# Interaction Term in Tomonaga-Luttinger Model

I am studying Tomonaga-Luttinger Model from Altland and Simon's textbook called Condensed Matter Field Theory. From the derivation, I am stuck with showing that the contribution to the interaction term comes only from $$(k, k', q) = (\pm k_F, \pm k_F, 0)$$ and $$(k, k', q) = (\pm k_F, \mp k_F, 2k_F)$$.

To elaborate more on this, the textbook starts from the 1-D jellium model (sorry if the terminology is wrong), namely, $$H=\sum_k a_k^\dagger(\epsilon_k-E_F)a_k+\frac{1}{2L}\sum_{k, k', q\neq0}V_{ee}(q)a_{k+q}^\dagger a_{k'-q}^\dagger a_{k'}a_k \equiv H_0+H_1$$ From the above Hamiltonian, the book first linearizes the first term by expanding in the vicinity of the Fermi energy such that $$H_0=\sum_{s,q}\sigma_sv_Fqa_{sq}^{\dagger}a_{sq}$$ , where s denotes whether $$q$$ is expanded near $$k_F$$ or $$-k_F$$.

Also, noting that $$\rho_{sq}=\sum_{k}a^\dagger_{s, k+q}a_{s,k}$$, $$H_1 = \frac{1}{2L}\sum_{qs}\left[g_4\rho_{s,q}\rho_{s,-q}+g_2\rho_{\bar{s},q}\rho_{s,-q}\right]$$, where $$\bar{s}$$ denotes the complement of $$s$$.

What I do not figure out is the statement of the textbook that only $$(k, k', q) = (\pm k_F, \pm k_F, 0)$$ and $$(k, k', q) = (\pm k_F, \mp k_F, 2k_F)$$ contribute to the summation. For, in the course of using the relation $$\rho_{sq}=\sum_{k}a^\dagger_{s, k+q}a_{s,k}$$, the summation over the entire k and k' seems to have been already conducted. Furthermore, the Fourier transform of 1/r in 1-D is $$-2\gamma_E+ln(1/q^2)$$, so I understand $$q \sim 0$$ mostly contributes to the summation but do not figure out how come $$q \sim 2 k_F$$ also contributes to the summation.

I tried to find any other literature dealing with this issue but in vain. Could anyone please help me understand the abovementioned statement?

In the Tomonaga-Luttinger Model we are considering an effective low energy model for a 1D Fermi system. The low energy excitations of a (non interacting) Fermi gas in any dimension take a particle from just below the Fermi surface to just above the Fermi surface. In 1D the Fermi surface consists of two points (the Fermi points) at $$\pm k_F$$. So there are really two qualitatively different low energy excitations in 1D: Ones that excite a Fermion from just below one Fermi point to just above it ($$q\sim 0$$) and ones that excite a Fermion from just below one Fermi point to just above the other Fermi point ($$q\sim \pm 2k_F$$). This is very different from higher dimensions where we can move continuously around the Fermi surface.