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 φ α 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 φ α under the Hamiltonian ℋ in (6.52).
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 φ α 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 φ α under the Hamiltonian ℋ in (6.52).
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.
This chapter turns to the models obtained by the quantum–classical mapping QC on the D -dimensional, N -component, classical ferromagnets in (3.3) with N ≥ 2. These are the O( N ) quantum rotor models in d = D − 1 dimensions, originally written down in (1.23). The quantum Ising model studied previously had a discrete Z 2 symmetry. An important new ingredient in the rotor models will be 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. We will introduce the important concept of the spin stiffness , which characterizes the rigidity of the ordered state and determines the dispersion spectrum of the low energy “spin-wave” excitations. Apart from this, much of the technology and the physical ideas introduced earlier for the d = 1 Ising chain will generalize straightforwardly, although we will no longer be able to obtain exact results for crossover functions. 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, will continue to be extremely useful and will again be the basis of our discussion. Because we will consider models in spatial dimensions d > 1, it will be possible to have a thermodynamic phase transition at a nonzero temperature. We shall be particularly interested in the interplay between the critical singularities of the finite-temperature transition and those of the quantum critical point.
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.
We have thus far looked at the classical statistical mechanics of a collection of Ising spins in terms of their partition function. We have seen how to extract the free energy per site and correlation functions of the d = 1 Ising model. We will now consider a second way to do this. While the results are, of course, going to be the same, the method exposes a deep connection between classical statistical mechanics in d = 1 and the quantum mechanics of a single spin-12 particle. This is the simplest way to learn about a connection that has a much wider range of applicability: the classical statistical mechanics of a problem in d dimensions may be mapped on to a quantum problem in d – 1 dimensions. The nature of the quantum variable will depend on the nature of the classical variable. In general, the allowed values of the classical variable ( s =±1 in our example) will correspond to the maximal set of simultaneous eigenvalues of the operators in the quantum problem ( σz in our example). Subsequently we will study a problem where the classical variable called x lies in the range –∞ < x < ∞ and will correspond to the eigenvalues of the familiar position operator X that appears in the quantum problem. The correlation functions will become the expectation values in the ground state of a certain transfer matrix . The different sites in the d = 1 lattice will correspond to different, discrete, times in the life of the quantum degree of freedom. By the same token, a quantum problem may be mapped into a statistical mechanics problem in one-higher dimensions, essentially by running the derivation backwards. In this case the resulting partition function is also called the path integral for the quantum problem. There is, however, no guarantee that the classical partition function generated by a legitimate quantum problem (with a Hermitian Hamiltonian) will always make physical sense: the corresponding Boltzmann weights may be negative, or even complex! In the quantum mechanics we first learn, time t is a real parameter. For our purposes we also need to get familiar with quantum mechanics in which time takes on purely imaginary values t = –iτ , where τ is real. It turns out that it is possible to define such a Euclidean quantum mechanics .
To set the stage for our subsequent thorough discussion of dynamic critical phenomena, we first review the theoretical description of second-order equilibrium phase transitions. (Readers already well acquainted with this material may readily move on to Chapter 2.) To this end, we compare the critical exponents following from the van-der-Waals equation of state for weakly interacting gases with the results from the Curie–Weiss mean-field approximation for the ferromagnetic Ising model. We then provide a unifying description in terms of Landau–Ginzburg theory, i.e., a long-wavelength expansion of the effective free energy with respect to the order parameter. The Gaussian model is analyzed, and a quantitative criterion is established that defines the circumstances when non-linear fluctuations need to be taken into account properly. Thereby we identify d c = 4 as the upper critical dimension for generic continuous phase transitions in thermal equilibrium. The most characteristic feature of a critical point turns out to be the divergence of the correlation length that renders microscopic details oblivious. As a consequence, not only the correlation functions, but remarkably the thermodynamics as well of a critical system are governed by an emergent unusual symmetry: scale invariance. A simple scaling ansatz is capable of linking different critical exponents; as an application, we introduce the basic elements of finite-size scaling. Finally, a brief sketch of Wilson's momentum shell renormalization group method is presented, intended as a pedagogical preview of the fundamental RG ideas. Exploiting the scale invariance properties at the critical point, the scaling forms of the free energy and the order parameter correlation function are derived.
The approach of Chapter 11, in which we have kept a large momentum or short-distance cut-off, is sufficient for the needs of computing the various universal indices and functions that characterize criticality. All calculations were made in the limit in which the ratio ξ/a of the correlation length ξ to the lattice spacing a is large. In the previous chapters, this was done by keeping a fixed and increasing ξ . The renormalized theory relies on the opposite strategy: it deals directly with the scaling theory in which one takes the limit of a vanishingly small a . I have included here a chapter on the renormalized theory for several reasons: • There is no ultraviolet (or lattice) cut-off any more and, getting rid of one dimensionful parameter considerably simplifies the integrations, especially at higher orders. • All the critical exponents and scaling functions can be computed at T c , i.e., in a massless theory, even when one considers temperature-dependent laws, such as those concerning the correlation length, the magnetic susceptibility or the spontaneous magnetization. Integrals in a massless theory without cut-off are by far simpler, and they are often given by mere dimensional analysis. • However, the main reason is to understand the meaning of renormalizability from the viewpoint both of critical phenomena and of high-energy physics. This will be done in the following section.