Topological phase in classical mechanics

Malykin, G. B.; Kharlamov, Sergei A.

doi:10.1070/pu2003v046n09abeh001635

Cited by 14 publications

(6 citation statements)

References 28 publications

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“…If the direction of the axis is altered, the gyroscopic device will change its orientation due to the effect of the parallel transport. This phenomenon was studied as an unwanted feature of the inertial navigation systems based on such gyroscopic devices [23].…”

Section: Parallel Transport In Classical Mechanicsmentioning

confidence: 99%

Topological insulators and geometry of vector bundles

Сергеев

2023

SciPost Phys. Lect. Notes

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For a long time, band theory of solids has focused on the energy spectrum, or Hamiltonian eigenvalues. Recently, it was realized that the collection of eigenvectors also contains important physical information. The local geometry of eigenspaces determines the electric polarization, while their global twisting gives rise to the metallic surface states in topological insulators. These phenomena are central topics of the present notes. The shape of eigenspaces is also responsible for many intriguing physical analogies, which have their roots in the theory of vector bundles. We give an informal introduction to the geometry and topology of vector bundles and describe various physical models from this mathematical perspective.

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Section: Parallel Transport In Classical Mechanicsmentioning

confidence: 99%

Topological insulators and geometry of vector bundles

Сергеев

2023

SciPost Phys. Lect. Notes

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“…This sheds new light on the dynamical approach introduced in Section II, with h eff (t) being in clear correspondence with the effective field defined by Eq. (7).…”

Section: Field Driven Curvature and Geometrical Torquementioning

confidence: 99%

“…Starting from these seminal works, the concept of geometric phase has been further developed, setting its relation with the area enclosed by the cyclic trajectory on the corresponding domain of the projective space. This approach has further led to the remarkable observation that there is a nontrivial geometric phase even for classical systems [6][7][8]. Alternative advancements have brought to the construction of the geometric phase in non-cyclic evolution [9][10][11] where, for an arbitrary quantum trajectory, it is also possible to show that the integral of the uncertainty of energy with respect to time is independent of the particular Hamiltonian used to transport the quantum system along a given curve in the projective Hilbert space [11].…”

Section: Introductionmentioning

confidence: 99%

Geometric driving of two-level quantum systems

Ying

Gentile

Baltanás

et al. 2020

Phys. Rev. Research

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We investigate a class of cyclic evolutions for driven two-level quantum systems (effective spin-1/2) with a particular focus on the geometric characteristics of the driving and their specific imprints on the quantum dynamics. By introducing the concept of geometric field curvature for any field trajectory in the parameter space we are able to unveil underlying patterns in the overall quantum behavior: the knowledge of the field curvature provides a non-standard and fresh access to the interrelation between field and spin trajectories, and the corresponding quantum phases acquired in non-adiabatic cyclic evolutions. In this context, we single out setups in which the driving field curvature can be employed to demonstrate a pure geometric control of the quantum phases. Furthermore, the driving field curvature can be naturally exploited to introduce the geometrical torque and derive a general expression for the total quantum phase acquired in a cycle. Remarkably, such relation allows to access the mechanisms controlling the changeover of the quantum phase across a topological transition and to disentangle the role of the spin and field topological windings. As for implementations, we discuss a series of physical systems and platforms to demonstrate how the geometric control of the quantum phases can be realized for pendular field drivings. This includes setups based on superconducting islands coupled to a Josephson junction and inversion asymmetric nanochannels with suitably tailored geometric shapes.

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“…In spite of the simplicity, this result had a great influence on the subsequent development of the theory of dynamical systems and found numerous applications [1,2,3,4,5]. However, until now the question, which systems can be described by the Hamiltonian of the GHO is still open.…”

Section: Generalized Harmonic Oscillatormentioning

confidence: 99%

“…The existence of one more remarkable property of this equation, the so-called geometric phase, was noticed only 80 years later. Historical aspects of the development of ideas related to the understanding of the properties of solutions of differential equations with slowly varying parameters as well as their theoretical, experimental and applied aspects one can find in many reviews and books (see, for example, [1,2,3,4,5]).…”

Section: Introductionmentioning

confidence: 99%

Adiabatic dynamics of one-dimensional classical Hamiltonian dissipative systems

2018

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We give an example of a simple mechanical system described by the generalized harmonic oscillator equation, which is a basic model in discussion of the adiabatic dynamics and geometric phase. This system is a linearized plane pendulum with the slowly varying mass and length of string and the suspension point moving at a slowly varying speed, the simplest system with broken T -invariance. The paradoxical character of the presented results is that the same Hamiltonian system, the generalized harmonic oscillator in our case, is canonically equivalent to two different systems: the usual plane mathematical pendulum and the damped harmonic oscillator. This once again supports the important mathematical conclusion, not widely accepted in physical community, of no difference between the dissipative and Hamiltonian 1D systems, which stems from the Sonin theorem that any Newtonian second order differential equation with a friction of general nature may be presented in the form of the Lagrange equation.

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Topological phase in classical mechanics

Cited by 14 publications

References 28 publications

Topological insulators and geometry of vector bundles

Topological insulators and geometry of vector bundles

Geometric driving of two-level quantum systems

Adiabatic dynamics of one-dimensional classical Hamiltonian dissipative systems

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