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This expression for (Formula presented) Eq. (2), may also be obtained by canonical transformation to the attractive Hubbard model of the corresponding result (Formula presented) obtained in the transverse spin channel for the repulsive model
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The renormalized-classical contribution to the self-energy equation (8) is the one that is most sensitive to the large correlation length (Formula presented). In two dimensions, we know that on an infinite-size lattice one can approximate this (Formula presented) contribution to (Formula presented) by (Formula presented) because q is constrained by (Formula presented) to be of order (Formula presented) and (Formula presented) (i.e., (Formula presented)). On a finite lattice, the same (Formula presented) dependence for all (Formula presented) will also follow when (Formula presented) i.e., when (Formula presented). While the prefactor, (Formula presented) could in principle be different on infinite and finite lattices, the fact that (Formula presented) depends little on system size ensures that (Formula presented) is basically system-size independent since finite (Formula presented) contributions to the sum rule equation (7) also are size independent
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The renormalized-classical contribution to the self-energy equation (8) is the one that is most sensitive to the large correlation length (Formula presented). In two dimensions, we know that on an infinite-size lattice one can approximate this (Formula presented) contribution to (Formula presented) by (Formula presented) because q is constrained by (Formula presented) to be of order (Formula presented) and (Formula presented) (i.e., (Formula presented)). On a finite lattice, the same (Formula presented) dependence for all (Formula presented) will also follow when (Formula presented) i.e., when (Formula presented). While the prefactor, (Formula presented) could in principle be different on infinite and finite lattices, the fact that (Formula presented) depends little on system size ensures that (Formula presented) is basically system-size independent since finite (Formula presented) contributions to the sum rule equation (7) also are size independent.
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Even though the attractive part of the Hamiltonian could be in the weak coupling regime, if there is a strongly repulsive term also in the Hamiltonian the details of the physics near the superconducting transition could be quite different from what is found in the attractive Hubbard model. Nevertheless, the fact that wewere able to treat the model up to intermediate coupling suggests that many of our qualitative results should hold in the more general case
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Even though the attractive part of the Hamiltonian could be in the weak coupling regime, if there is a strongly repulsive term also in the Hamiltonian the details of the physics near the superconducting transition could be quite different from what is found in the attractive Hubbard model. Nevertheless, the fact that wewere able to treat the model up to intermediate coupling suggests that many of our qualitative results should hold in the more general case.
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