Spin quantum Hall transition on random networks: exact critical exponents via quantum gravity

Abstract

We solve the problem of the spin quantum Hall transition on random networks using a mapping to classical percolation that focuses on the boundary of percolating clusters. Using tools of two-dimensional quantum gravity, we compute critical exponents that characterize this transition and confirm that these are related to the exponents for the regular (square) network through the KPZ relation. Our results demonstrate the relevance of the geometric randomness of the networks and support conclusions of numerical simulations of random networks for the integer quantum Hall transition.

Figures (8)

• Figure 1: A random network and its Manhattan lattice
• Figure 2: Loop configurations on flat and random networks.
• Figure 3: A colored piece on the boundary of a clockwise oriented random ML (left) and the equivalent boundary with convention mentioned in the text (right).
• Figure 4: Two ways of splitting a surface.
• Figure 5: Examples of a clockwise oriented surface split in two ways by removing the white face adjacent to the marked black boundary side. (a) The removal of the white face immediately gives two separate surfaces. This splitting contributes to $W_l^{(1)}$ in Fig. \ref{['fig:alak']}. (b) The removal of the white face follows cutting along the exposed inner loop (in blue). The rightmost surface gives another example of our boundary drawing convention, where we maximally reduce boundary sides of colored polygons.