Transparent lattices and their solitary waves.

Area of Science:

• Condensed Matter Physics
• Quantum Mechanics
• Mathematical Physics

Background:

• Tight-binding models are crucial for understanding electronic properties in materials.
• Nontrivial potentials and site-dependent hopping introduce complexity and novel phenomena.
• Supersymmetric quantum mechanics offers powerful tools for constructing exactly solvable models.

Purpose of the Study:

• To develop a family of transparent tight-binding models with tunable parameters.
• To explore the application of these models in physical systems like resonators and traps.
• To generalize supersymmetric quantum mechanics for discrete systems and construct new Hamiltonians.

Main Methods:

• Construction of transparent tight-binding models with nontrivial potentials and site-dependent hopping.
• Application of a generalized supersymmetric quantum mechanics framework.
• Utilizing a finite-difference Darboux transformation on scattering matrices of periodic arrays.
• Developing a procedure for hierarchical construction of discrete Hamiltonians.

Main Results:

• A family of transparent tight-binding models with specified potentials and hopping parameters was successfully derived.
• Feasibility of these models was demonstrated in electromagnetic resonators, dielectric slabs, and quantum traps.
• A novel procedure for constructing discrete Hamiltonians was established, yielding a biparametric family.
• Identified potentials and hopping functions as solitary waves, suggesting a discrete spinorial Korteweg-deVries generalization.

Conclusions:

• The developed tight-binding models offer a versatile platform for studying complex physical phenomena.
• The generalization of supersymmetric quantum mechanics provides a powerful method for designing discrete quantum systems.
• The discovery of solitary wave solutions highlights potential connections to nonlinear physics and integrable systems.