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A Van Hove singularity in the single particle DOS at the Fermi surface corresponds to a second derivative of the free energy with respect to filling which is negative, i.e., an unstable situation with a local maximum in the free energy. The way a system avoids such a situation when the Fermi level approaches a Van Hove singularity, is through reconstruction of the underlying lattice. This splits a 2D logarithmic singularity into two singularities, such that the Fermi level is located at a local minimum in the DOS in between the new singularities. Note that the fact that high-(Formula presented) superconductors self-dope to optimum filling clearly negates the notion that optimum doping corresponds to an unstable situation with a local maximum in the free energy.
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A Van Hove singularity in the single particle DOS at the Fermi surface corresponds to a second derivative of the free energy with respect to filling which is negative, i.e., an unstable situation with a local maximum in the free energy. The way a system avoids such a situation when the Fermi level approaches a Van Hove singularity, is through reconstruction of the underlying lattice. This splits a 2D logarithmic singularity into two singularities, such that the Fermi level is located at a local minimum in the DOS in between the new singularities. Note that the fact that high-(Formula presented) superconductors self-dope to optimum filling clearly negates the notion that optimum doping corresponds to an unstable situation with a local maximum in the free energy.
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On the other hand, it may be worth emphasizing that the physical origin of in-plane pairing is logically independent from the ILPT mechanism, as earlier recognized by Anderson (see, e.g., Ref. 20
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It is understood that normal Hartree terms, i.e., of the form (Formula presented) can be absorbed in the renormalization of the chemical potential, which is here treated as a free parameter.
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Note that it is the presence of a generally nonzero (Formula presented) in the pair susceptibility (Formula presented) which prevents the quantity (Formula presented) from being actually zero for (Formula presented)
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