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This chapter places the work presented in the previous chapters on the electron momentum density and band structure in condensed matter in the context of other probes. Specifically, studies by techniques such as positron annihilation, which is traditionally used to reconstruct Fermi surface topology and angle-resolved photo-emission (ARPES) used to reconstruct k-space density of states. Electron scattering coincidence spectroscopy, which reveals the three-dimensional momentum density in gaseous molecules for each shell of electrons, is then compared to Compton coincidence spectroscopy. The insight that is afforded by spin dependent x-ray Compton scattering is compared to resonant x-ray diffraction studies of magnetization. The chapter finishes with a comparison between x-ray Compton scattering studies of electron density distributions and Deep Inelastic Neutron Scattering (DINS) studies of nuclear momentum distributions.

This chapter studies two problems of statistical physics: the ferromagnet and the spin glass, on large random graphs with fixed degree profile. It describes the use of the replica symmetric cavity method in this context, and studies its stability. The analysis relies on physicists methods, without any attempt at being rigorous. It provides a complete solution of the ferromagnetic problem at all temperatures. In the spin glass case, the replica symmetric solution is asymptotically correct in the high temperature ‘paramagnetic’ phase, but it turns out to be wrong in the spin glass phase. The phase transition temperature can be computed exactly.

This chapter introduces the basic concepts of statistical physics. The restrictive point of view adopted here keeps to classical (non-quantum) statistical physics and treats it as a branch of probability theory. The mechanism of phase transitions is described in the context of magnetic systems: ferromagnets and spin glasses.

The energy momentum tensor for the vacuum field which represents gravity is non-covariant, since the effective gravitational field obeys hydrodynamic equations rather than Einstein equations. However, even for the fully covariant dynamics of gravity, in Einstein theory the corresponding quantity ‘the energy momentum tensor for the gravitational field’ cannot be presented in the covariant form. This is the famous problem of the energy momentum tensor in general relativity. One must sacrifice either covariance of the theory or the true conservation law. From the condensed matter point of view, the inconsistency between the covariance and the conservation law for the energy and momentum is an aspect of the much larger problem of the non-locality of effective theories. This chapter discusses the advantages and drawbacks of effective theory, non-locality in effective theory, true conservation and covariant conservation, covariance versus conservation, paradoxes of effective theory, Novikov–Wess–Zumino action for ferromagnets as an example of non-locality, effective versus microscopic theory, whether quantum gravity exists, what effective theory can and cannot do, and universality classes of effective theories of superfluidity.

This chapter explains how the exchange splitting between up- and down-spin bands in ferromagnets unexceptionally generates spin-polarised electronic states at the Fermi energy. The quantity of spin polarisation P in ferromagnets is one of the important parameters for application in spintronics, since a ferromagnet having a higher P is able to generate larger various spin-dependent effects such as the magnetoresistance effect, spin transfer torque, spin accumulation, and so on. However, the spin polarisations of general 3d transition metals or alloys generally limit the size of spin-dependent effects. Thus, ‘half-metals’ attract much interest as an ideal source of spin current and spin-dependent scattering because they possess perfectly spin-polarised conduction electrons due to the energy band gap in either the up- or down-spin channel at the Fermi level.

This chapter discusses the spin Hall effect that occurs during spin injection from a ferromagnet to a nonmagnetic conductor in nanostructured devices. This provides a new opportunity for investigating AHE in nonmagnetic conductors. In ferromagnetic materials, the electrical current is carried by up-spin and downspin electrons, with the flow of up-spin electrons being slightly deflected in a transverse direction while that of down-spin electrons being deflected in the opposite direction; this results in an electron flow in the direction perpendicular to both the applied electric field and the magnetisation directions. Since up-spin and downspin electrons are strongly imbalanced in ferromagnets, both spin and charge currents are generated in the transverse direction by AHE, the latter of which are observed as the electrical Hall voltage.

This chapter shows how the spin Hall effect (SHE) has been described as a source of spin-polarised electrons for electronic applications without the need for ferromagnets or optical injection. Because spin accumulation does not produce an obvious measurable electrical signal, electronic detection of the SHE proved to be elusive and was preceded by optical demonstrations. Several experimental schemes for the electronic detection of the SHE had been originally proposed, including the use of ferromagnetic electrodes to determine the spin accumulation at the edges of the sample. However, the difficulty of sample fabrication and the presence of spin-related phenomena such as anisotropic magnetoresistance or the anomalous Hall effect in the ferromagnetic electrodes could mask or even mimic the SHE signal in the sample layouts.

At a phase transition two or more different phases may coexist, such as vapour and liquid. Phase transitions can be classified according to their order. A phase transition is of first order if going from one phase to the other involves a discontinuous change in entropy, and, thus, a finite amount of latent heat; higher-order phase transitions do not involve latent heat but exhibit other types of discontinuities. This chapter investigates the necessary conditions for the coexistence of phases, and how phases are represented in a phase diagram. The order of a phase transition is defined with the help of the Ehrenfest classification. The chapter discusses the Clausius–Clapeyron relation which, for a first-order phase transition, relates the discontinuous changes in entropy and volume. Finally, this chapter considers the Ising ferromagnet as a simple model which exhibits a second-order phase transition. It also introduces the notion of an order parameter.

This chapter aims at showing that the features occurring in mean-field models, described in the previous chapters, can be found also in the other long-range systems. The first four sections are dedicated to generalizations of the models of chapter 4, in which either the mean-field interaction is augmented with a nearest neighbour interaction, or it is replaced by a slowly decaying interaction. It is shown that the long-range characteristics of the associated mean-field models are preserved, and in addition ensemble inequivalence, microcanonical negative specific heat and ergodicity breaking are induced in some cases. The final section introduces the dipolar interaction, a marginal long-range system. Dipolar systems are treated in details in chapter 15, and in this chapter few relevant properties are presented, focussing in particular on elongated ferromagnets and on ergodicity breaking.

The vast majority of materials are either paramagnetic or diamagnetic, and have no magnetic moment in the absence of an external magnetic field. However, a small set of materials can have a spontaneous magnetic structure (i.e., in the absence of an external magnetic field). The two most well known magnetic states pertain to ferromagnets and antiferromagnets, which are the focus of this chapter. It begins with a survey of magnetic properties. The discussions then turn to the density functional theory of band magnetism; the gradient energy; surface tension of a domain wall; ferromagnetic domains; total magnetic free energy of a body; single domain particles; the magnetization curve of an easy-axis uniaxial ferromagnet; and measuring the magnetization. Sample problems are also provided at the end of the chapter.

This chapter discusses the spin Hall effect that occurs during spin injection from a ferromagnet to a nonmagnetic conductor in nanostructured devices. This provides a new opportunity for investigating AHE in nonmagnetic conductors. In ferromagnetic materials, the electrical current is carried by up-spin and downspin electrons, with the flow of up-spin electrons being slightly deflected in a transverse direction while that of down-spin electrons being deflected in the opposite direction; this results in an electron flow in the direction perpendicular to both the applied electric field and the magnetization directions. Since up-spin and downspin electrons are strongly imbalanced in ferromagnets, both spin and charge currents are generated in the transverse direction by AHE, the latter of which are observed as the electrical Hall voltage.

This chapter shows how the spin Hall effect (SHE) has been described as a source of spin-polarized electrons for electronic applications without the need for ferromagnets or optical injection. Because spin accumulation does not produce an obvious measurable electrical signal, electronic detection of the SHE proved to be elusive and was preceded by optical demonstrations. Several experimental schemes for the electronic detection of the SHE had been originally proposed, including the use of ferromagnetic electrodes to determine the spin accumulation at the edges of the sample. However, the difficulty of sample fabrication and the presence of spin-related phenomena such as anisotropic magnetoresistance or the anomalous Hall effect in the ferromagnetic electrodes could mask or even mimic the SHE signal in the sample layouts.

In this chapter we present the solution to the problem of mass. It is based on the phenomenon of spontaneous symmetry breaking (SSB). We first give the example of buckling, a typical example of spontaneous symmetry breaking in classical physics. We extract the main features of the phenomenon, namely the instability of the symmetric state and the degeneracy of the ground state. The associated concepts of the critical point and the order parameter are deduced. A more technical exposition is given in a separate section. Then we move to a quantum physics example, that of the Heisenberg ferromagnet. We formulate Goldstone’s theorem which associates a massless particle, the Goldstone boson, to the phenomenon of spontaneous symmetry breaking. In the last section we present the mechanism of Brout–Englert–Higgs (BEH). We show that spontaneous symmetry breaking in the presence of gauge interactions makes it possible for particles to become massive. The remnant of the mechanism is the appearance of a physical particle, the BEH boson, which we identify with the particle discovered at CERN.

The general phenomenon of continuous phase transitions is discussed, and treated by the Landau mean field approach. First some illustrative examples are discussed, and the concepts of order parameter and critical exponents are introduced. The critical exponents for the liquid–vapour critical point of a van der Waals gas are obtained. Then the Landau mean field theory is described for a generic system, showing how the competition between entropy and internal energy affects the free energy. Critical exponents are obtained, and the static scaling hypothesis briefly mentioned. The results are then applied to ferromagnetism and to binary mixtures. Eutectic behaviour is introduced.

The subject of these lecture notes combines several different manifestations of topology in a condensed matter system. The most classical of these is through the notion of texture. By this is meant any non-singular and topologically non-trivial spatial configuration of some relevant order parameter. The goal of these notes is to provide a theory-oriented introduction to the physics of textures in quantum Hall ferromagnets. They do not attempt to provide a comprehensive review of this very rich subject, and many important aspects are not mentioned, but instead they focus on recent advances in this field.

This chapter explains how the exchange splitting between up- and down-spin bands in ferromagnets unexceptionally generates spin-polarized electronic states at the Fermi energy. The quantity of spin polarization P in ferromagnets is one of the important parameters for application in spintronics, since a ferromagnet having a higher P is able to generate larger various spin-dependent effects such as the magnetoresistance effect, spin transfer torque, spin accumulation, and so on. However, the spin polarizations of general 3d transition metals or alloys generally limit the size of spin-dependent effects. Thus,“‘half-metals” attract much interest as an ideal source of spin current and spin-dependent scattering because they possess perfectly spin-polarized conduction electrons due to the energy band gap in either the up- or down-spin channel at the Fermi level.

Some exceptional situations in fluid mechanics can be modeled by equations that are analytically solvable. The most famous example is the Korteweg–de Vries (KdV) equation for shallow water waves in a channel. The exact soliton solution of this equation is derived. The Lax pair formalism for solving the general initial value problem is outlined. Two hamiltonian formalisms for the KdV equation (Fadeev–Zakharov and Magri) are explained. Then a short review of the geometry of curves (Frenet–Serret equations) is given. They are used to derive a remarkably simple equation for the propagation of a kink along a vortex filament. This equation of Hasimoto has surprising connections to the nonlinear Schrödinger equation and to the Heisenberg model of ferromagnetism. An exact soliton solution is found.

The magnetic behavior of materials can be quite diverse with a wide range of susceptibilities. All materials exhibit diamagnetism with a very small, negative susceptibility and it is predominant in elements with completely filled shells, ionic and covalently bonded solids, organic compounds, water, and blood. In addition, superconductors expel the magnetic flux (Meissner effect) below the critical temperature and can be treated as ideal diamagnets. Paramagnetism, exhibited by elements with unpaired electrons, can be described classically by the Langevin function or quantum mechanically by the Brillouin function; the Curie law describes their temperature dependence. Ferromagnets are characterized by an internal “molecular” field, proportional to the magnetization, which explains the coupling between atomic magnetic moments and the spontaneous magnetization in regions called domains. The internal field has its origin in the quantum mechanical exchange interaction that is isotropic, short range, but can be of very large magnitudes. The Bethe–Slater curve describes the exchange interaction in terms of interatomic distances and correctly predicts ferromagnetism for intermediate distances; for smaller distance antiferromagnetism (§4) and for larger distances, an absence of order is also predicted. The strength of the exchange interaction determines the Curie temperature; above which the spontaneous magnetization vanishes and the material behaves as a simple paramagnet, as described by the Curie–Weiss law. In the vicinity of the Curie temperature, ferromagnets shows critical phenomena—four exponents characterize the magnetization and susceptibility and describe the order–disorder transition. When nearest and next-nearest exchange interactions are considered, the ground state can be a helical structure, an interesting spin order observed in some rare earth elements with hexagonal unit cells.