In this chapter we present a real-space approach to density functional calculations. Real-space calculations [28, 134, 207, 291, 4, 202, 118, 39, 116, 76, 123, 325, 122, 117, 326, 361, 124, 223, 257, 244, 125, 138, 245] are being rapidly developed as alternatives to plane wave calculations. In this chapter we will use a real-space grid with a finite difference representation for the kinetic energy operator. The advantage of real-space grid calculations is their simplicity and versatility (e.g., there are no matrix elements to be calculated and the boundary conditions are more easy imposed). As with plane wave basis sets, the accuracy can be improved easily and systematically. In fact, there exists a rigorous cutoff for the plane waves, which can be represented in a given grid without aliasing, that provides a convenient connection between the two schemes. Pseudopotentials, developed in the plane wave context, can be applied equally well in grid-based methods, resulting in an accurate and efficient evaluation of the electron–ion potential.

Unlike in the case of plane waves, the evaluation of the kinetic energy using finite differences is approximate, but it can be significantly improved by using high-order representations of the Laplacian operator. However, an important difference between finite difference schemes and basis set approaches is the lack of a Rayleigh–Ritz variational principle in the finite difference case.

Introduction

Two-dimensional few-electron systems have been the focus of extensive theoretical and experimental investigation. Recent advances in nanofabrication techniques have enabled experiments with 2D quantum dots having highly controlled parameters such as electron number, size, shape, confinement strength, and magnetic field. The possibility of fabricating these “artificial atoms” with tunable properties is a fascinating new development in nanotechnology. The principal motivations for these investigations are the variety of possible applications in quantum computing [298], spintronics [261], information storage [199], and nanoelectronics [15, 159, 315].

Theoretical calculations of quantum dot systems are based on the effective mass approximation [42, 131, 130, 350, 205, 101, 233, 356, 184, 139, 35, 127]. In these models the electrons move in an external confining potential and interact via the Coulomb interaction. The apparent similarity of “natural” atoms and quantum dots have motivated the application of sophisticated theoretical methods borrowed from atomic physics and quantum chemistry to calculate the properties of quantum dots. Parabolically confined 2D quantum dots have been studied by several different well-established methods: exact diagonalization techniques [131, 205], Hartree–Fock approximations [101, 233, 356], and density functional approaches [184, 139]. Quantum Monte Carlo (QMC) techniques have also been used for 2D [35, 127, 255, 85] as well as 3D structures. The strongly correlated low-electronicdensity regime has received much attention owing to the intriguing possibility of the formation of Wigner molecules [356, 85].

This chapter turns to the O(N) quantum rotor studied earlier in Chapter 6. We extend the earlier results to T > 0 aided by an exact solution obtained in the N → ∞ limit.

The quantum Ising model studied in Chapter 10 had a discrete Z2 symmetry. An important new ingredient in the rotor models is the presence of a continuous symmetry: the physics is invariant under a uniform, global O(N) transformation on the orientation of the rotors, which is broken in the magnetically ordered state. Thus we have to use ideas on the spin stiffness which were introduced in Chapter 8. Apart from this, much of the technology and the physical ideas introduced earlier for the d = 1 Ising chain generalize straightforwardly, although we are no longer able to obtain exact results for crossover functions at finite N. The characterization of the physics in terms of three regions separated by smooth crossovers, the high-T and the two low-T regions on either side of the quantum critical point, continues to be extremely useful and is again the basis of our discussion. Because we consider models in spatial dimensions d > 1, it is possible to have a thermodynamic phase transition at a nonzero temperature, as in Fig. 1.2b. We are particularly interested in the interplay between the critical singularities of the finite-temperature transition and those of the quantum critical point.

Research on quantum phase transitions has undergone a vast expansion since the publication of the first edition, over a decade ago. Many new theoretical ideas have emerged, and the arena of experimental systems has grown rapidly. The cuprates have been firmly established to be d-wave superconductors, with a massless Dirac spectrum for their electronic excitations; the latter spectrum has also been observed in graphene and on the surface of topological insulators. Such fermions play a key role in a variety of quantum phase transitions. The observation of quantum oscillations in the presence of strong magnetic fields in the underdoped cuprates has highlighted the relevance of competing orders, and their quantum critical points. Optical lattices of ultracold atoms now offer a realization of the boson Hubbard model, and exhibit the superfluid–insulator transition. And ideas on quantum criticality and entanglement have had an interesting interplay with developments in quantum information science.

The second edition does not present a fully comprehensive survey of these ongoing developments. I believe the core topics of the first edition had a certain coherence, and they continue to be central to the more modern developments; I did not wish to dilute the global perspective they offer in understanding both condensed matter and ultracold atom experiments. However, wherever possible, I have discussed important advances, or directed the reader to review articles.

The large-N limit of quantum rotor models for d = 2 was examined in Chapter 11 and led to the phase diagram shown in Fig. 11.2. There we claimed that the large-N results provided a satisfactory description of the crossovers in the static and thermodynamic observables for N ≥ 3. We establish this claim in this chapter and also treat the dynamic correlations of n at nonzero temperatures. The discussion of the dynamics takes place in a physical framework suggested by the modified version of Fig. 11.2 shown in Fig. 13.1. The low-T region on the quantum paramagnetic side can be described in an effective model of quasi-classical particles that is closely related to those developed in Sections 10.4.2 and 12.2. In the other low-T region on the magnetically ordered side, we obtain a “dual” model of quasi-classical waves, which is connected to that developed in Section 12.3. Finally, in the intermediate “quantum critical” or “continuum high-T” region, neither of these descriptions is adequate: quantum and thermal behavior, as well as particle- and wavelike behavior, all play important roles, and we use a melange of these concepts to obtain a complete picture in this and the following two chapters.

The past decade has seen a substantial rejuvenation of interest in the study of quantum phase transitions, driven by experiments on cuprate superconductors, heavy fermion materials, organic conductors, and related compounds. Although quantum phase transitions in simple spin systems, like the Ising model in a transverse field, were studied in the early 1970s, much of the subsequent theoretical work examined a particular example: the metal–insulator transition. While this is a subject of considerable experimental importance, the greatest theoretical progress was made for the case of the Anderson transition of non-interacting electrons, which is driven by localization of the electronic states in the presence of a random potential. The critical properties of this transition of noninteracting electrons constituted the primary basis upon which most condensed matter physicists have formed their intuition on the behavior of the systems near a quantum phase transition. However, it is clear that strong electronic interactions play a crucial role in the systems of current interest noted earlier, and simple paradigms for the behavior of such systems near quantum critical points are not widely known.

It is the purpose of this book to move interactions to center stage by describing and classifying the physical properties of the simplest interacting systems undergoing a quantum phase transition. The effects of disorder will be neglected for the most part but will be considered in the concluding chapters. Our focus will be on the dynamical properties of such systems at nonzero temperature, and it will become apparent that these differ substantially from the noninteracting case.

We have so far described our quantum phases and critical points in terms of the wave-functions and energies of the eigenstates of the Hamiltonian. However, as we saw in our treatment of D-dimensional classical statistical mechanics in Chapters 3 and 4, a more subtle and complete characterization is obtained by considering correlation functions of various observable operators. These correlation functions are also amenable to a Feynman graph expansion and the renormalization group transformation, which was crucial in our full treatment of the classical critical point. This chapter considers correlation functions of the d-dimensional quantum model, and applies them to obtain an improved understanding of the quantum phases and the quantum critical point.

Section 5.5.3 has already presented a detailed description of the connection between the correlation functions of the D = 1 classical Ising chain and the single-site (i.e. d = 0) quantum Ising model. This mapping is immediately extended to the general D case, following the reasoning in Sections 5.6 and 6.5. From this we obtain the fundamental result that the two-point correlation function, C, of φα in (3.39) of the D-dimensional classical field theory (2.11) is precisely the same as the time-ordered correlation function of the operator φα under the Hamiltonian ℋ in (6.52).