The approach to obtain a self-energy formula that takes into account both longitudinal and transverse fluctuations is detailed in [20]. Crossing symmetry, rotational symmetry and sum-rules and comparisons with QMC dictate the final formula for the improved self-energy 2(2) as we have explained in Sect. 13.2.3.

13.2.6 Pseudogap in the Renormalized Classical Regime

When we compared TPSC with QMC simulations and with FLEX in Fig. 13.4 above, perhaps you noticed that at the Fermi surface, the frequency dependent spectral weight has two peaks instead of one. In addition, at zero frequency, it has a minimum instead of a maximum. That is called a pseudogap. A cartoon explanation [29] of this pseudogap is given in Fig. 13.5. At high temperature we start from a Fermi liquid, as illustrated in panel I. Now, suppose the ground state has long-range antiferromagnetic order as in panel III, in other words at a filling between half-filling and nc. In the mean-field approximation we have a gap and the Bogoliubov transformation from fermion creation-annihilation operators to quasiparticles has weight at both positive and negative energies. In two dimensions, because of the Mermin-Wagner theorem, as soon as we raise the temperature above zero, longrange order disappears, but the antiferromagnetic correlation length £ remains large so we obtain the pseudogap illustrated in panel II. As we will explain analytically below, the pseudogap survives as long as £ is much larger than the thermal de Broglie wavelength £th = vf/(^T) in our usual units. At the crossover temperature TX, the relative size of £ and £th changes and we recover the Fermi liquid.

Fig. 13.5 Cartoon explanation of the pseudogap due to precursors of long-range order. When the antiferromagnetic correlation length £ becomes larger than the thermal de Broglie wavelength, there appears precursors of the T = 0 Bogoliubov quasiparticles for the long-range ordered antiferromagnet. This can occur only in the renormalized classical regime, below the dashed line on the left of the figure

Fig. 13.5 Cartoon explanation of the pseudogap due to precursors of long-range order. When the antiferromagnetic correlation length £ becomes larger than the thermal de Broglie wavelength, there appears precursors of the T = 0 Bogoliubov quasiparticles for the long-range ordered antiferromagnet. This can occur only in the renormalized classical regime, below the dashed line on the left of the figure

We now proceed to sketch analytically where these results come from starting from finite T. Details and more complete formulae may be found in [6,7,41,61].14 We begin from the TPSC expression (13.26) for the self-energy. Normally one has to do the sum over bosonic Matsubara frequencies first, but the zero Matsubara frequency contribution has the correct asymptotic behavior in fermionic frequencies i!n so that, as in Sect. 13.2.2, one can once more isolate on the right-hand side the contribution from the zero Matsubara frequency. In the renormalized classical regime then, we have

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