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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.

This chapter discusses quantum phase transitions (QPT). It starts with a brief review of thermal phase transitions, critical exponents and scaling laws. The scaling laws are then generalized to the QPT case which is also illustrated with two specific examples. The first example that of the one-dimensional Ising model in a transverse magnetic field; the second is that of the bosonic Hubbard model. Quantum Monte Carlo is described briefly and mean field theory is introduced with the help of several examples and exercises.

Superconductor-insulator (SI) transitions of homogeneously disordered ultrathin quench-condensed films in many instances appear to be direct, without any intervening metallic regime. This is in contrast with what has been found in some other systems. These direct transitions have been analyzed using finite size scaling. The products of the dynamical critical exponent and the coherence length exponent found vary, depending upon the tuning parameter. They are approximately 1.3 for the thickness tuned SI transition, and approximately 0.7 for perpendicular and parallel magnetic field tuning. Charge tuning also yields 0.7. Assuming that the dynamical critical exponent is unity as is anticipated for systems with long range interactions, all of the transitions, except the thickness-tuned transition would appear to belong to the 3D XY universality class. This behavior is different from that observed for magnetic field tuned transitions of compounds such as InO­_x or TiN, or other metallic systems. The source of these differences is not known but may be due to differences in carrier density or structural or chemical disorder on a mesoscopic scale.

This chapter discusses the quantum phase transition using the example of superconductor-insulator transition. For the 3D case, a version of the Ginzburg-Landau formalism is formulated from both normal and anomalous diffusion. The description of such transition in the case of 2D superconductors is very specific and strongly differs from the 3D case. The ideas of boson and fermion mechanisms of the T_c suppression in 2D cases are presented, and their predictions are compared with the experimental conditions.

Interacting many-particle systems may undergo phase transitions and exhibit critical phenomena in the limit of infinite system size, while the precursors of these phenomena are studied in the theory of finite-size scaling. After surveying the basic notions of phases, phase diagrams, and phase transitions, this chapter focuses on critical behaviour at a second-order phase transition. The Landau-Ginzburg theory and the concept of scaling prepare readers for an elementary introduction to the concepts of the renormalization group, followed by an introduction into the field of quantum phase transitions where quantum fluctuations take over the role of thermal fluctuations.

This chapter starts off with a discussion of the specifics of superconductivity in ultrasmall superconducting grains. The method of optimal fluctuations in the vicinity of T_c is then introduced, and applied to the study of the formation of superconducting drops in a system with quenched disorder or in strong magnetic fields. The exponential DOS tail in a superconductor with quenched disorder is calculated. Properties of Josephson coupled superconducting grains and drops are discussed. The XY-model for granular superconductor and the GL description of the granular superconductor are formulated. The broadening of superconducting transition by the quenched disorder is found. The final part of the chapter focuses on the specifics of the quantum phase transition in granular superconductors. It discusses Coulomb suppression of superconductivity in the array of tunnel coupled granules, properties of superconducting grains in the normal metal matrix, and phase transition in disordered superconducting film in strong magnetic field.

The equilibrium thermodynamics of bosons and fermions in optical lattices are considered in the single-band Hubbard regime, with an emphasis on interesting magnetic, superfluid, and spin liquid ground states. The parameters of the Hubbard model—the tunneling and interaction parameters—can be obtained quantitatively in terms of the strength and periodicity of the optical lattice potential, tuned by the laser intensity and wavelength. This direct link between the parameters of a theoretical model and the actual experimental optical lattice gives rise to the phase diagram of the Bose–Hubbard model. The definition of the order parameter as an expectation value of the annihilation operator in a coherent state is discussed. This phase-coherent superfluid state is contrasted with the phase-incoherent Mott state naturally defined in terms of number states. Following the repulsive Bose–Hubbard model, the phase diagram of the Fermi–Hubbard model is considered with both attractive and repulsive interactions.

This chapter explains the basic ideas and concepts of the modern quantum theory of condensed matter. It starts by discussing phase transition in classical systems, scaling, and renormalisation theory, both in real and in momentum space. The chapter then moves on to quantum systems and discusses Bose–Einstein condensation (BEC) in non-interacting gases. It presents the concept of quantum phase transitions, illustrating it with spin systems and BEC in weakly interacting gases. Finally, the chapter discusses specific properties of low dimensional Bose systems: the role of phase fluctuations in 1D and 2D gases, Tonks-Girardeau gas in 1D, and Berezinskii-Kosterlitz-Thouless transition in 2D.

A superconductor is a remarkable emergent state of matter in which electrons pair up and develop long range phase coherence resulting in zero resistance and perfect diamagnetism. How can a superconductor decohere? A thin superconducting film can be driven insulating in a remarkable number of ways: decreasing thickness, increasing disorder, changing the gate voltage, or applying a magnetic field. Such superconductor-insulator transitions (SIT) are quantum phase transitions of strongly correlated electrons occurring at very low temperatures. This chapter gives an overview of the field, with particular emphasis on recent developments. This chapter describes how the theoretical understanding of SITs has evolved over the years, and how the increasing quality of experimental data is beginning to reveal the importance of amplitude and phase fluctuations. Most importantly new paradigms have been developed to describe these phenomena. This chapter contains numerous references to the contributions by various authors in subsequent chapters

This chapter presents a theoretical treatment of two types of superconductor-insulator transitions — the disorder-tuned transition and the parallel-magnetic-field-tuned transition. This is performed within the framework of the attractive Hubbard model, which is a ‘minimal’ lattice model that nevertheless captures much of the essential physics. The effects of hopping, attraction, disorder, and parallel magnetic field are taken into account one by one in proressively more refined approximations, from pairing-of-exact-eigenstates to Bogoliubov-de Gennes to determinant quantum Monte Carlo. By examining the successes and failures of each approach, the chapter elucidates the role of amplitude and phase fluctuations. This pedagogical approach provides considerable details of the calculation of thermodynamic, transport, and spectral properties, such that a suitably inclined reader should be able to reproduce many of the results.

This chapter discusses the momentum space topology of 2+1 systems. In the D = 2 space the possible manifolds of gap nodes in the quasiparticle energy are point nodes and nodal lines. The nodal lines are described by the same invariant as Fermi surfaces, while point nodes are typically marginally stable: they may be topologically protected being described by the Z₂ topological charge. The chapter focuses on topologically non-trivial fully gapped vacua — vacua with fully non-singular Green's function. The topological invariant for the gapped 2+1 systems is introduced either in terms of Hamiltonian (where the relevant topological object in momentum space is the p-space skyrmion) or in terms of Green's function (the invariant is obtained by dimensional reduction from the invariant describing the point nodes in 3+1 space). Examples are provided by p-wave and d-wave superfluids/superconductors. Topological quantum phase transitions are discussed at which the integer topological invariant changes abruptly. Topological transition occurs via the intermediate gapless state, and the process represents the diabolical point — analog of magnetic monopole — the termination point of Dirac string at which the Berry phase has singularity. The chapter also discusses broken time reversal symmetry, families (generations) of fermions in 2+1 systems, and Dirac vacuum as marginal state with fractional topological charge.

This chapter presents phenomenological scaling theory of quantum metal-insulator transition of non-interacting electrons. It confirms the existence of the transition in 3D-system; and elucidates the relation between the correlation length and the conductance, the shape of the critical region, and the temperature dependence of the conductance inside it (T^1/3-law). For 2D-systems, it predicts that there is no transition, only a weak-to-strong localization crossover.

This chapter opens the part of the book devoted to the nucleation of excitation of the vacuum — quasiparticles and topological defects. The superfluid vacuum flows with respect to environment (the container walls) without friction until the relative velocity becomes so large that the Doppler-shifted energy of excitations becomes negative in the frame of the environment, and these excitations can be created from the vacuum. The threshold velocity at which excitations of a given type acquire for the first time the negative energy is called the Landau critical velocity. In terms of the effective metric in superfluids, the region where the quasiparticle energy is negative represents the ergoregion, and in some geometry of flow — the event horizon. Nucleation of quasiparticles is also similar to electron-positron pair production in strong electric fields. The chapter also discusses thermal nucleation of vortices, which corresponds to sphaleron in high-energy physics, vortex nucleation by hydrodynamic instability and by macroscopic quantum tunnelling (the vortex instanton). The macroscopic action for vortices is used, which is topological and leads to quantization of particle number in quantum vacuum.

Actual computations of fixed points and eigenvalues usually involve approximations, often crude ones, except for a very limited number of simple cases such as the one-dimensional Ising model of the previous chapter. In real- and momentum-space renormalization group theory, there are no general prescriptions to systematically improve the degree of the approximation with a modest amount of effort. There are established methods to systematically improve precision, but they usually need a large amount of numerical calculations. The scope of the present chapter is modest as we limit ourselves to basic examples, including the epsilon expansion about the Gaussian fixed-point of the Landau-Ginzburg-Wilson model. Finally, the last section illustrates the extension of the renormalization group framework to study quantum phase transitions.

DC and finite frequency transport measurements of thin films of amorphous indium oxide that were driven through the critical point of superconductor-insulator transition by the application of perpendicular magnetic field are presented. The observation of non-monotonic dependence of resistance on magnetic field in the insulating phase, novel transport characteristics near the resistance peak and finite superfluid stiffness in the insulating phase are all discussed from the point of view that suggests a possible relation between the conduction mechanisms in the superconducting and insulating phases. The results are summarized in the form of an experimental phase diagram for disordered superconductors in the disorder-magnetic field plane.

This chapter describes advances made with a variety of heavy fermion (HF) superconducting systems, adding new challenges to the physics of this growing field. Much of the understanding has evolved through a new class of phase transition, the quantum phase transition (QPT), occurring at a quantum critical point (QCP). Studies of QCPs have led to the realisation of complicated phase diagrams for various f-electron systems and, above all, an unconventional superconductivity. Various special features described include non-centrosymmetric materials, occurrence of multiple superconducting phases, FFLO state, hidden order and metamagnetic transitions. Similarly, the observation of exceptionally high T_c values in recently discovered Pu-based HF superconductors indicates that the field is wide open for unexpected developments resulting from the discovery of new systems. The phenomenon shows extreme sensitivity to defects, impurities and stoichiometry, and materials processing therefore poses a clear challenge to the discovery of new unconventional superconductors.

This chapter examines strongly correlated systems. The discussions of the quantum mechanics of electrons in solids up to this point have largely involved properties that can be treated within the framework of self-consistent, one-electron, band theory. This approach provides a good description of many metals, including those with d electrons, and various semiconductors with constituents having only s and p valence electrons. But this approach usually fails for the f electrons in rare earth metals, and is problematic for the so-called heavy-fermion inter metallic compounds, formed with certain 4f and 5f elements, as well as a host of chemical compounds containing atoms with d and f electrons, particularly oxides. To understand strongly correlated systems one must directly confront the physics that is left out of the one-electron models. The chapter discusses the following: the Mott insulator and the Mott transition; the local density approximation (LDA) implementation of density functional theory (DFT); strongly correlated electrons in metals; quantum phase transitions and marginal Fermi liquids; and possible mechanisms underlying heavy fermion superconductivity.