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).

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.