It was recently reported that, in the highly overdoped side of single-crystal 2− 4 films, the transition temperature and zero-temperature superfluid phase stiffness (0) will obey a parabolic scaling = • √ (0). Parabolic scaling indicates a quantum phase transition from a superconductor to a normal metal, for which there has been scant understanding [Nature 536, 309-311 (2016)]. The current study shows that, using the quantum critical model for zero-temperature Cooper pairs [EPL 118, 57007 (2017)], parabolic scaling can be exactly derived, where = ( , ) is uniquely determined by the Fermi energy and the minimal lattice constant of superconducting materials. For single-crystal 2− 4 films, we calculate the theoretical value of , which yields 4.29 • 1 2 ⁄ and is in accordance with an experimental measure value (4.2 ± 0.5) • 1 2 ⁄ with high accuracy. Our formula for can be further tested by investigating other BCS-like materials.

We show that an exponential income distribution will emerge spontaneously in a peer-to-peer economic network that shares the publicly available technology. Based on this finding, we identify the exponential income distribution as the benchmark structure of the well-functioning market economy. However, a real market economy may deviate from the well-functioning market economy. We show that the deviation is partly reflected as the invalidity of exponential distribution in describing the super-low income class that involves unemployment. In this regard, we find, theoretically, that the lower-bound u of exponential income distribution has a linear relationship with (per capita) unemployment compensation. In this paper, we test this relationship for the United Kingdom from 2001 to 2015. Our empirical investigation confirms that the income structure of low and middle classes (about 90% of populations) in the United Kingdom exactly obeys exponential distribution, in which the lower-bound u is exactly in line with the evolution of unemployment compensation.

Wilson’s quantum field theory in less than 4 dimensions has achieved a great success in the study of critical phenomenon, but is still not tested within the scope of particle physics. To guarantee the validity of Wilson’s quantum field theory in less than 4 dimensions, Newton-Leibniz’s differential-integral formulas must be extended to the non-integer dimensional situation. We show that this leads to a new prediction that Planck’s constant will be expressed in terms of three fundamental constants: critical time scale, dimension of time axis and total energy of universe. We propose the corresponding methods to measure these three constants. It will be thus interesting to compare the well-known value of Planck’s constant with the potential theoretical value consisting of three fundamental constants.

Recently, Bozovic et al. reported that [Nature 536, 309-311 (2016)], in the overdoped side of the single-crystal 2− 4 (LSCO) films, the transition temperature and zero-temperature superfluid phase stiffness (0) will obey a two-class scaling law: = • √ (0) for ≤ and ∝ (0) for ≥ , where = (4.2 ± 0.5) 1 2 ⁄ , ≈ 15 , and ≈ 12 . They further pointed out that the parabolic scaling observed in the highly overdoped side indicates a quantum phase transition from a superconductor to a normal metal. In this paper, we propose a quantum partition function (QPF) for zero-temperature Cooper pairs, by which one can effectively distinguish the mean-field and quantum critical behaviors. We theoretically show that the two-class scaling law can be exactly derived by using the QPF, and the theoretical values of , , and are well in accordance with experimental measure values. Our analyses indicate that the linear scaling ∝ (0) is a mean-field behavior, while the parabolic scaling = • √ (0) is a quantum critical behavior. [1]. Y. J. Uemura et al., Universal Correlations between and * ⁄ (Carrier Density over Effective Mass) in High-Cuprate Superconductors, Phys. Rev. Lett. 62, 2317 (1989)