Mechanism behind a possible shift of the critical momentum

Determine the physical mechanism that could cause the critical momentum of the long-range transverse-field Ising model to depart from its nearest-neighbor value at small decay exponents.

Background

The full momentum-dependent dispersion enabled by the graph-zeta method permits the authors to search for changes in the momentum at which the excitation gap is minimized. They note that such a change could occur at small decay exponents, although their calculations find the nearest-neighbor critical momentum for the cases studied. The physical origin of any possible departure remains unidentified.

References

This capability is particularly important when the critical momentum $\bm k_c$ deviates from its nearest-neighbor value at small decay exponents $\sigma$, potentially due to an as-yet-unknown physical mechanism.

— Exact and fast series expansions for quantum models with long-range interactions  (2609.18761 - Duft et al., 16 Sep 2026) in Section 7.3, “Physical quantities”

The precise long-range--to--short-range crossover remains unsettled despite early renormalization-group and large-$N$ analyses, subsequent numerical and field-theoretic studies supporting different scenarios, and recent large-scale studies of several classical long-range models together with a quantum $O(n)$ renormalization-group analysis.

— Universal Quantum Criticality from a Gaussian Variational Manifold  (2610.26246 - Khasseh et al., 19 Oct 2026) in Introduction; Supplemental Material, Section S2, paragraph “Long-range--to--short-range crossover”