This NATO Advanced Study Institute was the fourth in a series devoted to the subject of phase transitions and instabilities with particular attention to structural phase transforma~ions. Beginning wi th the first Geilo institute in 19'(1 we have seen the emphasis evolve from the simple quasiharmonic soft mode description within the Landau theory, through the unexpected spectral structure re presented by the "central peak" (1973), to such subjects as melting, turbulence and hydrodynamic instabilities (1975). Sophisticated theoretical techniques such as scaling laws and renormalization group theory developed over the same period have brought to this wide range of subjects a pleasing unity. These institutes have been instrumental in placing structural transformations clearly in the mainstream of statistical physics and critical phenomena. The present Geilo institute retains some of the counter cul tural flavour of the first one by insisting whenever possible upon peeking under the skirts of even the most successful phenomenology to catch a glimpse of the underlying microscopic processes. Of course the soft mode remains a useful concept, but the major em phasis of this institute is the microscopic cause of the mode softening. The discussions given here illustrate that for certain important classes of solids the cause lies in the electron phonon interaction. Three major types of structural transitions are considered. In the case of metals and semimetals, the electron phonon interaction relie6 heavily on the topology of the Fermi surface.

The Jahn-Teller effect is a consequence of the electron-phonon coupling in high symmetry systems. Its influence covers a wide range of physical and chemical properties and systems. As the biannual Jahn-Teller symposia bring together experimental and theoretical physicists and chemists from all over the world, this proceedings volume reports the latest scientific news on the effect.The contents of the volume range from the general aspects to some special topics, such as ultrafast processes, fullerenes, point defects, cooperative phenomena, HTSC and oxide properties. Some contributions are dedicated to the memory of Mary O'Brien, a globally honored specialist in the theory of the Jahn-Teller effect. Throughout personal reminiscences of O'Brien's enormous contributions to the subject, the recent history of the effect is summarized.

Electron-electron and electron-phonon interactions play fundamental roles in condensed matter physics. Strong correlations among electrons and between electrons and phonons lead to beautiful emergent phenomena both in materials and in the models used to describe them. Unfortunately, the complexity induced from the combination of interactions and large numbers of degrees of freedom makes analytically solving these models very difficult, even when greatly simplified. As a consequence, many important questions in many-body physics remain open. For example, the discoveries of charge density wave (CDW) in the pseudogap phase of the unconventional high-temperature cuprate superconductors motivate on-going research on electron-phonon interactions and its effects on the off-diagonal long-range order (ODLRO). In conventional superconductors, the attractive interaction between electrons which is mediated by the electron-phonon interaction is essential for the formation of Cooper pairs. However, if the electron-phonon interaction is sufficiently strong, charge order emerges near commensurate filling to compete with superconductivity. In this thesis, we use a combination of numerical and analytical methods to understand this sort of interplay between different types of order in the microscopic and macroscopic behavior of many-body systems. In Chapter 1, we introduce the Hubbard and Holstein Hamiltonians and the some of the exotic phases and phase transitions which they describe. We also build up some of the connections between numerical solutions of these models and experimental results for superconducting, charge, and spin order. In Chapter 2 and 3, we set up the frameworks of quantum Monte Carlo (QMC) algorithms and machine learning (ML) methods. We show how to translate a quantum-mechanical problem into an algorithm with analytical analysis encoded in it, which can be widely applied to various models and physics. In Chapter 4 and 5, we quantitatively determine the phase diagrams of one dimensional electron-phonon models where electrons have a long-range coupling to phonons as well as repulsive electron-electron interactions. We analyze the resulting metallic, Mott insulator, Peierls insulator phases, as well as the phase separation which we show often arises from momentum-dependent electron-phonon coupling. Although much work has been done on the extended Hubbard model, our research on including electron-phonon interactions pushes the field in a new direction. In Chapter 6, we describe the first study of the interplay between electron-phonon interaction and the effects of randomness. Our central result is a somewhat unexpected one: the suppression of the charge density wave correlations in the half-filled Holstein model by disorder can stabilize a superconducting phase. In Chapter 7, we use QMC and cutting-edge ML methods to identify phase transitions involving 'off-diagonal' order parameters using 'diagonal' order parameter descriptors. Our study has implications for the exploration of strong correlations using quantum gas microscopy (QGM). Chapter 8 summarizes some of the key results of this thesis, and points areas of investigation which would be important to pursue further. The material presented in Chapters 3, 4 and 5 of this dissertation is based on two published articles in Physical Review B, references [1, 2], and one manuscript which has been submitted and is under review at Physical Review Letters, reference [3]. Chapter 7 is based on reference [4], which is in preparation.

This monograph is a radical departure from the conventional quantum mechanical approach to electron-phonon interactions. It translates the customary quantum mechanical analysis of the electron-phonon interactions carried out in Fourier space into a predominantly classical analysis carried out in real space. Various electron-phonon interactions such as the polar and nonpolar optical phonons, acoustic phonons that interact via deformation potential and via the piezoelectric effect and phonons in metals, are treated in this monograph by a single, relatively simple “classical” model. This model is shown to apply to electron interactions with the deep lying X-ray levels of atoms, with plasmons and with Cerenkov radiation. The unifying concept that applies to all of these phenomena is a new definition of a coupling constant. The essentially classical interaction of an electron with its surrounding is clearly brought out to be the cause of spontaneous emission of phonons. The same concept also applies to the case of spontaneous emission of photons. While the bulk of this monograph deals with quanta of phonons and quanta of photons, a discussion of the acousto electric effect which is a purely classical phenomenon is presented. The newly defined coupling constant turns out to be valid too for this discussion. This universality of the coupling constant goes far beyond. It is equally applicable to amorphous materials. This significant application gives an analytic formulation of mobility in amorphous materials.

Phonons and electrons are two types of excitations which are responsible for many properties of condensed matter materials. The interaction between them plays an important role in condensed matter physics. In this thesis we present some theoretical investigations of the effects due to the interactions between phonons and electrons interactions. We show evidence that a structural martensitic transition is related to significant changes in the electronic structure, as revealed in thermodynamic measurements made in high magnetic fields. The effect of the magnetic field is considered unusual, as many influential investigations of martensitic transitions have emphasized that the structural transitions are primarily lattice dynamical and are driven by the entropy due to the phonons. We provide a theoretical frame-work which can be used to describe the effect of a magnetic field on the lattice dynamics in which the field dependence originates from the dielectric constant. The temperature-dependence of the phonon spectrum of alpha-uranium has recently been measured by Manley et al. using inelastic neutron scattering and x-ray scattering techniques. Although there is scant evidence of anharmonic interactions, the phonons were reported to show some softening of the optic modes at the zone boundary. The same group of authors later reported that an extra vibrational mode was observed to form at a temperature above 450 K. The existence of the proposed new mode is inconsistent with the usual theory of harmonic phonons, as applied to a structure composed of a monoclinic Bravais lattice with a two-atom basis. We investigate the effect that the f electron-phonon interaction has on the phonon spectrum and its role on the possible formation of a breathing mode of mixed electronic and phonon character. We examine the model by using Green's function techniques to obtain the phonon spectral density. Some materials undergo phase transitions from a high temperature state with periodic translational invariance to a state in which the electronic charge density is modulated periodically. The wave vector of the modulation may be either commensurate or incommensurate with the reciprocal lattice vectors of the high temperature structure. In the case of an incommensurate charge density wave, the system supports phason excitation. For an incommensurate state, the new ground state has a lower symmetry than the high temperature state since the charge density does not have long-ranged periodic translational order. If the metal is ideal (with no impurities), a charge density wave should be able to slide throughout the crystal without resistance, resulting in current flow similar to that of a superconductor. The phason is an excitation of the charge density wave which is related to the collective motion of electrons. We estimate the phason density of states, and the phason contribution to the specific heat. Angle-resolved photoemission experiments have been performed on USb2, and very narrow quasiparticle peaks have been observed in a band which local spin-density approximation (LSDA) predicts to osculate the Fermi energy. The observed band is found to be depressed by 17 meV below the Fermi energy. The experimentally observed quasiparticle dispersion relation for this band exhibits a kink at an energy of about 23 meV below the Fermi energy. The kink is not found in LSDA calculations and, therefore, is attributable to a change in the quasiparticle mass renormalization by a factor of approximately 2. The existence of a kink in the quasiparticle dispersion relation of a band which does not cross the Fermi energy is unprecedented. The kink in the quasiparticle dispersion relation is attributed to the effect of the interband self-energy involving transitions from the osculating band into a band that does cross the Fermi energy.

The study of electrons and holes confined to two, one and even zero dimensions has uncovered a rich variety of new physics and applications. This book describes the interaction between these confined carriers and the optic and acoustic phonons within and around the confined regions. Phonons provide the principal channel of energy transfer between the carriers and their surroundings and also the main restriction to their room temperature mobility. But they have many other roles; they provide, for example, an essential feature of the operation of the quantum cascade laser. Since their momenta at relevant energies are well matched to those of electrons, they can also be used to probe electronic properties such as the confinement width of 2D electron gases and the dispersion curve of quasiparticles in the fractional quantum Hall effect. The book describes both the physics of the electron-phonon interaction in the different confined systems and the experimental and theoretical techniques that have been used in its investigation. The experimental methods include optical and transport techniques as well as techniques in which phonons are used as the experimental probe. The aim of the book is to provide an up to date review of the physics and its significance in device performance. It is also written to be explanatory and accessible to graduate students and others new to the field.