τTOPOLOGY& QUANTUM PHENOMENA
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  1. 01 · Topology versus geometry
    Geometry measures distances and angles; topology tracks properties unchanged by continuous deformation.    Foundations of Topology
  2. 02 · Open sets, continuity and compactness
    Topology supplies a precise definition of continuity without first choosing a distance.    Foundations of Topology
  3. 03 · Connectedness and topological sectors
    Disconnected regions of a configuration space cannot be connected by an allowed continuous path.    Foundations of Topology
  4. 04 · Homotopy and winding
    A loop can wrap a circle an integer number of times; smooth deformations cannot change that integer without breaking continuity.    Foundations of Topology
  5. 05 · Manifolds, spheres and tori
    A manifold looks locally like Euclidean space, even when its global topology is different.    Foundations of Topology
  6. 06 · Euler characteristic and genus
    Counting vertices, edges and faces reveals a global invariant of a closed orientable surface.    Foundations of Topology
  7. 07 · Fiber bundles and quantum eigenstates
    A quantum eigenstate can be defined locally while a single smooth global phase convention may fail to exist.    Foundations of Topology
  8. 08 · Gauge transformations and observables
    The phase assigned to an eigenvector is conventional; measurable holonomy and curvature are not.    Foundations of Topology
  9. 09 · Projective Hilbert space
    Two state vectors differing only by a nonzero complex factor represent the same pure quantum state.    Geometric Quantum Mechanics
  10. 10 · Berry phase and adiabatic evolution
    A slowly transported eigenstate acquires a geometric phase in addition to its dynamical phase.    Geometric Quantum Mechanics
  11. 11 · Berry curvature as effective magnetic field
    Berry curvature plays the role of a magnetic field in parameter or momentum space.    Geometric Quantum Mechanics
  12. 12 · Holonomy and Wilson loops
    Parallel transport around a closed path can rotate a quantum state or a degenerate subspace.    Geometric Quantum Mechanics
  13. 13 · Aharonov–Bohm phase
    A charged quantum wavefunction responds to enclosed magnetic flux even along paths with zero local magnetic field.    Geometric Quantum Mechanics
  14. 14 · Fubini–Study quantum metric
    Quantum states have a natural notion of distance that measures how quickly distinguishable states change.    Geometric Quantum Mechanics
  15. 15 · Bloch theorem and Brillouin torus
    Periodic lattices let us label electronic states by crystal momentum.    Topological Band Theory
  16. 16 · Chern number from Berry curvature
    Integrating Berry curvature over a closed momentum-space surface yields an integer topological invariant.    Topological Band Theory
  17. 17 · TKNN quantization and Hall conductance
    Hall conductivity can be locked to integers because it is the integral of band geometry.    Topological Band Theory
  18. 18 · Landau levels and integer quantum Hall effect
    A perpendicular magnetic field organizes two-dimensional electron motion into quantized cyclotron levels.    Topological Band Theory
  19. 19 · Bulk–boundary correspondence
    A change of bulk topology across an interface requires gap-crossing boundary states under suitable assumptions.    Topological Band Theory
  20. 20 · Haldane model and Chern insulators
    Complex next-neighbor hopping can generate a quantum Hall-like phase without net magnetic flux per unit cell.    Topological Band Theory
  21. 21 · SSH model and winding number
    A one-dimensional dimerized chain has distinct gapped winding sectors.    Topological Band Theory
  22. 22 · Zak phase and polarization
    The Berry phase across a one-dimensional Brillouin zone is related to electric polarization.    Topological Band Theory
  23. 23 · Time-reversal symmetry and Z2 topology
    Time-reversal symmetry can protect helical edge modes even when total Chern number vanishes.    Topological Band Theory
  24. 24 · Topological insulators in three dimensions
    Three-dimensional time-reversal-invariant insulators can have robust surface Dirac states.    Topological Band Theory
  25. 25 · Weyl semimetals and monopoles
    Weyl nodes act as sources or sinks of Berry curvature in three-dimensional momentum space.    Topological Band Theory
  26. 26 · Disorder and topological robustness
    Topological properties can survive many imperfections because small perturbations cannot change a discrete invariant continuously.    Topological Band Theory
  27. 27 · Fractional quantum Hall effect
    Strongly interacting electrons can form incompressible liquids with fractionally charged excitations.    Interacting Topological Phases
  28. 28 · Anyons and braid statistics
    In two spatial dimensions, exchanges can have richer statistics than bosons or fermions.    Interacting Topological Phases
  29. 29 · Topological order and long-range entanglement
    Some many-body phases cannot be characterized by local symmetry-breaking order parameters.    Interacting Topological Phases
  30. 30 · Toric code and stabilizers
    Local commuting operators can produce a globally entangled ground space.    Interacting Topological Phases
  31. 31 · Majorana zero modes
    A Majorana operator is its own Hermitian conjugate and can describe a zero-energy boundary mode in a superconducting model.    Interacting Topological Phases
  32. 32 · Kitaev chain and topological superconductivity
    A simple one-dimensional p-wave superconducting model can host unpaired Majorana end modes.    Interacting Topological Phases
  33. 33 · Non-Abelian braiding and encoded gates
    Exchanging non-Abelian anyons can rotate a protected degenerate subspace.    Interacting Topological Phases
  34. 34 · Symmetry-protected topological phases
    Some short-range-entangled phases become distinct only when a symmetry is enforced.    Interacting Topological Phases
  35. 35 · Why topology emerges from quantum mechanics
    Quantum superposition, phase, and interference make eigenstates vary continuously across parameter spaces; global patterns can then become topological.    Quantum Phenomena and Applications
  36. 36 · Gap closing and topological phase transitions
    A topological index cannot change smoothly while its defining gap and symmetries remain intact.    Quantum Phenomena and Applications
  37. 37 · Topological photonics
    Light propagating through engineered structures can inherit topological band features.    Quantum Phenomena and Applications
  38. 38 · Topological quantum sensing
    Geometric phases and topological features can be used to encode or stabilize sensing signals.    Quantum Phenomena and Applications
  39. 39 · Cold atoms and synthetic dimensions
    Laser-assisted hopping can simulate gauge fluxes and topological band structures with neutral atoms.    Quantum Phenomena and Applications
  40. 40 · Quantum computing with topological codes
    Logical quantum information can be stored nonlocally in stabilizer-code degrees of freedom.    Quantum Phenomena and Applications
  41. 41 · Experimental diagnostics of topology
    Different topological phases demand different experimental evidence.    Quantum Phenomena and Applications
  42. 42 · Topology, decoherence and limitations
    Topological invariants can protect selected structures, but real quantum systems still couple to environments.    Quantum Phenomena and Applications
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