Information about the Integer Quantum Hall Transition Extracted from the Autocorrelation Function of Spectral Determinants

The autocorrelation function of spectral determinants (ASD) is used to probe the sensitivity of a system of two-dimensional disordered electrons to the system size $L$. The impurity-averaged ASD is represented by a functional integral with the action of the O(3) nonlinear sigma model. For weak magnetic fields its integrability at vanishing frequency $\omega =0\mathrm{}$ is used to prove a trivial dependence on $L$. This is shown to be a strong indication that all states are localized. For strong magnetic fields at a central energy $E_0$ with a Hall conductance ${\sigma }_{\mathrm{xy}}\left(E_0\right)=1\mathrm{}/2$ the ASD is related to a functional integral over the integrable sine-Gordon model with imaginary coupling proportional to the frequency $\omega$. It is shown that the ASD depends at the energy $E_0$ nontrivially on $L$, which is argued to be a proof for the existence of critical wave functions at $E_0$.