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12
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84931507059
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and references therein.
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20
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84931507055
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see also Ref. 9.
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25
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84931507052
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Notice that this critical value is smaller than that required to have the exponent of the wave function renormalization larger than 1, i.e., γ>0, which means g(0)>1/2=0.7071... .
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29
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84931507053
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The statement for any number of chains is the following: If the Fermi surface in presence of t⊥ is equal to that in the absence, i.e., if the Fermi surface is simply two parallel straight lines in the transverse direction, then the asymptotic behavior of the system is equivalent to that of completely uncoupled chains. In fact if the Fermi surface does not change by switching on a transverse hopping, this implies that on this state the average value of the transverse hopping operator O⊥, i.e., [ < O_⊥> = 2t_⊥ sumk_⊥in BZ cosk_⊥ a < hatnk⊥> , ] is zero, because < hatnk⊥ >, i.e., the total number of electrons with transverse momentum k⊥, is a constant independent of k⊥. Therefore only the term of the Hamiltonian which describes the set of uncoupled 1D chains contributes to the total energy. This contribution is clearly minimized if the ground state is exactly equal to the product of the ground states of each isolated chain, which in our case simply describes a Luttinger liquid. Thus we have demonstrated the initial statement.
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31
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84931507036
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Anticipating later results, our fixed point LL2 (see Fig. refdiagramma_di_fase) corresponds to F1 in Table I of Ref. onlinecitePencS; I to the first fixed point of Table II in the same reference, and II to the second fixed point on the same table.
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32
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84927374324
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We note that both ΠI in I and ΠIII in III are such that < ci,α,uparrow daggerci,α,downarrow >=0, consistent with strong coupling see
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(1987)
Science
, vol.235
, pp. 96
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Anderson, P.W.1
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33
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84931507037
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Notice that the value U=8t at which approximately phase II occurs is in quite good agreement with that we predicted, by means of very simple arguments, in the previous section, i.e., U simeq 0.21×8π vF simeq 7.5t.
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40
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84931507038
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Notice that in the text we have used a different mapping by which the spin labels correspond to the bonding and antibonding combination of the two chains. This equivalent representation is less transparent than the other one, but is more suited for a renormalization group calculation, especially if, as we believe, the transverse hopping does move the Fermi singularities.
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