Berry connection and curvature

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. These concepts were introduced by Michael Berry in a paper published in 1984 emphasizing how geometric phases provide a powerful unifying concept in several branches of classical and quantum physics. 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. These concepts were introduced by Michael Berry in a paper published in 1984 emphasizing how geometric phases provide a powerful unifying concept in several branches of classical and quantum physics. In quantum mechanics, the Berry phase arises in a cyclic adiabatic evolution. The quantum adiabatic theorem applies to a system whose Hamiltonian H ( R ) {displaystyle H(mathbf {R} )} depends on a (vector) parameter R {displaystyle mathbf {R} } that varies with time t {displaystyle t} . If the n {displaystyle n} 'th eigenvalue ε n ( R ) {displaystyle varepsilon _{n}(mathbf {R} )} remains non-degenerate everywhere along the path and the variation with time t is sufficiently slow, then a system initially in the eigenstate | n ( R ( 0 ) ) ⟩ {displaystyle ,|n(mathbf {R} (0)) angle } will remain in an instantaneous eigenstate | n ( R ( t ) ) ⟩ {displaystyle ,|n(mathbf {R} (t)) angle } of the Hamiltonian H ( R ( t ) ) {displaystyle ,H(mathbf {R} (t))} , up to a phase, throughout the process. Regarding the phase, the state at time t can be written as where the second exponential term is the 'dynamic phase factor.' The first exponential term is the geometric term, with γ n {displaystyle gamma _{n}} being the Berry phase. From the requirement that | Ψ n ( t ) ⟩ {displaystyle |Psi _{n}(t) angle } satisfy the time-dependent Schrödinger equation, it can be shown that