This chapter turns to a systematic analysis of transport of conserved charges in the quantum rotor model. We introduced some general concepts in Section 8.3, and these are illustrated here by explicit computations at higher orders.

For d = 1, we considered time-dependent correlations of the conserved angular momentum, L(x, t), of the O(3) quantum rotor model in Chapter 12. We found, using effective semiclassical models, that the dynamic fluctuations of L(x, t) were characterized by a diffusive form (see (12.26)) 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 Table 12.1). 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 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 14.3 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.

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been co-authored with T. Senthil, and adapted from the Ph.D. thesis of T. Senthil, submitted to Yale University (1997, unpublished).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 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 are therefore rather limited: we 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 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 also make some remarks in Section 21.4 on the effects of disorder on the ordering transitions of Fermi liquids considered in Chapter 18. Models with much stronger disorder and frustrating interactions that have new phases not found in ordered systems are considered in Chapter 22.

Part II analyzed the properties of quantum Ising and rotor models in some detail at T = 0. We related the quantum phase transitions in these models to the N-component relativistic field theory (2.11), and used it to understand the critical properties.

The purpose of Part III is to extend this understanding to T > 0. We will demonstrate that the T = 0 quantum critical point controls a wide “quantum critical” region at T > 0, as illustrated in Fig. 1.2. We are especially interested in dynamic properties in this region: an interesting feature is that many “friction” coefficients are universal and depend only on fundamental constants of nature. We also explore the other regions of the phase diagrams in Fig. 1.2, including behavior in the vicinity of the phase transition at T > 0.

We begin this chapter by extending results of the d = 1 quantum Ising model of Chapter 5 to T > 0. This model does not have any phase transition at any T > 0, and so the crossover structure of the phase diagram is in the class in Fig. 1.2a. Phase transitions at T > 0 appear in models to be studied in the following chapter.

This chapter finally moves beyond the quantum rotor models which have been the complete focus of our attention so far in Part II. Our motivation is two-fold: to introduce the coherent state path integral, which plays an important role in developing the field theory for many interesting quantum phase transitions; and to provide a deeper and more complete explanation of our claimed connection between the N = 2 rotor model and the experiments on ultracold bosonic atoms in an optical lattice which was claimed in Sections 1.3 and 1.4.3. We do this by studying the boson Hubbard model, which has a direct connection to the microscopic Hamiltonian of the ultracold atoms.

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 [160, 165].

In this chapter, we examine only the much simpler “boson Hubbard model,” following the analysis in an important paper by Fisher et al. [148]. As the name implies, the elementary degrees of freedom in this model are spinless bosons, which take the place of the spin-1/2 fermionic electrons in the original Hubbard model. These bosons could represent Cooper pairs of electrons undergoing Josephson tunneling between superconducting islands, helium atoms moving on a substrate, or ultracold atoms in an optical lattice.

The Fermi liquid is perhaps the most familiar quantum many-body state of solid state physics; we met it briefly in Section 16.2.2. It is the generic state of fermions at nonzero density, and is found in all metals. Its basic characteristics can already be understood in a simple free fermion picture. Noninteracting fermions occupy the lowest energy single-particle states, consistent with the exclusion principle. This leads to the fundamental concept of the Fermi surface: a surface in momentum space separating the occupied and empty single fermion states. The lowest energy excitations then consist of quasiparticle excitations which are particle-like outside the Fermi surface, and hole-like inside the Fermi surface. Landau's Fermi liquid theory is a careful justification for the stability of this simple picture in the presence of interactions between fermions. Just as we found in Chapters 5 and 7 for the quantum Ising and rotor models, interaction corrections modify the wavefunction of the quasiparticle and so introduce a quasiparticle residue A; however, they do not destabilize the integrity of the quasiparticle, as we review in Section 18.1.

The purpose of this chapter is to describe two paradigms of symmetry breaking quantum transitions in Fermi liquids. In the first class, studied in Section 18.2, the broken symmetry is related to the point-group symmetry of the crystal, while translational symmetry is preserved; consequently, the order parameter resides at zero wavevector. In the second class, studied in Section 18.3, the order parameter is at a finite wavevector, and so translational symmetry is also broken.

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.

This and the following chapter are at a more advanced level, and some readers may wish to skip ahead to Chapter 14.

In Chapter 11 we studied the O(N) quantum rotor model in the large-N limit for a number of values of the spatial dimensionality, including d = 1. We noted that the results provided an adequate description of the static properties when d = 1 for N ≥ 3. This is justified in the present chapter where we obtain a number of exact results for the same static observables. We also noted that the large-N limit did a very poor job of describing dynamical properties at nonzero temperatures. This is repaired in this chapter by simple physical arguments that lead to a fairly complete (and believed exact) description of the long-time behavior. Some of the discussion in this chapter is specialized to the O(N = 3) model, which is also the case of greatest physical importance; the properties of the O(N > 3) models are very similar, and many of our results are quoted for general N. Of the remaining cases, the d = 1, N = 1 model has already been considered in Chapter 10, and study of the d = 1, N = 2 model is postponed to Section 20.3.

In this appendix we list some of the commonly needed physical constants in spectroscopy. We also calculate the emitted field intensity directly from the third-order response functions for standard experimental conditions and molecular properties. The purpose of this calculation is not to obtain a highly accurate result, but to serve as a (somewhat interesting) exercise of units. In the following, we use units of meters-kilograms-seconds (SI units) unless otherwise noted.