Ultrasound Response in Quantum Critical β-YbAlB4 and α-YbAl0.986Fe0 .014B4

We analyze the key origin of quantum valence criticality in the heavy electron metal beta-YbAlB(4 )evidenced in the sister compound alpha-YbAl0.986Fe0.014B4. By constructing a realistic canonical model for beta-YbAlB4, we evaluate Coulomb repulsion between the 4f and 5d electrons at Yb U-fd approximate to 6.2eV realizing the quantum critical point (QCP) of the Yb-valence transition. To reveal the Yb 5d contribution to the quantum critical state, we propose ultrasound measurement. We find that softening of elastic constants of not only the bulk modulus but also the shear moduli is caused by electric quadrupole fluctuations enhanced by critical 4f and 5d charge fluctuations for low temperatures at the valence QCP. Possible relevance of these results to beta-YbAlB4 and also alpha-YbAl1-xFexB4 is discussed.