Microscopic validation of the quantum Griffiths rare-region mechanism in the field-driven superconductor–metal transition of disordered two-dimensional superconductors

Ultrathin crystalline Ga films driven through a superconductor–metal transition (SMT) by a perpendicular magnetic field show a diverging dynamical critical exponent, the 2015 hallmark of a quantum Griffiths singularity (QGS) caused by rare superconducting islands surviving above the mean-field critical field. In the decade since, the same phenomenology has been replicated in many two-dimensional (2D) superconductors and extended to three-dimensional, interface, cuprate, and graphene platforms, yet the underlying rare-region mechanism has not been validated microscopically. This problem asks for the missing microscopic evidence: direct space-resolved imaging of the superconducting rare regions and their statistics and coupling near the infinite-randomness quantum critical field, quantitative characterization of the quenched disorder that seeds them, and a determination of whether the activated (infinite-randomness) critical exponents are universal or disorder dependent, so that the rare-region mechanism can be distinguished from vortex-glass, superconducting-fluctuation, and dissipation-driven alternatives.

Background

In a two-dimensional superconducting film, a perpendicular magnetic field can destroy superconductivity at zero temperature, producing a field-tuned quantum phase transition — a superconductor–insulator transition (SIT) or, in sufficiently clean and highly conducting films, a superconductor–metal transition (SMT) into a weakly localized metallic state with $k_{\mathrm{F}}\ell\gg1$ (large product of Fermi wave vector and mean free path) and a sheet resistance well below the quantum resistance $h/e^{2}$. Near such a quantum critical point, transport obeys finite-size scaling: magnetoresistance isotherms $R(B,T)$ cross at a critical field, and the data collapse onto $R(B,T)=R_{c}F(|B-B_{c}|T^{-1/z\nu})$, where $z$ is the dynamical critical exponent relating the correlation time to the correlation length, $\nu$ the correlation-length exponent, and $F$ a universal scaling function. In a conventional quantum critical point $z\nu$ is a finite constant.

Quenched (frozen-in) disorder changes this picture qualitatively. In a disordered quantum system, statistical fluctuations can locally favor the ordered phase: rare regions (also called rare islands or puddles) — spatially isolated patches that are locally superconducting while the bulk is not — can appear on the disordered side of the transition. Because large rare regions flip their order parameter exponentially slowly, they produce a so-called Griffiths phase in which the effective dynamical exponent $z$ grows without bound as the transition is approached; in the strong-disorder limit the critical point becomes an infinite-randomness critical point at which dynamics is governed not by a power law but by activated scaling, $\ln\xi_{\tau}\propto\xi^{\psi}$ (the logarithm of the temporal correlation length scales as a power of the spatial correlation length, with exponent $\psi$). This physics was established theoretically in random transverse-field Ising-type models (lattice spin models whose exchange couplings and transverse fields are both frozen random variables), where in two spatial dimensions the infinite-randomness exponents are $\nu\approx1.2$ and $\psi\approx0.5$, and it is expected for the disordered $N=2$ order parameter (the two-component complex order parameter of a superconductor) with the additional possibility that Ohmic dissipation from the metallic bath suppresses the rare-region fluctuations (Vojta's strong-disorder theory, based on a renormalization group that iteratively eliminates the strongest coupling or field and becomes exact at infinite randomness). A crucial empirical precedent is that scanning tunneling spectroscopy — a technique in which the tunneling current between a sharp tip and the surface maps the local superconducting energy gap $\Delta$ with nanometer resolution — had already revealed strong spatial inhomogeneity of $\Delta$ and locally surviving superconductivity in disordered TiN films near the disorder-driven SIT (Sacépé et al., 2008), but not in the field-tuned, ultralow-temperature Griffiths regime.

Xing et al. (Science 2015) studied three-monolayer crystalline Ga films grown by molecular beam epitaxy (a technique that deposits elemental atomic beams onto a heated crystalline substrate in ultrahigh vacuum, yielding epitaxial films) and measured magnetoresistance from 2.8 K down to 25 mK. Instead of a single crossing point with constant $z\nu$, they found a continuous line of crossing points whose critical field follows the mean-field Werthamer–Helfand–Hohenberg (WHH) theory of the upper critical field above ~1 K but bends sharply upward at lower temperature, and a product $z\nu$ that diverges as $B^{*}$ is approached, consistent with $z\nu\propto|B-B^{*}|^{-\nu\psi}$ using the 2D infinite-randomness exponents $\nu\approx1.2$, $\psi\approx0.5$. They interpreted this as the quantum Griffiths singularity of the SMT, with the rare regions being superconducting islands that survive in a vortex-glass-like environment (a disordered, pinned-vortex phase; a vortex lattice is the ordered array of magnetic flux lines in a type-II superconductor). The authors and the accompanying Perspective by Markovic framed several limitations: the observation was unique at the time, the attribution of its earlier absence (thermal destruction of the vortex-glass-like phase, requiring ultralow temperatures) was plausible but unproven, the universality of the exponents was not established, alternative mechanisms were not excluded, the quenched disorder responsible (its correlation length $L_{\mathrm{P}}$ and statistical distribution) was not directly characterized, and no direct space-resolved probe of the predicted rare islands and their coupling near $B^{*}$ existed.

Problem Statement

Provide the microscopic experimental validation of the rare-region (infinite-randomness) mechanism of the magnetic-field-driven superconductor–metal transition, as proposed in the Ga-film quantum Griffiths singularity experiment of Xing et al. (Science 2015). The problem has three linked parts: (1) Directly image the superconducting rare regions: using a space-resolved low-temperature probe under magnetic field, observe the rare superconducting islands predicted to survive above the mean-field critical field, and measure their spatial density, size distribution, superconducting gap statistics, and inter-island coupling (the proximity/Josephson-like tunneling-mediated coupling of superconducting order between neighboring islands through the metallic host) as the infinite-randomness quantum critical field $B^{*}$ is approached, in the same regime in which the dynamical critical exponent $z\nu$ diverges. (2) Quantitatively characterize the quenched disorder that seeds the rare regions: measure its correlation length $L_{\mathrm{P}}$ and statistical distribution in the film and establish the quantitative connection between the measured disorder landscape and the observed rare-region statistics and Griffiths critical behavior. (3) Settle the universality question and exclude alternatives: determine whether the activated critical behavior is governed by universal 2D infinite-randomness exponents ($\nu\approx1.2$, $\psi\approx0.5$) or instead depends on disorder strength, dimensionality, and material — including whether the Griffiths regime gives way to multiple or single quantum criticality as disorder weakens — and discriminate the rare-region mechanism against alternative explanations of the diverging effective exponent, such as vortex-glass physics, superconducting fluctuations with strong spin-orbit coupling, thermal cloaking or smearing of the quantum critical point, and dissipation-driven scenarios. The phenomenon to be explained is the same phenomenology that motivated the original experiment — the continuous line of SMT crossing points, the activated scaling of the critical line, and the diverging $z\nu$ at ultralow temperatures — and the field, material, or geometry need not be restricted to the original Ga films. The question is also resolved if the rare-region mechanism is refuted specifically, by a null result for the space-resolved imaging and the disorder characterization measured under conditions matching the Griffiths regime, or by showing that the diverging $z\nu$ is reproduced by a non-rare-region mechanism.

References

1. Ying Xing, Hui-Min Zhang, Hai-Long Fu, Haiwen Liu, Yi Sun, Jun-Ping Peng, Fa Wang, Xi Lin, Xu-Cun Ma, Qi-Kun Xue, Jian Wang, X. C. Xie, 'Quantum Griffiths singularity of superconductor-metal transition in Ga thin films', Science 350, 542 (2015). DOI: 10.1126/science.aaa7154, https://doi.org/10.1126/science.aaa7154
2. Markovic, 'Randomness rules' (Perspective), Science 350, 509 (2015). DOI: 10.1126/science.aad4136, https://doi.org/10.1126/science.aad4136
3. B. Sacépé, C. Chapelier, T. I. Baturina, V. M. Vinokur, M. R. Baklanov, M. Sanquer, 'Disorder-Induced Inhomogeneities of the Superconducting State Close to the Superconductor-Insulator Transition', Physical Review Letters 101, 157006 (2008). DOI: 10.1103/PhysRevLett.101.157006, https://doi.org/10.1103/PhysRevLett.101.157006
4. Thomas Vojta, 'Strong-disorder magnetic quantum phase transitions: Status and new developments', Journal of Physics: Conference Series 529, 012016 (2014). DOI: 10.1088/1742-6596/529/1/012016, https://doi.org/10.1088/1742-6596/529/1/012016 (preprint arXiv:1311.7361, https://arxiv.org/abs/1311.7361)
5. Shengchun Shen, Ying Xing, Pengjie Wang, Haiwen Liu, Hailong Fu, Yangwei Zhang, Lin He, X. C. Xie, Xi Lin, Jiacai Nie, Jian Wang, 'Observation of quantum Griffiths singularity and ferromagnetism at the superconducting LaAlO3/SrTiO3(110) interface', Physical Review B 94, 144517 (2016). DOI: 10.1103/PhysRevB.94.144517, https://doi.org/10.1103/PhysRevB.94.144517
6. Ying Xing, Kun Zhao, Pujia Shan, Feipeng Zheng, Yangwei Zhang, Hailong Fu, Yi Liu, Mingliang Tian, Chuanying Xi, Haiwen Liu, Ji Feng, Xi Lin, Shuaihua Ji, Xi Chen, Qi-Kun Xue, Jian Wang, 'Ising Superconductivity and Quantum Phase Transition in Macro-Size Monolayer NbSe2', Nano Letters 17, 6802 (2017). DOI: 10.1021/acs.nanolett.7b03026, https://doi.org/10.1021/acs.nanolett.7b03026 (preprint arXiv:1707.05473, https://arxiv.org/abs/1707.05473)
7. Malcolm Durkin, Rita Garrido-Menacho, Sarang Gopalakrishnan, Narendra K. Jaggi, Ji-Hwan Kwon, Jian-Min Zuo, Nadya Mason, 'Rare-region onset of superconductivity in niobium nanoislands', Physical Review B 101, 035409 (2020). DOI: 10.1103/PhysRevB.101.035409, https://doi.org/10.1103/PhysRevB.101.035409
8. Yi Liu, Ziqiao Wang, Pujia Shan, Chaofei Liu, Yue Tang, Cheng Chen, Ying Xing, Qingyan Wang, Haiwen Liu, Xi Lin, X. C. Xie, Jian Wang, 'Anomalous quantum Griffiths singularity in ultrathin crystalline lead films', Nature Communications 10, 3633 (2019). DOI: 10.1038/s41467-019-11607-w, https://doi.org/10.1038/s41467-019-11607-w
9. Ce Huang, Enze Zhang, Yong Zhang, Jinglei Zhang, Faxian Xiu, Haiwen Liu, Xiaoyi Xie, Linfeng Ai, Yunkun Yang, Minhao Zhao, Junjie Qi, Lun Li, Shanshan Liu, Zihan Li, Runze Zhan, Ya-Qing Bie, Xufeng Kou, Shaozhi Deng, X. C. Xie, 'The Discovery of Tunable Universality Class in Superconducting beta-W Thin Films', arXiv:2010.12775 (2020). arXiv: 2010.12775, https://arxiv.org/abs/2010.12775
10. Konstantin S. Tikhonov, Mikhail V. Feigel'man, 'Strange metal state near quantum superconductor-metal transition in thin films', Annals of Physics 417, 168138 (2020). DOI: 10.1016/j.aop.2020.168138, https://doi.org/10.1016/j.aop.2020.168138
11. Yi Liu, Shichao Qi, Jingchao Fang, Jian Sun, Chong Liu, Yanzhao Liu, Junjie Qi, Ying Xing, Haiwen Liu, Xi Lin, Lili Wang, Qi-Kun Xue, X. C. Xie, Jian Wang, 'Observation of In-Plane Quantum Griffiths Singularity in Two-Dimensional Crystalline Superconductors', Physical Review Letters 127, 137001 (2021). DOI: 10.1103/PhysRevLett.127.137001, https://doi.org/10.1103/PhysRevLett.127.137001
12. Zihan Cui, Longxin Pan, Jingchao Fang, Shichao Qi, Ying Xing, Haiwen Liu, Yi Liu, Jian Wang, 'The activated scaling behavior of quantum Griffiths singularity in two-dimensional superconductors', Journal of Physics D: Applied Physics 56, 374002 (2023). DOI: 10.1088/1361-6463/acd560, https://doi.org/10.1088/1361-6463/acd560 (preprint arXiv:2211.11016, https://arxiv.org/abs/2211.11016)
13. Simrandeep Kaur, Hemanta Kumar Kundu, Sumit Kumar, Anjana Dogra, Rajesh Narayanan, Thomas Vojta, Aveek Bid, 'Novel emergent phases in a two-dimensional superconductor', New Journal of Physics 26, 083001 (2024). DOI: 10.1088/1367-2630/ad6800, https://doi.org/10.1088/1367-2630/ad6800
14. Sachin Yadav, M. P. Saravanan, Sangeeta Sahoo, 'Emergence of quantum Griffiths singularity in disordered TiN thin films', Communications Physics 7, 215 (2024). DOI: 10.1038/s42005-024-01709-3, https://doi.org/10.1038/s42005-024-01709-3
15. Shao-Bo Liu, Congkuan Tian, Yongqing Cai, Hang Cui, Xinjian Wei, Mantang Chen, Yang Zhao, Yuan Sui, Shuyue Guan, Shuang Jia, Yu Zhang, Ya Feng, Jiankun Li, Jian Cui, Yuanjun Song, Tingting Hao, Chaoyu Chen, Jian-Hao Chen, 'Three-dimensional quantum Griffiths singularity in bulk iron-pnictide superconductors', National Science Review 11, nwae220 (2024). DOI: 10.1093/nsr/nwae220, https://doi.org/10.1093/nsr/nwae220
16. Fan Chen, Yixin Liu, Wang Guo, Teng Wang, Ziao Tian, Miao Zhang, Zhongying Xue, Gang Mu, Xiaofu Zhang, Zengfeng Di, 'Quantum Griffiths Singularity in Ordered Artificial Superconducting-Islands-Array on Graphene', Nano Letters 24, 2444 (2024). DOI: 10.1021/acs.nanolett.3c03870, https://doi.org/10.1021/acs.nanolett.3c03870
17. Ziqiao Wang, Yi Liu, Chengcheng Ji, Jian Wang, 'Quantum phase transitions in two-dimensional superconductors: a review on recent experimental progress', Reports on Progress in Physics 87, 014502 (2024). DOI: 10.1088/1361-6633/ad14f3, https://doi.org/10.1088/1361-6633/ad14f3
18. Changshuai Lan, Chengyu Yan, Yihang Li, Qiao Chen, Huai Guan, Xinming Zhao, Dong Wu, Butian Zhang, Youwei Zhang, Shun Wang, 'Evolution of quantum criticality in underdoped cuprates', arXiv:2504.19535 (2025). arXiv: 2504.19535, https://arxiv.org/abs/2504.19535
19. Phanibhusan S. Mahapatra, Haining Pan, Kenji Watanabe, Takashi Taniguchi, J. H. Pixley, Eva Y. Andrei, 'Quantum criticality and tunable Griffiths phase in superconducting twisted trilayer graphene', arXiv:2507.10687 (2025). arXiv: 2507.10687, https://arxiv.org/abs/2507.10687
20. Zi-Yan Han, Li-Min Yu, Yu-Cheng Cong, Yang Yang, Zhi-Xiang Sun, Zhi-Qing Li, 'Two-dimensional quantum Griffith singularity in three-dimensional ZrN$_x$ superconducting films', arXiv:2608.28166 (2026). arXiv: 2608.28166, https://arxiv.org/abs/2608.28166
21. Chushan Li, Yiyu Liu, Shichao Qi, Hongkun Zhou, Mebrouka Boubeche, Ying Xing, Huixia Luo, Jian Wang, Huichao Wang, 'Quantum Griffiths singularity in quasi-2D bulk superconductor 1T-NbSeTe', Superconductor Science and Technology 39, 055011 (2026). DOI: 10.1088/1361-6668/ae6625, https://doi.org/10.1088/1361-6668/ae6625
22. Zi-Xiao Wang, Tian-Yu Jing, Zi-Yan Han, Kuang-Hong Gao, Songci Li, Zhi-Qing Li, 'Quantum Griffiths singularity in three-dimensional MoTiN superconducting films', Physical Review B 109, 224508 (2024). DOI: 10.1103/PhysRevB.109.224508, https://doi.org/10.1103/PhysRevB.109.224508