A Brief Introduction to Quantum Phase Transitions

Marek M. Rams

Achim Rosch

At a quantum critical point the ground state of a many-particle quantum system undergoes a phase transition when some control parameters such as pressure, magnetic field, or chemical composition is varied. The universal behaviour characteristic for such quantum phase transitions often affects a wide temperature range and gives rise to novel material properties. This chapter gives a short overview over five lectures held during the Les Houches summer school ‘Quantum theory from small to large scales’ in 2010. After an introductory chapter, field driven magnetic quantum phase transitions of insulators are used as an example to discuss some of the concepts underlying quantum phase transitions. Both experimentally and theoretically, quantum phase transitions in metals are much less understood compared to insulating systems. After a brief review of the standard approach to describe those systems, the importance of multiple time scales and associated multiple critical exponents z are discussed. Finally, emergent gauge theories close to critical points are investigated. As an example, it discusses why a gauge theory describes the (classical) phase transitions of a nematic, if topological defects are suppressed.

Immanuel Bloch

Ultracold atoms in optical lattices have proven to be a versatile model system in which strong correlation effects in a many-body quantum system can be studied [1]. The tunability and control over almost all underlying system parameters makes them rather unique and allows one to switch between different ground states of the underlying many-body quantum system. One of the most prominent effects in this respect is the quantum phase transition (QPT) from a superfluid (SF) to a Mott insulator (MI) that started the field of strong-correlation physics with ultracold gases. First theoretically proposed by Fisher et al. [2], the Bose-Hubbard model (BHM) was introduced into the field of ultracold atoms by Jaksch et al. [3], eventually being realized in subsequent experiments [4-7]. This article focuses on giving a brief summary of the main results achieved in the SF-MI transition with ultracold bosons in optical lattices. A brief introduction into the setup of optical lattices is followed by a discussion of key results such as the coherence properties, the density profiles, transport properties and SF characteristics across the QPT.

Antônio Sérgio Teixeira Pires

Salvish Goomanee

In this short paper, a very brief introduction of the quantum harmonic oscillator has been provided. The understanding of the transition from a classically damped oscillator and the quantum mechanical model of the latter is included via a clear mathematics involving simple calculus. Only the one dimensional time dependent Schrodinger’s equation is being studied. This paper is intended to simply provide an analytical description of the subject, there are no advance calculations implying further suggestions in quantum theory.

M. A. Contínentino