Introduction

Although it can seem paradoxical, the quantum magnetic states found in insulators offer an interesting route to study Bose-Einstein condensation (BEC) and more generally phenomena related to itinerant, interacting bosonic systems.

In many materials, the repulsion between the electrons causes the ground state to be a Mott insulator with one electron localized per site. In such a case, the charge degrees of freedom are completely locked by large charge gaps (usually of the order of tens of electron volts). However, one can study the fascinating physics of the spin degrees of freedom. As was shown a long time ago, the localized spins are coupled by a so-called superexchange which may be ferro- or antiferromagnetic. The properties of the exchange coupling between the spins depend mainly on the underlying atomic structure of the material. The resulting physics is extremely rich and such materials can have a host of phases ranging from ordered magnetic phases to so-called spin-liquid phases in which spin–spin correlations decay extremely fast with distance. The field of research concerning these materials, dubbed quantum magnetism, is thus an extremely active field in its own right [1].

Additionally, one can exploit the formal mapping that exists between spin operators and bosonic ones, as will be described in the next section, to connect the magnetic materials to itinerant bosonic systems with interactions between the bosons. This opens the way to use such quantum spin systems as quantum simulators. A quantum simulator means that the experimental system can be employed in order to “solve” in a controlled way a problem/Hamiltonian known from a different less controlled physical realization, e.g., inspired by fundamental many-body physics, quantum information or computational applications.

At the time of the first workshop in this series in 1993, the only experimentally realized Bose condensate (at least in the simple sense conjectured by Einstein) was liquid 4He. In the intervening twenty-plus years, much has happened in the world of Bose-Einstein condensation (BEC). Probably the most exciting development has been the attainment of condensation in ultracold bosonic atomic gases such as 87Rb and 23Na in 1995, followed a few years later by the achievement of degeneracy and eventually Bardeen-Cooper-Schrieffer (BCS) pairing in their fermionic counterparts, and the experimental realization of the theoretically long-anticipated “BEC-BCS crossover” by using the magnetic field degree of freedom to tune the system through a Feshbach resonance. One particularly fascinating aspect of the latter has been the realization of a “unitary gas” at the resonance itself – a system which prima facie has no characteristic length scale other than the interparticle separation, and is therefore a major challenge to theorists. Other systems in which BEC has been realized, sometimes transiently, include exciton-polariton complexes in semiconducting microcavities and, at least in a formal sense, the magnons in a magnetic insulator, as well as ultracold gases with a nontrivial and sometimes large “spin” degree of freedom.

As compared with our “traditional” Bose condensate, liquid 4He, these new systems typically have many more (and more rapidly adjustable) control parameters, and have therefore permitted qualitatively new types of experiment. One particularly fascinating development has been the use of optical techniques to generate “synthetic gauge fields” and thus mimic some of the topologically nontrivial systems which have recently been of such intense interest in a condensed-matter setting. At the same time, there remain long-standing issues from helium physics, such as the nature and consequences of “spontaneously broken U(1) symmetry,” the “Kibble-Zurek” mechanism, and more generally the relaxation of strongly nonequilibrium states to equilibrium; in some cases, the new systems have been used to address these more quantitatively than was possible with 4He. The chapters in this volume address all of these questions and more, and should be of intense interest to both the experimental and the theoretical sides of the BEC community.

Introduction

Magnons are quasiparticles corresponding to quantised spin waves that describe the collective motion of spins [1]. In recent years, it has become possible to realise a BEC of magnons in two remarkably different systems: in superfluid phases of an isotope of helium – Helium-3 at ultralow temperatures [2, 3] and in ferrimagnetic insulators [4, 5, 6, 7]. In the normal, noncondensed state, these magnetic materials exhibit a magnetically ordered state with a gas of magnons whose phases are not correlated. However, once condensation occurs, spins develop phase coherence resulting in a common global frequency and phase of precession. At room temperature, ferrimagnetic insulators, such as yttrium-iron garnet (YIG) films, together with a combination of an in-plane magnetic field and microwave radiation, are required for the magnon condensation [4, 5, 6, 7]. Therefore, in analogy with exciton– polariton condensates, this provides yet another example of a BEC of quasiparticle excitations in a solid-state system.

Experimental Realisation of a Bose-Einstein Condensateof Magnons in an Epitaxial YIG-film

A room-temperature Bose-Einstein condensate of magnons was created in an epitaxial YIG-film using an experimental setup shown in Fig. 25.1a, which, in general, is similar to that used in our previous studies [4, 5]. Given that magnons are quasiparticle excitations, their number is not conserved and, therefore, the chemical potential is zero.

Introduction

The field of Bose-Einstein condensation (BEC) has undergone an explosive expansion during the past twenty years. Newcomers to this field are now often introduced to this as a universal phenomenon, which nonetheless exhibits diverse (and sometimes strikingly different) manifestations. Despite such differences, the common underlying theme creates a unique identity across many different energy and length scales.

The study of BEC as a universal phenomenon was highlighted in a focused conference in 1993, leading to the publication of the well-known “green book” [1], which surveyed the breadth of the field of condensate physics at that time. The success of the conference led to a second meeting in 1995, at which Eric Cornell and Carl Wieman announced the achievement of Bose-Einstein condensation of ultracold 87Rb atoms in a harmonic trap. That work began an explosion of new research in the field of cold atoms, which has continued to this day, and this very success inevitably led many of those studying cold atoms to pay less attention to other types of condensates. Wolfgang Ketterle gives a historical overview of this exciting period of time in Chapter 3. Recently, various scientific meetings have worked to re-establish the physical connections across different BEC systems, and in 2013 a workshop was held with the focused goal of improving communications across disciplines. This present book is an outgrowth of that meeting.

The Situation Before the Revolution

Because of the great success of the cold atom BEC and other BEC systems in the past twenty years, it may be hard for young scientists to understand the climate of BEC research in the early 1990s. At that time, there was only one known example of BEC, namely liquid helium-4, and there was a small but vocal minority of scientists who questioned whether BEC was established even in that system.

This book marks the twentieth anniversary of the publication of the book Bose- Einstein Condensation by Cambridge University Press. The book was the result of the 1993 meeting in Levico-Terme, Italy, organized by Allan Griffin, David Snoke, and Sandro Stringari, with significant help from Andre Mysyrowicz. That meeting grew out of a desire by many theorists and experimentalists to discuss the general themes of Bose-Einstein condensation, to draw connections between different physical systems.

One of the major driving forces for that meeting was the desire to have another example of Bose-Einstein condensation besides liquid helium. There was serious discussion at the time of whether nature abhorred a condensate, and liquid helium was a special, anomalous case. Experiments on spin-polarized hydrogen, excitons in semiconductors, and optically trapped atoms had been going on for more than a decade, without success. To move the field forward, the organizers of the 1993 meeting brought together world experts on the general theory, and experimentalists of all types, to discuss the universal themes of Bose-Einstein condensation generally. There were fascinating and heated debates about such topics as the time scale for condensation (could it be possible that condensation will not occur in a system with finite lifetime?), the concept of spontaneous symmetry breaking (could there ever be a universal “phase standard” for condensates?), and how superfluidity and condensation are related.

The situation is quite changed now. We now have many experimental examples of Bose-Einstein condensation, most notably atoms at very low temperature in optical traps, which led to the Nobel Prize in Physics in 2001. This work of Eric Cornell and Carl Wieman was first announced at the second general meeting on Bose-Einstein condensation in 1995, in Mt. Ste. Odile, France. This led to the successful conference series on atomic Bose-Einstein condensation, which now takes place regularly in San Feliu de Guixols, Spain.

Because of this changed situation, the present book does not have the same form as the 1995 book. At that time, it was possible to survey a good fraction of all the experimental and theoretical efforts in the field. The field is now so large that no book can do that comprehensively, and this book leaves out a good many significant topics.

A Quantum Storm in a Teacup

A befitting title to this chapter could have been “a quantum storm in a teacup.” The storm refers to a turbulent state of a fluid, teeming with swirls and waves. Quantum refers to the fact that the fluid is not the classical viscous fluid of conventional storms but rather a quantum fluid in which viscosity is absent and the swirls are quantized. The quantum fluid in our story is a quantum-degenerate gas of bosonic atoms, an atomic Bose-Einstein condensate (BEC), formed at less than a millionth of a degree above absolute zero. And finally the teacup refers to the bowl-like potential used to confine the gas; this makes the fluid inherently inhomogeneous and finite-sized. A typical image of our quantum storm in a teacup is shown in Fig. 17.1a.

This chapter reviews quantum turbulence in atomic condensates, tracing its history (Section 17.2), introducing the main theoretical approach (Section 17.3) and the underyling quantum vortices (Section 17.4).We then turn to describing physical characteristics (Section 17.5), the experimental observations to date (Section 17.6), methods of generating turbulence (Section 17.7), and some exciting research directions (Section 17.8) before presenting an outlook (Section 17.9).

Origins

Turbulence refers to a highly agitated, disordered, and nonlinear fluid motion, characterized by the presence of eddies and energy across a range of length and time scales [3]. It occurs ubiquitously in nature, from blood flow and waterways to atmospheres and the interstellar medium, and is of practical importance in many industrial and engineering contexts. Since da Vinci's first scientific study of turbulent flow of water past obstacles, circa 1507, research into turbulence in classical viscous fluids continues with vigor; however, due to its rich complexities, the physical essence and mathematical description of turbulence remain a challenge.

Introduction

The theory of quantum criticality has made enormous progress in the last two decades. The reason is that many physical systems have been discovered which are close to, or can be driven to the proximity of, a quantum critical point. The nature of these QCP is diverse: they can be associated with magnetic transitions, with localisation phenomena, or with superconductivity, to name just a few cases treated in this book. The most common driving mechanisms are external pressure, physical or chemical, external magnetic fields and disorder. In spite of the variety of QCP, with fluctuations of the most different characters, we expect to find some universal behaviour and a unifying perspective for these phenomena. The scaling approach has sufficiently embracing concepts as to provide this type of description, but a deep understanding of quantum critical systems demands that it comes accompanied by microscopic models. These models are necessary to determine the critical exponents and fully characterise the different universality classes.

In this chapter we add some substance to the ideas presented so far, examining a class of materials that historically has played an important role in the study of quantum criticality. Heavy fermions are found by chance to be close to some instability such that small changes in external parameters can give rise to a new phase. This can be approached sufficiently smoothly so as to appear that the phase transition is continuous, of the second order. The theory of quantum criticality provides an elegant framework to understand many of the properties of these systems, as we show in this chapter.

Heavy fermion physics is associated with elements with unstable f-shells like Ce,U and Yb (Steglich et al., 1995, 1998; Kambe, Flouquet and Hargreaves, 1997; Thompson and Lawrence, 1994). The f-states in these materials have an ambiguous character, at the borderline between localised and itinerant. They form compounds with a variety of ground states, such as, magnetic (generally antiferromagnetic), superconductor or simply Fermi liquid (FL) down to the lowest temperatures. By Fermi liquids we mean here metallic systems which, below a characteristic temperature T*, exhibit a constant, i.e. temperature-independent, uniform susceptibility χ (Pauli susceptibility), a linear temperature-dependent specific heat C(T) and a resistivity ρ with a T2 temperature dependence. The localised nature of the f-electrons involved in the heavy fermion phenomenon, implies the existence of strong correlations in these materials.

The Simplest Density-Driven Transition

The simplest metal–insulator (MI) transition is that due to band filling (or emptying) in a system of non-interacting electrons. This is the fermionic equivalent of Bose–Einstein condensation in a system of non-interacting bosons. Let us consider a system of non-interacting electrons described by a tight-binding band in a hyper-cubic lattice in d-dimensional space. As the number of electrons per site n, for a given spin direction, approaches the value nc = 1 the system evolves from a metallic to an insulating state. This transition is a quantum critical phenomenon with which we can associate critical exponents. We can introduce a characteristic length χ, which can be identified with a screening length which diverges at the metal–insulator transition. The natural variable to describe this zero-temperature transition is the distance from the chemical potential or Fermi level μ, to the bottom, Eb = 0, or to the top of the band Et, i.e. g = μ − Eb,t. In terms of this variable, ξ ∝ |g|−ν and the ground state free energy density has a singular part that behaves as, f ∝ |g|2−α, close to the transition. This expression defines the critical exponent α. The critical exponents ν and α associated with this zero temperature transition are related through the quantum hyperscaling relation 2 − α = ν(d + z).

For this simple MI transition it is possible to determine its universality class for any dimension (Continentino, 1995a). One starts by noticing that the gap for excitations is actually a linear function of the distance g = μ − Et. This fixes the gap exponent at the value νz = 1. Furthermore, for a hypercubic lattice the density of states close to the band edges varies as, such that, the number where μ is the chemical potential.

Introduction

In this chapter we study metal–insulator transitions which occur due to correlations between the charge carriers. This quantum phase transitions results from the competition between kinetic energy and Coulomb repulsion where the latter has the tendency to localise the electrons. The simplest model which describes this competition is the one-band Hubbard model with its two energy scales, the bandwidth W and the Coulomb repulsion U between electrons of opposite spins on the same site. For a fixed density of electrons, n = 1; as the ratio U/W increases, we expect to find a zero-temperature phase transition from a metallic state to an insulator where the electrons are frozen at the lattice sites. The method employed here to investigate this transition is the Gutzwiller approach (Gutzwiller, 1965). This is a variational method which played an important role in providing a framework for the ideas of Mott on the problem of the metal–insulator transition (Mott, 1974). On the metallic side of the transition, this approach describes a metal with renormalised Fermi liquid parameters, essentially with enhanced effective mass and uniform susceptibility. This is the so-called highly correlated electron gas (Brinkman and Rice, 1970) for which the metal oxide V2O3 is the prototype system. In the insulating side, Gutzwiller's approximation yields a noninteracting system of localised electrons. This insulating phase is incompressible, i.e. has zero compressibility and infinite zero temperature susceptibility. The latter is not a consequence of interactions but simply due to the divergence of the Curie susceptibility, χ ∝ 1 /T, of non-interacting moments at T = 0. Then, in Gutzwiller's approach the transition is from a paramagnetic metal to a paramagnetic insulator. We will continue using the designation Mott transition to describe this type of phase transition where the insulating phase has no long-range magnetic order. In real systems, most frequently, localisation due to interactions is accompanied by the appearance of antiferromagnetic order.

Quantum phase transitions remain an exciting and vigorous scientific subject. This field has experienced continuous theoretical progress. Equally important is the intense experimental work the field has engendered, including the discovery of new materials exhibiting different forms of quantum criticality that in turn further advance the theory. Thus, the new, enlarged edition of this book is more than timely.

The layout maintains the general structure of the first edition. New material has been added to the original chapters, and some of them have been modified, even substantially – such as the one on first-order quantum phase transitions. Moreover, I have written entirely new chapters on the study of topological quantum phase transitions and superconducting quantum critical points.

The book emphasises a general scaling approach that considers the change of parameters of a system under length and timescales close to quantum criticality. This provides a unifying view to such diverse phenomena as zero temperature magnetic transitions, metal-insulator, topological, superconductor and first-order quantum phase transitions. As in the previous edition, the renormalisation group with its concepts such as fixed points associated with scale invariance, crossover, and relevant and irrelevant interactions, provides the mathematical framework for the scaling ideas. It is the basic tool used to study, understand and describe quantum critical phenomena, even when the usual Landau paradigm fails, as for topological quantum phase transitions.

I would like to thank the many colleagues with whom I have been discussing and learning the subjects of this book. Among them, G. Aeppli, Elisa Baggio-Saitovitch, Sergey Bud'ko, Premala Chandra, Piers Coleman, Mauricio D. Coutinho-Filho, Alvaro Ferraz, Zach Fisk, Jacques Flouquet, Magda Fontes, Alex Lacerda, Mireille Lavagna, Hilbert v. Löhneysen, Andy J. Millis, Luiz Nunes de Oliveira, Pascoal Pagliuso, Silke Paschen, Aline Ramires, Daniel Reyes-Lopez, Peter Riseborough, Subir Sachdev, Andre Schuwartz, Qimiao Si, Frank Steglich, Manuel Nunez-Regueiro, Jean-Louis Tholence, Joe Thompson, Amos Troper and the lates Bruno Elschner and Bernard Coqblin. My sincere thanks to Alfredo Ozorio de Almeida, Claudine Lacroix, Eduardo Marino, Eduardo Miranda, Enzo Granato, Heron Caldas, Jose A. Helayël-Neto, Jose Abel Hoyos-Neto, Pedro Sacramento and Tharnier Puel de Oliveira for enlightening discussions and several suggestions to improve the new manuscript. The author, however, is solely responsible for any mistakes or omissions in the book.

I would like also to express my recognition of the encouragement of my wife Sonia, my daughter Ilana and my son Marcelo.

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, m2, such that, at m2 = 0, the system presents a quantum phase transition from a superfluid state, for m2 < 0, φ ≠ 0, to a normal state for m2 > 0 where φ = 0. In the case of the superconductor with particles of charge q = 2e, 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.