OBSERVATION CHAMBER / ACTIVE

OBSERVATION ROUTE

CURRENT: 74 BERRY PHASE HOLONOMY OBSERVATORY

Open the full 4-chamber route

Berry Phase Holonomy Observatory

Carry a two-level eigenstate around closed loops on the Bloch sphere and compare its Berry phase with the enclosed curvature.

quantum / holonomy Observation Model

ROUTE 74 / GEOMETRY / QUANTUM HOLONOMY

TWO-LEVEL / BLOCH SPHERE

Carry one eigenstate around a closed loop

Move the direction of a two-level Hamiltonian on the Bloch sphere. Parallel transport leaves a gauge-invariant Berry phase that follows the oriented curvature.

READY

The sphere is Hamiltonian parameter space, not a measured physical trajectory. The wheel shows the selected band holonomy modulo 2π.

All calculations stay local in this browser.

Band: +
Loop: LATITUDE CAP
theta0: 0.90 rad
Delta: 1.00
Gap: 2.00
Berry phase gamma: —
Solid angle Omega: —
Local Ftheta phi: —
Min overlap: —

HOLONOMY CONTROLS

Shape the parameter loop

READY

CHAMBER STATUS

SYSTEM: ONLINE
MODEL: ADIABATIC LINE
BOUNDARY: CLOSED LOOP

PARALLEL TRANSPORT

0 / 512

TRACE walks the eigenstate around C. STEP advances one segment. No dynamic phase is included.

LOOP PRESET

LATITUDE CAP

The reverse cap changes only the orientation. Figure 8 shows local curvature with nearly zero net area.

EIGENBAND

UPPER +

Both bands share the same gap. Their Berry curvatures and phases have opposite signs.

CAP ANGLE theta0

0.90 rad

0.90

For CAP, Omega = 2 pi (1 − cos theta0). The same angle sets the meridian extent.

ENERGY SCALE Delta

1.00

H = Delta n dot sigma. Delta changes the gap, not the geometric phase.

KEYBOARD

Focus the canvas: 1 CAP · 2 RETRACE · 3 FIGURE 8 · 4 REVERSE · B BAND · Space TRACE · S STEP · R RESET · ←/→ theta0. Touch targets are 44px or larger.

GEOMETRIC READOUT

Phase from curvature

PASS

Berry phase gamma selected: —
Solid angle Omega: —
Curvature Ftheta phi: —
Energy gap g: —
Minimum overlap: —
Eigen residual: —
Phase residual: —
Trace progress: 0 / 512
Interpretation: The displayed phase is geometric holonomy of an eigenline. It is not a dynamic phase and it is not a physical measurement.

For the stated eigenvector convention, gamma plus = −Omega/2 and gamma minus = +Omega/2 modulo 2 pi. The gap protects the two bands in this finite numerical model.

BAND	GAMMA	CURVATURE	PHASE RESIDUAL	EIGEN RESIDUAL

DETERMINISTIC SELF-TEST

Numerical acceptance

PASS

CAP / PASS

CHECK	VALUE	LIMIT	RESULT

Observation report

OBSERVATION CONTRACT

One eigenline, one closed loop

The Hamiltonian direction n moves on the Bloch sphere. The selected eigenstate is parallel-transported along C, and the phase left after closure is the line-bundle holonomy. Dynamic time evolution is intentionally outside this chamber.

WHAT TO WATCH

Area, orientation, band

CAP grows the solid angle and the magnitude of gamma.
REVERSE changes the sign of the same geometric phase.
RETRACE cancels area; FIGURE 8 cancels opposite lobes while the local curvature remains visible.

MODEL NOTE

Finite two-level model

H = Delta n dot sigma, n = (sin theta cos phi, sin theta sin phi, cos theta)

F plus = −(sin theta / 2) dtheta wedge dphi, gamma plus = −Omega / 2. The lower band has the opposite signs.

This page observes a numerical eigenline bundle. It does not claim a laboratory measurement, a material property, or a visitor score.

Formula note

e^(i gamma) = product_j <u_j | u_(j+1)> / |<u_j | u_(j+1)>|. The implementation uses the negative argument of the overlap product because the transported vector removes each local overlap phase. The solid angle is computed independently from oriented spherical triangles.

OBSERVATION GUIDE

Touchpoints for observation

Start with CAP and the upper band. Press TRACE, then compare CAP with REVERSE and RETRACE. Lower Delta changes the gap only; changing the band reverses the curvature sign.

Use STEP to inspect the transported marker one segment at a time.
Use FIGURE 8 to see local curvature without a large net phase.
Use REPORT or PNG for the current screen. Reduced-motion keeps TRACE static and leaves STEP available.

This is a finite deterministic two-level numerical model. It excludes dynamic phase and physical measurement.

Runs inside the browser with no upload and no registration.

OBSERVATION POLICY

This observatory runs a deterministic two-level Hamiltonian model in your browser. Eigenstates, loops, calculations, reports, and images stay local. It does not measure visitors, devices, or physical matter.

Route

This chamber belongs to Shape & Space.

GEOMETRY Back to Shape & Space Route.