From Sect. 1 it is apparent that ab-initio calculations for strongly correlated electron systems are still at their beginning. Therefore, simplifying model Hamiltonians are very helpful in order to unravel physical effects caused by them. The one studied most is the Hubbard Hamiltonian 
It has one orbital associated with each site i and the Hamiltonian contains a hopping term between nearest-neighbor sites and an on-site Coulomb repulsion term U. The simplifications made by this ansatz as compared with a quantum chemical ab-initio Hamiltonian are enormous. But nevertheless, important insight can be gained by studying (1.9), in particular, when a system is at or close to half filling. Various techniques have been applied to study the Hubbard model with emphasis on two dimensions (2D). This is, because claims have been made that a 2D Hubbard model contains all of the important physics of high-Tc superconductivity.
A large role is being played by cluster approximations. They replace the infinite lattice by a finite cluster which is often embedded in an effective medium. In view of the short range of the correlation hole of an electron, that seems fine even for an infinite system, provided the correlation hole is properly constructed. In order not to limit for computational reasons the cluster to a too-small size, the method of increments can be used as a tool. It can be looked at as an expansion of a (cumulant) scattering matrix in terms of one, two, three etc. site clusters. But that is often not done and in that case translational invariance within a cluster is hij
hij missing. Established approaches within quantum cluster theory are the dynamical cluster approximation [16,17] (see Chap. 8.8), the cluster DMFT mentioned earlier  (see Chap. 10.7) and the cluster perturbation theory  (see Chap. 7.9). In the latter case a perturbation expansion in terms of the hopping matrix element t is combined with the exact diagonalization of small clusters. The dynamical cluster approximation has the advantage of ensuring translational invariance within a cluster. In that respect it differs from the cluster DMFT in which this symmetry is violated. Otherwise, both methods are similar in many aspects. An incremental cluster expansion of the self-energy or scattering matrix has been applied in a fully self-consistent projection operator approach .
One can try to find a formalism from which all the different cluster methods derive as special cases, at least as long as they are based on Feynman diagrams. The hope is that this allows for the development of improved cluster methods. Such an attempt has been made with the Self-Energy Functional Theory  (see Chap. 9.5). It starts from the generating functional 0[G, U] of the Green's function G. The self-energy is required to be given by the derivative of 0 with respect to G, i.e., E = 10[G]/1G. From the work of Baym and Kadanoff it is known that conservation laws are obeyed if 0[G, U] is calculated from all distinct, connected, and closed skeleton diagrams expressed in G and U .One may then specify to which approximations for 0 a given cluster method corresponds to.
Quantum cluster theories are one possibility to find approximate solutions for the Hubbard model. Of course, other methods have been developed too, which treat the infinite system. Thereby special attention is paid to conserving approximations, i.e., approximations which do not violate conservation laws  (see Chap. 12.6). Also the functional renormalization group has been applied to the two-dimensional Hubbard model. It works like a microscope with variable resolution. For understanding the low excitation energy sector of the Hubbard model, one eliminates all degrees of freedom of the system which are irrelevant for its behavior in that limit. For that purpose differential equations for the one-particle Green's function are derived which describe the flow of parameters as the degrees of freedom are reduced. Since it is difficult to calculate the renormalization flow for a strongly interacting system, one is starting out from a weakly interacting system in which case the energy scale is given by the kinetics. But as the energy scale decreases, the coupling function increases. This is indicative of possible instabilities, e.g., of magnetic or pair forming origin. In that case one has to switch to a modified description of the system which accounts for these changes .
A much studied example is the Mott-Hubbard metal-to-insulator phase transition at half filling when U t . But even now it is not known at which critical ratio of Uc/t this phase transition is taking place at T = 0. The value has been steadily increasing with improvements of the approximation schemes. At present the most accurate estimates are obtained when the self-energy is computed with a dynamical CPA in combination with self-consistent projection operator methods. That allows for the calculation of E(k, a>) with an accuracy, which includes effects up to the 12th nearest neighbors. Despite considerable efforts, the topic itself is still wide open. There are several reasons for this. One is that in the Hubbard model we include one orbital per site only. But in a realistic multiorbital system it is likely that criteria for localization are first fulfilled for a single orbital or for two of them (compare with the dual model of 5f electrons) before the system becomes an insulator. Also, there may be a redistribution of electrons among different orbitals as the phase transition is approached. The spectral density and in particular the low-energy peak as obtained, e.g., from a DMFT calculation behave quite different when the ground state is paramagnetic and when it has a long-range AF order. An open question is how AF correlations modify the spectral density as the correlation length increases continuously near a metal-insulator transition.
For small deviations from half filling the Hubbard Hamiltonian can be transferred into a t — J model Hamiltonian of the form
Ht-J = —t X (a+j + h.c.) + J x( Si • Sj — ^j)
where the operators a+o, aio are defined by a+r = a+0(1 — ni —a);
The spin operators are S, = (1/2) J2ap a+aoapa,p and nio = a+aaio. The coupling constant is J = 4t2/U and defines a low-energy scale caused by the strong correlations. At half filling Ht = 0 and we deal with an antiferromagnetic (AF) Heisenberg Hamiltonian. Its excitations involve only spin degrees of freedom and constitute the simplest example of spin-charge separation. The t — J model has been very successful in understanding the motion of doped holes or electrons in an antiferromagnetic surroundings. The bandwidth of a coherent hole motion is of order J and therefore strongly renormalized as compared with the bare hopping matrix element t. The energy dispersion of the coherent quasihole motion strongly resembles that found by ab initio calculations using quantum chemical methods. With the t — J model one can also show that two doped holes attract each other, a possible mechanism for high-Tc superconducting in hole-doped cuprates. One can also study the effects of small hole concentration back on the form of AF order, i.e., the development of spiral spin states .
Improved Hubbard Hamiltonians have been applied to the Cu-O planes of the high-Tc cuprates. For example, in addition to a 3dx2—y2 orbital for the Cu ions the 2p orbitals of the O ions which hybridize with the 3dx2—y2 Cu orbitals are included.This leads to three-band or five-band Hubbard Hamiltonian depending on whether one includes one or two of the 2p oxygen orbitals in the Cu-O plane. A new feature which is not contained in a one-band Hubbard description is the formation of a Zhang-Rice singlet state, when a hole is doped in an otherwise half-filled system. It is mainly located on the oxygen sites and the singlet is formed with the hole on the Cu2+ ion .
Model Hamiltonians have also been widely used in order to study the microscopic origins of different systems with heavy quasiparticle excitations. The Anderson lattice Hamiltonian has played a prominent role. It is of the form
There are conduction electrons with band index n which weakly hybridize with strongly correlated electrons. They are created (destroyed) by f+, f operators with an orbital index m and are positioned at lattice sites i. Various approximation schemes have been applied to that Hamiltonian. They are of diagrammatic nature like the noncrossing approximation [26,27], which describes the Anderson lattice as a system of independent impurities. Or they introduce auxiliary fields like slave boson fields [28-30] which allow for mean-field treatments. Also the so-called Gutzwiller approximation has been applied [31-33]. It renormalizes the hybridization matrix element and moves the orbital energy close to the Fermi energy.
The common feature of those approximations is to reproduce the low-energy scales, which strongly correlated electrons generate.
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