Discrete scale invariance and ln(B) periodic quantum oscillations in topological semimetals

Abstract

Quantum oscillations have played a major role in the history of condensed matter physics-and indeed in physics as a whole. In 1930, Shubnikov and de Haas (SdH) [1] discovered oscillations in the resistance of bismuth as a function of magnetic field, R(B). Because of theoretical work of Landau [2] in the same year, it was soon realized that the oscillations were periodic in 1/B: if a peak in R is observed at B = 1 T and another at B = 0.5 T, then we should see one at B = 0.33 T, B = 0.25 T and so on. This work is famous, but maybe not as much as it should be: it was the first discovery of a macroscopic effect that depends for its existence on the fact that an electron has a phase. The magnetic susceptibility also oscillates. This is the deHaas-van Alphen effect, also a product of the year 1930. The Hall resistance does as well, but in bulk materials the effect is not so big. All of these effects are periodic in 1/B, just like the SdH oscillations. It must be so, since all of these quantities depend on the density of electron states at the Fermi energy ρ(E F), and it is this underlying physical quantity that is periodic in 1/B. These discoveries were dramatic in that they were a beautiful demonstration of quantum mechanics in bulk materials. The observation of Onsager [3] in 1952 that the period in (1/B) is inversely proportional to the extremal cross section of the Fermi surface meant that SdH and dHvA effects could be used for the measurement of band structure,

Research progress on topological nodal line semimetals

“…此外, 一些非点式对称操作还可能导 致能带在整个面上是简并的, 可以形成"节面半金 属" [53][54][55] . 在实验上观测节线半金属最直接的方法是 通过角分辨电子谱(ARPES)测量体系的电子结构(鼓 膜型表面态) [56][57][58][59] , 其次还可以通过Shubnikov-de Haas (SdH)振荡探测节线相 [60][61][62] , 此外科学家还在节线半金 属中观察到了巨大的表面电声耦合效应 [63] . 拓扑节线…”