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.