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Assume that we have an Hamiltonian $$H=H_0+H_{int}$$ where $H_0$ has gappless excitations and $H_{int}$ has usual two body interactions.

If $H_0$ were gapped I would able to get low energy effective hamiltonian,

1.by using Hubbard stranatovich transformation to get rid of quintic terms,

2.I would integrate out the fermions

3.solve euler lagnrange equations to find classical solutions

4.expand the action around those let's say up to second order

So the this second order expansion would be my effective low energy action.

however, this procedure relies on the fact that $H_0$ is gapped it wont work if it is gapless I don't understand why? I assume that if $H_0$ is gappless i.e. degenerate than the step 2 can not be done because of the gaussian integral rules, is that the reason?

On the other hand let assume that now the system is bosonic so in that case the procedure is as follows;

  1. since the degrees of freedom are bosons, they have classical limit so we can just solve the euler lagrange equations without making hs transformation and without integratin gout anything.

  2. then we change variables in ginzburg landau case you can use vortex anzats

  3. you expand bosonic fields around classical solutions let's say up to second order and you get you effective action.

so I assume maybe in fermionic gapless case you can't integrate our fermions that's why you can't have the effective action.

however I don't understand why I can't have effective low energy action for gapless bosonic case, because the procedure does not involve integrating out.

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For the notion of effective theory to make sense you need a separation of scales. This is so that experiments at low energies (with respect to that scale) do not excite the heavy modes too much. In other words, you need to have a notion of "low energies", which is equivalent to a separation of scales or a gap.

Formally you could integrate out massless excitations, but you won't get anything useful as a result.

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