The past decade has seen a substantial rejuvenation of interest in the study of quantum phase transitions, driven by experiments on the cuprate superconductors, the 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 70's, 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 the localization of the electronic states in the presence of a random potential. The critical properties of this transition of non-interacting 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. On the other hand, it is clear that strong electronic interactions play a crucial 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 non-zero temperature, and it shall become apparent that these differ substantially from the non-interacting case. We shall also be considering inelastic collision-dominated quantum dynamics and transport: our results will apply to clean physical systems whose inelastic scattering time is much *shorter *than their disorder-induced elastic scattering time. This is the converse of the usual theoretical situation in Anderson localization or mesoscopic system theory, where inelastic collision times are conventionally taken to be much *larger *than all other time scales.

One of the most interesting and significant regimes of the systems we shall study is one in which the inelastic scattering and phase coherence times are of order *h*/*k*_{B} *T*, where *T* is the absolute temperature. The importance of such a regime was pointed out by Varma *et al.* [66,67] by an analysis of transport and optical data on the cuprate superconductors. Neutron scattering measurements of Hayden *et al. * [30] and Keimer *et al. * [37] also supported such an interpretation in the low doping region. It was subsequently realized [59,12,56] that the inelastic rates are in fact a *universal number* times *k*_{B} *T* /*h*, and are a robust property of the high temperature limit of renormalizable, interacting quantum field theories which are not asymptotically free at high energies. In the Wilsonian picture, such a field theory is defined by renormalization group flows away from a critical point describing a second order quantum phase transition. It is not essential for this critical point to be in an experimentally accessible regime of the phase diagram: the quantum field theory it defines may still be an appropriate description of the physics over a substantial intermediate energy and temperature scale. Among the implications of such an interpretation of the experiments was the requirement that response functions should have prefactors of anomalous powers of *T* and a singular dependence on the wavevector; recent observations of Aeppli *et al * [1], at somewhat higher dopings, appear to be consistent with this. These recent experiments also suggest that the appropriate quantum critical points involve competition between an insulating state in which the holes have crystallized into a striped arrangement, and a *d*-wave superconductor. There is no theory yet for such quantum transitions, but we shall discuss numerous simpler models here which capture some of the basic features.

It is also appropriate to note here theoretical studies [43,3,65,13,14] on the relevance of finite temperature crossovers near quantum critical points of Fermi liquids [31] to the physics of the heavy fermion compounds.

A separate motivation for the study of quantum phase transitions is simply the value in having another perspective on the physics of an interacting many body system. A traditional analysis of such a system would begin from either a weak coupling Hamiltonian, and then build in interactions among the nearly free excitations, or from a strong-coupling limit, where the local interactions are well accounted for, but their coherent propagation through the system is not fully described. In contrast, a quantum critical point begins from an intermediate coupling regime which straddles these limiting cases. One can then use the powerful technology of scaling to set up a systematic expansion of physical properties away from the special critical point. For many low-dimensional strongly correlated systems, I believe that such an approach holds the most promise for a comprehensive understanding. Many of the vexing open problems are related to phenomena at intermediate temperatures, and this is precisely the region over which the influence of a quantum critical point is dominant. Related motivations for the study of quantum phase transitions appear in a recent discourse by Laughlin [38].

The particular quantum phase transitions that are examined in this book are undoubtedly heavily influenced by my own research. However, I do believe that my choices can also be justified on pedagogical grounds, and lead to a logical development of the main physical concepts in the simplest possible contexts. Throughout, I have also attempted to provide experimental motivations for the models considered: this is mainly in the form of a guide to the literature, rather than in-depth discussion of the experimental issues. I have also highlighted some especially interesting experiments in a recent popular introduction to quantum phase transitions [57]. An experimentally oriented introduction to the subject of quantum phase transitions can also be found in the excellent review article of Sondhi, Girvin, Carini and Shahar [62]. Readers may also be interested in a recent introductory article [70], intended for a general science audience.

Many important topics have been omitted from this book due to the limitations of space, time and my expertise. The reader may find discussion on the metal insulator transition of electronic systems in the presence of disorder and interactions in a number of reviews [39,10,21,6,33]. The fermionic Hubbard model, and its metal insulator transition is discussed in most useful treatises by Georges, Kotliar, Krauth and Rozenberg [28] and Gebhard [27]. I have also omitted discussions of quantum phase transitions in quantum Hall systems: these are reviewed by Sondhi *et al.* [62] by Huckenstein [32], and also in the collections edited by Prange and Girvin [53] and Das Sarma and Pinczuk [17] (however, some magnetic transitions in quantum Hall systems [50,51] will be briefly noted). Quantum impurity problems are also not discussed, although these have been the focus of much recent theoretical and experimental interest; useful discussions of significant developments may be found in Refs [47,40,71,2,20,36,41,68,15,16,48,49].

Some recent books and review articles offer the reader a complementary perspective on the topics covered: I note the works of Fradkin [26], Auerbach [4], Continentino [13] ,Tsvelik [64] and Chakrabarti, Dutta and Sen [11], and I will occasionally make contact with some of them.

I wrote most of this book at a level which should be accessible to graduate students who have completed the standard core curriculum of courses required for a master's degree. In principle, I also do not assume a detailed knowledge of the renormalization group and its application to the theory of second-order phase transitions in classical systems at nonzero temperature. I provide a synopsis of the needed background in the context of quantum systems, but my treatment is surely too concise to be comprehensible to students who have not had a prior exposure to this well-known technology. I decided it would be counterproductive for me to enter into an in-depth discussion of topics for which numerous excellent texts are already available. In particular, the texts by Ma [42], Itzykson and Drouffe [34] and Goldenfeld [29], and the review article by Brézin *et al.* [9] can serve as useful companions to this book.

An upper level graduate course on quantum statistical mechanics can be taught on selected topics from this book, as I have done at Yale. I suggest that such a course begin by covering all of Part 1, followed by Chapters 4, 5, and 8 from Part 2. The material in Chapter 8 should be supplemented by some of the readings on the renormalization group mentioned above. Depending upon student interest and time, I would then pick from Chapters 10-12 (as a group), Chapter 13, and Chapter 14 from Part 3. A more elementary course should skip Chapters 8 and 10-12. The chapters not mentioned in this paragraph are at a more advanced level, and can serve as starting points for student presentations.

Readers who are newcomers to the subject of quantum phase transitions should read the chapters selected above first. More advanced readers should go through all the chapters in the order they appear.

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