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.
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.
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.
In this chapter, we introduce the analytical machinery to investigate the properties of many-body systems perturbatively. Specifically, employing the “ø 4 -theory” as an example, we learn how to describe systems that are not too far from a known reference state by perturbative means. Diagrammatic methods are introduced as a tool to efficiently implement perturbation theory at large orders. The new concepts are then applied to the analysis of various properties of the weakly interacting electron gas. In previous chapters we have emphasized repeatedly that the majority of many-particle problems cannot be solved in closed form. Therefore, in general, one is compelled to think about approximation strategies. One promising ansatz leans on the fact that, when approaching the low-temperature physics of a many-particle system, we often have some idea, however vague, of its preferred states and/or its low-energy excitations. One may then set out to explore the system by using these prospective ground state configurations as a working platform. For example, one might expand the Hamiltonian in the vicinity of the reference state and check that, indeed, the residual “perturbations” acting in the low-energy sector of the Hilbert space are weak and can be dealt with by some kind of approximate expansion. Consider, for example, the quantum Heisenberg magnet. In dimensions higher than one, an exact solution of this system is out of the question. However, we know (or, more conservatively, “expect”) that, at zero temperature, the spins will be frozen into configurations aligned along some (domain-wise) constant magnetisation axes.
In this chapter, we introduce the analytical machinery to investigate the properties of many-body systems perturbatively. Specifically, employing the “φ 4 -theory” as an example, we learn how to describe systems that are not too far from a known reference state by perturbative means. Diagrammatic methods are introduced as a tool to efficiently implement perturbation theory at large orders. The new concepts are then applied to the analysis of various properties of the weakly interacting electron gas. In previous chapters we have emphasized repeatedly that the majority of many-particle problems cannot be solved in closed form. Therefore, in general, one is compelled to think about approximation strategies. One promising ansatz leans on the fact that, when approaching the low-temperature physics of a many-particle system, we often have some idea, however vague, of its preferred states and/or its low-energy excitations. One may then set out to explore the system by using these prospective ground state configurations as a working platform. For example, one might expand the Hamiltonian in the vicinity of the reference state and check that, indeed, the residual “perturbations” acting in the low-energy sector of the Hilbert space are weak and can be dealt with by some kind of approximate expansion. Consider, for example, the quantum Heisenberg magnet. In dimensions higher than one, an exact solution of this system is out of the question. However, we know (or, more conservatively, “expect”) that, at zero temperature, the spins will be frozen into configurations aligned along some (domain-wise) constant magnetisation axes.
In the previous chapters DMFT has been developed as a methodology for calculation of dynamical and thermodynamic properties of many-body systems, with applications to model systems. In order for DMFT to be a general method for quantitative calculations, there must be systematic procedures to derive all the ingredients from first principles. In this chapter we consider the most prominent examples, materials with partially filled 3 d , 4 f , or 5 f localized, atomic-like states, and set up the characteristic form of the hamiltonian. This is the basis for DFT+DMFT methods discussed here and used in the examples in Ch. 20, and for the combination of DMFT with many-body perturbation theory in Ch. 21. Simpler techniques, including self-interaction correction, DFT+U, and DFT+Gutzwiller, are also useful and are summarized in the last sections of this chapter. Up to this point, dynamical mean-field theory has been developed for model systems with Hubbard-type hamiltonians, where the hopping matrix elements and interactions are considered as parameters. For problems with strong interactions, DMFT leads to renormalized bands, satellites in the one-particle spectra, magnetic phase transitions and local moments, metal–insulator transitions, and many other phenomena observed in real materials, as brought out in Ch. 2. The purpose of this chapter is to provide the framework for a quantitative theory of these materials. These are difficult problems, experimentally, theoretically, and computationally, and the first step is to identify the aspects of the materials that must be treated explicitly as the local strongly interacting system, and the “rest” that are treated by a different method. This division of the problem is useful because a real material involves many degrees of freedom, i.e., the electronic states must be described in a basis sufficiently large to describe all the relevant states. However, the part that can be treated by non-perturbative solvers (Ch. 18) is limited by the computational cost that scales rapidly with the number of states included in the embedded site or cluster. Thus one must choose a small number of localized orbitals that inevitably are non-unique. This may seem like a step backward for quantitative methods; however, it opens a window for methods to treat these interesting problems and it presents a challenge to overcome the limitations.
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.
This chapter addresses a range of problems for which functional integrals provide unique insights and approximations. It shows how to generate a variety of alternative functional integrals for the evolution operator which gives rise to different physical approximations in the stationary-phase approximation. The chapter aims to perform the general saddle point approximation around a static mean field solution and derive an expansion in terms of the fundamental vibrational excitations of the system. It also shows how time-dependent mean field solutions may be used to calculate transition amplitudes between specific states, approximate quantum eigenstates of large amplitude collective motion, study barrier penetration and spontaneous fission. The chapter evaluates the asymptotic behavior of large orders of perturbation theory and explores the freedom available in formulating functional integrals for the evolution operator and how to exploit this freedom in making physical approximations.