Introduction

As in the case of quasidegenerate perturbation theory (Chapter 8), multireference coupled-cluster (MRCC) theory is designed to deal with electronic states for which a zero-order description in terms of a single Slater determinant does not provide an adequate starting point for calculating the electron correlation effects. As already discussed in Chapters 8 and 13, these situations arise primarily for certain open-shell systems that are not adequately described by a high-spin single determinant (such as transitionmetal atoms), for excited states in general and for studies of bond breaking on potential-energy surfaces; they arise usually because of the degeneracy or quasidegeneracy of the reference determinants. While single-reference coupled-cluster (SRCC) methods are very effective in treating dynamic electron correlation, the conditions discussed here involve nondynamic correlation effects that are not described well by truncated SRCC at practical levels of treatment.

As shown in Section 13.4, many open-shell and multireference states can be treated by EOM-CC methods, including a single excitation from a closed shell state to an open-shell singlet state, which normally requires two equally weighted determinants in its zero-order description. Furthermore, doubleionization and double-electron-attachment EOM-CC, as well as spin-flip CC (Krylov 2001), allow the treatment of many inherently multireference target states. These methods have the advantage of being operationally of single-reference form, since then the only choices that need to be made are of the basis set and the level of correlation treatment. Although, they require an SRCC solution for an initial state (not necessarily the ground state) to initiate the procedure, once initiated multireference target states are available by the diagonalization of an effective Hamiltonian matrix in a determinantal representation.

In this chapter, we discuss the static and dynamic properties of the spin chain which is often referred to as the Haldane–Shastry model [77, 161]. In contrast with more familiar spin chains with the nearest-neighbor exchange interaction, the Haldane–Shastry model has a particular form of the longrange exchange interaction. In spite of its peculiar shape, it has turned out that the Haldane–Shastry model is the most fundamental one-dimensional spin system. We shall start with a discussion of how the model is solved for the ground state in Sections 4.1 to 4.3, and proceed to static correlation functions in Sections 4.4 and 4.5. The excitation spectrum is interpreted by magnons in Section 4.6, and by spinons in Section 4.7. The spinon picture gives the correct degeneracy of the energy levels in Section 4.8, and leads to the derivation of thermodynamics in Section 2.4, and the dynamical correlation function in Section 4.11. Most results have been obtained in an analytic form without any approximation.

There is a close connection between the spectrum of the spin chain and that of the Sutherland model. The most remarkable fact is that the spin chain has a correspondence with two different values of the coupling parameter λ in the Sutherland model. Namely, the spectrum is mapped to the case with either λ = 2 or λ = ∞. In the latter case, particles crystallize with equal spacing, and a small oscillation from the equilibrium, as well as exchange interactions, make up the spectrum. In both cases of λ, however, the degeneracy of each level in the spin chain is not reproduced. Hence, the mapping as it stands cannot be used for thermodynamics.

A ballistic MOSFET is a hypothetical device in which the mobile carriers suffer no collisions in the channel. This may happen, in principle, when the channel length is shorter than the mean free path, the average distance carriers travel between collisions. In an ordinary MOSFET, carriers moving from the source to the drain under the influence of the applied field (Vds) collide with the silicon lattice, impurity (dopant) atoms, and surfaces. These collisions limit the velocity they can acquire from the field (Appendix 3), resulting in a reduced drain current. Under low field conditions, the effect of these collisions is lumped into a mobility factor proportional to the mean free time between successive collisions (Appendix 3). For long-channel MOSFETs, the drain current is proportional to the mobility (Section 3.1.2). For short-channel MOSFETs under high drain bias conditions, high-field scattering becomes important. This is usually modeled by velocity saturation (Section 3.2.2). In the absence of any scattering, carriers entering the channel from the source are accelerated by the applied field ballistically toward the drain. They can attain very high speeds especially in the high-field region near the drain. However, such high speeds (velocity overshoot) do not necessarily translate into large currents. Since current must be continuous from source to drain, it is bounded by the rate at which carriers are injected from the source. In a ballistic MOSFET then, the bottleneck is near the source where carriers move into the channel at relatively low velocities (before field acceleration). Current continuity is satisfied by a decreased carrier density near the drain such that the product of carrier density and velocity at the drain is the same as that at the source (see Fig. 3.31). This appendix describes the drain current model for a ballistic MOSFET published by Natori in 1994 (Natori, 1994).

This book is concerned primarily with the exact dynamical properties of one-dimensional quantum systems. As a crucial property of exactly soluble models, we assume that the interaction decays as the inverse square of the distance. The family of these models is called the inverse-square interaction (1/r2) models. In the one-dimensional continuum space, the model is often referred to as the Calogero–Sutherland model. In the one-dimensional lattice, on the other hand, the first 1/r2 models appeared as a spin model, which is now called the Haldane–Shastry model. Soon after the discovery of the Haldane–Shastry model, it was recognized that the imposition of supersymmetry allows the model to acquire the charge degrees of freedom, while keeping the exactly soluble nature. The resultant one-dimensional electron model is called the supersymmetric t–J model. Various generalizations of these models have been proposed.

Recent experimental progress in quasi-one-dimensional electron systems, especially by neutron scattering and photoemission spectroscopy, has enhanced the theoretical motivation for exploring the dynamics over a wide frequency and momentum range. The 1/r2 models are ideally suited to meet this situation, since the model allows derivation of exact dynamical information most easily and transparently. In spite of the special appearance of the 1/r2 models, the intuition thus obtained contributes greatly to understanding low-dimensional physics in general. This kind of approach to dynamics is complementary to another powerful approach using the bosonization and conformal field theory. The latter is especially suitable to asymptotics of correlation functions at long spatial and temporal distances.

In Chapter 7, the design of the individual regions and parameters of a bipolar transistor was discussed. It was noted that, during device operation, an individual device region is not isolated from and independent of the other device regions. Optimization of one device parameter often adversely affects the other device parameters. Thus, optimization of the design of a bipolar transistor is a tradeoff process. This design tradeoff should be done at the circuit and/or chip level, for the optimum design of a transistor is a function of its application and environment. In this Chapter, we will first discuss some figures of merit for evaluating a bipolar transistor for typical digital and analog circuit applications, and then discuss the tradeoffs in the design of a bipolar transistor for these applications.

When we consider the performance of a circuit, the wires connecting the transistors and elements that make up the circuit and connecting the output of the circuit to the input of another circuit must be included. The resistance and capacitance as well as the signal propagation delays associated with the interconnect wires have been discussed in Section 5.2.4 in connection with CMOS circuits. The reader is referred to that subsection for details. In this Chapter, the wire capacitance which acts as a load on a bipolar circuit is included when we consider the performance and optimization of bipolar transistors and circuits.

The spectrum of the Sutherland model for fermions with spin has a degeneracy. The orthogonal set of the basis of the Fock space of degenerate states is usually related to a Lie algebra such as su2 and su3. In our case, however, the degeneracy extends to states with different values of total spin, and the ordinary Lie algebra is insufficient. The relevant algebra is called Yangian [43, 44], and is the algebra controlling those operators satisfying the Yang–Baxter relation [118, 178]. In this chapter we describe how the Yangian naturally emerges in the many-particle states described by Jack polynomials, and constructs an orthogonal basis of degenerate states. We concentrate here on the simplest case of the gl2 Yangian, corresponding to spin 1/2 particles. More general cases will be discussed in Chapter 9.

We introduce a variant of the R-matrix in Section 8.1, and its standard form in Section 8.2. We then introduce a monodromy matrix and discuss its algebraic properties, such as the Yang–Baxter relation and the intertwining property. In Section 8.4, we consider the action of the monodromy matrix on the set of states constituting the degenerate eigenstates. We will see that an element of the monodromy matrix plays the role of a creation or annihilation operator of orthogonal eigenfunctions. This means that all eigenfunctions can be generated algebraically from the highest-weight state, for example, by successive action of an element of the monodromy matrix. Thus, the degeneracy beyond the Lie algebra is accounted for by the symmetry of the monodromy matrix, which is nothing but the Yangian.

A demonstration of this construction will be given in Section 8.5 for the two-particle and three-particle systems.

Background

There are two stages in the study of perturbation theory and related techniques (although they are mixed intimately in most derivations in the literature). The first is the formal development, carried out in terms of the total Hamiltonian and total wave function (and total zero-order wave function), without attempt to express anything in terms of one- and two-body quantities (components of Ĥ, orbitals, integrals over orbitals etc.). We can make a considerable amount of progress in this way before considering the detailed form of Ĥ. The second is the many-body development, where all expressions are obtained in terms of orbitals (one-electron states) and oneand two-electron integrals. We shall try to keep these separate for a while and begin with a consideration of formal perturbation theory.

Another aspect of the study of many-body techniques is the large variety of approaches, notations and derivations that have been used. Each different approach has contributed to the lore and the language of many-body theory, and each tends to illuminate some aspects better than the other approaches. If we want to be able to read the literature in this field, we should be familiar with several alternative formulations. Therefore, we shall occasionally derive some results in more than one way and, in particular, we shall derive the basic perturbation-theory equations and their many-body representations in several complementary ways.

Classical derivation of Rayleigh–Schrödinger perturbation theory

The perturbation Ansatz

We begin with a classical textbook derivation of formal Rayleigh–Schrödinger perturbation theory (RSPT).