We develop a dynamical approach based on the Schwinger-Keldysh formalism to derive a field-theoretic description of disordered and interacting electron systems. Within this formalism we calculate the perturbative renormalization-group (RG) equations for interacting electrons expanded around a diffusive Fermi liquid fixed point, as obtained originally by Finkelstein (Zh. \'Eksp. Teor. Fiz. 84, 168 (1983) [Sov. Phys. JETP 57, 97 (1983)]) using replicas. The major simplifying feature of this approach, as compared to Finkelstein's, is that instead of $\stackrel{\ensuremath{\rightarrow}}{N}0$ replicas, we only need to consider
N=2 species. We compare the dynamical Schwinger-Keldysh approach and the replica methods, and we present a simple and pedagogical RG procedure to obtain Finkelstein's RG equations.
Based on a functional-integral formalism, a generalization of the self-energy-functional theory (SFT) is proposed which is applicable to systems of interacting electrons with disorder. Similar to the pure case without disorder, a variational principle is set up which gives the physical (disorder) self-energy as a stationary point of the (averaged) grand potential. Although the resulting self-energy functional turns out to be more complicated, the formal structure of the theory can be retained since the unknown part of the functional is universal. This allows one to construct nonperturbative and thermodynamically consistent approximations via searching for a stationary point on a restricted domain of the functional. The theory and the possible approximations are worked out for models with local interactions and local disorder. This results in a derivation of different mean-field approaches and various cluster extensions, including well-known concepts such as the statistical dynamical mean-field theory, the molecular coherent-potential approximation, and the dynamical cluster approximation. Due to the common formal framework provided by the SFT, one achieves a general systematization of dynamical approaches, i.e., approaches based on the spectrum of one-particle excitations. Different mean-field and different cluster schemes naturally appear in this framework and complement the existing ones. Their prospects for future applications are discussed.
A general system composed of many weakly interacting bosons and/or fermions is considered. The object is to develop a procedure for renormalizing the single-particle creation operators, so as to remove the interactions to successively higher orders of perturbation. The basic idea is that creation operators ${{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ must satisfy the Hamiltonian commutator equations $[\mathcal{H},{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}]={\ensuremath{\omega}}_{i}{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$, where
H is the Hamiltonian and ${\ensuremath{\omega}}_{i}$ are the particle energies. For the many-weakly-interacting-particle system, the zeroth-order boson and fermion creation operators satisfy these equations to zeroth order. It is shown that if the creation operators are renormalized so as to satisfy the Hamiltonian commutator equations to order
m, and also to satisfy the appropriate boson commutators and fermion anticommutators to order
m, then the problem is solved to order
m. In particular, the vectors formed by operating on the ground state with the renormalized creation operators, according to the usual boson and fermion occupation-number representation, are eigenvectors of
H and are orthonormal, all to order
m. A procedure is otained for finding the (
m+1)-order contributions to the particle creation energies, in terms of the
m-order operators. Explicit first-order calculations of these general results are provided for a system of bosons and a system of fermions, and these first-order results are shown to include similar results of Rayleigh-Schr\"odinger perturbation theory, the random-phase approximation, and the method of thermodynamic Green's functions. The problem of anharmonic lattice dynamics is studied in detail, and a method of undetermined coefficients is used to renormalize the phonon creation operators to first order. The phonon energies are calculated to second order, and this calculation shows that the interactions between renormalized phonons cannot be removed in second order. Statistical averages of the phonon energies give the energy shifts and lifetimes which have been calculated previously by various propagator techniques. In addition, the renormalized energies are used to calculate the temperature-dependent part of the Helmholtz free energy correct to second order. As a final example, electron-phonon interactions in a normal metal are studied. The electron and phonon creation operators are renormalized to first order, the particle energies are calculated to second order, and statistical averages of the particle energies recover the usual thermodynamic Green's-function results for energy shifts and lifetimes. These examples show the simplicity by which the renormalization procedure obtains a great amount of detailed information about the single-particle nature of a many-particle system.
We introduce an extension of a variationally optimized perturbation method, by combining it with renormalization group properties in a straightforward (perturbative) form. This leads to a very transparent and efficient procedure, with a clear improvement of the nonperturbative results with respect to previous similar variational approaches. This is illustrated here by deriving optimized results for the mass gap of the
O(2N) Gross-Neveu model, compared with the exactly known results for arbitrary
N. At large
N, the exact result is reproduced already at the very first order of the modified perturbation using this procedure. For arbitrary values of
N, using the original perturbative information only known at two-loop order, we obtain a controllable percent accuracy or less, for any
N value, as compared with the exactly known result for the mass gap from the thermodynamical Bethe Ansatz. The procedure is very general and can be extended straightforwardly to any renormalizable Lagrangian model, being systematically improvable provided that a knowledge of enough perturbative orders of the relevant quantities is available.
The supersymmetry technique has proven to be an efficient method for studying disordered and chaotic systems as well as random matrix models. So far, it has been used for noninteracting systems only. Here, contrary to general belief, the robust methodology of supersymmetry to handle disordered and chaotic many-body systems is shown to work in effectively zero space dimensions. Within this framework, nonperturbative results are obtained for the celebrated Sachdev-Ye-Kitaev non-Fermi-liquid theory.