Asymptotic Freedom of Elastic Strings and Barriers

Details Asymptotic Freedom of Elastic Strings and Barriers Asymptotic Freedom of Elastic Strings and Barriers

2005-03-31

arxiv.org/abs/cond-mat/0503728v3

Phys.Rev. B72 (2005) 052503

Physics

Condensed Matter

Statistical Mechanics

Text slightly improved, bibilography corrected, more typos corrected, still Latex 7 pages

Scientific paper

10.1103/PhysRevB.72.052503

We study the problem of a quantized elastic string in the presence of an impenetrable wall. This is a two-dimensional field theory of an N-component real scalar field $\phi$ which becomes interacting through the restriction that the magnitude of $\phi$ is less than $\phi_{\rm max}$, for a spherical wall of radius $\phi_{\rm max}$. The N=1 case is a string vibrating in a plane between two straight walls. We review a simple nonperturbative argument that there is a gap in the spectrum, with asymptotically-free behavior in the coupling (which is the reciprocal of $\phi_{\rm max}$) for N greater than or equal to one. This scaling behavior of the mass gap has been disputed in some of the recent literature. We find, however, that perturbation theory and the 1/N expansion each confirms that these theories are asymptotically free. The large N limit coincides with that of the O(N) nonlinear sigma model. A theta parameter exists for the N=2 model, which describes a string confined to the interior of a cylinder of radius $\phi_{\rm max}$.

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