Instantons and the spectral function of electrons in the half-filled Landau level

We calculate the instanton–anti-instanton action ${\mathit{S}}_{\mathit{MM}\mathit{¯}}$(τ) in the gauge theory of the half-filled Landau level. It is found that ${\mathit{S}}_{\mathit{MM}\mathit{¯}}$(τ)=(3-η)[${\mathrm{\Omega }}_0$(η)τ${]}^{1/\left(3\mathrm{-}\mathrm{\eta }\right)}$ for a class of interactions v(q)=${\mathit{V}}_0$/${\mathit{q}}^{\mathrm{\eta }}$ (0≤η<2) between electrons. This means that the instanton–anti-instanton pairs are confining so that a well-defined ‘‘charged’’ composite fermion can exist. It is also shown that ${\mathit{S}}_{\mathit{MM}\mathit{¯}}$(τ) can be used to calculate the spectral function of electrons from the microscopic theory within a semiclassical approximation. The resulting spectral function varies as ${\mathit{e}}_0^{\mathrm{-}[\mathrm{\Omega }}$(η)/ω${]}^{1/\left(2\mathrm{-}\mathrm{\eta }\right)}$ at low energies.