Quantum Phase Transition and Critical Phenomena

J. Wosnitza

Hans-Peter Eckle

Abstract 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.

Masatoshi Imada

D. Belitz, T. R. Kirkpatrick

M. A. Contínentino

Quantum Phase Transitions Quantum phase transitions, in contrast to temperature-driven critical phenomena, occur due to a competition between different parameters describing the basic interactions of the system. Their specific feature is the quantum character of the critical fluctuations. This implies, through the uncertainty principle, that energy fluctuations and time are coupled. Then at zero temperature time plays a crucial and fundamental role, the static properties being entangled with the dynamics (Continentino, 1994a; Sondhi et al ., 1997; Sachdev, 1999). In this book we are mainly interested in quantum phase transitions that occur in electronic many-body systems and how scaling concepts can be useful to understand their properties close to these transitions (Continentino, 1994a). Even though a similar approach has been used in the case of interacting bosons (Fisher et al ., 1989) the fermionic problem has its own idiosyncrasies and difficulties. For example, there is no natural order parameter associated with the localisation transition in the electronic case, while bosons at zero temperature are either localised or superfluid so that the superfluid order parameter can be used to distinguish between both phases. If the results of the study of quantum phase transitions were restricted to zero temperature, this would be an interesting but purely academic area of research. What is really exciting about this subject is the effect of quantum critical points (QCP) in the finite temperature phase diagram of actual physical systems, even far away from the QCP (Freitas, 2015). As we will show, there is a special line in this phase diagram, the quantum critical trajectory , where the temperature dependence of the thermodynamic and transport properties is governed by the quantum critical exponents, i.e. those associated with the QCP. This and the observed crossover effects induced by temperature in the non-critical side of the phase diagram of a material with a QCP are sufficient to make the study of quantum phase transitions an inevitable subject. We start this chapter by introducing the scaling theory of quantum critical phenomena, since scaling concepts are used throughout this book.

Christine Cartwright

This thesis focuses on identifying different types of critical phenomena within one-dimensional quantum lattice spin systems. People are exposed to classical critical phenomena everyday in the form of phase transitions, such as from water to ice. Surrounding quantum phase transitions, there is a number of exciting phenomena still to be explored, enhancing our knowledge of the quantum effects. We will examine the features of quantum phase transitions within spin chains, where our focus will be on the use of entanglement measures, and we will explore an extension of the standard Bose-Hubbard model on a one-dimensional lattice with a non-trivial elementary cell. Our first main focus will be on the use of entanglement scaling as an indicator of when a phase transition has occurred, more specifically for first-order quantum phase transitions in the vicinity of second-order quantum phase transitions. In this context, we show the dramatic importance of finite-size effects when a first-order quantum phase transition occurs in the nearby region of multi-critical point containing also a second-order quantum phase transition. Through the use of finite-size scaling we highlight that a bipartite measure of entanglement can correctly identify a given first-order quantum phase transition. Motivated by the plethora of diverse features that geometric frustration may lead to, we will also examine the Bose-Hubbard model on a geometrically frustrated lattice. The lattice we will examine is a one-dimensional chain of rhombi and causes the particles to form pairs in the presence of repulsive interactions. We perform a detailed numerical analysis of the different phases emerging from the model, including the block entropy and entanglement spectrum. Additionally, we perform an analysis of the model under dynamics with a single, a pair of particles and for unit filling.