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Cambridge University Press eBooks

The d = 1, O(N ≥ 3) rotor models

Subir Sachdev A

Published Apr 7, 2011• 0 citations
Publisher

Abstract

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.

DOI
10.1017/cbo9780511973765.014

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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.
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.
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.
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 part of the book is fully devoted to the physics of (1 + 1)-dimensional quantum or twodimensional classical models. As usual, I shall distinguish between classical and quantum systems only if it is necessary. Those models where the difference between quantum and classical is not relevant will be called ‘two-dimensional’; those where for some reason or other I want to emphasize the quantum aspect will be called ‘one-dimensional’. The reason I spend so much time discussing two-dimensional physics is that here one finds a kind of paradise for strong interactions and nonperturbative effects. Those effects which are tricky or even impossible to achieve in higher dimensions appear in almost every two-dimensional model. To our greatest satisfaction there are theoretical tools at our disposal which allow us to solve many of the corresponding problems and describe these phenomena. The availability of these strong mathematical tools is a unique feature of two-dimensional physics. Even if it turns out to be impossible to generalize these tools for higher dimensions, I hope that the reader will be rewarded for the time spent by the pleasure obtained from contemplating their beauty. The world we are about to enter has certain distinct features which are worth mentioning in the introduction. The first one concerns the second-order phase transitions. In (1 + 1)-dimensional systems such transitions may occur only at zero temperature.
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.