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.
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 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 Z 2 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.
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.
A systematic method for the computation of finite-temperature (T) crossover functions near quantum-critical points close to, or above, their upper-critical dimension is devised. We describe the physics of the various regions in the T and critical tuning parameter (t) plane. The quantum-critical point is at T=0, t=0, and in many cases there is a line of finite-temperature transitions at T=Tc(t), t0, with Tc(0)=0. For the relativistic, n-component ${\mathrm{\ensuremath{\varphi}}}^{4}$ continuum quantum field theory [which describes lattice quantum rotor (n\ensuremath{\geqslant}2) and transverse field Ising (n=1) models] the upper-critical dimension is d=3, and for d3, \ensuremath{\epsilon}=3-d is the control parameter over the entire phase diagram. In the region |T-Tc(t)|\ensuremath{\ll}Tc(t), we obtain an \ensuremath{\epsilon} expansion for coupling constants which then are input as arguments of known classical, tricritical, crossover functions. In the high-T region of the continuum theory, an expansion in integer powers of $\sqrt{\ensuremath{\epsilon}}$, modulo powers of ln \ensuremath{\epsilon}, holds for all thermodynamic observables, static correlators, and dynamic properties at all Matsubara frequencies; for the imaginary part of correlators at real frequencies (\ensuremath{\omega}), the perturbative $\sqrt{\ensuremath{\epsilon}}$ expansion describes quantum relaxation at \ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\sim}kBT or larger, but fails for \ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\sim}$\sqrt{\ensuremath{\epsilon}}$kBT or smaller. An important principle, underlying the whole calculation, is the analyticity of all observables as functions of t at t=0, for T>0; indeed, analytic continuation in t is used to obtain results in a portion of the phase diagram. Our method also applies to a large class of other quantum-critical points and their associated continuum quantum field theories.
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.
Quantum Phase Transitions is the first book to describe in detail the fundamental changes that can occur in the macroscopic nature of matter at zero temperature due to small variations in a given external parameter. The subject plays a central role in the study of the electrical and magnetic properties of numerous important solid state materials. The author begins by developing the theory of quantum phase transitions in the simplest possible class of non-disordered, interacting systems - the quantum Ising and rotor models. Particular attention is paid to their non-zero temperature dynamic and transport properties in the vicinity of the quantum critical point. Several other quantum phase transitions of increasing complexity are then discussed and clarified. Throughout, the author interweaves experimental results with presentation of theoretical models, and well over 500 references are included. The book will be of great interest to graduate students and researchers in condensed matter physics.
Quantum Phase Transitions Quantum phase transitions, in contrast to temperature-driven critical phenomena, occur due to a competition between different parameters describing the basic interactions of the system. Their specific feature is the quantum character of the critical fluctuations. This implies, through the uncertainty principle, that energy fluctuations and time are coupled. Then at zero temperature time plays a crucial and fundamental role, the static properties being entangled with the dynamics (Continentino, 1994a; Sondhi et al ., 1997; Sachdev, 1999). In this book we are mainly interested in quantum phase transitions that occur in electronic many-body systems and how scaling concepts can be useful to understand their properties close to these transitions (Continentino, 1994a). Even though a similar approach has been used in the case of interacting bosons (Fisher et al ., 1989) the fermionic problem has its own idiosyncrasies and difficulties. For example, there is no natural order parameter associated with the localisation transition in the electronic case, while bosons at zero temperature are either localised or superfluid so that the superfluid order parameter can be used to distinguish between both phases. If the results of the study of quantum phase transitions were restricted to zero temperature, this would be an interesting but purely academic area of research. What is really exciting about this subject is the effect of quantum critical points (QCP) in the finite temperature phase diagram of actual physical systems, even far away from the QCP (Freitas, 2015). As we will show, there is a special line in this phase diagram, the quantum critical trajectory , where the temperature dependence of the thermodynamic and transport properties is governed by the quantum critical exponents, i.e. those associated with the QCP. This and the observed crossover effects induced by temperature in the non-critical side of the phase diagram of a material with a QCP are sufficient to make the study of quantum phase transitions an inevitable subject. We start this chapter by introducing the scaling theory of quantum critical phenomena, since scaling concepts are used throughout this book.