Anomalous Quantum Criticality at a Continuous Metal-Insulator Transition

The Falicov-Kimball model (FKM) is long known to be the simplest model of correlated fermions exhibiting a novel Mott-like quantum critical point (QCP) assocaited with a {\it continuous} MIT in dimensions $D \geq 3$. It is also known to be isomorphic to an {\it annealed} binary-alloy disorder model. Notwithstanding extensive numerical studies for the FKM, analytic insight into the microscopic processes spawning novel Mott-like quantum criticality is scarce. Here, we develop a fully analytic theory for the Mott-like quantum criticality in the FKM on a hierarchical Cayley tree (Bethe lattice) by utilizing a single input from a 2-site cluster-dynamical mean-field theory (CDMFT). We find that density fluctuation modes acquire anomalous dimensions, originating from infra-red power-law singular cluster self-energies. Interestingly, we uncover, at $T=0$, that this {\it sub-diffusive} metal with glassy dynamics separating a weakly ergodic metal from a non-ergodic insulator shrinks to a single point, namely the Mott-like QCP, at least on the Bethe lattice. We detail the consequences of this anomalous quantum criticality for a range of thermal and dynamical responses in a variety of physical systems that can be effectively modelled by the FKM.

💡 Research Summary

The paper presents a fully analytic treatment of the Falicov‑Kimball model (FKM) on a Bethe lattice, focusing on the continuous metal‑insulator transition (MIT) that occurs in dimensions D ≥ 3. While the FKM is known to host a Mott‑like quantum critical point (QCP) associated with a continuous band‑splitting transition, previous work has largely relied on numerical DMFT or cluster‑DMFT (CDMFT) calculations, leaving the microscopic origin of the critical behavior poorly understood.

The authors start from a two‑site CDMFT calculation, using the cluster self‑energies ΣS(ω) and ΣP(ω) as the sole input to construct a self‑consistent theory on the hierarchical Cayley tree. The key CDMFT result is that the imaginary part of the local self‑energy near the critical interaction Uc behaves as Im Σc(ω) ∝ |ω|α with α ≈ 1/3. This power‑law singularity implies a frequency‑dependent quasiparticle residue zc(ω) ≈ ω1‑α and a three‑leg vertex Λc(ω) ∝ ω‑(1‑α), which diverges in the infrared.

Because the vertex enters the charge‑density response χch(q, ω) through the Ward identity, the usual diffusion pole (∝ iDq²/ω) is replaced by a branch‑point structure χch(q, ω) ≈