arXiv:1504.05885v1 [math-ph] 22 Apr 2015 TIME-DEPENDENT BOGOLUBOV–DE-GENNES EQUATIONS AND NON-VALIDITY OF GINZBURG–LANDAU THEORY RUPERT L. FRANK, CHRISTIAN HAINZL, BENJAMIN SCHLEIN, AND ROBERT SEIRINGER Abstract. We study the time-dependent Bogolubov–de-Gennes equations for generic translation-invariant fermionic many-body systems, and show that even slightly above the critical temperature the order parameter does not de- cay in time, in contrast to what was predicted previously in the literature on the basis of Ginzburg–Landau theory. The full non-linear structure of the equations is necessary to understand this behavior. 1. Introduction The Ginzburg–Landau (GL) model [1] is a paradigm for the phenomenological description of phase transitions in physical systems. In the static case, its relation to the microscopic BCS theory [2] was clarified by Gor’kov [3] (see also [4] and [5] for alternative approaches), and a mathematically rigorous derivation of the GL model from BCS theory was recently given in [6, 7]. The present work is concerned with the question of the validity of the time-dependent GL equation. Several heuristic derivations of this equation can be found in the literature, starting with [8, 9, 10] in the case of superconductors. These attempts were critically analyzed in [11] where it was argued that the equation can only hold in a gapless regime, however; we refer to [12] for a thorough discussion. By applying similar arguments in the study of superfluid cold gases, it was stated in [14, 13] that the non-linear time-dependent GL equation can be derived from the Bogolubov–de-Gennes (BdG) equations for temperatures T slightly above the critical temperature. The main message of our present work is that this assertion is wrong, in general. We shall consider the BdG equations for a translation-invariant system, which is initially close to a thermal equilibrium state close to the critical temperature, with non-vanishing order parameter. We then show that, in contrast to what would be expected from GL theory, the order parameter does not decay in time. Interestingly, this is a purely non-linear effect. If, instead, we consider the corresponding linear equation, then the solution indeed decays exponentially in time, on a time scale that can be calculated via the imaginary part of the corresponding resonance pole. Our claims are mathematically rigorous and are not based on any ad-hoc approx- imations; they are confirmed numerically in [15] in the case of a one-dimensional system with contact interactions. The present work can be viewed as a continuation of a recent series of studies of the mathematical properties of the BCS theory of superconductivity and su- perfluidity [16, 17, 18, 19]. It is motivated by the current interest concerning the applicability of the theory to cold gases, in particular concerning the BCS–BEC crossover [20]. In the BCS regime [21, 22], a rigorous proof of the emergence of 1