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Aspects of Berry phase in QFT

Marco Baggio, Vasilis Niarchos, Kyriakos Papadodimas

TL;DR

This work develops a comprehensive Berry-phase framework for quantum field theories, showing that adiabatic parameter variations induce non-trivial geometric phases even in simple QFTs like electromagnetism with a theta term. By leveraging the operator-state correspondence, conformal perturbation theory, and SUSY nonrenormalization properties, the authors connect Berry connections to the geometry of conformal manifolds and Seiberg–Witten moduli spaces, yielding exact results (tt* equations) in 2d N=(2,2) and 4d N=2 SCFTs. They derive explicit curvature formulas for various settings, including Coulomb-branch photons and massive N=1 theories on R × T^3, and establish a general equivalence between Berry curvature and conformal perturbation theory data. The results illuminate how operator mixing under adiabatic deformations encodes deep geometric information about the moduli spaces of QFTs and offer potential experimental probes in systems with tunable theta or coupling parameters.

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.

Aspects of Berry phase in QFT

TL;DR

This work develops a comprehensive Berry-phase framework for quantum field theories, showing that adiabatic parameter variations induce non-trivial geometric phases even in simple QFTs like electromagnetism with a theta term. By leveraging the operator-state correspondence, conformal perturbation theory, and SUSY nonrenormalization properties, the authors connect Berry connections to the geometry of conformal manifolds and Seiberg–Witten moduli spaces, yielding exact results (tt* equations) in 2d N=(2,2) and 4d N=2 SCFTs. They derive explicit curvature formulas for various settings, including Coulomb-branch photons and massive N=1 theories on R × T^3, and establish a general equivalence between Berry curvature and conformal perturbation theory data. The results illuminate how operator mixing under adiabatic deformations encodes deep geometric information about the moduli spaces of QFTs and offer potential experimental probes in systems with tunable theta or coupling parameters.

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.

Paper Structure

This paper contains 28 sections, 216 equations.

Table of Contents

  1. Introduction
  2. Review of Berry phase
  3. Abelian Berry connection
  4. Non-abelian generalization
  5. Systems with time-reversal or CPT invariance
  6. Berry phase in QFT
  7. A warmup example: free electromagnetism
  8. Berry phase of photon states
  9. A global effect
  10. Rotation of the polarization plane
  11. Potential realization of the $U(1)$ Berry phase
  12. 4d ${\cal N}=2$ QFTs on the Coulomb branch
  13. Coulomb branch as the parameter space of effective field theory
  14. Pure ${\cal N}=2$ SU(2) gauge theory
  15. Generalization to higher rank Coulomb branch
  16. ...and 13 more sections