# Curvature and edge state

If the boundary of quantum hall fluid has non-constant curvature, how will it affect the edge state which is usually described in chiral Luttinger fluid?

-

## 1 Answer

A naive guess would be that there isn't any real difference.

The theoretical logic behind my guess is that at low-energies FQH states are described by 2+1D Chern-Simons theories, which are topological gauge theories. Although the bulk does not have any local degrees of freedom, the boundary does. This is because in the presence of a boundary $\partial M$ one has to impose boundary conditions and reduce the set of gauge transformations to those that respect this BC, therefore there will be an infinite number of states which are not gauge equivalent anymore and therefore correspond to physical degrees for freedom. More formally, the boundary dynamics are described by a Wess-Zumino-Witten theory which I think is nothing but a chiral Luttinger liquid in the simplest case. Now this is a conformal field theory and only depend on the conformal class of the boundary metric, not the metric itself. 2D manifolds, like the boundary $\partial M$, are however all conformally flat and therefore the boundary dynamics are insensitive to the curvature.

This robustness against boundary curvature, impurity scattering and so on is a general feature of quantum Hall states. If you have access to Nature, see (here and here) the simulations done for photonic crystal analogs of IQHE boundary states. It is here seen that the light wave goes around any boundary defect, curvature or impurity without any reflections at all. It is quite non-intuitive that light can go around a mirror without any reflection!

-

1
It's a good comment, but I do think this is indeed too naive. We already know that edge reconstruction comes into play when the boundary potential is not that steep (see e.g. arxiv.org/abs/cond-mat/0302344 ). What this effectively does is introduce higher order, non-linear terms in the otherwise linear CFT of the boundary. So coupling to the boundary potential really does matter. The experiment you mention really shows that the chirality of the system is not affected by these defects. You can't really make the same strong claim about the Hamiltonian though. – Olaf Oct 15 '12 at 10:09
@Olaf: Thanks for the very nice comment. Would that then mean that (assuming smooth but curved boundary) my argument is only correct if the length scales on which the boundary potential changes is very long? Since the arguments relies on Chern-Simons theory, which is only valid at very long wave-lengths? (I am not sure what "edge reconstruction" means here). And very good point, robustness of the chirality of the edge modes does not imply the detailed dynamics are robust! This is of course true. – Heidar Oct 15 '12 at 14:40
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9426307678222656, "perplexity_flag": "head"}