We see that there are two contributions. The first term is due to the explicit B dependence in the SFT grand potential while the second is due to the implicit B dependence via the B dependence of the stationary point. Equation (10.63) also demonstrates (see [31]) that for the calculation of the paramagnetic susceptibility X one may first consider spin-independent variational parameters only to find a stationary point. This strongly reduces the computational effort [32]. Once a stationary point is found, partial derivatives according to (10.63) have to be calculated with spin-dependent parameters X in a single final step.

Spontaneous symmetry breaking is obtained at B = 0 if there is a stationary point with Bopt ^ 0. Figure 10.7 gives an example for the particle-hole symmetric Hubbard model on the square lattice at half-filling and zero temperature. As a reference system a cluster with Lc = 10 sites is considered, and the ficticious staggered

Fig. 10.7 (Taken from [33].) SFT grand potential as a function of the strength of a ficticious staggered magnetic field B'. VCA calculation using disconnected clusters consisting of Lc = 10 sites each for the two-dimensional Hubbard model on the square lattice at half-filling, zero temperature, U = 8 and nearest-neighbour hopping t = 1

Fig. 10.7 (Taken from [33].) SFT grand potential as a function of the strength of a ficticious staggered magnetic field B'. VCA calculation using disconnected clusters consisting of Lc = 10 sites each for the two-dimensional Hubbard model on the square lattice at half-filling, zero temperature, U = 8 and nearest-neighbour hopping t = 1

magnetic field is taken as the only variational parameter. There is a stationary point at B' = 0, which corresponds to the paramagnetic phase. At B' = 0 the usual CPT is recovered. The two equivalent stationary points at finite B' correspond to a phase with spontaneous antiferromagnetic order - as expected for the Hubbard model in this parameter regime. The antiferromagnetic ground state is stable as compared to the paramagnetic phase. Its order parameter m is the conjugate variable to the ficticious field. Since the latter is a variational parameter, m can either be calculated by integration of the spin-dependent spectral density or as the derivative of the SFT grand potential with respect to the physical field strength B with the same result. More details are given in [33].

The possibility to study spontaneous symmetry breaking using the VCA with suitably chosen Weiss fields as variational parameters has been exploited frequently in the past. Besides antiferromagnetism [33-36], spiral phases [37], ferromagnetism [32], d-wave superconductivity [38-44], charge order [45,46] and orbital order [47] have been investigated. The fact that an explicit expression for a thermodynamical potential is available allows us to study discontinuous transitions and phase separation as well.

Continuous symmetries of a Hamiltonian imply the existence of conserved quantities: The conservation of total energy, momentum, angular momentum, spin and particle number is enforced by a not explicitly time-dependent Hamiltonian which is spatially homogeneous and isotropic and invariant under global SU(2) and U(1) gauge transformations. Approximations may artificially break symmetries and thus lead to unphysical violations of conservation laws. Baym and Kadanoff [48, 49] have analyzed under which circumstances an approximation respects the mentioned macroscopic conservation laws. Within diagrammatic perturbation theory it could be shown that approximations that derive from an explicit but approximate expression for the LW functional 0 (0-derivable approximations) are "conserving." Examples for conserving approximations are the Hartree-Fock or the fluctuation-exchange approximation [48,50].

The SFT provides a framework to construct 0-derivable approximations for correlated lattice models which are non-perturbative, i.e., do not employ truncations of the skeleton-diagram expansion. Like in weak-coupling conserving approximations, approximations within the SFT are derived from the LW functional, or its Legendre transform FU [Z]. These are 0-derivable since any type-III approximation can also be seen as a type-II one, see Sect. 10.2.6.

For fermionic lattice models, conservation of energy, particle number and spin have to be considered. Besides the static thermodynamics, the SFT concentrates on the one-particle excitations. For the approximate one-particle Green's function, however, it is actually simple to prove directly that the above conservation laws are respected. A short discussion is given in [51].

At zero temperature T = 0 there is another non-trivial theorem which is satisfied by any 0-derivable approximation, namely Luttinger's sum rule [23,52]. This states that at zero temperature the volume in reciprocal space that is enclosed by the Fermi surface is equal to the average particle number. The original proof of the sum rule by Luttinger and Ward [23] is based on the skeleton-diagram expansion of 0 in the exact theory and is straightforwardly transferred to the case of a 0 -derivable approximation. This also implies that other Fermi-liquid properties, such as the linear trend of the specific heat at low T and Fermi-liquid expressions for the T = 0 charge and the spin susceptibility are respected by a 0-derivable approximation.

For approximations constructed within the SFT, a different proof has to be found. One can start with (10.39) and perform the zero-temperature limit for an original system (and thus for a reference system) of a finite size L. The different terms in the SFT grand potential then consist of finite sums. The calculation proceeds by taking the ^-derivative, for T = 0, on both sides of (10.39). This yields the following result (see [51] for details):

(N) = (N)' C 2 X ®(G*(0)) - 2 X ®(G¿(0)). (10.64)

Here, (N) ((N)') is the ground-state expectation value of the total particle number N in the original (reference) system, and Gk(0) (Gk(0)) is the diagonal elements of the one-electron Green's function G at! = 0. As Luttinger's sum rule reads

k this implies that, within an approximation constructed within the SFT, the sum rule is satisfied if and only if it is satisfied for the reference system, i.e., if (N)0 = 2 k ® (Gk (0)). This demonstrates that the theorem is "propagated" to the original system irrespective of the approximation that is constructed within the SFT. This propagation also works in the opposite direction. Namely, a possible violation of the exact sum rule for the reference system would imply a violation of the sum rule, expressed in terms of approximate quantities, for the original system.

There are no problems to take the thermodynamic limit L ! i (if desired) on both sides of (10.64). The k sums turn into integrals over the unit cell of the reciprocal lattice. For a D-dimensional lattice the D — 1-dimensional manifold of k points with Gk (0) = i or Gk (0) = 0 form Fermi or Luttinger surfaces, respectively. Translational symmetry of the original as well as the reference system may be assumed but is not necessary. In the absence of translational symmetry, however, one has to re-interprete the wave vector k as an index which refers to the elements of the diagonalized Green's function matrix G. The exact sum rule generalizes accordingly but can no longer be expressed in terms of a Fermi surface since there is no reciprocal space. It is also valid for the case of a translationally symmetric original Hamiltonian where, due to the choice of a reference system with reduced translational symmetries, such as employed in the VCA, the symmetries of the (approximate) Green's function of the original system are (artificially) reduced. Since with (10.64) the proof of the sum rule is actually shifted to the proof of the sum rule for the reference system only, we are faced with the interesting problem of the validity of the sum rule for a finite cluster. For small Hubbard clusters with non-degenerate ground state this has been checked numerically with the surprising result that violations of the sum rule appear in certain parameter regimes close to half-filling, see [51]. This leaves us with the question where the proof of the sum rule fails if applied to a system of finite size. This is an open problem that has been stated and discussed in [51,53] and that is probably related to the breakdown of the sum rule for Mott insulators [54].

10.5 Bath Degrees of Freedom

10.5.1 Motivation and Dynamical Impurity Approximation

Within SFT, an approximation is specified by the choice of the reference system. The reference system must share the same interaction part with the original model and should be amenable to a (numerically) exact solution. These requirements restrict the number of conceivable approximations. So far we have considered a decoupling of degrees of freedom by partitioning a Hubbard-type lattice model into finite Hubbard clusters of Lc sites each, which results in the VCA.

Another element in constructing approximation is to add degrees of freedom. Since the interaction part has to be kept unchanged, the only possibility to do that consists in adding new uncorrelated sites (or "orbitals"), i.e., sites where the Hubbard U vanishes. These are called "bath sites." The coupling of bath sites to the correlated sites with finite U in the reference system via a one-particle term in the Hamiltonian is called "hybridization."

Figure 10.8 shows different possibilities. Reference system A yields a trial self-energy which is local Sijaiw) = lijSiw) and has the same pole structure as the self-energy of the atomic limit of the Hubbard model. This results in a variant of the Hubbard-I approximation [2]. Reference systems B and C generate VCA. In reference system D an additional bath site is added to the finite cluster. Reference system E generates a local self-energy again but, as compared to A, allows to treat more variational parameters, namely the on-site energies of the correlated and of the bath site and the hybridization between them. We call the resulting approximation a "dynamical impurity approximation" (DIA) with Lb = 1.

The DIA is a mean-field approximation since the self-energy is local which indicates that non-local two-particle correlations, e.g., spin-spin correlations, do not feed back to the one-particle Green function. It is, however, quite different from static mean-field (Hartree-Fock) theory since even on the Lb = 1 level it includes retardation effects that result from processes / V2 where the electron b c

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