For 1+1D systems at long wavelength, it is known that the (Tomonaga)-Luttinger Liquid can be rewritten in terms of bosonic parameters, where the Hamiltonian densities can be written \begin{align*} H =& \left[\psi_L i\partial_x \psi_L - \psi_R i \partial_x \psi_R + 2\pi g_2 \rho_R \rho_L + \pi g_4 (\rho_R^2 + \rho_L^2) \right]\\ \rightarrow H_\text{boson} =& \frac{v}{2}\left[\frac{1}{g}\Pi^2 + g(\partial_x \phi)^2\right] \end{align*} where $\Pi = \partial_x \theta$ is the momentum conjugate to $\phi$.

I have been confused about the relationship of $g$ to the parameters $g_2, g_4$, where either \begin{align*} g = \sqrt{\frac{1 + g_4 \pm g_2}{1+ g_4 \mp g_2}} \end{align*} In particular, for repulsive interactions (i.e. $g_2,g_4 > 0$), the choice of sign determines whether or not $g>1$ or $g<1$. For instance, in the seminal work by Kane and Fisher, they write that $g>1$ for attractive interactions, while in the lecture notes by Senechal and lecture notes by Fradkin, they write the opposite.

Which one is correct, or am I just missing something silly here? Perhaps a more physical question is if there can be an unambiguous physical meaning assigned to g?


Unfortunately the literature on Luttinger liquid theory and bosonization suffers from a lack of consistent notation. This is just one of many inconsistencies you will find between different authors. Generally any single author will (hopefully) be self consistent, but comparing between different authors can require quite some care and effort. The situation is so bad that I know of at least 2 authors (Giamarchi's textbook Quantum Physics in One Dimension and this by von Delft and Schoeller) which have sections explaining how to convert their notation into the notation found in a number of other notable works. (Not the same notable works naturally).

So to answer your question both are correct and you just need to pick a convention and stick to in in your own work, whilst being very careful when comparing to others.

In terms of the physical interpretation, $g$ can be directly related to the compressibility of the system. Obviously I cannot say the exact nature of this relationship without going through exactly how $g$ and the bosonic fields have been defined, but it should be either directly proportional or inversely proportional (notice that your 2 sign conventions for $g$ amount to swapping $g$ with $\frac{1}{g}$)


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