Renormalized Perturbation Theory

In chapter 5 where the Fermi liquid theory for magnetic impurity models was developed we considered two approaches. One was the phenomenological approach, based on the work of Landau and Nozières, where we conjectured a specific form for a quasi-particle Hamiltonian with a local interaction term. This Hamiltonian was essentially equivalent to the effective Hamiltonian near the strong coupling fixed point obtained by Wilson in his numerical renormalization group calculation (section 4.5, equation (4.49)). In the later chapters, where we used this approach for the N-fold degenerate Anderson model (section 7.4) and the n-channel Kondo model for n = 2S (section 9.3), the conjectured form for the quasi-particle Hamiltonian was not backed up by any first principles renormalization group calculation. The other approach developed in chapter 5 was the microscopic Fermi liquid theory based largely on the work of Luttinger (1960, 1961) and Yamada & Yosida (1975) using a conventional perturbation expansion in powers of U. This microscopic treatment confirmed all the results based on the conjectured quasi-particle Hamiltonian. Here we develop a synthesis of the two approaches which we will refer to as ‘renormalized perturbation theory’ (Hewson, 1992). It is based on the general idea of renormalization used in quantum field theory. The results at low temperatures correspond essentially to our earlier calculations with the conjectured quasi-particle Hamiltonian. These are obtained from first and second order perturbation theory in powers of the renormalized interaction Ũ.

The Linear Dispersion s-d Model

In 1980 exact solutions for the s-d model were found independently by Andrei (1980) and Wiegmann (1980), and later also exact solutions for the Anderson model (Wiegmann, 1981; Kawakami & Okiji, 1981). This was a rather surprising development, more particularly, because the methods of solution were based on the ansatz used by Bethe as early as 1931 in his solution of the one dimensional Heisenberg model. Though a quantitative understanding of the Kondo problem was obtained in the 70s using the renormalization group and Fermi liquid approaches, the exact solutions have produced new results, analytic formulae for the behaviour in weak and strong magnetic fields, the form of the electronic specific heat over the full temperature range. The results give complete confirmation of the physical picture that emerged with the earlier work. The approach has proved to be generalizable to some of the more physically relevant models, the s-d model with spin greater than ½ the degenerate models for rare earth impurities (we will discuss these in the next chapter), and models including crystal fields, providing a greater range of predictions for comparison with experiment. The solutions have proved to be immensely valuable in testing some of the approximate methods, ones which can also be used to calculate dynamic as well as thermodynamic properties (it has not proved possible to calculate dynamics directly from the Bethe ansatz), methods which may prove useful in tackling multiple impurity and lattice problems.

Rules for Diagrams

Here we summarize the rules for drawing and evaluating the diagrams for the perturbational contributions to the resolvent Rα(z), defined in equation (7.8), for the degenerate Anderson model. The expansion is in powers of the hybridization parameter V. We give rules for the U = ∞ model and general N.

(1) Each diagram has a base line running from left to right which corresponds to the impurity state. A full line with an arrow directed from left to right and a label m indicates an occupied state f, nf = 1, m. A dashed line represents the unoccupied state nf = 0. The initialand final lines correspond to the state |α〉.

(2) The vertices are associated with the interaction (7.10), at which a single conduction electron (full line) is created or annihilated. When the arrow is directed away from the vertex the electron is created, when directed towards the vertex it is annihilated. Contributions of the order |V|2n correspond to all possible diagrams with 2n vertices consistent with the direction of the arrows.

(3) Quantum numbers k, m, are assigned to each conduction line and have an associated factor, for lines running from left to right, and a factor, for lines running in the reverse direction. The z-component of angular momentum, m, is conserved at each vertex.

The foundations of the physics of liquid crystals were laid in the 1920s but, surprisingly, interest in these substances died down almost completely during the next three decades. The situation was summarized by F. C. Frank in his opening remarks at a Discussion of the Faraday Society in 1958: ‘After the Society's successful Discussion on liquid crystals in 1933, too many people, perhaps, drew the conclusion that the major puzzles were eliminated, and too few the equally valid conclusion that quantitative experimental work on liquid crystals offers powerfully direct information about molecular interactions in condensed phases.’ In the last few years there has been a resurgence of activity in this field, owing partly to the realization that liquid crystals have important uses in display technology.

An exposition of the physics of liquid crystals involves many disciplines: continuum mechanics, optics of anisotropic media, statistical physics, crystallography etc. In covering such a wide field it is difficult to define what precisely the reader is expected to know already. An attempt is made to present as far as possible a self-contained treatment of each of these different aspects of the subject. Naturally, discussion of some topics has had to be curtailed for reasons of space. For example, we have not dealt with lyotropic systems, whose complex structures are only just beginning to be elucidated; or the special applications of magnetic resonance techniques, as these have been adequately reviewed elsewhere; or the very recent results of neutron scattering experiments.

Description of the liquid crystalline structures

Since the early investigations of Lehmann and others elucidating the relationship between liquid crystalline behaviour and chemical constitution, the accepted principle was that the molecule has to be rod-like in shape for thermotropic mesomorphism to occur, but it has emerged during the last decade that compounds composed of disc-like molecules may also exhibit stable mesophases. Some typical discotic molecules are shown in fig. 6.1.1. Generally speaking, they have flat (or nearly flat) cores with six or eight (or sometimes four) long chain substituents, commonly with ester or ether (but rarely more complex) linkage groups. Available experimental evidence indicates that the presence of these side chains is crucial to the formation of discotic liquid crystals. The mesophases whose structures are clearly identified fall into two distinct categories, the columnar and the nematic. A smectic-like (lamellar) phase has also been reported but the precise arrangement of the molecules in each layer is not yet fully understood.

The basic columnar structure is as illustrated in fig. 1.1.8(a); it is somewhat similar to the hexagonal phase of soap–water and other lyotropic systems (fig. 1.2.2). However, a number of variants of this structure have been found. Fig. 6.1.2 presents the different two-dimensional lattices of columns that have been identified; here the ellipses denote discs or, more precisely, cores that are tilted with respect to the column axis. Table 6.1.1 gives the space groups of the columnar structures formed by some derivatives of triphenylene. (These are planar space groups that constitute the subset of the 230 space groups when symmetry elements relating to translations along one of the axes, in this case the column axis, are absent.)