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 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 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.
As we noted in the preface, this and the following chapter are at a more advanced level, and some readers may wish to skip ahead to Chapter 8. In Chapter 5 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 in d = 1 for N ≥ 3. This will be justified in the present chapter where we will 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 will be 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 will be 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 will be quoted for general N . Of the remaining cases, the d = 1, N = 1 model has been already considered in Chapter 4, and study of the d = 1, N = 2 model is postponed to Section 14.3. The physical picture of the T = 0, N = 3 state that emerged in Chapter 5 was very simple.
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.
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 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.
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.