Exactly solvable toy model for the pseudogap state

We present an exactly solvable toy model which describes the emergence of a pseudogap in an electronic system due to a fluctuating off-diagonal order parameter. In one dimension our model reduces to the fluctuating gap model (FGM) with a gap Delta (x) that is constrained to be of the form Delta (x) = A e^{i Q x}, where A and Q are random variables. The FGM was introduced by Lee, Rice and Anderson [Phys. Rev. Lett. {\bf{31}}, 462 (1973)] to study fluctuation effects in Peierls chains. We show that their perturbative results for the average density of states are exact for our toy model if we assume a Lorentzian probability distribution for Q and ignore amplitude fluctuations. More generally, choosing the probability distributions of A and Q such that the average of Delta (x) vanishes and its covariance is < Delta (x) Delta^{*} (x^{prime}) > = Delta_s^2 exp[ {- | x - x^{\prime} | / \xi}], we study the combined effect of phase and amplitude fluctutations on the low-energy properties of Peierls chains. We explicitly calculate the average density of states, the localization length, the average single-particle Green's function, and the real part of the average conductivity. In our model phase fluctuations generate delocalized states at the Fermi energy, which give rise to a finite Drude peak in the conductivity. We also find that the interplay between phase and amplitude fluctuations leads to a weak logarithmic singulatity in the single-particle spectral function at the bare quasi-particle energies. In higher dimensions our model might be relevant to describe the pseudogap state in the underdoped cuprate superconductors.