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Quantum phase transitions and geometric phases

Zhu Shi

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

Quantum phase transition is one of main interests in the field of condensed matter physics. Similarly, geometric phases arrest considerable interest in the field of quantum mechanics. However, no any relevant relation was recognized before recent work. In this communication, we mainly introduce the recent results on the connections between quantum phase transitions and geometric phases: a non-contractible geometric phase is a signature of quantum phase transitions, especially all key ingredients of quantum criticality are present in geometric phases of the ground state. These results may arrest interest from both communities.

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Quantum phase transition is one of main interests in the field of condensed matter physics. Similarly, geometric phases arrest considerable interest in the field of quantum mechanics. However, no any relevant relation was recognized before recent work. In this communication, we mainly introduce the recent results on the connections between quantum phase transitions and geometric phases: a non-contractible geometric phase is a signature of quantum phase transitions, especially all key ingredients of quantum criticality are present in geometric phases of the ground state. These results may arrest interest from both communities.
Quantum phase transition is one of the main interests in the field of condensed matter physics, while geometric phase is a fundamental concept and has attracted considerable interest in the field of quantum mechanics. However, no relevant relation was recognized before recent work. In this paper, we present a review of the connection recently established between these two interesting fields: investigations in the geometric phase of the many-body systems have revealed the so-called "criticality of geometric phase", in which the geometric phase associated with the many-body ground state exhibits universality, or scaling behavior in the vicinity of the critical point. In addition, we address the recent advances on the connection of some other geometric quantities and quantum phase transitions. The closed relation recently recognized between quantum phase transitions and some of the geometric quantities may open attractive avenues and fruitful dialogue between different scientific communities.
In recent years, quantum phase transitions have attracted the interest of both theorists and experimentalists in condensed matter physics. These transitions, which are accessed at zero temperature by variation of a non-thermal control parameter, can influence the behaviour of electronic systems over a wide range of the phase diagram. Quantum phase transitions occur as a result of competing ground state phases. The cuprate superconductors which can be tuned from a Mott insulating to a d -wave superconducting phase by carrier doping are a paradigmatic example. This review introduces important concepts of phase transitions and discusses the interplay of quantum and classical fluctuations near criticality. The main part of the article is devoted to bulk quantum phase transitions in condensed matter systems. Several classes of transitions will be briefly reviewed, pointing out, e.g., conceptual differences between ordering transitions in metallic and insulating systems. An interesting separate class of transitions is boundary phase transitions where only degrees of freedom of a subsystem become critical; this will be illustrated in a few examples. The article is aimed at bridging the gap between high-level theoretical presentations and research papers specialized in certain classes of materials. It will give an overview on a variety of different quantum transitions, critically discuss open theoretical questions, and frequently make contact with recent experiments in condensed matter physics.
During recent years the interest to dynamics of quantum systems has grown considerably. Quantum many body systems out of equilibrium often manifest behavior, different from the one predicted by standard statistical mechanics and thermodynamics in equilibrium. Since the dynamics of a many body quantum system typically involve many excited eigenstates, with a non-thermal distribution, the time evolution of such a system provides an unique way for investigation of non-equilibrium quantum statistical mechanics. Last decade such new subjects like quantum quenches, thermalization, pre-thermalization, equilibration, generalized Gibbs ensemble, etc. are among the most attractive topics of investigation in modern quantum physics. One of the most interesting themes in the study of dynamics of quantum many-body systems out of equilibrium is connected with the recently proposed important concept of dynamical quantum phase transitions. During the last few years a great progress has been achieved in studying of those singularities in the time dependence of characteristics of quantum mechanical systems, in particular, in understanding how the quantum critical points of equilibrium thermodynamics affect their dynamical properties. Dynamical quantum phase transitions reveal universality, scaling, connection to the topology, and many other interesting features. Here we review the recent achievements of this quickly developing part of low temperature quantum physics. The study of dynamical quantum phase transitions is especially important in context of their connection to the problem of the modern theory of quantum information, where namely non-equilibrium dynamics of many-body quantum system plays the major role.
Geometric phases play a central role in a variety of quantum phenomena, especially in condensed matter physics. Recently, it was shown that this fundamental concept exhibits a connection to quantum phase transitions where the system undergoes a qualitative change in the ground state when a control parameter in its Hamiltonian is varied. Here we report the first experimental study using the geometric phase as a topological test of quantum transitions of the ground state in a Heisenberg XY spin model. Using NMR interferometry, we measure the geometric phase for different adiabatic circuits that do not pass through points of degeneracy.
The connection between the geometric phase and quantum phase transition has been discussed extensively in the two-band model. By introducing the twist operator, the geometric phase can be defined by calculating its ground-state expectation value. In contrast to the previous numerical examinations, our discussion presents an exact calculation for the determination of the geometric phase. Through two representative examples, our calculation shows the intimate connection between the geometric phase and phase transition: different behaviors of the geometric phase can be identified in this paper, which are directly related to the energy gap above the ground state.