Laboratory 15 · Geometric phase · Independent self-study

How much phase is enclosed by a path on a sphere?

Move an adiabatic spin eigenstate around a latitude circle. Compare the solid-angle prediction against a discrete gauge-invariant product of eigenstate overlaps, including finite-resolution errors.

Self-paced explorationAdiabatic Berry holonomyGauge-invariant overlapsAnalytic + numerical predictions

Question and first prediction

Why can a closed Hamiltonian cycle leave a detectable phase?

A spin Hamiltonian can return to exactly the same matrix after a slow cycle of control parameters, while an eigenstate follows a closed path on the Bloch sphere. Adiabatic evolution produces both dynamical phase and a geometric contribution controlled by the enclosed solid angle. This geometry must be handled carefully: the Berry connection depends on a local eigenvector gauge, while the closed-loop phase modulo 2π is invariant. Finite overlap products expose the invariant directly, without numerically differentiating phase-discontinuous eigenvectors.

Predict before exploring. For a loop at the equator, the north-gauge spin-up eigenstate encloses solid angle 2π. Predict its Berry phase modulo 2π. Then shrink the cone to zero and infer the limiting phase.

Recommended background: Instantaneous eigenvectors, adiabatic evolution, complex inner products and phases modulo 2π.

Results that must agree with the equations

Computed observables

Geometric signature

Phase as a complex unit-circle holonomy

A phase is intrinsically defined modulo 2π; comparing cosγ and sinγ avoids false numerical jumps at branch cuts.

Exact holonomy exp[iγ] for one or several oriented closed circuits. Orange marks the selected polar angle. The phase itself is understood modulo 2π.

Independent discrete calculation

How many overlap segments are sufficient?

The polygonal Pancharatnam product approaches the continuous Berry phase as sampling along the path is refined.

The plotted error is |Arg exp[i(γdiscrete−γexact)]|, where γdiscrete is obtained from the cyclic product of eigenstate overlaps. Single-valued integer-winding gauge factors cancel.

Mathematical explanation

Closed-loop Berry phase is a holonomy

Let H(t)=−(Δ/2)n(t)·σ, Δ>0. Its ground eigenstate is aligned with n. Choose n=(sinθ cosφ,sinθ sinφ,cosθ) and the north-patch normalized eigenspinor |n+〉=(cosθ/2,eiφsinθ/2)ᵀ. By the convention A=i〈n+|∂φ n+〉,

Aφ=−sin²(θ/2)=−(1−cosθ)/2.

A closed latitude loop φ:0→2πw gives γ=∮Aφdφ=−πw(1−cosθ), equal to minus one half the oriented enclosed solid angle Ω=2πw(1−cosθ). The physically gauge-invariant object is exp(iγ).

For sampled eigenstates u₀,…,uN−1 along the loop, form Z=∏ⱼ〈uⱼ|uⱼ₊₁〉, closing the last link to u₀. The discrete phase is γN=−Arg Z. Each local phase factor eiχⱼ occurs once conjugated and once unconjugated in the cyclic product, and therefore cancels exactly.

This approximation converges as the step in φ decreases, provided adjacent eigenvectors retain nonzero overlaps and the gap remains open throughout the adiabatic cycle.

Test an exactly solvable limit

At θ=0 the path collapses to a point, so the Berry phase vanishes. At θ=90° a single loop yields γ=−π, equivalent to +π modulo 2π. Reversing the loop orientation reverses the phase. A 2π integer gauge winding changes the connection integral by multiples of 2π but preserves the holonomy.

Specify the model's validity

Adiabatic following of the isolated lower-energy eigenstate of H=−(Δ/2)n·σ with fixed Δ>0, and closed parameter-space cycles at fixed θ. The calculation isolates geometric phase after subtraction of the dynamical phase; finite-time nonadiabatic transitions are outside this experiment.

Follow a self-directed learning path

Three quantitative investigations

  1. Trace the solid angle. Increase θ from zero to the equator. Relate γ directly to the area swept on the Bloch sphere and check the trigonometric holonomy.
  2. Change the eigenvector gauge. Hold the same physical path fixed while varying integer m. Explain why the numerical overlap product is unchanged even though eigenvector phases at intermediate points vary.
  3. Check numerical convergence. Choose a large cone and compare N=8 with N=256. Record the difference in wrapped phase from the analytic solid-angle result and assess its decrease.

Check your understanding: For a spin-up adiabatic eigenstate transported around the equator once, what is the Berry phase modulo 2π?

Extension: work through two overlapping gauge patches on the Bloch sphere and track their transition function around a closed loop.

Connect back to quantum theory

What you can conclude

The Berry phase is a geometric property of a closed adiabatic eigenstate path, captured by a cyclic phase-product that is invariant under arbitrary local rephasing. This concept feeds directly into Zak phases, band polarization and Chern numbers.

Lecture notes: Berry phase and gauge · Band Berry curvature.