At a quantum critical point the ground state of a many-particle quantum system undergoes a phase transition when some control parameters such as pressure, magnetic field, or chemical composition is varied. The universal behaviour characteristic for such quantum phase transitions often affects a wide temperature range and gives rise to novel material properties. This chapter gives a short overview over five lectures held during the Les Houches summer school ‘Quantum theory from small to large scales’ in 2010. After an introductory chapter, field driven magnetic quantum phase transitions of insulators are used as an example to discuss some of the concepts underlying quantum phase transitions. Both experimentally and theoretically, quantum phase transitions in metals are much less understood compared to insulating systems. After a brief review of the standard approach to describe those systems, the importance of multiple time scales and associated multiple critical exponents z are discussed. Finally, emergent gauge theories close to critical points are investigated. As an example, it discusses why a gauge theory describes the (classical) phase transitions of a nematic, if topological defects are suppressed.
At a quantum critical point the ground state of a many-particle quantum system undergoes a phase transition when some control parameters such as pressure, magnetic field, or chemical composition is varied. The universal behaviour characteristic for such quantum phase transitions often affects a wide temperature range and gives rise to novel material properties. This chapter gives a short overview over five lectures held during the Les Houches summer school ‘Quantum theory from small to large scales’ in 2010. After an introductory chapter, field driven magnetic quantum phase transitions of insulators are used as an example to discuss some of the concepts underlying quantum phase transitions. Both experimentally and theoretically, quantum phase transitions in metals are much less understood compared to insulating systems. After a brief review of the standard approach to describe those systems, the importance of multiple time scales and associated multiple critical exponents z are discussed. Finally, emergent gauge theories close to critical points are investigated. As an example, it discusses why a gauge theory describes the (classical) phase transitions of a nematic, if topological defects are suppressed.
In recent years, quantum phase transitions have attracted the interest of both theorists and experimentalists in condensed matter physics. These transitions, which are accessed at zero temperature by variation of a non-thermal control parameter, can influence the behaviour of electronic systems over a wide range of the phase diagram. Quantum phase transitions occur as a result of competing ground state phases. The cuprate superconductors which can be tuned from a Mott insulating to a d -wave superconducting phase by carrier doping are a paradigmatic example. This review introduces important concepts of phase transitions and discusses the interplay of quantum and classical fluctuations near criticality. The main part of the article is devoted to bulk quantum phase transitions in condensed matter systems. Several classes of transitions will be briefly reviewed, pointing out, e.g., conceptual differences between ordering transitions in metallic and insulating systems. An interesting separate class of transitions is boundary phase transitions where only degrees of freedom of a subsystem become critical; this will be illustrated in a few examples. The article is aimed at bridging the gap between high-level theoretical presentations and research papers specialized in certain classes of materials. It will give an overview on a variety of different quantum transitions, critically discuss open theoretical questions, and frequently make contact with recent experiments in condensed matter physics.
In the previous chapter, we discussed both the insulator–superconductor and the insulator–metal transitions. As such transitions are disorder or magnetic-field tuned, thermal fluctuations play no role. Phase transitions of the insulator–superconductor or insulator–metal type are called quantum phase transitions. Such phase transitions are not controlled by changing the temperature, as in the melting of ice or the λ-point of liquid helium, but rather by changing some system parameter, such as the number of defects or the concentration of charge carriers. In all such instances, the tuning parameter transforms the system between quantum mechanical states that either look different topologically (as in the transition between localized and extended electronic states) or have distinctly different magnetic properties. As quantum mechanics underlies such phase transitions, all quantum phase transitions obtain at the absolute zero of temperature and thus are governed by a T = 0 quantum critical point. While initially surprising, this state of affairs is expected, because quantum mechanics is explicitly a zero-temperature theory of matter. Of course, this is of no surprise to chemists who have known for quite some time that numerous materials can exhibit vastly distinct properties simply by changing the chemical composition and, most importantly, that such transformations persist down to zero temperature. Common examples include turning insulators such as the layered cuprates into superconductors simply by chemical doping or semiconductors into metals once again by doping, or ferromagnets such as Li(Ho, Tb) x Y 1- x F 4 into a spin glass (AR1998) by altering x .
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.
Introduction In this chapter we provide a comprehensive and unified treatment of the physical concepts and quantities relevant to phase transitions and critical phenomena. Some of these concepts and quantities have already been discussed in the particular contexts of magnetism (Chapter 7), or liquid crystals (Chapter 9, We conclude the chapter with a brief reference to the application of renormalisation group ideas to critical phenomena. The resulting reconciliation of the predictions of classical theories with experiments and exact or quasi-exact solutions of model systems is a major triumph of twentieth-century physics. Phases and phase transitions Thermodynamics habitually deals with equilibrium systems that are uniform on the macroscopic scale – i.e., where the structure, density and composition are the same throughout the bulk of the system, which is many times greater than a single atom or molecule. Because the system is in equilibrium, intensive thermodynamic variables such as the temperature or chemical potential are also spatially uniform. Such a system is said to consist of a single phase – it is a one-phase system. Phases are what used to be called the states of matter. Solids, liquids, gases and plasmas are four commonly occurring phases – they correspond to the four elements of Aristotelian physics: earth, water, air and fire, respectively. There are, however, many other, less easily identified phases, some of which we have already encountered in this book, such as the ferromagnetic, antiferromagnetic, nematic or superconducting phases. On varying the temperature, pressure and chemical potential(s) of a system, properties such as its structure, density, composition, compressibility, specific heat, etc., in general change smoothly. Yet under certain conditions the changes may be abrupt – some of these properties may exhibit discontinuities, or even divergences, at a phase transition . Phase transitions are among the most spectacular phenomena of nature. Consider, for instance, the phase sequence of water on heating: from ice, an opaque, slippery, rock-like solid; to colourless, incompressible, liquid water, which nevertheless easily flows under gravity; to water vapour, an invisible and very low-density (greenhouse!) gas, hence of ill-defined volume.
Introduction The quantum phase transitions considered so far in this book are mostly continuous, second order. They are associated with the divergence of a correlation length and of a characteristic time that renders the quantum critical point scale invariant. However, there are many important phase transitions at which the order parameter goes abruptly to zero and, more important, at which the characteristics of length and critical time do not diverge. In general there is also a latent heat , or work required, to transform one phase into another. In this and the next chapter we study these discontinuous quantum phase transitions. We proceed by considering initially some important cases of what is known as fluctuation induced quantum phase transitions to next treat a more general case of a first-order quantum phase transition. We will show through the study of these problems that the concepts of critical exponents and scaling remain useful, even though these systems do not exhibit scale invariance at their critical point. In Chapter 15 we present the scaling theory of first-order quantum phase transitions and use its results to treat some general problems. One of the most interesting cases of fluctuation-induced transitions occurs in superconductors due to the coupling of the charged carriers to the electromagnetic field. We consider initially a wider class of systems with macroscopic wave functions, the superfluids, of which superconductors are the charged version. We adopt a macroscopic approach to this problem and describe the superfluid by a complex order parameter φ , such that | φ | 2 is proportional to the density of condensed particles. The zero temperature phase diagram is expressed in terms of a control parameter, m 2 , such that, at m 2 = 0, the system presents a quantum phase transition from a superfluid state, for m 2 < 0, φ ≠ 0, to a normal state for m 2 > 0 where φ = 0. In the case of the superconductor with particles of charge q = 2 e , we will include the coupling of these particles to the electromagnetic field and study its effect in the phase diagram of these systems. We show that in the neutral superfluid, the zero temperature transition to the normal state is continuous and occurs at a quantum critical point.
Quantum phase transitions occur at zero temperature driven by quantum fluctuations instead of thermal fluctuations. They take place due to competing ground state phases that are accessible for different values of certain non-thermal parameters such as coupling constants, pressure or magnetic field. The chapter starts with a discussion of the main phenomenological features of this class of non-thermal transitions. In particular, it is argued that traces of these transitions can be detected at finite (but low) temperature. Then, examples of materials that show this behaviour are provided. Finally, the quantum Ising model is discussed and it is shown that a quantum model in d dimensions can be mapped to a classical model in d+1 dimensions.