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Beyond Homes Scaling: Disorder, the Planckian Bound, and a New Universality

D. M. Broun1, Vivek Mishra2,3, J. S. Dodge1, and P. J. Hirschfeld3

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Phys. Rev. X 15, 041005 – Published 8 October, 2025

DOI: https://doi.org/10.1103/xbv7-3s3h

Abstract

Beginning with high-Tc cuprate materials, it has been observed that many superconductors exhibit so-called “Homes scaling,” in which the zero-temperature superfluid density ρs0 is proportional to the product of the normal-state dc conductivity and the superconducting transition temperature σdcTc. For conventional, s-wave superconductors, such scaling has been shown to be a natural consequence of elastic-scattering disorder, not only in the extreme dirty limit, but across a broad range of scattering parameters. Here we show that when an analogous calculation is carried out for elastic scattering in d-wave superconductors, a stark contrast emerges, with ρs0∝(σdcTc)2 in the dirty limit, in apparent violation of Homes scaling. Within a simple approximate Migdal-Eliashberg treatment of inelastic scattering, we show how the observed Homes scaling is recovered. The normal-state behavior of near-optimally-doped cuprates is dominated by inelastic scattering, but significant deviations from Homes scaling occur for disorder-dominated cuprate systems, such as underdoped YBa2Cu3O6.333 and overdoped La2−xSrxCuO4, and in very clean materials with little inelastic scattering, such as Sr2RuO4. We present a revised analysis where both axes of the original Homes scaling plot are normalized by the Drude plasma weight ωp,D2 and show that a new universal scaling emerges, in which the superfluid fractions of dirty s-wave and dirty d-wave superconductors coalesce to a single point at which normal-state scattering is occurring at the Planckian bound. The combined result is a new tool for classifying superconductors in terms of order parameter symmetry, as well as scattering strength and character. Although our model starts from a Fermi-liquid assumption, it describes underdoped cuprates surprisingly well.

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Popular Summary

In condensed matter physics, understanding universal behavior across different materials helps reveal fundamental principles. One such universal relation is Homes scaling, which links the zero-temperature superfluid density of a superconductor to the product of its normal-state conductivity and critical temperature. While this relation holds for many materials, its theoretical origin has remained unclear, with existing explanations working only for certain classes of superconductors. In this study, we uncover a more fundamental form of universality by considering the effects of chemical impurities and inelastic scattering, showing that all superconductors can be classified in a unified framework when carrier density is properly accounted for.

We analyze superconductors with different symmetries of their electron pair wave functions, including those with zero and nonzero angular momentum. By incorporating both impurity scattering and inelastic scattering from fluctuations in the electronic liquid, we construct a revised scaling plot. This plot removes the influence of carrier density and reveals that all superconductors fall on one of two branches. Notably, Planckian-limit superconductors, where the scattering rate is comparable to the critical temperature, all collapse to the same point on the new plot. This approach highlights the role of scattering and symmetry in determining superfluid behavior across diverse materials.

Our findings provide a powerful classification scheme for superconductors, which can guide the discovery and characterization of new superconducting materials and may help identify which systems approach the Planckian limit.

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Phys. Rev. X 15, 041005– Published 8 October, 2025
  • Received 18 February 2025
  • Revised 15 May 2025
  • Accepted 19 August 2025
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