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Critical Dynamics in Short-Range Quadratic Hamiltonians

Miroslav Hopjan1,2 and Lev Vidmar1,3

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Phys. Rev. Lett. 135, 060401 – Published 5 August, 2025

DOI: https://doi.org/10.1103/5vyx-k877

Abstract

We investigate critical transport and the dynamical exponent through the spreading of an initially localized particle in quadratic Hamiltonians with short-range hopping in lattice dimension dl. We consider critical dynamics that emerges when the Thouless time, i.e., the saturation time of the mean-squared displacement, approaches the typical Heisenberg time. We establish a relation, z=dl/ds, linking the critical dynamical exponent z to dl and to the spectral fractal dimension ds. This result has notable implications: it says that superdiffusive transport in dl≥2 and diffusive transport in dl≥3 cannot be critical in the sense defined above. Our findings clarify previous results on disordered and quasiperiodic models and, through Fibonacci potential models in two and three dimensions, provide nontrivial examples of critical dynamics in systems with dl≠1 and ds≠1.

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Phys. Rev. Lett. 135, 060401– Published 5 August, 2025
  • Received 4 March 2025
  • Revised 4 June 2025
  • Accepted 9 July 2025
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