Universal scaling behavior of resistivity under two-dimensional superconducting phase fluctuations

npj | quantum materials Article Published in partnership with Nanjing University https://doi.org/10.1038/s41535-025-00822-y Universal scaling behavior of resistivity under two-dimensional superconducting phase fluctuations Check for updates 1 1,2 1,3 1,2 1234567890():,; 1234567890():,; Zongsheng Zhou , Kang Wang , Hai-Jun Liao , Zi-Xiang Li 1,2,4 & Tao Xiang In superconductors with low superfluid density, superconducting phase fluctuations can significantly influence many physical properties. A quantitative microscopic description of electrical transport under thermal phase fluctuations has remained scarce. Using Monte Carlo simulations of the classical XY model, we investigate the numerically exact transport properties arising from thermal phase fluctuations. We demonstrate that the quasiparticle lifetime is determined by the correlation length of the superconducting order parameter. This leads to a universal scaling of the electrical resistivity governed by Berezinskii-Kosterlitz-Thouless criticality. While it is consistent with the Halperin-Nelson framework, our results reveal a scaling exponent smaller than one for the inverse correlation length. We also discuss the dependence of resistivity on pairing amplitude and implications for related experiments. Superconductivity (SC) is a macroscopic quantum phenomenon stemming from the condensation of paired electrons, which is characterized by a complex pairing order parameter with both amplitude and phase1,2. The superconducting transition temperature Tc is governed by the lower energy scale between Cooper pair formation and phase coherence establishment. In BCS mean-field theory, it is generally assumed that the phase coherence temperature significantly exceeds the Cooper pair formation temperature. Consequently, fluctuations of the order parameter are negligible for determining important physical properties such as transition temperature Tc, which generically holds for SC in many conventional metals3. However, in systems with low superfluid density due to their low charge carrier density, such as unconventional superconductors, including cuprates4–7, organic superconductors8,9, and twisted bilayer graphene10–13, this relationship can be reversed. Additionally, superfluid density in conventional superconductors could be heavily suppressed by disorder14–16. Even if Cooper pairs form, the system can remain in a non-superconducting state due to the incoherence of the superconducting phases. Therefore, superconducting fluctuations can become significant above the critical temperature Tc. Generally, low-energy physics is governed by superconducting phase fluctuations17, while pairing amplitude fluctuations dominate at higher temperatures18. When superconducting fluctuations are remarkable, superconducting phase fluctuations can drive many intriguing phenomena19. In the past decades, phase fluctuations in superconductors have been extensively investigated from both theoretical20–38 and experimental5,39–50 perspectives. In unconventional superconductors, the pairing order parameter usually has a more complex structure, such as nodal lines for d-wave pairing. Strong phase fluctuations can lead to the opening of the spectral gap above superconducting Tc19–22,25–27,36. It has been proposed theoretically that phase fluctuations could trigger novel phenomena, such as the Fermi arc26–29,51 observed in underdoped cuprates52–54, and the possibility of charge-4e SC55–62 as a vestigial order. Additionally, the phase fluctuation picture can be employed to explain the light-enhanced superconductivity63. Nevertheless, a qualitative study of the transport properties arising from superconducting phase fluctuations of a microscopic model in the temperature regime above Tc is rare. Recently, armed with the development of thin-film technology, strange-metal electrical transport behaviors, which were unveiled in underdoped and optimally doped cuprates soon after the discovery of high-Tc superconductors64–66, have been observed in a wide doping range of electron-doped cuprates La2−xCexCuO
467,68 and 

iron-pnictide  super conductors FeSe69, along with Ca10 Pt4 As8 Fe0:97 Pt0:03 2 As2 5 70. More remarkably, the coefficient of linear-T resistivity exhibits a quadratic dependence on the superconducting transition temperature Tc68–70, which implies the possible relation between anomalous metallic transport and the SC or other ingredients associated with superconducting pairing mechanisms. Therefore, a qualitative investigation of the electrical resistivity driven by superconducting phase fluctuations above the critical temperature Tc is highly desirable. This study could provide valuable insights into the relationship between resistivity and the parameters of superconductivity. Additionally, understanding 1 Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing, China. 2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing, China. 3Songshan Lake Materials Laboratory, Dongguan, Guangdong, China. 4Beijing Academy of Quantum e-mail: ; Information Sciences, Beijing, China. npj Quantum Materials | (2025)10:106 1 Article https://doi.org/10.1038/s41535-025-00822-y the role of pairing symmetries in resistivity under phase fluctuations is of significant interest. By considering different pairing symmetries, it could deepen our understanding of the interplay between anisotropic scattering and transport properties in superconducting systems. In this work, we systematically investigate the properties of electrical transport in a microscopic Bogoliubov-de Gennes (BdG) Hamiltonian that incorporates superconducting pairing and accounts for thermal phase fluctuations above the superconducting critical temperature Tc. As the pairing amplitude fluctuations are insignificant when temperatures are much lower than the pairing temperature, the amplitude is fixed as a spatially independent value. Distinct from the BCS mean-field Hamiltonian, the model under consideration incorporates the spatial fluctuations of phase. The phase fluctuations are governed by a classical XY model, characterizing the thermal fluctuations of the order parameter’s phase with varying temperatures. Employing large-scale Monte Carlo simulation, we study the transport properties of the model at finite temperatures. The numerical results reveal that the electrical resistivity induced by the thermal phase fluctuations in the low-temperature regime above Tc obeys the scaling pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi behavior ρðT Þ / A expðB= T=T c  1Þ, which is consistent with the critical scaling relation of the correlation length for the BerezinskiiKosterlitz-Thouless (BKT) transition71. Thanks to the special form of the BKT correlation length, our results are formally in accord with the phenomenological theory17. However, a notable deviation is revealed by our numerical results, we find that the resistivity is inversely proportional to th (...truncated)