I am reading this paper(pdf) and on page 11, the chiral boson theory on a cylinder is studied when both edges of the cylinder are brought in close proximity so that electron is allowed.
Why is it that for large tunneling amplitudes, $g$, the field $\phi$ can be treated 'classically' in the sense that it gets pinned to the classical minima of the cosine to obtain the ground state energy?
Could someone give a more detailed explanation?
It is simply the fact that if the $g$ is large, $$ g \cos(\hat{\phi}) \to g(1- \frac{1}{2}\hat{\phi}^2 +\dots) $$ the ground state will be trapped in one of the vacuum sectors, at the minimum of the potential energy. (Please check S Coleman's book $\it{Aspects\; of\; symmetry}$ or his Phys Rev D paper on quantum sine-Gordon eq. It may offer differ views than what I said. If so, please report to me. :-))
It is NOT the only way to treat in the $\phi$ space and counting the minimum vacuum energy, and further count (topological) ground state degeneracy, and module out the equivalence classes. You can also treat in the dual conjugate momentum of the field $\phi$ variable. In a compact 1+1D boundary, the mode expansion of $\phi$ is
$$ \hat{\Phi}_I(x) ={\hat{\phi}_{0}}_{I}+K^{-1}_{IJ} \hat{P}_{\phi_J} \frac{2\pi}{L}x+i \sum_{n\neq 0} \frac{1}{n} \hat{\alpha}_{I,n} e^{-in x \frac{2\pi}{L}} $$
In the sense of large coupling limit, from the view points of the mode expansion of $\phi$, the topological features is controlled by the nontrivial zero modes: $\phi_0$. The dual conjugate variable of $\phi_0$ variable is the nontrivial winding mode, $p_\phi$.
Please see this paper arxiv-1212.4863. You can construct the Hilbert space of either zero modes $\phi_0$, or the Hilbert space of winding mode, $p_\phi$. Both approaches should give the same consistent result, though constructing the Hilbert space of winding modes $p_\phi$ has some better advantages than $\phi_0$ in the case of modding out equivalence classes. See Appendix B of this paper arxiv-1212.4863 for some highly relevant discussions.
ps. everything listed above can be treated as operator levels, not just classical fields. i.e. $\phi \to \hat{\phi}$
