This chapter examines different problems that apply the field theory in condensed matter physics. It considers bosonic as well as fermionic systems and applies the techniques of the quantum field theory. The chapter provides a model describing electrons in a metal known as the electron gas model that considers a system of electrons interacting with each other via the instantaneous Coulomb force e2r. The electron gas model provides a framework from which introduces standard many-body approaches such as the Gell-Mann-Bruckner theory or random phase approximation and show how they are formulated within diagrammatic or functional integral techniques. The chapter also considers a quantum three-dimensional coulomb gas and apply the field-theoretic techniques to systems of interacting electrons. The terminology of the order parameter for the field ϕ is to indicate the treatment of the Coulomb interaction within a mean-field theory with the inclusion of Gaussian fluctuations around the mean-field value of the order parameter.
This chapter examines different problems that apply the field theory in condensed matter physics. It considers bosonic as well as fermionic systems and applies the techniques of the quantum field theory. The chapter provides a model describing electrons in a metal known as the electron gas model that considers a system of electrons interacting with each other via the instantaneous Coulomb force e2r. The electron gas model provides a framework from which introduces standard many-body approaches such as the Gell-Mann-Bruckner theory or random phase approximation and show how they are formulated within diagrammatic or functional integral techniques. The chapter also considers a quantum three-dimensional coulomb gas and apply the field-theoretic techniques to systems of interacting electrons. The terminology of the order parameter for the field ϕ is to indicate the treatment of the Coulomb interaction within a mean-field theory with the inclusion of Gaussian fluctuations around the mean-field value of the order parameter.
The methods that have been used to deal with a many-particle system can be basically sorted into three types: Hamiltonian, field theory and phenomenological method. The first two methods are more popular. Traditionally, the Hamiltonian method has been widely adopted in the conventional electronic theory for metals, alloys and semiconductors. Basically, the mean-field approximation (MFA) that has been working well for a weakly coupled system like a metal is employed to simplify a Hamiltonian corresponding to a particular electron system. However, for a strongly coupled many-particle system like a cuprate superconductor MFA should in principle not apply. Therefore, the field theory on the basis of Green’s function and the Feynman diagrams must be invoked. In this method, one is however more familiar with the random phase approximation (RPA) that gives rise to the same results as MFA because of being short of the information for higher-order terms of interaction. For a strongly coupled electron system, it is obvious that one has to deal with higher-order terms of a pair interaction to get a correct solution. Any ignorance of the higher-order terms implies that the more sophisticated information contained in those terms is discarded. However, to date one has not reached a consensus on how to deal with the higher-order terms beyond RPA. We preset here a method that is termed the diagrammatic iteration approach (DIA) and able to derive higher-order terms of the interaction from the information of lower-order ones on the basis of Feynman diagram, with which one is able to go beyond RPA step by step. It is in principle possible that all of higher-order terms can be obtained, and then sorted to groups of diagrams. It turns out that each of the groups can be replaced by an equivalent one, forming a diagrammatic Dyson-equation-like relation. The diagrammatic solution is eventually “translated” to a four-dimensional integral equation. The method can be applied to a layered 2D system that is a model system of cuprate superconductors and others such as atomic, nuclear, heavy-fermion systems, etc.
A general field-theoretical description of many-fermion systems, both with and without quenched disorder, is developed. Starting from the basic Grassmannian action for interacting fermions, we first bosonize the theory by introducing composite matrix variables that correspond to two-fermion excitations and integrating out the fermion degrees of freedom. The saddle-point solution of the resulting matrix field theory reproduces a disordered Hartree-Fock approximation, and an expansion to Gaussian order about the saddle point corresponds to a disordered random-phase approximation (RPA)-like theory. In the clean limit they reduce to the ordinary Hartree-Fock and random-phase approximations. We first concentrate on disordered systems, and perform a symmetry analysis that allows for a systematic separation of the massless modes from the massive ones. By treating the massive modes in a simple approximation, one obtains a technically satisfactory derivation of the generalized nonlinear $\ensuremath{\sigma}$ model that has been used in the theory of metal-insulator transitions. The theory also allows for the treatment of other phase transitions in the disordered Fermi liquid. We further use renormalization group techniques to establish the existence of a disordered Fermi-liquid fixed point, and show that it is stable for all dimensions d>2. The so-called weak-localization effects can be understood as corrections to scaling near this fixed point. The general theory also allows for studying the clean limit. For that case we develop a loop expansion that corresponds to an expansion in powers of the screened Coulomb interaction, and that represents a systematic improvement over RPA. We corroborate the existence of a Fermi-liquid fixed point that is stable for all dimensions d>1, in agreement with other recent field-theoretical treatments of clean Fermi liquids.
Abstract We present a general approach to justify the random phase approximation for the homogeneous Fermi gas in three dimensions in the mean-field scaling regime. We consider a system of N fermions on a torus, interacting via a two-body repulsive potential proportional to N−31 . In the limit $N\rightarrow \infty $ , we derive the exact leading order of the correlation energy and the bosonic elementary excitations of the system, which are consistent with the prediction of the random phase approximation in the physics literature.
Think globally, act locally. Summary Dynamical mean-field theory is designed to treat systems with local effective interactions that are strong compared with the independent-particle terms that lead to delocalized band-like states. Interactions are taken into account by a many-body calculation for an auxiliary system, a site embedded in a dynamical mean field, that is chosen to best represent the coupling to the rest of the crystal. The methods are constructed to be exact in three limits: interacting electrons on isolated sites, a lattice with no interactions, and the limit of infinite dimensions d → ∞ where mean-field theory is exact. This chapter is devoted to the general formulation, the single-site approximation where the calculation of the self-energy is mapped onto a self-consistent quantum impurity problem, and instructive examples for the Hubbard model on a d → ∞ Bethe lattice. Further developments and applications are the topics of Chs. 17–21. One of the most rewarding features of condensed matter theory is the ability to address difficult problems from different points of view. The preceding Chs. 9–15 present an approach based on perturbation expansions in the Coulomb interaction. In particular, the GW approximation for the self-energy has proven to be extremely successful in describing electronic spectra of many materials, as described in Ch. 13. The methods can be applied to the ordered states of materials with d and f states, for example, ferromagnetic Ni and anti-ferromagnetic NiO, as described in Secs. 13.4 and 20.7. However, the GW and related approximations have difficulties in treating cases with degenerate or nearly degenerate states and low-energy excitations. Present-day methods do not describe phenomena like the fluctuations of local moments in a ferromagnetic material above the Curie temperature or the insulating character of NiO in the paramagnetic phase; more generally, they have difficulty describing strong correlation. The topic of this chapter and Chs. 17–21 is dynamical mean-field theory, which is also a Green's function method in which the key quantity is the self-energy. However, it is designed to treat strong interactions for electrons in localized atomic-like states, such as the d and f states in transition metals, lanthanide and actinide elements and compounds.
It is a characteristic of wisdom not to do desperate things. Henry David Thoreau Summary The essence of a mean-field method is to replace an interacting many-body problem with a set of independent-particle problems having an effective potential. It can be chosen either as an approximation for effects of interactions in an average sense or as an auxiliary system that can reproduce selected properties of an interacting system. The effective potential can have an explicit dependence on an order parameter for a system with a broken symmetry such as a ferro- or antiferromagnet. Mean-field techniques are vital parts of many-body theory: the starting points for practical many-body calculations and often the basis for interpreting the results. This chapter provides a summary of the Hartree–Fock approximation, the Weiss mean field, and density functional theory that have significant roles in the methods described in this book. Mean-field methods denote approaches in which the interacting many-body system is treated as a set of non-interacting particles in a self-consistent field that takes into account some effects of interactions in some way. In the literature such methods are often called “one-electron”; however, in this book we use “non-interacting” or “independent-particle” to refer to mean-field concepts and approaches. We reserve the terms “one-electron” or “one-body” to denote quantities that involve quantum mechanical operators acting independently on each body in a many-body system. Mean-field approaches are relevant for the study of interacting, correlated electrons because they lead to approximate formulations that can be solved more easily than more sophisticated approaches; when judiciously chosen, mean-field solutions can yield useful, physically meaningful results, and they can provide the basis and conceptual structure for investigating the effects of correlation. The particles that are the “bodies” in a many-body theory can be the original particles with their bare masses and interactions, or, most often, they may be the solutions of a set of mean-field equations chosen to facilitate the solution of the many-body problem. A large part of many-body theory in condensed matter involves the choice of the most appropriate independent particles. Hence, it is essential to define clearly the particles that are created and annihilated by the operators and in which the many-body theory is formulated.
Previously, we have seen how the field integral method can be deployed to formulate perturbative approximation schemes to explore weakly interacting theories. In this chapter, we will learn how elements of the perturbative approach can be formulated more efficiently by staying firmly within the framework of the field integral. More importantly, in doing so, we will see how the field integral provides a method for identifying and exploring non-trivial reference ground states – “mean-fields.” A fusion of perturbative and mean-field methods will provide us with analytical machinery powerful enough to address a spectrum of rich applications ranging from superfluidity and superconductivity to metallic magnetism and the interacting electron gas. As mentioned in Chapter 5, the perturbative machinery is but one part of a larger framework. In fact, the diagrammatic series already contained hints indicating that a straightforward expansion of a theory in the interaction operator might not always be an optimal strategy: all previous examples that contained a large parameter “ N ” – and usually it is only problems of this type that are amenable to controlled analytical analysis – shared the property that the diagrammatic analysis bore structures of “higher complexity.” (For example, series of polarization operators appeared rather than series of the elementary Green functions, etc.) This phenomenon suggests that large- N problems should qualify for a more efficient and, indeed, a more physical formulation. While these remarks appear to be largely methodological, the rationale behind searching for an improved theoretical formulation is, in fact, much deeper.