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Semi-Classical Approach to the Two-Level Pairing Model Various Aspects of Phase Change

Yasuhiko Tsue, Constança Providência

Published Jan 1, 2006• 1 citation
OpenAlex

Abstract

Aspects of the phase change of the two-level pairing model are investigated in the semi-classical treatment by using the variational approch with the mixed-mode coherent state. In the classical limit, � → 0, the sharp phase transition appears and the two phases exist in the region where the force strength is larger than a certain critical value. However, it is shown that, in the semi-classical treatment, the abovementioned behavior of the phase change disappears in the analytical and the numerical treatments. As a result, new understanding about aspects of the phase change in this model is given. The phase structure and the phase transitions between various phases in condensed matter systems, such as nuclear and/or quark matter as well as the condensed matter in atoms and molecules, are important concepts to understand the nature of the matter under consideration. In the infinite systems, the clear phase separation is often realized and, in that

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Aspects of the phase change of the two-level pairing model are investigated in the semi-classical treatment by using the variational approch with the mixed-mode coherent state. In the classical limit, � → 0, the sharp phase transition appears and the two phases exist in the region where the force strength is larger than a certain critical value. However, it is shown that, in the semi-classical treatment, the abovementioned behavior of the phase change disappears in the analytical and the numerical treatments. As a result, new understanding about aspects of the phase change in this model is given. The phase structure and the phase transitions between various phases in condensed matter systems, such as nuclear and/or quark matter as well as the condensed matter in atoms and molecules, are important concepts to understand the nature of the matter under consideration. In the infinite systems, the clear phase separation is often realized and, in that
Properties of the phase change of the two-level pairing model are investigated in the semi-classical treatment by using the variational approach with a mixed-mode coherent state. In the classical limit, ħ → 0, a sharp phase transition appears, and the two phases exist in the region where the force strength is larger than a certain critical value. However, it is shown that, in the semi-classical treatment, the above-mentioned behavior of the phase change disappears in both analytical and numerical treatments. This leads to a new understanding of the properties of the phase change in this model.
We consider the pairing-plus-quadrupole model Hamiltonian within the framework of a many-fermion shell model. The Hamiltonian exhibits a transition from a superconducting to a rotational phase with variation of a parameter. The model has analytical solutions in its two limits due to the presence of dynamical symmetries. However, the symmetries are basically incompatible with one another which means that no simple solution exists in intermediate situations. Exact (numerical) solutions are nevertheless possible on a computer and enable one to study the behavior of competing but incompatible symmetries and the phase transitions that result in a semirealistic situation. This mixed-symmetry dynamics generates a rich structure which gives new insight for the study of phase transitions.
Various classical counterparts to the two-level pairing model in a many-fermion system are presented in the Schwinger boson representation. It is shown that one of the key ingredients giving the classical descriptions for quantal systems is the use of various trial states in addition to the su(2) ⊗ su(2)-coherent state, which may be a natural selection for the two-level pairing model governed by the su(2) ⊗ su(2)-algebra. It is pointed out that a spurious behavior such as a sharp phase transition can be avoided by using the other states including the su(2) ⊗ su(1,1)- and su(1,1) ⊗ su(1,1)-coherent states. This contrasts with the situation regarding the usual Hartree-Fock-Bogoliubov and quasi-particle random phase approximations applied to the original fermion system, which yield a sharp phase transition.
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.
Quantum and classical results are often presented as being dependent upon separate postulates as if the two are distinct and unrelated, and there is little attempt to show how the quantum implies the classical. The transformation to classical phase space gives researchers access to a range of algorithms derived from classical statistical mechanics that promise results on much more favourable numerical terms. Quantum Statistical Mechanics in Classical Phase Space offers not just a new computational approach to condensed matter systems, but also a unique conceptual framework for understanding the quantum world and collective molecular behaviour. A formally exact transformation, this revolutionary approach goes beyond the quantum perturbation of classical condensed matter to applications that lie deep in the quantum regime. It offers scalable computational algorithms and tractable approximations tailored to specific systems. Concrete examples serve to validate the general approach and demonstrate new insights. For example, the computer simulations and analysis of the λ-transition in liquid helium provide a new molecular-level explanation of Bose–Einstein condensation and a quantitative theory for superfluid flow. The intriguing classical phase space formulation in this book offers students and researchers a range of new computational algorithms and analytic approaches. It offers not just an efficient computational approach to quantum condensed matter systems, but also an exciting perspective on how the classical world that we observe emerges from the quantum mechanics that govern the behaviour of atoms and molecules. The applications, examples, and physical insights foreshadow new discoveries in quantum condensed matter systems. Key features • A computationally efficient approach to quantum condensed-matter many-particle systems based on a formally exact transformation of quantum statistical mechanics to classical phase space. • Shows how the observed classical world emerges from the underlying quantum mechanics of atoms and molecules. • Describes successive quantum corrections to classical statistical mechanics. • Derives a number of expansions that account for the non-commutativity of quantum position and momentum operators when transformed to classical phase space. • Tests various computer simulation algorithms against benchmark results for model condensed matter systems.
We study the normal modes and the stability of two component condensates in a phase separated regime. In such a regime, the system can undergo a quantum phase-transition upon the change of interaction strength between bosons of the same species or the variation of the trap frequencies. In this transition, the distribution of the two components changes from a symmetric to an asymmetric shape. We discuss the nature of the phase transition, the role of the interface tension, and the phase diagram. The symmetric to asymmetric transition is the simplest quantum phase transition that one can imagine. We found new branches of normal modes which are specific for trapped multicomponent condensates and are analogous to the waves at the interface between two layers of immiscible fluids under gravity. At the point of the phase transition, the frequencies of those modes go to 0 and become imaginary which causes an instability. The interface tension shifts the normal-mode frequencies and changes the stability region of the system.