Beyond Homes Scaling: Disorder, the Planckian Bound, and a New Universality
Bibliographic record
Abstract
Beginning with high- <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi>T</a:mi> <a:mi>c</a:mi> </a:msub> </a:math> cuprate materials, it has been observed that many superconductors exhibit so-called “Homes scaling,” in which the zero-temperature superfluid density <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msub> <c:mi>ρ</c:mi> <c:mrow> <c:mi>s</c:mi> <c:mn>0</c:mn> </c:mrow> </c:msub> </c:math> is proportional to the product of the normal-state dc conductivity and the superconducting transition temperature <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi>σ</e:mi> <e:mi>dc</e:mi> </e:msub> <e:msub> <e:mi>T</e:mi> <e:mi>c</e:mi> </e:msub> </e:math> . For conventional, <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>s</g:mi> </g:math> -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 <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>d</i:mi> </i:math> -wave superconductors, a stark contrast emerges, with <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msub> <k:mi>ρ</k:mi> <k:mrow> <k:mi>s</k:mi> <k:mn>0</k:mn> </k:mrow> </k:msub> <k:mo>∝</k:mo> <k:msup> <k:mrow> <k:mo stretchy="false">(</k:mo> <k:msub> <k:mi>σ</k:mi> <k:mi>dc</k:mi> </k:msub> <k:msub> <k:mi>T</k:mi> <k:mi>c</k:mi> </k:msub> <k:mo stretchy="false">)</k:mo> </k:mrow> <k:mn>2</k:mn> </k:msup> </k:math> 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 <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mrow> <o:msub> <o:mrow> <o:mi>YBa</o:mi> </o:mrow> <o:mrow> <o:mn>2</o:mn> </o:mrow> </o:msub> <o:msub> <o:mrow> <o:mi>Cu</o:mi> </o:mrow> <o:mrow> <o:mn>3</o:mn> </o:mrow> </o:msub> <o:msub> <o:mrow> <o:mi mathvariant="normal">O</o:mi> </o:mrow> <o:mrow> <o:mn>6.333</o:mn> </o:mrow> </o:msub> </o:mrow> </o:math> and overdoped <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"> <r:mrow> <r:msub> <r:mrow> <r:mi>La</r:mi> </r:mrow> <r:mrow> <r:mn>2</r:mn> <r:mo>−</r:mo> <r:mi>x</r:mi> </r:mrow> </r:msub> <r:mrow> <r:msub> <r:mrow> <r:mi>Sr</r:mi> </r:mrow> <r:mrow> <r:mi>x</r:mi> </r:mrow> </r:msub> </r:mrow> <r:mrow> <r:msub> <r:mrow> <r:mi>CuO</r:mi> </r:mrow> <r:mrow> <r:mn>4</r:mn> </r:mrow> </r:msub> </r:mrow> </r:mrow> </r:math> , and in very clean materials with little inelastic scattering, such as <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"> <t:mrow> <t:msub> <t:mrow> <t:mi>Sr</t:mi> </t:mrow> <t:mrow> <t:mn>2</t:mn> </t:mrow> </t:msub> </t:mrow> <t:mrow> <t:msub> <t:mrow> <t:mi>RuO</t:mi> </t:mrow> <t:mrow> <t:mn>4</t:mn> </t:mrow> </t:msub> </t:mrow> </t:math> . We present a revised analysis where both axes of the original Homes scaling plot are normalized by the Drude plasma weight <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"> <v:msubsup> <v:mi>ω</v:mi> <v:mrow> <v:mi>p</v:mi> <v:mo>,</v:mo> <v:mi>D</v:mi> </v:mrow> <v:mn>2</v:mn> </v:msubsup> </v:math> and show that a new universal scaling emerges, in which the superfluid fractions of dirty <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"> <x:mi>s</x:mi> </x:math> -wave and dirty <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"> <z:mi>d</z:mi> </z:math> -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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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".