Berry Phase: The Phase a Quantum State Picks Up on a Loop

Walk around a field, holding an arrow so that it never twists relative to your path, and come back to where you started. On flat ground the arrow points the same way as before. On the surface of a sphere it does not: it has turned by an angle set by the area you enclosed. Quantum states do something similar. Taken slowly around a closed loop, a state returns with an extra phase that remembers the geometry of the trip. This is the geometric phase, or Berry phase Geometric phase, Wikipedia^[1], and it is the idea on our Berryphase sticker.

What is the Berry phase?

Take a quantum system whose Hamiltonian depends on some parameters R, for example the direction of a magnetic field. Start in an eigenstate |n(R)⟩ and change R so slowly that the system stays in that eigenstate (the adiabatic condition). Bring R back to its starting value around a closed circuit C. Berry showed in 1984 that the state then acquires a geometric phase factor exp(iγ(C)) on top of the familiar dynamical phase, which grows with the energy and the elapsed time Berry 1984^[2]. In modern notation Xiao, Chang and Niu 2010^[3]:

γ_n(C) = ∮_C A_n · dR, A_n = i⟨n(R)|∇_Rn(R)⟩

Here A_n is called the Berry connection, i is the imaginary unit and ∇_R is the gradient with respect to the parameters. γ is in radians. The time taken never appears: only the path in parameter space matters Berry 1984^[2]. By Stokes’s theorem the same phase can be written as a flux through any surface S spanning the loop Geometric phase, Wikipedia^[1]:

γ_n(C) = ∬_S Ω_n · dS, Ω_n = ∇_R × A_n

Ω_n is the Berry curvature, which acts like a magnetic field living in parameter space Xiao, Chang and Niu 2010^[3]. Berry derived a general formula for γ(C) in terms of the spectrum and eigenstates over that surface, and showed it takes a simple form when C passes near a degeneracy, a point where two energy levels meet Berry 1984^[2].

Who discovered the geometric phase?

Berry’s paper unified an idea that had appeared several times. S. Pancharatnam found it in 1956 in classical optics, studying how the phase of polarised light changes as its polarisation is cycled Pancharatnam 1956^[4] Geometric phase, Wikipedia^[1]. H. C. Longuet-Higgins found it in 1958 in molecular physics, and Aharonov and Anandan extended it in 1987 to cycles that are not slow Geometric phase, Wikipedia^[1].

What is the Berry phase of a spin in a magnetic field?

The cleanest example is a spinning particle in a magnetic field whose direction is turned slowly around a loop. Berry worked this case out explicitly Berry 1984^[2]. If m is the spin component along the field (m = ±½ for an electron), and the field direction sweeps out a solid angle Ω(C) in steradians, the geometric phase is

γ_m(C) = −m Ω(C)

For a spin-½ particle aligned with the field this is γ = −Ω/2, the formula printed on the sticker. Turn the field once around the equator of directions, a solid angle of 2π, and the phase is −π: the state changes sign. Berry pointed out that the sign reversal of spinors on rotation is a special case of this result, and that the effect also occurs for bosons Berry 1984^[2]. Why electrons need a full 720° to come back to themselves is the subject of why electrons have spin one-half.

How was the Berry phase first measured?

In 1986 Tomita and Chiao sent light through a single-mode optical fibre wound into a helix Tomita and Chiao 1986^[5]. Along the helix, the direction of the light’s momentum traces a cone, a closed loop on the sphere of directions. They found that linearly polarised light came out rotated by an angle that gives a direct measure of Berry’s phase Tomita and Chiao 1986^[5]. Deforming the fibre did not change the rotation, as long as the solid angle enclosed in momentum space stayed the same Tomita and Chiao 1986^[5]. For light the phase equals the enclosed solid angle times the photon spin, which is 1 Geometric phase, Wikipedia^[1]. For a cone of half-angle θ that solid angle is 2π(1 − cos θ).

Where else does the geometric phase appear?

The Aharonov–Bohm effect

In 1959 Aharonov and Bohm argued that electromagnetic potentials affect charged particles even where all fields, and so all forces, vanish Aharonov and Bohm 1959^[6]. A particle of charge q passing either side of a solenoid picks up a phase difference set by the magnetic flux Φ_B (in webers) that it encloses Aharonov–Bohm effect, Wikipedia^[7]:

Δφ = qΦ_B/ħ

Here ħ = h/2π is the reduced Planck constant. Berry showed that this effect can be read as a geometric phase Berry 1984^[2]. The effect was confirmed first by Robert Chambers, and then with smaller uncertainty by Akira Tonomura Aharonov–Bohm effect, Wikipedia^[7].

Foucault’s pendulum

The swing plane of a Foucault pendulum turns at 2π sin φ radians per day at latitude φ, and this rotation is a classical geometric phase: after one day the precession equals the solid angle enclosed by the pendulum’s circle of latitude, modulo 2π Geometric phase, Wikipedia^[1] Foucault pendulum, Wikipedia^[8]. Our Foucault pendulum guide and Pendulux sticker tell that story.

Electrons in solids

The Berry phase of electron wavefunctions shapes material properties, including electric polarisation, orbital magnetism and several Hall effects Xiao, Chang and Niu 2010^[3], among them the quantum Hall plateaus on our Stairohm sticker.

Quick FAQ

Is the Berry phase observable?

A global phase alone is not, but the difference between two paths, or a rotated polarisation, is. That is why experiments use interference or polarisation Tomita and Chiao 1986^[5] Aharonov–Bohm effect, Wikipedia^[7].

Does it need quantum mechanics?

No. The same geometry appears in polarised light and in pendulums Geometric phase, Wikipedia^[1]. Browse more physics stickers for the classical side.

References

1. Wikipedia, “Geometric phase”. https://en.wikipedia.org/wiki/Geometric_phase
2. Michael Victor Berry (1984), “Quantal phase factors accompanying adiabatic changes,” Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 392, 45–57 (DOI 10.1098/rspa.1984.0023; metadata verified via Crossref). https://doi.org/10.1098/rspa.1984.0023
3. Di Xiao, Ming-Che Chang and Qian Niu (2010), “Berry phase effects on electronic properties,” Reviews of Modern Physics 82, 1959–2007 (DOI 10.1103/RevModPhys.82.1959; metadata verified via Crossref). https://doi.org/10.1103/RevModPhys.82.1959
4. S. Pancharatnam (1956), “Generalized theory of interference, and its applications,” Proceedings of the Indian Academy of Sciences - Section A 44, 247–262 (DOI 10.1007/BF03046050; metadata verified via Crossref). https://doi.org/10.1007/BF03046050
5. Akira Tomita and Raymond Y. Chiao (1986), “Observation of Berry's Topological Phase by Use of an Optical Fiber,” Physical Review Letters 57, 937–940 (DOI 10.1103/PhysRevLett.57.937; metadata verified via Crossref). https://doi.org/10.1103/PhysRevLett.57.937
6. Y. Aharonov and D. Bohm (1959), “Significance of Electromagnetic Potentials in the Quantum Theory,” Physical Review 115, 485–491 (DOI 10.1103/PhysRev.115.485; metadata verified via Crossref). https://doi.org/10.1103/PhysRev.115.485
7. Wikipedia, “Aharonov–Bohm effect”. https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect
8. Wikipedia, “Foucault pendulum”. https://en.wikipedia.org/wiki/Foucault_pendulum

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.