In physics, Berry connection and Berry curvature are related concepts which can be viewed, respectively, as a local gauge potential and gauge field associated with the Berry phase or geometric phase. The concept was first introduced by S. Pancharatnam[1] as geometric phase and later elaborately explained and popularized by Michael Berry in a paper published in 1984[2] emphasizing how geometric phases provide a powerful unifying concept in several branches of classical and quantum physics.
Berry phase and cyclic adiabatic evolution
In quantum mechanics, the Berry phase arises in a cyclic adiabatic evolution. The quantum adiabatic theorem applies to a system whose Hamiltonian depends on a (vector) parameter that varies with time . If the 'th eigenvalue remains non-degenerate everywhere along the path and the variation with time t is sufficiently slow, then a system initially in the normalized eigenstate will remain in an instantaneous eigenstate of the Hamiltonian , up to a phase, throughout the process. Regarding the phase, the state at time t can be written as[3]
In the case of a cyclic evolution around a closed path such that , the closed-path Berry phase is
Gauge transformation
A gauge transformation can be performed
Berry connection
The closed-path Berry phase defined above can be expressed as
Berry curvature
The Berry curvature is an anti-symmetric second-rank tensor derived from the Berry connection via
For a closed path that forms the boundary of a surface , the closed-path Berry phase can be rewritten using Stokes' theorem as
Finally, by using for , the Berry curvature can also be written as a summation over all the other eigenstates in the form
Example: Spinor in a magnetic field
The Hamiltonian of a spin-1/2 particle in a magnetic field can be written as[3]
The Berry curvature per solid angle is given by . In this case, the Berry phase corresponding to any given path on the unit sphere in magnetic-field space is just half the solid angle subtended by the path. The integral of the Berry curvature over the whole sphere is therefore exactly , so that the Chern number is unity, consistent with the Chern theorem.
Applications in crystals
The Berry phase plays an important role in modern investigations of electronic properties in crystalline solids[5] and in the theory of the quantum Hall effect.[6] The periodicity of the crystalline potential allows the application of the Bloch theorem, which states that the Hamiltonian eigenstates take the form
References
- ^ Pancharatnam, S. (November 1956). "Generalized theory of interference, and its application". Proc. Indian Acad. Sci. 44 (5): 247–262. doi:10.1007/BF03046050. S2CID 118184376.
- ^ Berry, M. V. (1984). "Quantal Phase Factors Accompanying Adiabatic Changes". Proceedings of the Royal Society A. 392 (1802): 45–57. Bibcode:1984RSPSA.392...45B. doi:10.1098/rspa.1984.0023. S2CID 46623507.
- ^ a b Sakurai, J.J. (2005). Modern Quantum Mechanics. Vol. Revised Edition. Addison–Wesley.[permanent dead link]
- ^ Resta, Raffaele (2000). "Manifestations of Berry's phase in molecules and in condensed matter". J. Phys.: Condens. Matter. 12 (9): R107–R143. Bibcode:2000JPCM...12R.107R. doi:10.1088/0953-8984/12/9/201. S2CID 55261008.
- ^ a b c d e Xiao, Di; Chang, Ming-Che; Niu, Qian (Jul 2010). "Berry phase effects on electronic properties". Rev. Mod. Phys. 82 (3): 1959–2007. arXiv:0907.2021. Bibcode:2010RvMP...82.1959X. doi:10.1103/RevModPhys.82.1959. S2CID 17595734.
- ^ Thouless, D. J.; Kohmoto, M.; Nightingale, M. P.; den Nijs, M. (Aug 1982). "Quantized Hall Conductance in a Two-Dimensional Periodic Potential". Phys. Rev. Lett. 49 (6). American Physical Society: 405–408. Bibcode:1982PhRvL..49..405T. doi:10.1103/PhysRevLett.49.405.
- ^ Chang, Ming-Che; Niu, Qian (2008). "Berry curvature, orbital moment, and effective quantum theory of electrons in electromagnetic fields". Journal of Physics: Condensed Matter. 20 (19): 193202. Bibcode:2008JPCM...20s3202C. doi:10.1088/0953-8984/20/19/193202. S2CID 35936765.
- ^ Resta, Raffaele (2010). "Electrical polarization and orbital magnetization: the modern theories". J. Phys.: Condens. Matter. 22 (12): 123201. Bibcode:2010JPCM...22l3201R. doi:10.1088/0953-8984/22/12/123201. PMID 21389484. S2CID 18645988.
External links
- The quantum phase, five years after. by M. Berry.
- Berry Phases and Curvatures in Electronic Structure Theory A talk by D. Vanderbilt.
- Berry-ology, Orbital Magnetolectric Effects, and Topological Insulators - A talk by D. Vanderbilt.