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The $g(\mathbf r_1, \mathbf r_2)$ is defined as $$g(\mathbf{r}) = \frac{\rho^{(2)}(\mathbf{r}_1,\mathbf{r}_2)}{\rho^{(1)}(\mathbf{r}_1) \rho^{(1)}(\mathbf{r}_2)}$$ where $$\rho^{(n)} (\mathbf r_1, \dots, \mathbf r_n) = \frac{N!}{(N-n)!} \frac 1 Z \int e^{-\beta V} d \mathbf r^{(N-n)}$$ If the system is homogeneous, $$\rho^{(1)} (\mathbf r) = \rho \ \ \ \... 2 Any free fermion Hamiltonian where the Fermi energy is chosen such that it is exactly at the bottom (or top) of the band (in the case of a single band) is of this form: It is obviously gapless, and its ground state is the vacuum, i.e., short-range correlated (or rather uncorrelated). One such example would be the 1D XX model,$$ H=-\tfrac12\sum (\sigma_x^i\...
Comments to the question (v3): The time-ordered product of operators $$T[A_1(t_1)\ldots A_n(t_n)] ~:=~\sum_{\pi\in S_n} \theta \left(t_{\pi(1)} > \ldots > t_{\pi(n)} \right) (-1)^{\varepsilon_{\pi,A}} A_{\pi(1)}(t_{\pi(1)})\ldots A_{\pi(n)}(t_{\pi(n)}) \tag{1}$$ is graded symmetric. [Here $(-1)^{\varepsilon_{\pi,A}}$ is a sign factor in the case ...