Logarithmic effects on the critical behavior of superfluids in random media

The effects of logarithmic corrections on the critical behavior of systems near a phase transition with marginally irrelevant quenched disorder are examined, specifically in the context of superfluid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ in porous media. It is argued that these corrections are likely to change the critical behavior of the specific heat to an inverse logarithmic cusp, and alter, to a new universal constant, the magnitude of the amplitude ratio relating the critical specific heat to the superfluid density. An ɛ expansion is carried out along the line on which the randomness is marginal. The results also have implications for the behavior of dirty high-temperature superconductors.