Fractional Disclination Charge and Discrete Shift in the Hofstadter Butterfly

Area of Science:

• Condensed Matter Physics
• Topological Phases of Matter
• Quantum Mechanics

Background:

• Topological phases of matter possess invariants linked to crystalline symmetries.
• These invariants lead to quantized responses in materials.
• The discrete shift (S) is a key invariant for topological phases.

Purpose of the Study:

• Investigate the discrete shift (S) invariant in the square lattice Hofstadter model.
• Understand the classification (Z_M) associated with M-fold rotational symmetry.
• Quantify the impact of (S) on fractional charge and angular momentum.

Main Methods:

• Numerical computation of the discrete shift (S) for the Hofstadter model.
• Analysis of the Hofstadter butterfly spectrum.
• Development of an empirical formula for (S).

Main Results:

• The discrete shift (S) exhibits its own Hofstadter butterfly structure.
• Quantized contributions of (S) to fractional charge and ground state angular momentum were identified.
• An empirical formula relating (S) to density and flux per plaquette was proposed.
• Distinct values of (S) were found for bands with the same Chern number.

Conclusions:

• The discrete shift (S) provides a new classification scheme for topological phases.
• Odd and even Chern number bands exhibit distinct (integer or half-integer) values of (S).
• This work refines the understanding of the Hofstadter model's phase diagram and topological properties.