Critical Dynamics in Short-Range Quadratic Hamiltonians.

Area of Science:

• Condensed Matter Physics
• Quantum Dynamics
• Statistical Mechanics

Background:

• Investigating critical transport phenomena is crucial for understanding complex quantum systems.
• The interplay between localization and transport dictates system behavior.

Purpose of the Study:

• To establish a theoretical framework for critical transport and the dynamical exponent in quadratic Hamiltonians.
• To explore critical dynamics emerging from the Thouless time approaching the Heisenberg time.

Main Methods:

• Analysis of particle spreading in lattice systems with short-range hopping.
• Derivation of a relationship between the dynamical exponent (z), lattice dimension (d_l), and spectral fractal dimension (d_s).

Main Results:

• A new relation z = d_l / d_s is established for critical transport.
• Superdiffusive transport in d_l >= 2 and diffusive transport in d_l >= 3 are shown to be non-critical under these definitions.
• Nontrivial examples of critical dynamics are demonstrated in 2D and 3D Fibonacci potential models.

Conclusions:

• The findings clarify previous results in disordered and quasiperiodic models.
• The established relation provides a new perspective on critical dynamics in various dimensions.
• This work offers a deeper understanding of transport phenomena in quantum systems.