Renormalization Group for anisotropic "Gaussian" model

Question

I'm considering an "anisotropic" Hamiltonian of the form $$\beta H = \int d^n r_{||} d^{d-n} r_{\bot} \frac{K}{2} (\nabla_{||} m)^2 + \frac{L}{2} (\nabla^2_\bot m)^2 + \frac{t}{2}m^2 - hm$$

which in momentum space reads $$\beta H = \frac{1}{(2\pi)^d} \int_{|q_{||}|<\Lambda_{||}} d^n q_{||} \int_{|q_\bot|<\Lambda_\bot} d^{d-n} q_\bot \left[ \frac{K}{2}q_{||}^2 + \frac{L}{2} q_\bot^4 + \frac{t}{2}\right] |m(q_{||},q_\bot|^2$$

First, are my integration limits correct? I wonder, because if I'd imagine that the cutoffs for the different directions were the same, then shouldn't I be able to write the two integrals as an integral over all momenta $|q| < \Lambda$ for some $\Lambda$?

Second, when I do a renormalization group treatment, the idea is to integrate out the higher momenta and I'm not sure how to split the field in order to do so. I can imagine that I will get four integrals: One where both $q_{||}$ and $q_{\bot}$ are small, one where both are large, and two where one is small and the other large. Which of these can I ignore as an additive constant and which do I keep? Only the small-small one, or do the small-large ones also affect the renormalization?

Answer 1

These are all free field models, and the scaling is determined by dimensional analysis. This case is more interesting, because the dimensional analysis involves two different types of dimensions, with different scaling behavior, like for nonrelativistic models, where time and space scale independently.

You can make one of the coefficients, say K, equal to 1 by rescaling the field. Then when you rescale the longitudinal r by b, the field $m^2$ gets a factor of $b^{2-n}$ by dimensional analysis (so as to keep K fixed).

Then when you change the scale of longitudinal r, the coefficient L changes by $b^{2-n}$, because of the $m^2$, but if you compensate by scaling the transverse r by $b^q$, you get a factor $ b^{q(d-n-4)}$. So if you choose $q = - {2-n\over d-n-4}$ the L term is fixed. This is the scaling law relating the transverse and longitudinal coordinates, determined by holding the kinetic term fixed. The q is called a "dynamic exponent".

Then the dimensional analysis exponent of t determines how it changes under rescaling. The renromalization group flow is just the first order expansion of the scaling law you get by dimensional analysis.

Specific questions

You can't make the cutoffs for the two directions equal, because the two directions scale differently, in order to keep L fixed. The shape of the integration domain is not particularly important, but you want two different cutoffs because the scalings are independent.

There is no need to say you are integrating out the modes at high k, because they do not interact with the modes at low k, so there is no meaning to integrating out. All the regions only act to add to the total energy function. If there were actual interaction terms, then you want to think of b as $1+\epsilon$ where $\epsilon$ is infinitesimal, so that the regions which are small-large are less infinitesimal than small-small, which can be neglected.