Instanton in sine-Gordon equation

Question

This is a statement from Giamarchi's book on Quantum Physics in 1D:

"For a single-particle in a cosine potential, the slightest amount of tunneling between two cosine minima leads to conduction bands, for example, and restores the translational symmetry. However, our sine-Gordon problem is a two-dimensional (one space one time) problem. In that case it is well-known that instantons with a finite action (instanton) that would connect two cosine minima cannot exist (Rajaraman, 1982). There is thus no restoring of symmetry and the field is truly locked in one of the minima. This is of course related to the Mermin-Wagner theorem stating that in two (classical) dimensions it is impossible to break a continuous symmetry but one can break a discrete one."

The potential concerns us here of a scalar field $\Phi$ is: $$ g \cos(\beta \;\Phi) $$

I wonder how to show this statement:

"It is well-known that instantons with a finite action (instanton) that would connect two cosine minima cannot exist (Rajaraman, 1982) ... the field is truly locked in one of the minima."

Questions:

1. Are "there" any criteria or conditions when the field will be locked in one of the minima? Such as $g>g_c$ and certain values of $\beta$?

2. how to show this statement? How is the instanton analysis done here?

PS. I read Rajaraman book and S Coleman on instantons. So please do not post an answer for recommending just the Refs.

NEW Edit NOTE: I suppose we are talking about this kind of 1+1D bosonic action: $$ \frac{1}{4\pi} \int_{ \mathcal{M}^2} dt \; dx \; k\, \partial_t \Phi \partial_x \Phi - v \,\partial_x \Phi \partial_x \Phi + g \cos(\beta_{}^{} \cdot\Phi_{}) $$

Ref:

1. Rajaraman 1982, Solitons and Instantons, Volume 15: An Introduction to Solitons and Instantons in Quantum Field Theory (North-Holland Personal Library)

2. Thierry Giamarchi, Quantum Physics in One Dimension

3. S Coleman, Aspects of Symmetry

Answer 1

1) NO

2) The usual way of doing it is to first solve the instanton solution in euclidean time, which is equivalent to obtaining soliton solution of a given potential. Since you have read the book, I am not going to explain how it is done for this case.

Then, plugin your instanton solution to the euclidean action and evaluate it. Since

$$<q_f|e^{-iHt/\hbar}|q_i> = \int[dq]e^{-S_E/\hbar} $$

or equation 10.13 of Rajaraman 1982, the tunneling amplitude is proportional to $e^{-S_E/\hbar}$ and the instanton solution is the dominating contribution, evaluating $S_E$ of the instanton solution already give you a lot of information.

In your case, you will find that $S_E$ is infinite since the integration on spatial direction diverges. However, if you have a system with finite spatial dimension, $0<x<L$, the action will be finite thus tunneling can happen.

In chapter 10.1 Rajaraman 1982, the last few paragraphs argued that, instanton configuration in (1+1) dimension is equivalent to soliton in (2+1) dimension, which has infinite energy (stated by the virial theorem in 3.2), thus forbid the existence of such solution.

However, with the existence of gauge fields, these infinite energy configuration will become finite energy solution due to some amazing cancellation coming from the configuration of the gauge field. For example, the vortex solution of the Mexican hat potential has infinite energy if the scalar fields exist alone but the vortex solution become finite in energy with the companion of the $U(1)$ gauge field.