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**."

**Question**:

(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. Thanks. :)

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

This post imported from StackExchange Physics at 2014-06-15 18:05 (UCT), posted by SE-user Idear