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The Journal of Physical Chemistry A

Quantum and Classical Connections and Topological Phases: A Study of a Perturbed Rotator

Michael Baer, Gert D. Billing

Published Nov 17, 2000• 1 citation
Publisher

Abstract

In this paper are discussed the quantum mechanical connection and its classical counterpart, as well as the quantum and classical topological angles. The two were calculated for a model system to see whether they show any resemblance. The most interesting result we found is that the quantum and the classical topological phases are each formed due to a single discontinuous jump that occurs at some time during the external cycle. However, whereas the quantum mechanical topological phase is formed in the usual adiabatic limit, we had to establish a new parameter which, by allowing it to approach its limiting value, forms the classical topological phase.

DOI
10.1021/jp002316z

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In this paper are discussed the quantum mechanical connection and its classical counterpart, as well as the quantum and classical topological angles. The two were calculated for a model system to see whether they show any resemblance. The most interesting result we found is that the quantum and the classical topological phases are each formed due to a single discontinuous jump that occurs at some time during the external cycle. However, whereas the quantum mechanical topological phase is formed in the usual adiabatic limit, we had to establish a new parameter which, by allowing it to approach its limiting value, forms the classical topological phase.
A semiclassical connection is established between quantal and classical properties of a system whose Hamiltonian is slowly cycled by varying its parameters round a circuit. The quantal property is a geometrical phase shift gamma n associated with an eigenstate with quantum numbers n=(n l ); the classical property is a shift Delta theta l (I) in the lth angle variable for motion round a phase-space torus with actions I=(I l ); the connection is Delta theta l =- delta gamma / delta n l . Two applications are worked out in detail: the generalised harmonic oscillator, with quadratic Hamiltonian whose parameters are the coefficients of q 2 , qp and p 2 ; and the rotated rotator, consisting of a particle sliding freely round a non-circular hoop slowly turned round once in its own plane.
The work presented in this thesis meanly addresses two topics in theoretical physics which are quantum thermodynamics and topological order. In the first case, physicists are trying to build up a theory able to describe quite in general phenomena involving heat and energy exchanges in quantum systems. The second topic, instead, is related to exotic phenomena and states of matter like the quantum Hall effect (QHE) or topological insulators and topological superconductors. In the first part od the thesis we define the quantum dynamics for closed and open systems. This is a key ingredient to address the field of quantum thermodynamics. Then, after an introductory part about the quantum thermodynamic transformations, we move toward the field of nonequilibrium fluctuation relations. We address the problem of irreversibility in classical as well as quantum mechanics. Here we present one of our main result. We characterize the "thermodynamic" irreversible adiabatic evolution of a quantum system starting such branch in a thermal equilibrium state at inverse temperature ßi. We give the amount of thermodynamic entropy growth for the process. As direct application of the preceding result we then address a quantum Otto cycle (QOC) working at finite power. We saw that the increasing of irreversible character of the evolution affects the main figures of merit of the cycle. The second part of the thesis addresses the field of topological order. At first we introduce the concept of topological orders, classes and invariants. Then we introduce the well known Kitaev model for 1 D superconductors. This model predicts Majorana zero mode at the ends of the wire (the 1 D system). MZM are topological states showing great resistance against disorder, local perturbations and any dissipative element. Then we consider a generalized Kitaev model where long range interactions are accounted. We get rich topological phase diagrams showing the presence of several MZM per edge. We study the appearing/disappearing dynamics of the modes according to the time reversal symmetry, that is fundamental in the study of topological phase. The phase diagrams we obtained also show the presence of massive edge modes. In this last case the topological invariants do not well describe any transition. At last we focused on a very limit cases where MZM are obtained at finite length of the wire. Such cases are really interesting since the great advance we can get from the finiteness of the wire in an experimental setup. The last part is about single electron tunneling devices. Here we got a different ability to work as "heat-to-current harvester" for a device using quantum dots respect to an analogue one using metallic dots. These different arguments find their unity by considering recent scientific works in which heat transport is addressed in single electron transistor devices where some element of the circuit shows a topological behaviour. This is a perfect system from hich we can get new transport phenomena.
Using the symplectic formalism of classical mechanics, a smooth holonomy effect is identified in a special class of classical ergodic Hamiltonian systems (generalized canonical families) that are cycled adiabatically. It is a smooth shift in a time-resembling observable `tempus' that is canonically conjugated to the phase space volume contained in the energy shell. The curvature 2-form of the adiabatic connection obtained is closely related to Hannay's and the Robbins - Berry 2-form. The measurability of the `tempus' is discussed.
We study analytically the superfluid flow of a Bose-Einstein condensate in a ring geometry in the presence of a rotating barrier. We show that a phase transition breaking a parity symmetry among two topological phases occurs at a critical value of the height of the barrier. Furthermore, a discontinuous (accompanied by hysteresis) phase transition is observed in the ordered phase when changing the angular velocity of the barrier. At the critical point where the hysteresis area vanishes, the chemical potential of the ground state develops a cusp (a discontinuity in the first derivative). Along this path, the jump between the two corresponding states having a different winding number shows analogies with a topological phase transition. We finally study the current-phase relation of the system and compare some of our calculations with published experimental results.
The Berry phase arises naturally from the adiabatic dynamics of a pure quantum state, whereas the Uhlmann phase is usually formulated kinematically for a prescribed path of mixed states. For a qubit subject to a uniformly gapped conical drive, we obtain the Uhlmann connection and holonomy in closed form for the equilibrium Gibbs cycle. We then solve the corresponding Lindblad equation exactly in the rotating frame and show that its steady state forms a nonequilibrium limit cycle that lags the instantaneous Gibbs cycle. As the driving becomes slow, the Uhlmann phase approaches the equilibrium Uhlmann phase, which cooling then reduces to the Berry phase: in the joint adiabatic and low-temperature limit, the dynamical Uhlmann phase therefore converges to the Berry phase. This grounds the Uhlmann--Berry correspondence in the physical dynamics of an open system. We also characterize the finite-driving corrections, together with the discontinuous πππ jump that the Uhlmann phase undergoes on the equatorial cycle as the temperature is varied. Finally, we examine an isolated transversal gap closing, where this zero-temperature correspondence can break down: the pure-state path becomes open, its closure is ambiguous, and gap-opening regularizations close it along a geodesic set by the direction of the bias field, splitting the Berry phase into a one-parameter family of values. The Gibbs path instead closes smoothly through the maximally mixed state, so the Uhlmann holonomy requires no regularization and remains unique and continuous at every finite temperature. In the low-temperature limit the Uhlmann phase selects a single member of the Berry family---the geodesic closure in the osculating plane, fixed by the velocity and acceleration of the driving field. The Uhlmann--Berry correspondence of the gapped regime thus survives the gap closing as a geodesic selection rule.
The purpose of this paper is to give a somewhat expanded argument of the concept of the canonical phase recently investigated. This newly identified phase is considered to be a nonintegrable phase defined over the generalized phase space which is regarded as an integral realization of a “quantum (Hilbert) connection.” We formulate the theory in terms of the coherent state path integral. A particular interest is focused on the topological quantization in connection with the representation of compact Lie group and its application to the many-particle problem. We also examine the geometric characteristics of the canonical phase by using a spin model.