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We found a new relation for critical transport dynamics in quantum systems. This links the dynamical exponent to lattice and spectral fractal dimensions, clarifying transport behaviors in various dimensions.

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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.