Abstract
In this review paper, we illustrate a possible route to obtain a reliable solution of the 2D Hubbard model and an explanation for some of the unconventional behaviours of underdoped high-$T_\text{c}$ cuprate superconductors within the framework of the composite operator method. The latter is described exhaustively in its fundamental philosophy, various ingredients and robust machinery to clarify the reasons behind its successful applications to many diverse strongly correlated systems, controversial phenomenologies and puzzling materials.
Highlights
The tale of strongly correlated electronic systems (SCES) [1,2,3,4,5,6] is deeply intertwined to that of cuprate high-Tc superconductors (HTcS) [7,8,9,10,11,12,13,14,15,16,17,18,19] because the prototypical model for the former is exactly the same as the very minimal model to describe the latter [20]: the 2D Hubbard model [21,22,23]. This very fact has enormously increased the number of studies performed in the last thirty years, that is since the discovery of HTcS, on this model and its extensions and derivatives
The composite operator method (COM) has been designed and devised and is still currently developed with the aim of providing an analytical method that would seamlessly account for the natural emergence in SCES of elementary excitations, that is quasi-particles, whose operatorial description can only be realized in terms of fields whose commutation relations are inherently non-canonical
The presence in the Hamiltonian describing the system under analysis of strongly correlated terms — terms not quadratic in the canonical fields {aλ, bμ, . . .} describing theparticles building up the system and their internal degrees of freedom λ = {λi } — immediately and naturally leads to the emergence, in the equations of motion (EM) of the canonical fields, of products of the same canonical fields which is not possible to recast as linear combinations of the canonical fields themselves
Summary
The tale of strongly correlated electronic systems (SCES) [1,2,3,4,5,6] is deeply intertwined to that of cuprate high-Tc superconductors (HTcS) [7,8,9,10,11,12,13,14,15,16,17,18,19] because the prototypical model for the former is exactly the same as the very minimal model to describe the latter [20]: the 2D Hubbard model [21,22,23]. When it is not possible to close the hierarchy of the EM, one can always choose where to stop being sure that the processes up to those defined by the CO taken into account will be correctly described in such an approximation This is only one necessary step towards a complete solution because to compute the GF G and the correlators C of such CO under the effect of the Hamiltonian H it is necessary to calculate explicitly their weights and overlaps appearing in the normalization matrix I and the connections among them appearing in the matrix m [4, 46]. J κijψ (j, ti) where κij can be any function of the two sites i and j
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