,\mathrm{tr}[c_i c_j] $$ with $c_i$, $c_j$ all fermionic operators on the whole system (what $a$ and $c_i$ are exactly depends on the convention chosen by the respective paper), and the correlation matrix of the subsystem is the matrix you get by taking only operators $c_i$, $c_j$ acting on the modes in the subsystem - so it is just the corresponding submatrix of $C$.