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arXiv:1809.10303 · Condensed Matter Physics

The composite operator method route to the 2D Hubbard model and the cuprates

Di Ciolo, Adolfo Avella

Published Sep 1, 2018• 5 citations
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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 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.

DOI
10.5488/cmp.21.33701

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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 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.
The microscopical analysis of the unconventional and puzzling physics of the underdoped cuprates, as carried out lately by means of the composite operator method (COM) applied to the 2D Hubbard model, is reviewed and systematized. The 2D Hubbard model has been adopted as it has been considered the minimal model capable of describing the most peculiar features of cuprates held responsible for their anomalous behavior. COM is designed to endorse, since its foundation, the systematic emergence in any SCS of new elementary excitations described by composite operators obeying noncanonical algebras. In this case (underdoped cuprates—2D Hubbard model), the residual interactions—beyond a 2-pole approximation—between the new elementary electronic excitations, dictated by the strong local Coulomb repulsion and well described by the two Hubbard composite operators, have been treated within the noncrossing approximation. Given this recipe and exploiting the few unknowns to enforce the Pauli principle content in the solution, it is possible to qualitatively describe some of the anomalous features of high-Tc cuprate superconductors such as large versus small Fermi surface dichotomy, Fermi surface deconstruction (appearance of Fermi arcs), nodal versus antinodal physics, pseudogap(s), and kinks in the electronic dispersion. The resulting scenario envisages a smooth crossover between an ordinary weakly interacting metal sustaining weak, short-range antiferromagnetic correlations in the overdoped regime to an unconventional poor metal characterized by very strong, long-but-finite-range antiferromagnetic correlations leading to momentum-selective non-Fermi liquid features as well as to the opening of a pseudogap and to the striking differences between the nodal and the antinodal dynamics in the underdoped regime.
The two-dimensional Hubbard model is studied within the composite operator method (COM) with the self-energy computed in the self-consistent Born approximation (SCBA). The COM describes interacting electrons in terms of the new elementary excitations that appear in the system owing to strong correlations; residual interactions among these excitations are treated within the SCBA. On decreasing the doping (from the overdoped to underdoped region), anomalous features develop in the spectral function $A(\mathbf{k},\ensuremath{\omega})$, the Fermi surface, the momentum distribution function n(k)n(\mathbf{k})n(k), the dispersion, and the density of states $N(\ensuremath{\omega})$ in the intermediate-coupling regime (U=8)(U=8)(U=8) at low temperatures (T=0.01−−0.02)(T=0.01--0.02)(T=0.01−−0.02). At high doping (n=0.7)(n=0.7)(n=0.7), the system resembles an ordinary weakly interacting metal. At low doping (n=0.92)(n=0.92)(n=0.92), a pseudogap opens, hot and cold spots appear, and non-Fermi-liquid features develop. This behavior, together with the presence of kinks in the calculated electronic dispersion, is in agreement with angle-resolved photoemission spectroscopy data for high-Tc{T}_{c}Tc​ cuprates superconductors.
The two-dimensional Hubbard model is studied within the composite operator method (COM) with the electronic self-energy computed in the self-consistent Born approximation (SCBA). The main idea of the COM is to describe interacting electrons in terms of the composite elementary excitations appearing in the system owing to strong correlations; the residual interactions among these excitations are treated within the SCBA. By analysing the spectral function A(k,ω) on varying the filling, we find, at high doping, the ordinary Fermi-liquid behaviour of a weakly interacting metal and, at low doping, the opening of a pseudogap, as reported for cuprates superconductors.
The discovery of new materials with properties dominated by strong correlations among electrons has opened the problem of new appropriate calculation schemes.In the case of 1D models the Bethe ansatz provides an exact evaluation of many relevant physical quantities, but not a complete framework.By way of developing new methods appropriate to strongly correlated systems, we study the 1D Hubbard model by means of the Composite Operator Method.We investigate various local quantities.A comparison with exact results and other analytical approaches show a reasonable agreement and determine the applicability range of our approximate scheme.
Since the discovery of high Tc superconductors in cuprate oxides, physics of highly correlated electron systems has become one of the central issues in condensed matter physics. We can find many examples of highly correlated particle systems not only in electron systems, such as high Tc superconductors and transition metal oxides, but also in bosonic systems, such as alkaline atomic gases in an optical lattice and mesoscopic Josephson junction arrays. In this lecture, we illustrate several models of highly correlated particle systems of fermionic and bosonic systems, and clarify common features of these systems. Then theoretical approaches to analyze these systems will be reviewed; special emphasis is on the composite operator method, which has been developed by the present author and the Salerno group.
Strongly correlated electron systems with planar structure, like high{Tc superconductors, are currently under intensive study.In the search for a realistic model to describe the anomalous behaviors of these systems, we have studied the 2D Extended Hubbard model by means of the Composite Operator Method.The local properties, the density of states and the Fermi surface have been calculated as functions of the model parameters.Particular attention has been paid to the role played by the intersite interaction V which drives important changes in the electronic properties of the system.