CDW correlation function for 1D Dirac fermion in condensed matter

Question

I am following Shankar's lecture notes on bosonization, specifically the theory of left-/right-moving fields for a low-energy 1D fermionic chain. For now, I ignore the Heisenberg time dependence of the field operators because I am only interested in calculating equal-time correlation functions. The mode expansion is given by Eqn 17.14, $$\psi_{\pm}(x)=\int_{-\infty}^{\infty}\frac{dp_{\alpha}}{2\pi}\ \psi_{\pm}(p)\ e^{ipx}$$ where $$dp_{\alpha} = e^{-\frac{1}{2}\alpha|p|}dp$$ is introduced so that the integral converges (i.e., we first integrate with finite $\alpha$, then take the limit as $\alpha\rightarrow 0$). The mode expansion coefficients have an additional property that for the ground state $|GS\rangle$, $$\psi_+(p_+)|GS\rangle = \psi_-^{\dagger}(p_+)|GS\rangle = \psi_+^{\dagger}(p_-)|GS\rangle = \psi_-(p_-)|GS\rangle = 0$$ for any $p_+>0$ and $p_-<0$. Furthermore, they obey the anti-commutation relations $$\{\psi_{\alpha}(p),\psi_{\alpha'}^{\dagger}(p')\}=2\pi\delta(p-p')$$ and hence it is very easy to show (see Eqns. 17.16-17.18) that $$\left<\psi_+(x)\psi_+^{\dagger}(0)\right>_{\alpha}=\frac{1}{2\pi}\cdot\frac{1}{\alpha-ix}$$ This is basically the fermionic Green's function/propagator. So far, so good. My issue is in the next part, where Shankar calculates the correlation function for the charge density wave (CDW) operator, defined as $$\mathcal O_{\text{CDW}}(x) = \bar\psi\psi = i \psi_-^{\dagger}(x) \psi_+(x) + \text{h.c.}$$ The correlation function is $\left<\mathcal O(x)\mathcal O (0)\right>_{\alpha}$ and hence looks like a sum of four expectation values, each containing four field operators. Apparently the final answer is $$\frac{1}{2\pi^2}\cdot\frac{1}{x^2+\alpha^2}$$

and Shankar's claim is that it can be derived using only the propagators and the anti-commutation relations. This leads me to believe that we can manipulate the expression using $\{\psi_{\alpha}(x),\psi_{\alpha'}(y)\}=\delta(x-y)$ and it will somehow yield an expression that looks like the product of two Green's functions in Eqns 17.19 and 17.20. Is it possible to obtain the final answer using this strategy? Would it require something else, like Wick's theorem?

Answer 1

You are doing calculations for the free fermion theory (without interactions), so you have a Gaussian path integral in the fermion fields $\psi_\pm$. You need to calculate the expectation value of a four-fermion term in this Gaussian theory, and this can be reduced to a product of propagators using Wick's theorem, as you correctly guessed. I think this is what Shankar is also suggesting.

Note that Wick's theorem is not necessary to calculate this correlation function as you are just calculating a fourth moment of a multidimensional Gaussian integral, but it greatly simplifies the calculation. You could also evaluate it using source fields, for example.