Parabolic Scaling in Overdoped Cuprate: a Statistical Field Theory Approach

Abstract

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)

Relativistic Ginzburg–Landau equation: An investigation for overdoped cuprate films

“…Different from the -driven transitions, where 2 is linear in ( − ), equation (17) indicates that 2 ( ) is quadratic in . Now, due to this difference, we show that equation (17) leads to a novel scaling relationship between the transition temperature and the zero-temperature correlation length (0). To this end, we assume that quantum fluctuations with wavelengths larger than 2 Λ ⁄ cannot be averaged out.…”

“…Therefore, as → 0, quantum fluctuations are expected to be amplified so that the mean-field approximation breaks down. Based on this observation, by applying the renormalization group approach into the quantum partition function (7), we have proved that [15][16][17] when → 0, 4 ( ) yields a fixed point depending on the momentum cut-off Λ (up to the one-loop correction):…”

“…We have clarified that [15][16][17] the parabolic scaling (11) holds if and only if the mean-field approximation breaks down. To identify the scope of application of the mean-field approximation, we have proposed a quantum Ginzburg number to derive the following result [17]:…”

“…As possible evidence, Bozovic et al have observed that, with increased doping (𝑇 𝑐 → 0), LSCO becomes more metallic, and there is induction of a quantum phase transition from a superconductor to a normal metal [12][13]. To understand the parabolic scaling in equation ( 1), by introducing the imaginary time [1,14], we [15][16][17] extended Gor'kov's Ginzburg-Landau equation at zero temperature to an exact relativistic form, which is intended to describe the zerotemperature overdoped cuprate. Our calculation shows that [15][16][17], as 𝑇 𝑐 → 0 , quantum fluctuations play an important role in inducing the parabolic scaling, and that the two-class scaling (1) can be derived exactly.…”

“…To understand the parabolic scaling in equation ( 1), by introducing the imaginary time [1,14], we [15][16][17] extended Gor'kov's Ginzburg-Landau equation at zero temperature to an exact relativistic form, which is intended to describe the zerotemperature overdoped cuprate. Our calculation shows that [15][16][17], as 𝑇 𝑐 → 0 , quantum fluctuations play an important role in inducing the parabolic scaling, and that the two-class scaling (1) can be derived exactly. As evidence justifying the relativistic Ginzburg-Landau equation, we have obtained the correct theoretical values of 𝛾, 𝑇 𝑀 , and 𝑇 𝑄 [16][17].…”