  In a recent Letter by the authors [I.S. Burmistrov and A.M.M. Pruisken, Phys.
Rev. Lett. 101, 056801 (2008)] it was shown that single-electron devices
(single electron transistor or SET) display "macroscopic charge quantization"
which is completely analogous to the quantum Hall effect observed on very
different electron systems. In this investigation we present more detail on
these new findings. Based on the Ambegaokar-Eckern-Schoen (AES) theory of the
Coulomb blockade we introduce a general response theory that probes the
sensitivity of SET to changes in the boundary conditions. This response theory
defines a new set of physical observables and we establish the contact with the
standard results obtained from ordinary linear response theory. The response
parameters generally define the renormalization behavior of the SET in the
entire regime from weak coupling with large values of the tunneling conductance
all the way down to the strong coupling phase where the system displays the
Coulomb blockade. We introduce a general criterion for charge quantization that
is analogous to the Thouless criterion for Anderson localization. We present
the results of detailed computations on the weak coupling side of the theory,
i.e. both perturbative and non-perturbative (instantons). Based on an effective
theory in terms of quantum spins we study the quantum critical behavior of the
AES model on the strong coupling side. Consequently, a unifying scaling diagram
of the SET is obtained. This diagram displays all the super universal
topological features of the theta-angle concept that previously arose in the
theory of the quantum Hall effect.
