Euclidean action of Fermi-system with “hidden order”

Spontaneous break down of translational invariance of Fermi-system along Matsubara's time axis may lead to emergency of a new kind of “hidden order”. An exact solution is found that minimizes Landau–Ginzburg–Wilson (LGW) action of the fermionic system close to a usual spin density wave (SDW) instability. The solution has the form of SDW with amplitude that is analytic double-periodic function of the Matsubara and real time. This kind of order parameter behaves like a “hidden order”, since, e.g. in the limit of zero temperature, T=0 K, the state is characterized by a bound state of two antiferromagnetic SDW's propagating in the opposite directions in real space and forming “standing wave” with zero expectation value for the magnetization. There is finite energy gap between this state and lowest excited states corresponding to SDW's propagating in the opposite directions with slightly different amplitudes. Green's function of the hidden order field (HOF) is finite and periodic along Matsubara's axis, but Wick-rotated to the axis of real frequencies, it reveals periodic “chain” of second order poles. Hence, the scattering cross section of the HOF is zero in the lowest order approximation and, therefore, HOF is indeed “invisible”. Despite this, HOF induces a gap in the fermionic quasiparticle spectrum at Fermi-level, and also possesses several “visible” spin-modes with discrete frequencies above a finite gap. Relevance of the picture to high-Tc “hidden order” and pseudo-gap state is briefly discussed.