Dynamical field theory for glass-forming liquids, self-consistent resummations andtime-reversal symmetry

Abstract

We analyse the symmetries and the self-consistent perturbative approaches of dynamicalfield theories for glass-forming liquids. In particular, we focus on the time-reversalsymmetry, which is crucial to obtain fluctuation–dissipation relations (FDRs). Previousfield theoretical treatment violated this symmetry, whereas others pointed out thatconstructing symmetry-preserving perturbation theories is a crucial and open issue. In thiswork we solve this problem and then apply our results to the mode-coupling theory of theglass transition (MCT).We show that in the context of dynamical field theories for glass-forming liquidstime-reversal symmetry is expressed as a nonlinear field transformation that leaves theaction invariant. Because of this nonlinearity, standard perturbation theories generically donot preserve time-reversal symmetry and in particular fluctuation–dissipationrelations. We show how one can cure this problem and set up symmetry preservingperturbation theories by introducing some auxiliary fields. As an outcome we obtainSchwinger–Dyson dynamical equations that automatically preserve FDR and that serveas a basis for carrying out symmetry-preserving approximations. We apply ourresults to the mode-coupling theory of the glass transition, revisiting previousfield theory derivations of MCT equations and showing that they genericallyviolate FDR. We obtain symmetry-preserving mode-coupling equations and discusstheir advantages and drawbacks. Furthermore, we show, contrary to previousworks, that the structure of the dynamic equations is such that the ideal glasstransition is not cut off at any finite order of perturbation theory, even in thepresence of coupling between current and density. The opposite results found inprevious field theoretical works, such as the ones based on nonlinear fluctuatinghydrodynamics, were only due to an incorrect treatment of time-reversal symmetry.