Scaling behavior of superconductors

TL;DR

The paper argues that universal scaling in both conventional and unconventional superconductors arises from flat-band physics associated with a topological fermion-condensation quantum phase transition (FCQPT). It develops a fermion-condensation (FC) framework in which two quasiparticle subsystems coexist: a heavy, FC-driven sector with mass $M^*_{FC}$ and a lighter sector with mass $M^*_L$, linked by an energy scale $E_0 \approx 2\Delta_1$ and yielding a universal scaling $E_{\Delta}/\gamma \propto T_c^2$. This approach explains Homes' law $\rho_{s0} \propto T_c\sigma(T_c)$ and the linear-in-$T$ resistivity with a universal scattering rate $1/(\tau T) \sim k_B/\hbar$, across HF metals, cuprates, and ordinary metals, while predicting phenomena such as reduced superfluid density in overdoped cuprates ($n_s \ll n_{el}$) and asymmetrical conductivity tied to broken C/T symmetries in the NFL regime. The FCQPT framework thus provides a robust, topological mechanism underlying the physics of strongly correlated Fermi systems, with broad applicability to graphene, high-$T_c$ cuprates, and heavy-fermion superconductors, and explains a wide set of experimental scaling observations. These results suggest that a BCS-like FC state can describe superconductivity in both conventional and unconventional systems, connecting microscopic details to universal macroscopic behavior.

Abstract

In our brief review, we will consider the general universal scaling properties of superconductors. The physics of superconductors, represented by both conventional and unconventional superconductors, has been the main topic of high-$T_c$ superconductor physics for over thirty years, revealing some of the properties of high-$T_c$ (or unconventional) superconductors. Scaling relationships lead to the identification of fundamental laws of nature and reveal the essence of superconductor physics. Advances in experimental technology allow us to collect important data, which in turn allow us to make definitive statements about the physical processes underlying strongly correlated Fermi systems. Basing on this observation, we analyze experimental facts that reveal the general scaling properties of both high-$T_c$ and ordinary superconductors, and theoretically explain that the Homes' law $ρ_{s0}= (1/2πλ_D)^2= T_cσ(T_c)$ is applicable to the both types of superconductors. Here $ρ_{s0}$ is the superconducting electron density, $λ_D$ is the zero-$T$ penetration depth, $σ$ is the normal state conductivity, $T$ is temperature and $T_c$ is the temperature of superconducting phase transition. Overall, these scaling relationships lead to the identification of fundamental laws of nature and reveal the essence of superconductor physics. All these observations support the theory of fermion condensation. Our theoretical results agree well with a body of diverse and seemingly unrelated experimental facts. They show that the topological fermion condensation quantum phase transition, generating flat bands, is an intrinsic property of strongly correlated Fermi systems and can be considered as a universal agent explaining their basic physics.

Figures (10)

• Figure 1: Experimental results (shown by the squares) for the average Fermi velocity $V_F$ versus the critical temperature $T_c$ for graphene (MATBG) mac. The downward arrows depict that $V_F\leq V_0$, with $V_0$ is the maximal value shown by the red square. Theory is displayed by the solid straight line.
• Figure 2: The figure is adapted from nat01, and shows experimental dependence of the superconducting gap $\Delta$ versus the integrated local density of states collected on the high-$T_c$ superconductor $\rm Bi_2Sr_2CaCu_2O_{8+x}$. Here $\rm x$ is oxygen doping concentration. Darker color represents more data points with the same integrated local density of states and the same size gap $\Delta$nat01. The straight blue line shows average value $\Delta$ versus the integrated local density of states.
• Figure 3: Condensation energy $E_\Delta/\gamma\propto T_c^{2}$ divided by the specific heat $\gamma$ as a function of $T_c$ for a wide range of superconductors, with the slope $\alpha_{s}=2$prb2015, see Eq. \ref{['30']}. Deviations from the line of best fit, spanning six orders of magnitude for $E_\Delta/\gamma$ and almost three orders of magnitude for $T_c$, are relatively small.
• Figure 4: Scattering rates of different strongly correlated metals like HF metals, high-$T_c$ superconductors, organic metals, and conventional metals bruin. All these metals exhibit $\rho(T)\propto T$ and demonstrate two orders of magnitude variations in their Fermi velocities $v_F$. The parameter $a_2\simeq 1$ corresponds to the Planckian limit, and gives the best fit shown by the solid line, see Eq. \ref{['vf']}. The region occupied by the common metals is displayed by the two arrows, and the arrow shows the region of strongly correlated metals
• Figure 5: Schematic $T-B$ phase diagram of a strongly correlated Fermi system. The vertical and horizontal arrows crossing the transition region (hatched area) depict the LFL-NFL and NFL-LFL transitions at fixed $B$ and $T$, respectively. At $B<B_{c2}$ the system is in its possible superconducting (SC) state, with $B_{c2}$ is shown by the solid line denoting a critical magnetic field destroying the SC state. The hatched area with the solid curve $T_{\rm cross}(B\sim T)$ represents the crossover region separating NFL and LFL domains. A part of the crossover region is hidden in the possible SC state.