Nonequilibrium Theory of the Optical Stark Effect in the Excitonic Range of the Spectrum

One of the most important applications of the theory of Bose condensation of excitons is the optical Stark effect, or “AC Stark effect, ” first demonstrated for excitons in Cu2O [1] and since seen in several semiconductors and semiconductor heterostructures (e.g., Refs. 2-5). This effect is a promising tool for the field of optical communications. One of the major interests in this field is the development of all-optical-switching methods, by which light signals are switched on or off directly by other light signals, just as in electronics, electrical signals switch other electrical signals. At the present, most optical communications systems use electrical signals (e.g., electro-optic or acousto-optic devices) to switch the light signals, which means that the limiting bandwidth of the system is controlled by that of the electrical signals, not the optical bandwidth. The optical Stark effect offers this possibility. When one laser beam impinges on a sample, it can drastically alter the excitonic absorption line shape. Therefore a second beam can see either a transparent or an absorbing medium, depending on the presence of the control beam. This is the optical equivalent of a transistor.

The basic effect is shown in Fig. 6.1. When an intense laser is tuned to a photon energy that is just below the excitonic ground state, the exciton ground state is shifted in frequency and the oscillator strength is altered. This leads to a strong reduction in the optical absorption at the original wavelength of the exciton ground state.

This chapter has been co-authored with T. Senthil, and adapted from the Ph.D. thesis of T. Senthil, submitted to Yale University (1997, unpublished).

The last two chapters of this book will move beyond the study of regular Hamiltonians that have the full translational symmetry of an underlying crystalline lattice and consider the physically important case of disordered systems described by Hamiltonians with couplings that vary from point to point in space. By the standards of the regular systems we have already discussed, the quantum phase transitions of disordered systems are very poorly understood, and only a few well-established results are available. A large amount of theoretical effort has been expended toward unraveling the complicated phenomena that occur, and they remain active topics of current research. The aims of our discussion here will therefore be rather limited: We will highlight some important features that are qualitatively different from those of nondisordered systems, make general remarks about insights that can be drawn from our understanding of the finite-T crossovers in Part II, and discuss the properties of some simple solvable models.

In keeping with the general strategy of this book, we will introduce some basic concepts by studying the effects of disorder on the magnetic ordering transitions of quantum Ising/rotor models studied in Part II; we will also make some remarks in Section 15.3.1 on the effects of disorder on the ordering transitions of Fermi liquids considered in Chapter 12.

The Hubbard model was originally introduced as a description of the motion of electrons in transition metals, with the motivation of understanding their magnetic properties. This original model remains a very active subject of research today, and important progress has been made in recent years by examining its properties in the limit of large spatial dimensionality.

In this chapter, we shall only examine the much simpler “boson Hubbard model,” following the analysis in an important paper by Fisher, Weichman, Grinstein, and Fisher. As the name implies, the elementary degrees of freedom in this model are spinless bosons, which take the place of the spin-½ fermionic electrons in the original model. These bosons could represent Cooper pairs of electrons undergoing Josephson tunneling between super conducting islands or helium atoms moving on a substrate. Processes in which the Cooper pair boson decays into a pair of electrons are neglected in this simple model, and this caveat must be kept in mind while discussing experimental applications.

Many of the results discussed in this chapter were also obtained in early literature on quantum transitions in anisotropic magnets in the presence of an applied magnetic field. These are reviewed by Kaganov and Chubukov, who also gave an extensive discussion of experimental applications. We will, however, not use their formulation here.

Apart from its direct physical applications, the importance of the boson Hubbard model lies in providing one of the simplest realizations of a quantum phase transition that does not map onto a previously studied classical phase transition in one higher dimension.

We considered time-dependent correlations of the conserved angular momentum, L(x, t), of the O(3) quantum rotor model in d = 1 in Chapter 6. We found, using effective semiclassical models, that the dynamic fluctuations of L(x, t) were characterized by a diffusive form (see (6.26)) at long times and distances, and we were able to obtain values for the spin diffusion constant Ds at low T and high T (see Fig. 6.5). The purpose of this chapter is to study the analogous correlations in d = 2 for N ≥ 2; the case N = 1 has no conserved angular momentum, and so there is no possibility of diffusive spin correlations. Rather than thinking about fluctuations of the conserved angular momentum in equilibrium, we shall find it more convenient here to consider instead the response to an external space and time dependent “magnetic” field H(x, t) and to examine how the system transports the conserved angular momentum under its influence.

In principle, it is possible to address these issues in the high-T region using the nonlinear classical wave problem developed in Section 8.3 in the context of the ∈ = 3 − d expansion. However, an attempt to do this quickly shows that the correlators of L contain ultraviolet divergences when evaluated in the effective classical theory.

We will take the low-T Fermi liquid state of Section 11.2.2 in dimensions d ≥ 2 (or its spinful generalization) and examine the nature of its instabilities to other ground states of a dense gas of fermions. Possibilities include ferromagnets, states in which there is spin or charge density wave order (to be defined more precisely below), or various types of superconductors. All of these cases are of considerable practical importance and have numerous experimental applications.

A theoretical treatment of the quantum transition between a Fermi liquid and a magnetically or charge ordered state was given in a paper by Hertz, although many important points were anticipated in earlier work. We shall present Hertz's basic arguments in Section 12.1 for the case of a transition between a Fermi liquid and a spin density wave state. We shall not treat the other cases here and will, instead, refer the reader to the literature. There are a number of reasons for this neglect:

1. (i) Many aspects of these transitions are not fully understood (we will note some below) and are the subject of considerable debate in the literature; it is therefore inappropriate to include them in this introductory treatment.

2. (ii) We shall only consider systems in spatial dimensions d ≥ 2 here (the d = 1 case requires a separate treatment appropriate to Tomonaga–Luttinger liquids and will be addressed in Chapter 14).

3. […]

What Is a Quantum Phase Transition?

Consider a Hamiltonian, H(g), whose degrees of freedom reside on the sites of a lattice, and which varies as a function of a dimensionless coupling g. Let us follow the evolution of the ground state energy of H(g) as a function of g. For the case of a finite lattice, this ground state energy will generically be a smooth, analytic function of g. The main possibility of an exception comes from the case when g couples only to a conserved quantity (i.e., H(g) = H0 + gH1, where H0 and H1 commute). This means that H0 and H1 can be simultaneously diagonalized and so the eigenfunctions are independent of g even though the eigenvalues vary with g; then there can be a level-crossing where an excited level becomes the ground state at g = gc (say), creating a point of nonanalyticity of the ground state energy as a function of g (see Fig. 1.1). The possibilities for an infinite lattice are richer. An avoided level-crossing between the ground and an excited state in a finite lattice could become progressively sharper as the lattice size increases, leading to a nonanalyticity at g = gc in the infinite lattice limit. We shall identify any point of nonanalyticity in the ground state energy of the infinite lattice system as a quantum phase transition: The nonanalyticity could be either the limiting case of an avoided level-crossing or an actual level-crossing.

The past decade has seen a substantial rejuvenation of interest in the study of quantum phase transitions, driven by experiments on the cuprate superconductors, the 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 noninteracting electrons, which is driven by the 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.

This chapter will discuss the reason for the central importance of the quantum Ising and rotor models in the theory of quantum phase transitions, quite apart from any experimental motivations. It turns out that the quantum transitions in these models in d dimensions are intimately connected to certain well-studied finite-temperature phase transitions in classical statistical mechanics models in D = d + 1 dimensions. We will then be able to transfer much of the sophisticated technology developed to analyze these classical models to the quantum models of interest here. (It is important not to confuse this very general and formal mapping with the fact that some d-dimensional quantum systems in the vicinity of finite temperature phase transitions are described by effective d-dimensional classical models, as in the shaded region of Fig. 1.2).

We will discuss this mapping here in the simplest context of d = 0, D = 1: We will consider single-site quantum Ising and rotor models and explicitly discuss their mapping to classical statistical mechanics models in D = 1 (the cases d > 0 will then be discussed in Chapter 3). These very simple classical models in D = 1 actually do not have any phase transitions. Nevertheless, it is quite useful to examine them thoroughly as they do have regions in which the correlation “length” ξ becomes very large; the properties of these regions are very similar to those in the vicinity of the phase transition points in higher dimensions.

The large-N limit of quantum rotor models in d = 2 was examined in Chapter 5 and led to the phase diagram shown in Fig. 5.2. 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 shall establish this claim in this chapter and also treat the dynamic correlations of n at nonzero temperatures. The discussion of the dynamics shall take place in a physical framework suggested by the modified version of Fig. 5.2 shown in Fig. 7.1 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 4.5.2 and 6.2. In the other low-T region on the magnetically ordered side, we shall obtain a “dual” model of quasi-classical waves, which is connected to that developed in Section 6.3. 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 shall use a menage of these concepts to obtain a complete picture in this and the following two chapters.

The results for the quasi-classical wave regime described in this chapter will be obtained by a combination of analytical and numerical techniques, which become exact in the low-T limit.