Two-dimensional conformal field theory for disordered systems at criticality

Using a Kac-Moody current algebra with $U(1/1)\times U(1/1)$ graded symmetry, we describe a class of (possibly disordered) critical points in two spatial dimensions. The critical points are labelled by the triplets $(l,m,k^{\ }_j)$, where $l$ is an odd integer, $m$ is an integer, and $k^{\ }_j$ is real. For most such critical points, we show that there are infinite hierarchies of relevant operators with negative scaling dimensions. To interpret this result, we show that the line of critical points $(1,1,k^{\ }_j>0)$ is realized by a field theory of massless Dirac fermions in the presence of $U(N)$ vector gauge-like static impurities. Along the disordered critical line $(1,1,k^{\ }_j>0)$, we find an infinite hierarchy of relevant operators with negative scaling dimensions \{\Delta^{\ }_q|q\in {\rm I}\hskip -0.08 true cm{\bf N}\}, which are related to the disorder average over the $q$-th moment of the single-particle Green function. Those relevant operators can be induced by non-Gaussian moments of the probability distribution of a mass-like static disorder.