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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.

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    $\begingroup$ There's too many questions here. Also, I find it is a bit unclear what you really want to know (i.e. what are you given, what can you compute, and where are you stuck? You say "How do I get some correlations", but it is unclear which ones, and it is unclear what you are given. Also, the number of different H's seems a bit confusing ... ) $\endgroup$ – Norbert Schuch Apr 8 at 19:27
  • $\begingroup$ I have edited my question above. Thank you for your comment, @NorbertSchuch. Currently I want to know how to find the correlation function given a physical hamiltonian of the whole system. When I look at Peschel's paper, I am confused since I cannot find the relation between $C_{ij}$ and $\hat{C}_{nm}$. However, when I was looking for the relative posts in Stack Exchange, I found an interesting argument which was the two-point correlation function is independent whether one use full density matrix or reduced density matrix. Therefore,it should have a way to relate $C_{ij}$ with $\hat{C}_{ij}$ $\endgroup$ – Ricky Pang Apr 9 at 15:39
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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 $C$.

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