Instantons in the Schwinger model

The known calculations of the fermion condensate $〈\overline{\psi }\psi 〉$ and the correlator $〈\overline{\psi }\psi \mathrm{}\left(x\right)\mathrm{} \overline{\psi }\psi \mathrm{}\left(0\right)\mathrm{}〉$ have been interpreted in terms of localized instanton solutions presenting the minima of path integrals with quantum corrections being taken into account. Their size is of the order of the massive photon Compton wavelength ${\mu }^{-1\mathrm{}}$. At high temperature, these instantons become quasistatic and present the two-dimensional analog of the "walls" found recently in four-dimensional gauge theories. In spite of the static nature of these solutions, they should not be interpreted as "thermal solitons" living in Minkowski space: the mass of these would-be solitons does not display itself in the physical correlators. At small but nonzero fermion mass, the high-$T$ partition function of two-dimensional QED is saturated by the rarified gas of instantons and antiinstantons with density $\propto m \exp \left\{{-S}^{\mathrm{inst}}\right\}=m \exp \left\{\frac{-\pi T}{\mu }\right\}$ to be confronted with the dense strongly correlated instanton-antiinstanton liquid saturating the partition function at $T=0\mathrm{}$.