Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

International Journal of Modern Physics A

THE GEOMETRICAL PROPERTIES OF BERRY’S PHASE

S. Giler, Piotr Kosiński, L. Szymanowski

Published Apr 10, 1989• 16 citations
Publisher

Abstract

Using the spectral decomposition of the Hamiltonian, we clarify the relationship of Berry’s effect with the parallel transport in the parameter space of this Hamiltonian. We derive the necessary and sufficient conditions under which this parallel transport is described by a natural, Riemannian connection. The considerations are illustrated by several examples.

DOI
10.1142/s0217751x89000613

Related papers

GEOMETRIC METHODS AND THE GAUGE STRUCTURE OF BERRY'S PHASE

Bernard Francis Bednarz

The use of geometric methods to obtain the connection associated with Berry's phase is discussed. Failures and successes in past attempts at using only geometry are presented. It is argued that this approach is most useful for Hamiltonians rotated on the parameter space.

International Journal of Modern Physics A•Oct 10, 1989•2 citations

Interacting Topological Phases

Tudor D. Stanescu

T HE BERRY PHASE, A CENTRAL CONCEPT IN QUANTUM physics, is rooted in the state space structure of quantum theory and has a profound observable impact within a wide range of phenomena. The remarkably elegant interpretation of this geometric phase as the holonomy of a fiber bundle provides it with a mathematical framework capable of accounting for a broad spectrum of classical and quantum phenomena, such as, for example, the precession of a Foucault pendulum, the parallel transport of a vector on a sphere, the Aharonov-Bohm effect, the gauge theory of molecular physics, the electric polarization and orbital magnetization of solids, and the integer and fractional quantum Hall effects. The Berry phase is the central concept in topological band theory and provides the mechanism responsible for anyonic statistics in correlated many-body systems, a key element that could be used to achieve fault-tolerant quantum computation. In other words, the Berry phase is not just some fashionable idea useful for explaining a few exotic phenomena, but a generic feature of quantum mechanics and an essential ingredient necessary for a coherent understanding of basic quantum phenomena, such as, for example, the electronic properties of materials. It is therefore rather surprising that the full significance of geometric phases was practically ignored for more than fifty years prior to the publication of Berry’s work in 1984 [50]. In this chapter we discuss the basic physics that gives rise to these geometric phases, point out the elements that allow us to distinguish different types of holonomies, summarize the basic concepts relevant to understanding the mathematical structure of geometric phases, and mention a few relevant applications. The concepts and the mathematical framework introduced here will be used in subsequent chapters as the basis for describing the topological properties of condensed matter systems.

CRC Press eBooks•Dec 13, 2016•0 citations

Some geometrical considerations of Berry’s phase

J. Anandan, Leo Stodolsky

A group-theoretic treatment is given of the new phase found by Berry in the adiabatic evolution of a quantum-mechanical system in a finite-dimensional Hilbert space. It is shown how the Berry phases for the various eigenstates of the Hamiltonian are obtained from a set of angles associated with a group element. For the special case of a two-level system there is just one such angle which corresponds to the holonomy transformation associated with parallel transport around a closed curve on a sphere.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Apr 15, 1987•111 citations

TOPOLOGICAL PROPERTIES OF BERRY'S PHASE

KAZUO FUJIKAWA

By using a second quantized formulation of level crossing, which does not assume adiabatic approximation, a convenient formula for geometric terms including off-diagonal terms is derived. The analysis of geometric phases is reduced to a simple diagonalization of the Hamiltonian in the present formulation. If one diagonalizes the geometric terms in the infinitesimal neighborhood of level crossing, the geometric phases become trivial for any finite time interval T. The topological interpretation of Berry's phase such as the topological proof of phase-change rule thus fails in the practical Born–Oppenheimer approximation, where a large but finite ratio of two time scales is involved.

arXiv:quant-ph/0411006•Modern Physics Letters A•Feb 20, 2005•19 citations

The geometric and topological interpretation of Berry phase on a torus

Da-Bao Yang, Kun Meng, Yi-Zhi Wu +1 more

Illustration of the geometric and topological properties of Berry phase is often in an obscure and abstract language of fiber bundles. In this article, we demonstrate these properties with a lucid and concrete system whose parameter space is a torus. The instantaneous eigenstate is regarded as a basis. And the corresponding connection and curvature are calculated respectively. Furthermore, we find the magnitude of curvature is exactly the Gaussian curvature, which shows its local property. The topological property is reflected by the integral over the torus due to Gauss-Bonnet theorem. When we study the property of parallel transportation of a vector over a loop, we make a conclusion that the Berry phase is just the angle between the final and initial vectors. And we also illuminate the geometric meaning of gauge transformation, which is just a rotation of basis.

arXiv:1904.07671•Apr 16, 2019•1 citation

Berry’s phase on SU(n)/S(U(1)×U(n−1)) manifolds

Stefan Giller, Cezary Gonera, Piotr Kosiński +1 more

It is shown that the Berry phase/matrix for Hamiltonians parametrized by the points of M≡SU(n)/S(U(1)×U(n−1)) and having no accidental degeneracy is expressible in terms of Riemannian connection on M and representations of S(U(1)×U(n−1)).

Journal of Mathematical Physics•Feb 1, 1996•1 citation
Using the spectral decomposition of the Hamiltonian, we clarify the relationship of Berry’s effect with the parallel transport in the parameter space of this Hamiltonian. We derive the necessary and sufficient conditions under which this parallel transport is described by a natural, Riemannian connection. The considerations are illustrated by several examples.
T HE BERRY PHASE, A CENTRAL CONCEPT IN QUANTUM physics, is rooted in the state space structure of quantum theory and has a profound observable impact within a wide range of phenomena. The remarkably elegant interpretation of this geometric phase as the holonomy of a fiber bundle provides it with a mathematical framework capable of accounting for a broad spectrum of classical and quantum phenomena, such as, for example, the precession of a Foucault pendulum, the parallel transport of a vector on a sphere, the Aharonov-Bohm effect, the gauge theory of molecular physics, the electric polarization and orbital magnetization of solids, and the integer and fractional quantum Hall effects. The Berry phase is the central concept in topological band theory and provides the mechanism responsible for anyonic statistics in correlated many-body systems, a key element that could be used to achieve fault-tolerant quantum computation. In other words, the Berry phase is not just some fashionable idea useful for explaining a few exotic phenomena, but a generic feature of quantum mechanics and an essential ingredient necessary for a coherent understanding of basic quantum phenomena, such as, for example, the electronic properties of materials. It is therefore rather surprising that the full significance of geometric phases was practically ignored for more than fifty years prior to the publication of Berry’s work in 1984 [50]. In this chapter we discuss the basic physics that gives rise to these geometric phases, point out the elements that allow us to distinguish different types of holonomies, summarize the basic concepts relevant to understanding the mathematical structure of geometric phases, and mention a few relevant applications. The concepts and the mathematical framework introduced here will be used in subsequent chapters as the basis for describing the topological properties of condensed matter systems.
A group-theoretic treatment is given of the new phase found by Berry in the adiabatic evolution of a quantum-mechanical system in a finite-dimensional Hilbert space. It is shown how the Berry phases for the various eigenstates of the Hamiltonian are obtained from a set of angles associated with a group element. For the special case of a two-level system there is just one such angle which corresponds to the holonomy transformation associated with parallel transport around a closed curve on a sphere.
By using a second quantized formulation of level crossing, which does not assume adiabatic approximation, a convenient formula for geometric terms including off-diagonal terms is derived. The analysis of geometric phases is reduced to a simple diagonalization of the Hamiltonian in the present formulation. If one diagonalizes the geometric terms in the infinitesimal neighborhood of level crossing, the geometric phases become trivial for any finite time interval T. The topological interpretation of Berry's phase such as the topological proof of phase-change rule thus fails in the practical Born–Oppenheimer approximation, where a large but finite ratio of two time scales is involved.
Illustration of the geometric and topological properties of Berry phase is often in an obscure and abstract language of fiber bundles. In this article, we demonstrate these properties with a lucid and concrete system whose parameter space is a torus. The instantaneous eigenstate is regarded as a basis. And the corresponding connection and curvature are calculated respectively. Furthermore, we find the magnitude of curvature is exactly the Gaussian curvature, which shows its local property. The topological property is reflected by the integral over the torus due to Gauss-Bonnet theorem. When we study the property of parallel transportation of a vector over a loop, we make a conclusion that the Berry phase is just the angle between the final and initial vectors. And we also illuminate the geometric meaning of gauge transformation, which is just a rotation of basis.