but this is absolutely not the only issue with the above expression (4.26).

If CT(1)(k) is not identically zero for all values of k where !;(k) vanishes, the correlation function C(k,!) diverges as 1/^!; (k) for the same values of k in the limit ! ! 0 (i.e., at all times). Such a behavior of C(k,!) usually manifests the establishing of long-range spatial correlations in the system, but it is admissible iff the corresponding correlation function in real space C(r, t) stays always finite. Accordingly, the divergence should be integrable; this immediately excludes the possibility to have long-range order in finite systems and, at finite temperatures, in infinite systems (i.e., in the thermodynamic limit) with too low spatial dimension.

Actually, the lowest spatial dimension allowed to host long-range order will simply depend on the actual functional form of the vanishing !;(k) (e.g., if !;(k) / |k|a, then the spatial dimension has to be strictly larger than a). No restriction applies to infinite systems of any dimension at zero temperature. This is just the content of Mermin-Wagner theorem [26].

For bosonic basis too, it is necessary noticing that the correlation function C can be computed by means of (4.26) only if the basis f is closed or if we neglect the residual self-energy. Otherwise, we should use the more general expression

The use of the casual Green's function, for a bosonic basis, is strictly necessary in order to properly take into account the ergodicity issue that does not manifest in neither the advanced nor the retarded propagators, but only in the causal propagator and in the correlation function.

Obviously, we do not need to compute the residual self-energy at all (it is just identically zero) if we choose a closed basis; this is one of the main reasons why a closed basis is the best one we can choose. Accordingly, as much as a truncated basis is large enough less relevant will be the contribution of the residual self-energy to the description of the system under analysis. It is worth noticing that even if we would completely neglect the residual self-energy, the Green's function of the original interacting particles constituting the system, expressed in terms of the relevant entries of the propagator G0, will anyway feature a fully momentum and frequency dependent irreducible self-energy with a (n — 1)-polar structure, but sufficient to describe all scales of energy caught by the chosen basis.

Nevertheless, one cannot neglect the residual self-energy without being aware of the main drawback: one is actually promoting to the rank of true particle (i.e., with an infinite life-time) objects that, being still subject to all virtual processes not properly taken into account by a truncated basis, have finite life-times (i.e., they are quasiparticle) roughly inversely proportional to the largest neglected energy scale involving them (i.e., the transition described by a composite operator can or cannot be one of those necessary to construct the relevant virtual process). Clearly, this can be systematically controlled by enlarging the basis, although only on a quantitative level. Again, not-analytical energy scales can change so dramatically the properties of particles and quasiparticles to require a qualitative change of perspective also regarding the residual self-energy determination and role, as described in the Sect. 4.2.1.

As a matter of fact, it is sometimes convenient keeping the operatorial basis somewhat simpler than what is actually handleable in order to get simpler expressions for the residual currents too and effectively compute, starting from the latter, the residual self-energy. Such a procedure can lead to a description of the system under analysis featuring both some of the relevant energy scales and the decay effects inherent to the quasiparticle nature of composite operators belonging to a truncated basis.

In these years, we have been developing and testing different ways to compute residual self-energy; among others: the two-site resolvent approach [27,28] and the non-crossing approximation (NCA) [29-36]. Any of them has its pros and cons and specific problems can be better tackled by one or the other (see Sect. 4.3).

The intrinsic complexity of the operatorial algebra obeyed by composite operators (for instance, see Table 4.1) hides a noteworthy possible exploitation of the same algebra. Composite operators, whose lattice sites overlap (any composite operator may span a certain number of lattice sites), obey algebra constraints directly coming from the Pauli exclusion principle. For instance, £ and r satisfy the following exact relation: £o(i(i) = 0 that can be easily traced back to co(i)co(i) = 0.

The relevance of such algebra constraints resides, on one side, in their capability to enforce the Pauli principle and its derivatives, and, on the other side, in the not-so-trivial request that they should be obeyed at the level of thermal averages too. In fact, the related thermal averages (e.g., {^(i)^'(i)> = 0), through the well-known relation existing between correlation functions and Green's functions (fluctuaction-dissipation theorem; see Sect. 4.2.4), depend on all unknown parameters appearing in the relevant Green's function (in I, s, and E plus r(k) for bosonic basis) and, in turn, can be used to fix them:

The l.h.s of (4.28), for i and j such that elements of the basis f span on, at least, one common site, would be fixed by the algebra, which imposes contractions (e.g., £o(i)ro'(i) = 0, n(i)n(i) = n(i) + 2r\(i)r"(i)). The r.h.s. of (4.28) is given by the actual expression of the Green's function and contains all unknowns of the theory (for bosonic basis, it also contains T(i — j)).

This deceptively simple conclusion has enormous implications on both the capability to solve, either exact or approximately, strongly correlated systems, and the quality of the solution. Not only algebra constraints allow us to find a solution, but they make this solution as closer as possible to the exact one because they embody the primary cause of electronic correlations: the Pauli principle.

It is also worth noticing that the symmetries enjoined by the Hamiltonian imply the existence of constants of motion and the possibility to formulate relations among matrix elements of the relevant Green's functions known as Ward-Takahashi identities [37,38]. These identities can/should also be used to fix the unknowns and to constrain the theory.

4.2.7 Summary

Summarizing, COM framework envisages four main steps:

1. Choose a composite operator basis according to the system under analysis and all information we can gather from relevant numerical and exact solutions (see Sect. 4.2.1)

2. Compute I and m matrices and obtain e matrix and propagator G0 in terms of unknown correlators and unknown r function (see Sect. 4.2.1 and 4.2.4).

3. Choose a recipe to compute E or just neglect it (see Sect. 4.2.5).

4. Self-consistently compute the unknowns through algebra constraints and Ward-Takahashi identities (see Sect. 4.2.6).

In the last 15 years, COM has been applied to several models and materials: Hubbard [4,5,9,17,28,32,36,39-42], p-d [43-45], t-J [9,27], t-t0-U [46-48], extended Hubbard (t-U-V) [49,50], Double-exchange [22], Kondo [18], Anderson [19], Kondo-Heisenberg [51], two-orbital Hubbard [52], Kondo lattice [53], Ising [10-14,16], BEG [54], Heisenberg [55], Hubbard-Kondo [56], singlet-hole [57], cuprates [31,58-60]. A comparison with the results of numerical simulations has been systematically carried on. The interested reader may refer to the works cited in [5] and, for the last years, at the web page: http://scs.physics.unisa.it.

In the second part of this chapter, as relevant application of the formalism, we will consider the Hubbard model and we will go through the different approximation schemes illustrated in the previous sections in a systematic way. A comprehensive comparison with the results of numerical simulations and with the experimental data for high-Tc cuprates will be also reported.

4.3 Case Study: The Hubbard Model 4.3.1 The Hamiltonian

The Hubbard model reads

H = J2 (—^lij — 2dt«ij) ct(i)c(j) C U^nt(«>#(/). (4.29) ij i

The notation is the same as used in Sect. 4.1 with the following additions: t is the hopping and the energy unit; d is the dimensionality of the system; ay is the projector on the nearest-neighbor sites, whose Fourier transform, for a d-dimensional cubic lattice with lattice constant a, is a(k) = 1 cos(kna). The electronic operators c(i) and c^(/), as well as all other fermionic operators, are expressed in the spinorial notation [e.g., c^(/) = (c"(/) . A detailed and comprehensive summary of the properties of Hamiltonian (4.29) are given in Sect. 1 of [5]. We here report a study of this model by means of the COM as formulated in Sect. 4.2. In order to proceed in a pedagogical way, we will go through different stages. In Sect. 4.3.1.1, we consider a truncated fermionic basis (two-pole approximation) given by the two Hubbard operators £(/) and ^(2). In Sect. 4.3.1.2, we complement the analysis considering the related bosonic sector (charge and spin). In Sect. 4.3.2, we implement the study by considering the contribution of the residual self-energy. In Sect. 4.3.3, we enlarge the truncated basis by including higher-order composite fields (four-pole approximation). At all these stages, we will present COM results and compare them with experimental and/or simulation data. Due to the pedagogical nature of this manuscript, we will restrict the analysis to the paramagnetic state. Ordered phases (ferro and antiferromagnetic phases) have been also analyzed and the related results can be found in [5]. Finally, in Sect. 4.3.4, we will discuss the superconducting solution of the model in the d-wave channel and compare COM results with high-Tc cuprates experimental data.

4.3.1.1 Two-Pole Solution: Fermionic Sector

On the basis of the discussions reported in Sects. 4.1 and 4.2.1, we will adopt as fermionic basis f(i) = ( !$)), (430)

where £(/) and ^(2) are the Hubbard operators defined in Sect.4.1. This field satisfies the equation of motion

with n^(/) = c^(/)a^c(/) and = 10^^(2)^(2) + c(/)c^a(/)c(/). ct^ = (—1, a) and ct^ = (1, a), where a are the Pauli matrices. Hereafter, given a generic operator 0(/), we will use the notation $a(i) = Xy ay$(j,t). The current /(/) is projected on the basis (4.30) and the residual current 1J(i) is neglected. The normalization and energy matrices have the expressions r-1

0 V ( 1 " "/2 M Pitt - fmn(k)/-1 mi2(k)/- . I(k) I 0 I2J ^ 0 «/2j Ê(K) mi2(k)I-1m22(k)I-V'

where the entries of the m matrix are given by mn(k) = -MI11 - 2dt [A C a(k)(1 - n + p)] (4.33)

n = 1/N J2¡(n(i)) is the particle number per site; the parameters A and p cause a constant shift of the bands and a bandwidth renormalization, respectively, and are defined as

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