We present a general formalism that allows for the computation of large-order renormalized expansions, effectively doubling the numerically attainable perturbation order of renormalized Feynman diagrams. We show that this formulation compares advantageously to the currently standard techniques due to its high efficiency, simplicity, and broad range of applicability. Our formalism permits to easily complement perturbation theory with non-perturbative information, which we illustrate by implementing expansions renormalized by the addition of a gap or the inclusion of Dynamical Mean-Field Theory. As a result, we present numerically exact results for the square-lattice hole-doped Fermi-Hubbard model in the low-temperature non-Fermi-liquid regime, relevant to study the pseudogap of cuprate superconductors, and show the momentum-dependent suppression of fermionic excitations in the antinodal region.
We present a general formalism that allows for the computation of large-order renormalized expansions, effectively doubling the numerically attainable perturbation order of renormalized Feynman diagrams. We show that this formulation compares advantageously to the currently standard techniques due to its high efficiency, simplicity, and broad range of applicability. Our formalism permits to easily complement perturbation theory with non-perturbative information, which we illustrate by implementing expansions renormalized by the addition of a gap or the inclusion of Dynamical Mean-Field Theory. As a result, we present numerically exact results for the square-lattice hole-doped Fermi-Hubbard model in the low-temperature non-Fermi-liquid regime, relevant to study the pseudogap of cuprate superconductors, and show the momentum-dependent suppression of fermionic excitations in the antinodal region.
We present a technique that enables the evaluation of perturbative expansions based on one-loop-renormalized vertices up to large expansion orders. Specifically, we show how to compute large-order corrections to the random phase approximation in either the particle-hole or particle-particle channels. The algorithm's efficiency is achieved by the summation over contributions of all symmetrized Feynman diagram topologies using determinants, and by integrating out analytically the two-body long-range interactions in order to yield an effective zero-range interaction. Notably, the exponential scaling of the algorithm as a function of perturbation order leads to a polynomial scaling of the approximation error with computational time for a convergent series. To assess the performance of our approach, we apply it to the nonperturbative regime of the square-lattice fermionic Hubbard model away from half-filling and report, as compared to the bare interaction expansion algorithm, significant improvements of the Monte Carlo variance as well as the convergence properties of the resulting perturbative series.
We give a Hamiltonian-based interpretation of microscopic Fermi liquid theory within a renormalization group framework. The Fermi liquid fixed-point Hamiltonian with its leading-order corrections is identified and we show that the mean field calculations for this model correspond to the Landau phenomenological approach. This is illustrated first of all for the Kondo and Anderson models of magnetic impurities which display Fermi liquid behaviour at low temperatures. We then show how these results can be deduced by a reorganization of perturbation theory, in close parallel to that for the renormalized φ4 field theory. The Fermi liquid results follow from the two lowest order diagrams of the renormalized perturbation expansion. The calculations for the impurity models are simpler than for the general case because the self-energy depends on frequency only. We show, however, that a similar renormalized expansion can be derived also for the case of a translationally invariant system. The parameters specifying the Fermi liquid fixed-point Hamiltonian are related to the renormalized vertices appearing in the perturbation theory. The collective zero sound modes appear in the quasiparticle-quasihole ladder sum of the renormalized perturbation expansion. The renormalized perturbation expansion can in principle be used beyond the Fermi liquid regime to higher temperatures. This approach should be particularly useful for heavy fermions and other strongly correlated electron systems, where the renormalization of the single-particle excitations are particularly large. We briefly look at the breakdown of Fermi liquid theory from a renormalized perturbation theory point of view. We show how a modified version of the approach can be used in some situations, such as the spinless Luttinger model, where Fermi liquid theory is not applicable. Other examples of systems where the Fermi liquid theory breaks down are also briefly discussed.
Divergencies appearing in perturbation expansions of interacting many-body systems can often be removed by expanding around a suitably chosen renormalized (instead of the noninteracting) Hamiltonian. We describe such a renormalized perturbation expansion for interacting Fermi systems, which treats Fermi surface shifts and superconductivity with an arbitrary gap function via additive counterterms. The expansion is formulated explicitly for the Hubbard model to second order in the interaction. Numerical solutions of the self-consistency condition determining the Fermi surface and the gap function are calculated for the two-dimensional case. For the repulsive Hubbard model close to half-filling we find a superconducting state with d-wave symmetry, as expected. For Fermi levels close to the van Hove singularity a Pomeranchuk instability leads to Fermi surfaces with a broken square lattice symmetry, whose topology can be closed or open. For the attractive Hubbard model the second-order calculation yields an s-wave superconductivity with a weakly momentum dependent gap, whose size is reduced compared to the mean-field result.
This paper derives the Feynman rules for the diagrammatic perturbation expansion of the effective action around an arbitrary solvable problem. The perturbation expansion around a Gaussian theory is well-known and composed of one-line irreducible diagrams only. For the expansions around an arbitrary, non-Gaussian problem, we show that a more general class of irreducible diagrams remains in addition to a second set of diagrams that has no analogue in the Gaussian case. The effective action is central to field theory, in particular to the study of phase transitions, symmetry breaking, effective equations of motion, and renormalization. We exemplify the method on the Ising model, where the effective action amounts to the Gibbs free energy, recovering the Thouless–Anderson–Palmer mean-field theory in a fully diagrammatic derivation. Higher order corrections follow with only minimal effort compared to existing techniques. Our results show further that the Plefka expansion and the high-temperature expansion are special cases of the general formalism presented here.
Despite its apparent simplicity, the two-dimensional Hubbard model for locally interacting fermions on a square lattice is widely considered as a promising approach for the understanding of Cooper pair formation in the quasi two-dimensional high-Tc cuprate materials. In the present work this model is investigated by means of the functional renormalization group, based on an exact flow equation for the effective average action. In addition to the fermionic degrees of freedom of the Hubbard Hamiltonian, bosonic fields are introduced which correspond to the different possible collective orders of the system, for example magnetism and superconductivity. The interactions between bosons and fermions are determined by means of the method of “rebosonization” (or “flowing bosonization”), which can be described as a continuous, scale-dependent Hubbard-Stratonovich transformation. This method allows an efficient parameterization of the momentumdependent effective two-particle interaction between fermions (four-point vertex), and it makes it possible to follow the flow of the running couplings into the regimes exhibiting spontaneous symmetry breaking, where bosonic fluctuations determine the types of order which are present on large length scales. Numerical results for the phase diagram are presented, which include the mutual influence of different, competing types of order.
In this chapter, we introduce and explain one of the fundamental tools in the study of dynamic critical phenomena, namely dynamic perturbation theory. Inevitably, large parts of Chapter 4 need to be rather technical. Complementary to the straightforward iterative method for the solution of non-linear Langevin equations presented in Section 3.2.3, we now describe the more elegant and efficient field-theoretic techniques. Yet both for the derivation of general properties of the perturbation series and for the sake of practical calculations, the response functional and the following elaborations on Feynman diagrams, cumulants, and vertex functions prove indispensable. As is the case with every efficient formalism, once one has become acquainted with the Janssen–De Dominicis response functional for the construction of dynamic field theory, and therefrom the perturbation expansion in terms of vertex functions, it serves to save a considerable amount of rather tedious work. For the purely relaxational O ( n )-symmetric models A and B, we derive the fluctuation-dissipation theorem within this formalism, and then systematically construct the perturbation series and its diagrammatic representation in terms of Feynman graphs. In order to reduce the required efforts to a minimum, the generating functionals first for the cumulants, and then, motivated by Dyson's equation for the propagator, for the one-particle irreducible vertex functions are examined. As an example, the relevant vertex functions for models A and B are explicitly evaluated to two-loop order. Furthermore, alternative formulations of the perturbation expansion are discussed, and the Feynman rules are given in both the frequency and the time domain. The results of this chapter provide the foundation for the renormalization group treatment in the subsequent Chapter 5.