Anholonomy of a moving space curve and applications to classical magnetic chains
Abstract
The subject of space curves finds many applications in physics such as optical fibers, magnetic spin chains, and vortex filaments in a fluid. We show that the time evolution of a space curve is associated with a geometric phase. Using the concept of Fermi-Walker parallel transport, we show that this phase arises because of the path dependence of the rotation of the natural Frenet-Serret triad as one moves along the curve. We employ Lamb's formalism for space-curve dynamics to derive an expression for the anholonomy density and the geometric phase for a general time evolution. This anholonomy manifests itself as a gauge potential with a monopolelike structure in the space of the tangent vector to the space curve. Our classical approach is amenable to a quantum generalization, which can prove useful in applications. We study the application of our constructive formalism to ferromagnetic and antiferromagnetic (classical) spin chains by first presenting certain classes of exact, physically interesting solutions to these nonlinear dynamical systems and then computing the corresponding geometric phases.
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The Berry phase is analogous to the Aharonov-Bohm (or Dirac) phase in quantum mechanics.
“At the same time, it shows that the Berry phase is analogous to the Aharonov -Bohm (or Dirac) phase in quantum mechanics [94 -96].”
From Geometrodynamics of polarized light: Berry phase and spin Hall effect in a gradient-index medium (2009) · cites this paper as [94]
Cites
Quantal phase factors accompanying adiabatic changes
Berry, M. V. (1984) · Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences · doi:10.1098/rspa.1984.0023 · Cited by 9,252 · see citing works
Haldane, F. D. M. (1983) · Physical Review Letters · doi:10.1103/physrevlett.50.1153 · Cited by 3,678 · see citing works
Phase change during a cyclic quantum evolution
Aharonov, Y., & Anandan, J. (1987) · Physical Review Letters · doi:10.1103/physrevlett.58.1593 · Cited by 2,201 · see citing works
Rajaraman, R. (1982) · Cited by 1,934 · see citing works
Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase
Simon, B. (1983) · Physical Review Letters · doi:10.1103/physrevlett.51.2167 · Cited by 1,884 · see citing works
The soliton: A new concept in applied science
Scott, A. C., Chu, F. Y. F., & McLaughlin, D. W. (1973) · Proceedings of the IEEE · doi:10.1109/proc.1973.9296 · Cited by 1,686 · see citing works
A soliton on a vortex filament
Hasimoto, H. (1972) · Journal of Fluid Mechanics · doi:10.1017/s0022112072002307 · Cited by 1,006 · see citing works
General Setting for Berry's Phase
Samuel, J., & Bhandari, R. (1988) · Physical Review Letters · doi:10.1103/physrevlett.60.2339 · Cited by 997 · see citing works
Observation of Berry's Topological Phase by Use of an Optical Fiber
Tomita, A., & Chiao, R. Y. (1986) · Physical Review Letters · doi:10.1103/physrevlett.57.937 · Cited by 810 · see citing works
A Treatise on the Differential Geometry of Curves and Surfaces
Eisenhart, L. P. (1960) · Cited by 598 · see citing works
Haldane, F. D. M. (1988) · Physical Review Letters · doi:10.1103/physrevlett.61.1029 · Cited by 511 · see citing works
Continuum spin system as an exactly solvable dynamical system
Lakshmanan, M. (1977) · Physics Letters A · doi:10.1016/0375-9601(77)90262-6 · Cited by 479 · see citing works
Integration of the continuous Heisenberg spin chain through the inverse scattering method
Takhtajan, L. A. (1977) · Physics Letters A · doi:10.1016/0375-9601(77)90727-7 · Cited by 450 · see citing works
Solitons on moving space curves
Lamb, G. L. (1977) · Journal of Mathematical Physics · doi:10.1063/1.523453 · Cited by 317 · see citing works
Solitons in the continuous Heisenberg spin chain
Tjon, J. A., & Wright, J. (1977) · Physical review. B, Solid state · doi:10.1103/physrevb.15.3470 · Cited by 277 · see citing works
On the dynamics of a continuum spin system
Lakshmanan, M., Ruijgrok, T. W., & Thompson, C. (1976) · Physica A Statistical Mechanics and its Applications · doi:10.1016/0378-4371(76)90106-0 · Cited by 229 · see citing works
Solitons and magnons in the classical Heisenberg chain
Fogedby, H. C. (1980) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/13/4/035 · Cited by 156 · see citing works
Geometric phase in the classical continuous antiferromagnetic Heisenberg spin chain
Balakrishnan, R., Bishop, A. R., & Dandoloff, R. (1990) · Physical Review Letters · doi:10.1103/physrevlett.64.2107 · Cited by 135 · see citing works
Self-citationBerry’s phase, locally inertial frames, and classical analogues
Kugler, M., & Shtrikman, S. (1988) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.37.934 · Cited by 106 · see citing works
On the inhomogeneous Heisenberg chain
Балакришнан, Р. (1982) · Journal of Physics C Solid State Physics · doi:10.1088/0022-3719/15/36/007 · Cited by 78 · see citing works
Parallel transport along a space curve and related phases
Dandoloff, R., & ZAKRZEWSKI, W. J. (1989) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/22/11/003 · Cited by 73 · see citing works
Self-citationHoles in a quantum antiferromagnet: A formalism and some exact results
Shankar, R. (1990) · Nuclear Physics B · doi:10.1016/0550-3213(90)90118-w · Cited by 64 · see citing works
GEOMETRIC PHASE IN THE CLASSICAL CONTINUOUS ANTIFERROMAGNETIC HEISENBERG SPIN CHAIN
Balakrishnan, R., Bishop, A. R., & Dandoloff, R. (1990) · Modern Physics Letters B · doi:10.1142/s0217984990001264 · Cited by 2 · see citing works
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Geometrodynamics of polarized light: Berry phase and spin Hall effect in a gradient-index medium
Bliokh, K. Y. (2009) · Journal of Optics A: Pure and Applied Optics · doi:10.1088/1464-4258/11/9/094009 · Cited by 217 · see citing works
- Authors
- Balakrishnan, R., Bishop, A. R., & Dandoloff, R.
- Affiliations
- Institute of Mathematical Sciences (IN)
- Venue
- Physical Review B
- Publisher
- American Physical Society (APS)
- Published
- 1993-02-01
- Volume
- 47
- Issue
- 6
- Pages
- 3108-3117
- ISSN
- 0163-1829 (ISSN-L)
- 1095-3795
- OpenAlex
- W2050137475
- Cited by
- 122
- References
- 23
- Citations by year
- 2025: 3 · 2024: 4 · 2023: 28 · 2022: 22 · 2021: 32 · 2020: 2 · 2019: 3 · 2018: 1 · 2017: 2
- FWCI
- 3.60
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Physical review. B, Condensed matter (publishedVersion)
- PubMed (publishedVersion)
- Topics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Advanced Differential Geometry Research
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Nonlinear Waves and Solitons
- Keywords
- anholonomy, space curves, Berry phase, gauge potential, monopoles, classical spin chain, ferromagnetic spin chain, antiferromagnetic chains, exact solutions, nonlinear dynamical systems
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