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16.3 Much ado about zeros: Luttinger surfaceConsequently, the moments simplify to(3) =&+("浮以D +h.c).(16.55)The criterion for the zeros of the Green function now reduces to a condition on the parity of the right-handside of Eq. (16.55). Consider the case ofhalf-filling, particle-hole symmetry and nearest-neighbor hopping.Under these conditions,(nia}= 1/2 and by particle-hole symmetry,= U/2.The expressions fbranand ft,lays plain that the resultant coefficients+ (-1~2vanish fornodd. Consequently. Go(k. w) is an even function if the second term in Eq. (16.55) vanishes. InFourier space, the second term is proportional to the non-interacting band structuref(k).The momenta atwhich r(k) = 0 define the Fermi surface of the noninteracting system. Note that the condition /(k) = 0.which defines the surface of zeros, is independent of the anisotropy of the hopping. We conclude thatwhen particle-hole symmetry is present. G(0, k = kF) = 0 fbr a Mott insulator, where kp is the Fermi
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