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Abstract We prove that there are$$\gg \frac{X^{\frac{1}{3}}}{(\log X)^2}$$ imaginary quadratic fieldskwith discriminant$$|d_k|\le X$$ and an ideal class group of 5-rank at least 2. This improves a result of Byeon, who proved the lower bound$$\gg X^{\frac{1}{4}}$$ in the same setting. We use a method of Howe, Leprévost, and Poonen to construct a genus 2 curveCover$$\mathbb {Q}$$ such thatChas a rational Weierstrass point and the Jacobian ofChas a rational torsion subgroup of 5-rank 2. We deduce the main result from the existence of the curveCand a quantitative result of Kulkarni and the second author.
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Softening of a flat phonon mode in the kagome ScV6Sn6
Abstract Geometrically frustrated kagome lattices are raising as novel platforms to engineer correlated topological electron flat bands that are prominent to electronic instabilities. Here, we demonstrate a phonon softening at thekz = πplane in ScV6Sn6. The low energy longitudinal phonon collapses at ~98 K andq = $$\frac{1}{3}\frac{1}{3}\frac{1}{2}$$ due to the electron-phonon interaction, without the emergence of long-range charge order which sets in at a different propagation vectorqCDW = $$\frac{1}{3}\frac{1}{3}\frac{1}{3}$$ . Theoretical calculations corroborate the experimental finding to indicate that the leading instability is located at$$\frac{1}{3}\frac{1}{3}\frac{1}{2}$$ of a rather flat mode. We relate the phonon renormalization to the orbital-resolved susceptibility of the trigonal Sn atoms and explain the approximately flat phonon dispersion. Our data report the first example of the collapse of a kagome bosonic mode and promote the 166 compounds of kagomes as primary candidates to explore correlated flat phonon-topological flat electron physics.
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- Award ID(s):
- 2011750
- PAR ID:
- 10508186
- Publisher / Repository:
- Nature Communication
- Date Published:
- Journal Name:
- Nature Communications
- Volume:
- 14
- Issue:
- 1
- ISSN:
- 2041-1723
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1038/s41467-023-42186-6
