We address the problem of finding self-consistent parquet equations for two-body correlated Greens functions that arise out of a cumulant expansion. The general theory as developed for non-relativistic electron systems reproduces the conventional two-body parquet diagrammatic technique for the vertex functions. As an application, the crossing-invariant linearized basic parquet equations are treated for the correlated two-particle Greens function. It is pointed out that besides the Cooperon there are two additional prominent crossing-symmetric two-particle channels in many-particle systems with pair interaction.
In many-body theory, the single-particle self-energy can be constructed from the two-body vertex, the single-particle Green's function, and the interaction in a well-known way. Functional differentiation of the self-energy leads back to the two-body vertex. Given a set of two-body vertex diagrams, the process of constructing self-energy diagrams and functionally differentiating them generates a second set of two-body vertex diagrams. If the initial set of diagrams includes the noninteracting two-body Green's function, the two sets will be equal if and only if the initial set includes all diagrams. We discuss this result from the perspective of the parquet theory.
The ways of introducing and handling renormalizations in the many-body perturbation theory are reviewed.We stress the indispensable role the technique of Green functions plays in extrapolating the weak-coupling perturbative approaches to intermediate and strong couplings.We separately discuss mass and charge renormalizations.The former is incorporated in a self-consistent equation for the self-energy derived explicitly from generating Feynman diagrams within the Baym and Kadanoff approach.The latter amounts to self-consistent equations for two-particle irreducible vertices.We analyze the charge renormalization initiated by De Dominicis and Martin and demonstrate that its realization via the parquet approach may become a powerful and viable way of using the many-body diagrammatic approach reliably in non-perturbative regimes with cooperative phenomena induced by either strong interaction or strong static randomness.
Abstract The main task of microscopic many-body theory is to provide an understanding and an explanation of properties of macroscopic systems from no other information than the properties of the underlying Hamiltonian, the particle statistics, and the macroscopic geometry of the system. Variational wave functions have been a very successful technique for examining the ground state and dynamic properties of dense quantum fluids. Specifically the “optimized (Fermi)-hypernetted chain” ((F)HNC-EL) hierarchy of approximations reproduces, for simple interactions like electrons and quantum fluids, the most basic features of a self-bound many-body system, namely binding, saturation, and spinodal decomposition. The relationships between Green’s functions based perturbation theory and variational wave functions have been clarified in much detail in the past. In the language of Feynman diagrams, a self-consistent summation of ring- and ladder-diagrams is the minimum requirement for a satisfactory microscopic description of these basic features. Realistic nucleon–nucleon interactions pose further problems: The most difficult aspect of the nuclear many-body problem is caused by the form of the microscopic nucleon–nucleon interaction which depends at least on the spin, isospin, the relative orientation and angular momentum of the interacting particles. Simple variational methods as used for electrons and quantum fluids are inadequate. Correlation functions that have the same structure as the interactions lead to so-called “commutator diagrams” that have, so far, been mostly neglected. We have shown in the past that these corrections can be very important if the two-body interactions are very different in different spin- or isospin channels. To deal with the problem of realistic nuclear interactions we have combined techniques of the Jastrow–Feenberg variational method and the local parquet-diagram theory. In the language of diagrammatic perturbation theory, “commutator diagrams” can be identified with non-parquet diagrams. We examine the physical processes described by these terms and include the relevant diagrams in a way that is suggested by the Jastrow–Feenberg approach. We show that the corrections from non-parquet contributions are, at short distances, larger than all other many-body effects. We examine here neutron matter as a prototype of systems with state-dependent interactions. Calculations are carried out for neutrons interacting via the so-called
v8 v 8 version of four popular interactions. We determine the structure and effective interactions and apply the method to the calculation of the energetics, structure and dynamic properties such as the single-particle self-energy and the dynamic response functions as well as BCS pairing in both singlet and triplet states. We find that many-body correlations lead to a strong reduction of the spin-orbit interaction, and, therefore, to a suppression of the
3P2 3 P 2 and
3P2 3 P 2 -
3F2 3 F 2 gaps. We also find pairing in
3P0 3 P 0 states; the strength of the pairing gap depends sensitively on the potential model employed.
I review the way the many-body Green functions are used to renormalize the perturbation theory of correlated fermions. The Green functions are introduced to implement systematically dynamical corrections to the static mean-field theory. The renormalizations enter the perturbation theory via self-consistent evaluations of one-particle and two-particle Green functions. The one-particle self-consistency is discussed within the Baym-Kadanoff construction. We point to an inherent dichotomy of the Baym-Kadanoff approach in the relation between one and two-particle Green functions. They are the Schwinger-Dyson equation, directly derived from the generating Luttinger-Ward functional, and the Ward identity. The latter imposes a severe restriction on the two-particle irreducible vertex and is not guaranteed in the Baym-Kadanoff approach. No approximate theory is capable to obey both relations. I further discuss the two-particle approach with two-particle renormalizations that offers a way to reconcile the Schwinger-Dyson equation and the Ward identity. The two-particle irreducible vertex from the critical Bethe-Salpeter equation becomes a generating functional in the two-particle renormalization of the perturbation theory. The irreducible vertex is used in the Ward identity to determine the order parameter beyond the critical point and in the Bethe-Salpeter equation representing the vertex of the Schwinger-Dyson equation to determine the spectral self-energy in such a way that the thermodynamic critical behavior from both equations is qualitatively the same. Finally, I discuss a static mean-field-like approximation with a renormalized effective interaction determined self-consistently that leads to a qualitatively correct description of strong electron correlations in impurity and lattice models.
This thesis presents the development of new numerical methods for the treatment of strongly correlated electron systems based on self-consistent approaches at both the one and the two-particle level such as the parquet formalism. The parquet formalism was solved for the first time on a two-dimensional cluster. When the fully irreducible vertex is approximated by the bare vertex, we obtain the parquet approximation. Its validity was investigated by comparing results that it produces to those of other conserving approximations such as the FLuctuation EXchange (FLEX) approximation or the Second Order Perturbation Theory (SOPT). We found that it provides a significant improvement of FLEX or SOPT and a satisfactory agreement with Quantum Monte Carlo results despite instabilities in the self-consistency at low temperatures and for strong Coulomb interaction. We use the parquet formalism to study the Quantum Critical Point at finite doping in the Hubbard model by decomposing the vertex into its contributions from different channels. We apply this decomposition to the pairing channel and we find that the dominant contribution to the vertex originates in the spin channel even at the quantum critical doping. Furthermore, we explore the divergence of the two parts of the pairing matrix at optimal doping and observe that the irreducible vertex decreases monotonically as the doping is increased while the bare susceptibility exhibits an algebraic divergence at the quantum critical doping supporting the Quantum Critical BCS scenario proposed by She and Zaanen. To circumvent the instabilities in the iteration of the parquet formalism, we explored the dual fermion approach introduced by Rubtsov et al. Here, we extended the formalism to the Dynamical Cluster Approximation, in the process introducing a small parameter in the dual fermions perturbation theory. We demonstrate the quality of the resulting Dual Fermion DCA through a systematic study of the cluster size dependence and of the different perturbative approximations. These efforts represent the initial steps in the development of the Multi-Scale Many Body approach that appropriately treats correlations at different length scales.