Delocalization transition via supersymmetry in one dimension

We use supersymmetric (SUSY) methods to study the delocalization transition at zero energy in a one-dimensional tight-binding model of spinless fermions with particle-hole symmetric disorder. Like the McCoy-Wu random transverse-field Ising model to which it is related, the fermionic problem displays two different correlation lengths for typical and mean correlations. Using the SUSY technique, mean correlators are obtained as quantum-mechanical expectation values for a U(2$|$1,1) “superspin.” In the scaling limit, this quantum mechanics is closely related to a $0+1$-dimensional Liouville theory, allowing an interpretation of the results in terms of simple properties of the zero-energy wave functions. Our primary results are the exact two-parameter scaling functions for the mean single-particle Green’s functions. We also show how the Liouville quantum-mechanics approach can be extended to obtain the full set of multifractal scaling exponents $\tau \mathrm{}\left(q\right)\mathrm{}$, $y\left(q\right)\mathrm{}$ at criticality. A thorough understanding of the unusual features of the present theory may be useful in applying SUSY to other delocalization transitions.