Interactive Topological Quantum Vacuum Explorer

This simulation suite provides four interconnected 3D views of the core QVC theoretical framework, built on Three.js with real-time parameter control. Use the sidebar sliders to tune TEVC lattice parameters, QVC drive fields, and hopfion topology — all views update simultaneously.

Simulation Tabs

1. Hopfion Spinor Field — Visualizes the Bloch vector field of a Hopf fibration (pi_3(S^2) = Z). Color maps to spinor phase arg(psi_1), height to the n_z Bloch component. Purple tori mark the n_z=0 isosurface. The Hopf invariant Q_H counts linked preimage circles.

2. Catalyzon Dispersion Surface — Plots the catalyzon quasiparticle dispersion omega_cat(k, A_ZPF) as a 3D surface. Color shifts from blue (gapped) through purple (catalyzon regime) to red (near quantum critical point). The teal reference surface shows the in-phase Bogoliubov mode omega_+(k).

3. TEVC Topochiral Lattice — Renders the twisted-layer moire hexagonal lattice in 3D. Amber spheres mark AA-stacking Josephson nodes, purple tori show hopfion fields at each site, green rings indicate BEC coherence perimeters, and red cylinders appear as Abrikosov vortex cores when the gap collapses.

4. Bogoliubov Vacuum — Displays the Bogoliubov quasiparticle dispersion omega_-(k_x, k_y) with Rashba spin-orbit coupling breaking rotational symmetry. The purple plane marks the QVC gap level, and the pink sphere at k=0 is the ZPF coupling locus.

Physics & Parameter Guide

TEVC Parameters

• Twist angle theta — Controls moire superlattice period lambda_M = a/theta. Near magic angles (~1.1 degrees), flat bands emerge enabling strong correlation effects.
• Tunneling J — Interlayer hopping amplitude. Governs the excitonic BEC gap via Delta = sqrt(4J^2 - 2JA).
• Layers N — Number of van der Waals layers in the heterostructure stack.
• Chern |C| — Topological invariant of the occupied bands. Non-zero Chern number implies quantized Hall response and chiral edge states.

QVC Drive

• A_ZPF — Zero-point field amplitude. As A_ZPF approaches 2J, the excitonic gap collapses toward the quantum critical point (QCP), entering the deep catalyzon regime.
• Drive omega_0/2J — Ratio of external drive frequency to twice the tunneling energy. Resonance at omega_0/2J = 1 maximizes vacuum-matter coupling (Rabi splitting).
• Temperature T — Controls thermal decoherence. The BKT transition temperature T_BKT = (pi/2) rho_s sets the superfluid coherence boundary.

Hopfion / Topology

• Hopf charge Q_H — Integer topological invariant from pi_3(S^2). Each unit adds a linked torus to the spinor texture.
• Core size a_H/xi — Hopfion core radius in units of coherence length. Larger cores are more stable but carry less topological energy density.
• Rashba alpha_R — Rashba spin-orbit coupling strength. Breaks rotational symmetry in the Bogoliubov dispersion, splitting spin-up/down branches.
• Tilt v_t/v_F — Type-II tilt of the Dirac cone. At v_t/v_F = 1, the cone tips over (type-II Weyl point), creating electron/hole pockets.

Readout Panel

• Gap Delta_cat — Effective catalyzon gap after Rabi subtraction: Delta_cat = Delta_ex^QVC - Omega_R/2
• Omega_R/2 — Half the Rabi frequency from the three boson-mediated couplings (Josephson, polariton, hopfion)
• rho_s (BKT) — Superfluid stiffness; vanishes at the BKT transition
• Hopf Phi_H — Topological flux from hopfion texture
• T_BKT — Berezinskii-Kosterlitz-Thouless transition temperature
• Schwinger x — Effective Schwinger pair-production enhancement factor

Regime Badges

• GAP COLLAPSED - QCP: The excitonic gap has closed; system is at the quantum critical point
• Deep catalyzon regime: Delta_cat < Delta/2; strong vacuum-matter hybridization
• Resonance ON: Drive frequency matches the natural gap (omega_0/2J ~ 1)
• Normal: Standard gapped excitonic insulator regime

Key Equations

The catalyzon dispersion bridges three QVC mechanisms:

Delta_ex^QVC = sqrt(max(0, 4J^2 - 2JA_ZPF))     [Excitonic gap]
Omega_R = 2 sqrt(g_J^2 + g_pl^2 + g_H^2 + ...)  [Rabi splitting]
Delta_cat = max(0, Delta_ex - Omega_R / 2)         [Catalyzon gap]

Hopf invariant:  Q_H = (1/16pi^2) integral F . A  d^3r
Bogoliubov:      omega_-(k) = sqrt[eps_k(eps_k + 2g_-n) + Delta^2/4]
BKT threshold:   T_BKT = (pi/2) rho_s

Suggested Explorations

1. Magic angle catalyzon: Set theta = 1.10, J = 0.30, A_ZPF = 0.20, then slowly increase A_ZPF toward 0.50. Watch the gap collapse and the regime badge shift to "Deep catalyzon."

2. Hopfion linking: On the Hopfion tab, increase Q_H from 0 to 3. Each increment adds a linked torus — these are topologically protected and cannot be removed by continuous deformation.

3. Rashba symmetry breaking: On the Bogoliubov tab, set alpha_R = 0 (symmetric), then increase to 2.0. The circular dispersion surface distorts, splitting into spin branches.

4. Lattice vortex nucleation: On the TEVC tab, set J low (0.10) and A_ZPF high (0.50). Red Abrikosov vortex cores appear through the layer stack as the condensate gap collapses.

5. Resonant drive: Set omega_0/2J = 1.0 exactly with moderate A_ZPF ~ 0.25. The "Resonance: ON" badge appears, indicating maximal vacuum-matter Rabi coupling.