Establish logarithmic scaling violation at the upper critical dimension

Demonstrate the logarithmic violation of scaling at the upper critical dimension d=4 for the disorder-driven Weyl semimetal-to-metal quantum phase transition, using a computational framework capable of resolving the required energy and momentum ranges.

Background

At d=4, the paper predicts logarithmic corrections to scaling because four dimensions constitute the upper critical dimension of the disorder-driven Weyl semimetal-to-metal transition. The finite-range local hopping in the tight-binding model restricts the linear Weyl dispersion to a small region near zero energy, preventing the numerical analysis from accessing the broad energy and system-size ranges needed to observe these corrections.

The authors suggest SLAC fermions as a possible platform because they exhibit Weyl dispersion over the entire reciprocal space. However, their nonlocal hopping makes kernel polynomial method calculations computationally expensive, leaving the direct demonstration of logarithmic scaling violation unresolved.

References

Unfortunately, here we could not demonstrate the logarithmic violation of scaling at $d=4$ as the tight-binding model for WSMs with finite-range (local) hopping produces a linear energy-momentum relation only over a small segment of the Brillouin zone near $E=0$. SLAC fermions, featuring Weyl dispersion over the entire reciprocal space, on the other hand, can be the ideal platform to demonstrate logarithmic violation of scaling at $d=4$. However, the nonlocal hopping amplitudes in the construction of SLAC fermions make KPM, ideally suited for sparse matrices, numerically expensive. We, thus, leave this issue for a future study.

— Upper critical dimension for dirty Weyl semimetal-to-metal quantum phase transitions  (2609.30265 - Li et al., 24 Sep 2026) in Summary and discussions