Abstract 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.
Abstract 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.
This introduction presents an overview of the key concepts discussed in the subsequent chapters of this book. The book highlights quantum field theory techniques that were developed in parallel in high energy physics and condensed matter physics. It explains the relevance of quantum field theory methods developed in the context of high energy physics, to many particle systems, even though, in principle, quantum mechanics may suffice to understand these systems. The book discusses the modern perspective of phase transitions and use a length scale based approach. It deals with the application of the ideas of bosonization in simple one-dimensional models such as spin models and the Hubbard model, where the calculational tools of bosonization and Renormalisation group are used to compute physically relevant quantities like conductances. The book also highlights the idea that topological concepts play an important role in a wide range of phenomena.
The realization of Bose-Einstein Condensation (BEC) in dilute gases of ultracold atoms has revolutionized quantum physics by offering an unprecedented platform to study quantum phenomena on macroscopic scales. Ultracold atomic systems, cooled to temperatures near absolute zero, exhibit collective behaviors governed by quantum statistics, coherence, and interactions. This article discusses the theoretical and experimental frameworks underpinning the formation of BECs, focusing on the trapping, cooling, and coherence properties of ultracold bosonic gases. Further, we delve into the role of quantum phase transitions, atom-atom interactions, and external field manipulation in controlling condensate dynamics. The implications of BECs extend into precision measurements, quantum simulation, and quantum computing, positioning them as a cornerstone in modern atomic and condensed matter physics.
Bose-Einstein condensation is a unique phase transition in that it is not driven by interparticle interactions, but can theoretically occur in an ideal gas, purely as a consequence of quantum statistics. This chapter addresses the question, ‘ How is this ideal Bose gas condensation modified in the presence of interactions between the particles?’ This seemingly simple question turns out to be surprisingly difficult to answer. Here we outline the theoretical background to this question and discuss some recent measurements on ultracold atomic Bose gases that have sought to provide some answers. Introduction Unlike the vast majority of phase transitions, Bose-Einstein condensation (BEC) is not driven by interparticle interactions but can theoretically occur in an ideal (noninteracting) gas, purely as a consequence of quantum statistics. However, in reality, interactions are needed for a Bose gas to remain close to thermal equilibrium. It is thus interesting to discuss if something close to ideal gas BEC can be observed in a real system and what happens in the vicinity of the BEC transition in the presence of interparticle interactions. These simple questions have not been easy to answer, either theoretically or experimentally. The theoretical foundations for studying the effect of interactions on Bose condensed systems were laid over half a century ago by Bogoliubov [1], Penrose and Onsager [2], and Belieav [3], among others. These works initially focused on zerotemperature properties and were extended to nonzero temperature in the pioneering papers of Lee, Huang, and Yang [4, 5, 6, 7]. At that time, the main experimental system was liquid He-4 in which the inter-particle interactions are strong, making connections with theory difficult. The realisation in 1995 of BEC in weakly interacting ultracold atomic gases [8, 9] thus opened up the possibility to experimentally revisit some of these long-discussed questions. This was further aided by, among other advances, the use of Feshbach resonances to tune the interaction strength in atomic gases [10, 11]. Thus, in the last twenty years the study of ultracold Bose gases has been very successful (see Chapter 3 for an insightful account of selected topics and surprising developments over this exciting period). However, the fact that, until recently, ultracold atoms were confined using harmonic potentials has hindered the study of the BEC transition itself.
Bose–Einstein condensation (BEC) in cold atomic gases was first achieved experimentally in 1995. Since then there has been a surge of activity in this field, with ingenious experiments putting forth more and more astonishing results about the behaviour of matter at very cold temperatures. The theoretical investigation of BEC goes back much further, and even predates the modern formulation of quantum mechanics. It was investigated in two papers by Einstein in 1924 and 1925, respectively, following up on a work by Bose on the derivation of Planck's radiation law. Einstein's result, in its modern formulation, can be found in any textbook on quantum statistical mechanics, and was concerned with ideal, i.e., non-interacting gases. The understanding of BEC in the presence of interparticle interactions poses a formidable challenge to mathematical physics. Some progress has been made in the last ten years or so, and the purpose of this chapter is to explain part of what was achieved and how it is related to the actual experiments on cold gases.
Introduction Bose–Einstein condensation (BEC) is one of the most intriguing phenomena in physics. The basic idea goes back to the 1920s, when Einstein made predictions for the behavior of atoms at sufficiently low temperatures and high densities. However, it took seven decades for experimental physicists to realize this phase under the low-density conditions envisioned by Einstein. In the intermediate period there were many experiments in which aspects of BEC played important roles. Therefore, phenomena such as superfluidity and superconductivity were connected to BEC long before its 1995 realization in ultra-cold, low-density atomic vapors. Since BEC can be discussed with very many different backgrounds and applications in mind, the subject can sometimes be overwhelming for interested students. There is a very good reason why a chapter on BEC does not belong in this book. In all the other chapters here, the processes described involve one or at most two atoms, and the interactions that play a role are between the particles that constitute the atom, namely between the nucleus (or nuclei) and their electrons. The interactions are described quantum mechanically by their Hamiltonian that can always be written exactly in non-relativistic terms, and in some cases can even be solved exactly. This makes atoms and molecules a branch of physics where comparisons between theory and experiments become very meaningful since theoretical predictions are often confirmed to very high accuracy. In BEC, it is the interactions among many atoms that are the key for understanding the phenomenon. Although in principle one could write a Hamiltonian for a Bose condensed gas, its large number of particles makes this not very practical. The starting point for the description of a BEC is in the area of many-body physics that belongs to the expertise of condensed-matter physicists. The interactions between ~10 23 electrons in a solid state with a lattice formed by ~10 23 ions resemble the kind of interactions in a BEC much more closely than the simple fewbody interactions in an atom or molecule. Thus all the theoretical models that have been described up to now are not applicable for the contents of this chapter.
The properties of phase transitions and the types of order present in the low-temperature states of matter are fundamentally dependent on the dimensionality of physical systems. Generally, highly ordered states are more robust in higher dimensions, while thermal and quantum fluctuations, which favour disordered states, play a more important role in lower dimensions. The case of a two-dimensional (2d) Bose fluid is particularly fascinating because of its "marginal" behaviour. In an infinite uniform 2d fluid thermal fluctuations at any non-zero temperature are strong enough to destroy the fully ordered state associated with Bose--Einstein condensation, but are not strong enough to suppress superfluidity in an interacting system at low, but non-zero temperatures. Further, the presence of residual "quasi-long-range" order at low temperatures leads to an interesting interplay between superfluidity and condensation in all experimentally relevant finite-size systems. In these notes we give an introduction to the physics of 2d Bose fluids from an atomic physics perspective. Our goal is to summarize the recent progress in theoretical understanding and experimental investigation of ultra-cold atomic gases confined to 2d geometry, and we also hope to provide a useful introduction to these systems for researchers working on related topics in other fields of physics.