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Incompatibility of Time-Dependent Bogoliubov–de-Gennes and Ginzburg–Landau Equations


We study the time-dependent Bogoliubov–de-Gennes equations for generic translation-invariant fermionic many-body systems. For initial states that are close to thermal equilibrium states at temperatures near the critical temperature, we show that the magnitude of the order parameter stays approximately constant in time and, in particular, does not follow a time-dependent Ginzburg–Landau equation, which is often employed as a phenomenological description and predicts a decay of the order parameter in time. The full non-linear structure of the equations is necessary to understand this behavior.


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Correspondence to Robert Seiringer.

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Frank, R.L., Hainzl, C., Schlein, B. et al. Incompatibility of Time-Dependent Bogoliubov–de-Gennes and Ginzburg–Landau Equations. Lett Math Phys 106, 913–923 (2016).

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Mathematics Subject Classification

  • 82D50
  • 46N50


  • superconductivity
  • quasi-free states
  • critical temperature
  • BCS theory