# Berry connection and curvature
The Berry connection is a gauge potential defined on a space of parameters (or of quantum states) that determines the geometric phase a quantum system acquires when carried around a closed loop, and the Berry curvature is its curl, the gauge-invariant field strength from which that phase can be computed. Both arise when a system's Hamiltonian depends on parameters that vary slowly, and both have become standard tools in condensed-matter physics, optics, and quantum information science.
| Key fact | Detail |
|---|---|
| Definitions | Berry connection \( \mathbf{A} = i\langle u|\nabla_{\mathbf{k}}|u\rangle \); Berry curvature \( \boldsymbol{\Omega} = \nabla_{\mathbf{k}} \times \mathbf{A} \)[1](https://www.nature.com/articles/s41524-026-02155-1_reference.pdf) |
| Gauge behavior | The Berry phase (modulo 2π) and the curvature are gauge invariant; the connection is not, shifting by \( \nabla_{\mathbf{R}}\beta \) under a phase redefinition of the eigenstates[2](https://escholarship.org/content/qt99k708n7/qt99k708n7_noSplash_b634d8815f3bf112f481e4ee107c886f.pdf) |
| Spin example | For a spin-\(s\) particle in a magnetic field of fixed magnitude \( B_0 \) whose direction traces a curve at polar angle \( \theta \), the Berry phase is \( \gamma_m = 2\pi m(\cos\theta - 1) = -m\Omega_B \), with \( \Omega_B \) the enclosed solid angle[2](https://escholarship.org/content/qt99k708n7/qt99k708n7_noSplash_b634d8815f3bf112f481e4ee107c886f.pdf) |
| Topological invariant | The Chern number \( C_n = \frac{1}{2\pi}\int_{T^2} d^2k\, F_{xy}^{n}(\mathbf{k}) \) is an integer; it is always zero for time-reversal-invariant materials[3](https://doi.org/10.1088/2515-7639/abf0b5) |
| Hall conductance | The Hall conductance equals the sum of the Chern numbers of the occupied bands in units of \( e^2/h \); a usual band has Chern number zero and a Landau level has unity[4](https://academic.oup.com/nsr/article/1/1/38/1505089) |
| Non-Abelian form | For degenerate eigenspaces, the geometric phase becomes a matrix-valued (non-Abelian) holonomy, described by Wilczek and Zee[5](https://onlinelibrary.wiley.com/doi/10.1002/qua.24941) |
## Definitions
When a quantum system is driven by parameters \( \mathbf{R} \) that change slowly around a loop, an instantaneous eigenstate \( |n(\mathbf{R})\rangle \) returns to itself multiplied by a phase. Part of this phase is dynamical, set by the energy and elapsed time; the remainder depends only on the geometry of the path. That remainder is the geometric, or Berry, phase.[5](https://onlinelibrary.wiley.com/doi/10.1002/qua.24941)
The geometric phase is computed from the Berry connection, the inner product of the eigenstate with its gradient in parameter space,[1](https://www.nature.com/articles/s41524-026-02155-1_reference.pdf)
\[ \mathbf{A}^{(n)}(\mathbf{R}) = i\langle n(\mathbf{R})|\nabla_{\mathbf{R}}|n(\mathbf{R})\rangle, \]
and the phase around a closed curve \( C \) is the line integral of \( \mathbf{A} \) along \( C \). By [Stokes' theorem](https://www.edgepedia.org/stokes-theorem) this equals the flux of the Berry curvature,
\[ \mathbf{B}^{(n)}(\mathbf{R}) = \nabla_{\mathbf{R}} \times \mathbf{A}^{(n)}(\mathbf{R}), \]
through any surface bounded by \( C \).[2](https://escholarship.org/content/qt99k708n7/qt99k708n7_noSplash_b634d8815f3bf112f481e4ee107c886f.pdf) The notation deliberately parallels electromagnetism: the Berry connection plays the role of the magnetic vector potential \( \mathbf{A}(\mathbf{r}) \), the curvature plays the role of the magnetic field \( \mathbf{B}(\mathbf{r}) \), and the [Berry phase](https://www.edgepedia.org/berry-phase) plays the role of the magnetic flux through the loop.[2](https://escholarship.org/content/qt99k708n7/qt99k708n7_noSplash_b634d8815f3bf112f481e4ee107c886f.pdf)
**Gauge structure.** The eigenstates can be rephased by an arbitrary parameter-dependent factor \( e^{-i\beta(\mathbf{R})} \). Under this change the Berry phase shifts by \( 2\pi m \) for an integer \( m \), so it is invariant modulo \( 2\pi \); the curvature is unchanged because the curl of a gradient vanishes; but the connection shifts by \( \nabla_{\mathbf{R}}\beta \) and is not itself an observable.[2](https://escholarship.org/content/qt99k708n7/qt99k708n7_noSplash_b634d8815f3bf112f481e4ee107c886f.pdf) This mirrors electromagnetism, where the field and flux are physical while the vector potential depends on the gauge.
## Origin
The adiabatic geometric phase was first found in quantum chemistry by Longuet-Higgins and coworkers and by Stone; it underlies the molecular [Aharonov–Bohm effect](https://www.edgepedia.org/aharonov-bohm-effect), a topological phase shift when a molecule reshapes slowly around a conical intersection.[5](https://onlinelibrary.wiley.com/doi/10.1002/qua.24941) Michael V. Berry, a mathematical physicist at the [University of Bristol](https://www.edgepedia.org/university-of-bristol), demonstrated in 1984 that the phase is a general consequence of the geometry of the parameter space driving cyclic adiabatic evolution of energetically nondegenerate states, realizable in photons, nuclear magnetic resonance, neutrons, Jahn–Teller molecules, condensed-matter systems, and cold atoms.[5](https://onlinelibrary.wiley.com/doi/10.1002/qua.24941) [Barry Simon](https://www.edgepedia.org/barry-simon) later showed that Berry's phase is the holonomy in a Hermitian line bundle, establishing its geometric character in the language of fiber bundles.[5](https://onlinelibrary.wiley.com/doi/10.1002/qua.24941) In this framing, the geometric phase is a holonomy: the failure of parallel transport around a closed cycle to preserve the geometrical information being transported.[6](https://www.nature.com/articles/s42254-019-0071-1)
## Canonical example: spin in a rotating field
A spin-\( s \) particle in a magnetic field of constant magnitude \( B_0 \) whose direction precesses at a fixed polar angle \( \theta \) is the standard worked example. The eigenstates \( |m\rangle \) along the instantaneous field direction carry energies \( E_m = -g\mu B_0 m \). Computing the curvature from the ladder operators gives \( \mathbf{B}^{(m)} = -(m/B_0^2)\hat{\mathbf{z}} \), and the Berry phase around the loop is
\[ \gamma_m = 2\pi m(\cos\theta - 1) = -m\Omega_B, \]
where \( \Omega_B \) is the solid angle traced by the field direction on the unit sphere.[2](https://escholarship.org/content/qt99k708n7/qt99k708n7_noSplash_b634d8815f3bf112f481e4ee107c886f.pdf) For the two-level (spin-1/2) case, the Berry curvature is the field strength of a magnetic monopole of strength 1/2 living on the manifold traced by the field direction.[7](https://www.sciencedirect.com/topics/mathematics/berry-phase)
## Generalizations
Several extensions widen the original adiabatic, nondegenerate, cyclic setting. Wilczek and Zee showed that when the relevant eigenspace is degenerate, slowly varying parameters produce matrix-valued geometric phases; because matrices need not commute, these are called non-Abelian geometric phases or quantum holonomies.[5](https://onlinelibrary.wiley.com/doi/10.1002/qua.24941) For an \( N \)-fold degenerate eigenspace the connection and curvature lie in the [Lie algebra](https://www.edgepedia.org/lie-algebra) \( u(N) \) of the gauge group \( U(N) \), and the most prominent non-Abelian Berry curvature appears in materials with combined time-reversal and inversion symmetry.[8](https://www-old.mpi-halle.mpg.de/mpi/publi/pdf/10666_12.pdf) Aharonov and Anandan removed the adiabatic restriction, defining a geometric phase for arbitrary cyclic evolution, of which the Berry phase is the slow-limit special case.[5](https://onlinelibrary.wiley.com/doi/10.1002/qua.24941) • [9](https://www.sciencedirect.com/science/article/abs/pii/S0370157323002065) Building on Pancharatnam's work on polarized light, Samuel and Bhandari defined geometric phases for open paths, and mixed-state generalizations follow from Uhlmann's holonomy of density operators.[5](https://onlinelibrary.wiley.com/doi/10.1002/qua.24941)
## Condensed-matter physics
In crystals the parameters are the crystal momentum \( \mathbf{k} \), and the Berry connection and curvature are defined for the periodic part \( u_n(\mathbf{r}, \mathbf{k}) \) of the Bloch wave functions.[8](https://www-old.mpi-halle.mpg.de/mpi/publi/pdf/10666_12.pdf) The curvature acts as an emergent magnetic field in momentum space: once a finite Berry curvature is present, a solid acquires an intrinsic anomalous Hall conductivity, and the Hall conductivity is the integral of the Berry curvature over occupied states,[10](https://www.jstage.jst.go.jp/article/pjab/95/6/95_PJA9506B-05/_html/-char/ja) • [11](https://doi.org/10.1073/pnas.1816904116)
\[ \sigma_H = \frac{e^2}{\hbar}\sum_n \int \frac{d\mathbf{k}}{(2\pi)^d} f(\varepsilon_n(\mathbf{k}))\, b_{nz}(\mathbf{k}). \]
**Chern numbers.** In two dimensions the integral of the Berry curvature over the [Brillouin zone](https://www.edgepedia.org/brillouin-zone) torus is quantized,[3](https://doi.org/10.1088/2515-7639/abf0b5)
\[ C_n = \frac{1}{2\pi}\int_{T^2} d^2k\, F_{xy}^{n}(\mathbf{k}) \in \mathbb{Z}, \]
an integer that equals the number of times the Bloch vector wraps around the sphere and indicates whether the Bloch bundle is twisted.[3](https://doi.org/10.1088/2515-7639/abf0b5) • [7](https://www.sciencedirect.com/topics/mathematics/berry-phase) The Hall conductance of an insulator is the sum of the Chern numbers of the occupied bands in units of \( e^2/h \); a [Chern number](https://www.edgepedia.org/chern-number) can only change where the band gap closes, which forces conducting edge states at the boundary of a quantum Hall sample.[4](https://academic.oup.com/nsr/article/1/1/38/1505089) Because the curvature is an odd function of \( \mathbf{k} \) under time reversal, the Chern number is always zero for time-reversal-invariant materials, so breaking that symmetry is a necessary condition for a nonzero value.[3](https://doi.org/10.1088/2515-7639/abf0b5) Berry curvature also enters electron dynamics beyond the Hall response, producing the anomalous velocity and contributing to orbital magnetization, and its multiband generalization underlies spin transport phenomena.[12](https://wuli.iphy.ac.cn/en/article/doi/10.7693/wl20240401)
## Applications
**Quantum computation.** Holonomic quantum computation, proposed by Zanardi, Rasetti, and coworkers, uses non-Abelian Wilczek–Zee holonomies rather than conventional dynamical operations to implement quantum gates; universal computation is possible by holonomies alone, and the approach is conjectured to tolerate certain errors because the holonomy is a global feature of the path rather than a local accumulation.[13](https://arxiv.org/abs/quant-ph/0111019) • [5](https://onlinelibrary.wiley.com/doi/10.1002/qua.24941) Schemes have been proposed and demonstrated in nuclear magnetic resonance and proposed for superconducting nanocircuits and Josephson-junction networks.[13](https://arxiv.org/abs/quant-ph/0111019)
**Optics and beyond.** In optics the two main types of geometric phase come from spin redirection, when light of fixed polarization changes direction, and from a slow change of polarization in an anisotropic medium, the [Pancharatnam–Berry phase](https://www.edgepedia.org/pancharatnam-berry-phase).[6](https://www.nature.com/articles/s42254-019-0071-1) More broadly, the geometric phase appears in the quantum [Hall effect](https://www.edgepedia.org/hall-effect), electric polarization, and exchange statistics in condensed matter, and has been generalized to degenerate, open, non-adiabatic, non-cyclic, and non-unitary settings.[6](https://www.nature.com/articles/s42254-019-0071-1)
## References
1. [Berry curvature and quantum geometric tensor relations, npj Computational Materials reference list](https://www.nature.com/articles/s41524-026-02155-1_reference.pdf)
2. [Berry phase, Berry connection, and gauge invariance, University of California thesis (Appendix A)](https://escholarship.org/content/qt99k708n7/qt99k708n7_noSplash_b634d8815f3bf112f481e4ee107c886f.pdf)
3. [Topological and geometrical aspects of band theory, IOPscience](https://doi.org/10.1088/2515-7639/abf0b5)
4. [Quantum anomalous Hall effect, National Science Review](https://academic.oup.com/nsr/article/1/1/38/1505089)
5. [Erik Sjöqvist (2015). Geometric phases in quantum information. International Journal of Quantum Chemistry 115, 1311–1326.](https://onlinelibrary.wiley.com/doi/10.1002/qua.24941)
6. [Cohen, Larocque, Bouchard, Nejadsattari, Gefen, Karimi (2019). Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond. Nature Reviews Physics 1, 437–449.](https://www.nature.com/articles/s42254-019-0071-1)
7. [Berry Phase, ScienceDirect Topics overview](https://www.sciencedirect.com/topics/mathematics/berry-phase)
8. [Non-Abelian Berry connection and curvature in degenerate eigenspaces, Max Planck Institute Halle preprint](https://www-old.mpi-halle.mpg.de/mpi/publi/pdf/10666_12.pdf)
9. [Geometric and holonomic quantum computation, ScienceDirect review](https://www.sciencedirect.com/science/article/abs/pii/S0370157323002065)
10. [Emergent electromagnetism in condensed matter, Proceedings of the Japan Academy](https://www.jstage.jst.go.jp/article/pjab/95/6/95_PJA9506B-05/_html/-char/ja)
11. [Unraveling materials Berry curvature and Chern numbers from real-time evolution of Bloch states, PNAS](https://doi.org/10.1073/pnas.1816904116)
12. [NIU Qian, CHANG Ming-Che, XIAO Di (2024). The topology and geometry of Bloch electrons. 物理 53, 215–225.](https://wuli.iphy.ac.cn/en/article/doi/10.7693/wl20240401)
13. [Mahn-Soo Choi (2001). Geometric Quantum Computation on Solid-State Qubits. arXiv:quant-ph/0111019.](https://arxiv.org/abs/quant-ph/0111019)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Quantum physics › Quantum information science › Quantum computing and algorithms › Quantum computational models › Adiabatic quantum computation › Adiabatic theorem and its conditions*
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