Aspects of Berry phase in QFT

Abstract

When continuous parameters in a QFT are varied adiabatically, quantum states typically undergo mixing -a phenomenon characterized by the Berry phase. We initiate a systematic analysis of the Berry phase in QFT using standard quantum mechanics methods. We show that a non-trivial Berry phase appears in many familiar QFTs. We study a variety of examples including free electromagnetism with a theta angle, and certain supersymmetric QFTs in two and four spacetime dimensions. We also argue that a large class of QFTs with rich Berry properties is provided by CFTs with non-trivial conformal manifolds. Using the operator-state correspondence we demonstrate in this case that the Berry connection is equivalent to the connection on the conformal manifold derived previously in conformal perturbation theory. In the special case of chiral primary states in 2d N = (2, 2) and 4d N = 2 SCFTs the Berry phase is governed by the tt * equations. We present a technically useful rederivation of these equations using quantum mechanics methods.

Conformal manifolds: ODEs from OPEs

“…Subtleties arise when there is degeneracy and especially when there is more than one marginal operator. In this case, there is a non-trivial Zamolodchikov metric and the curvatures built up from it become interesting observables that affect how the equations for d∆ i dg and dλ ijk dg must be defined [14,15]. Even after we limit ourselves to a single marginal operator, these equations can only be written down once the appropriate conformal block expansions are known.…”

“…While it is still JHEP03(2018)127 possible to use a scheme that removes such mixing at a point, there is no guarantee that we may do so globally [14,15].…”

“…Our ability to quickly take the second derivative is only guaranteed because we are considering a manifold that has only one marginal operator and therefore a flat connection [14,15]. Three values of ∆ lead to small denominators in (3.16) but the only one that can dominate the sum is ∆ = ∆ 2 .…”

Geometry of Higgs-branch superconformal primary bundles

“…Ref. [6] showed that the curvature of the Berry connection associated to the variation of the exactly marginal couplings is identical to the curvature of the connection that was used in [14] in conformal perturbation theory. In the rest of the paper we will adopt the Hamiltonian approach in radial quantization and will work primarily with the corresponding Berry connection.…”

“…Alternatively, a natural connection arises in radial quantization as the Pancharatnam-Berry connection [3][4][5] on the vector bundle of Hilbert spaces. This connection, which has a specific counterpart in conformal perturbation theory [6], will play a key rôle in the discussion of this paper. We will review the specifics of the Berry connection that we need in section 2.…”