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Area of Science:

  • Condensed matter physics
  • Quantum mechanics
  • Materials science

Background:

  • Bloch's theorem traditionally requires crystal periodicity for bandgap formation.
  • Disordered systems like amorphous media and quasicrystals have challenged this requirement.
  • Previous models for disordered media failed to deterministically create bandgaps.

Purpose of the Study:

  • To reveal a deterministic pathway for creating bandgaps in disordered potentials.
  • To challenge the necessity of long-range order for bandgap formation.
  • To develop a method for generating "Bloch-like eigenstates" in random media.

Main Methods:

  • Applying supersymmetry to the wave equation.
  • Utilizing isospectrality to transform ordered systems into disordered ones while preserving bandgaps.
  • Developing random-walk potentials analogous to Brownian motion.

Main Results:

  • Demonstrated deterministic bandgap formation in extremely disordered potentials.
  • Showcased the ability to tune correlations in potentials without altering bandgaps.
  • Created a family of potentials exhibiting "Bloch-like eigenstates".

Conclusions:

  • Supersymmetry offers a deterministic route to bandgaps in disordered systems.
  • Bandgaps can be formed in potentials lacking traditional long-range order.
  • The developed method allows for controlled disorder while maintaining essential electronic properties.