Nature of the quantum criticality at the pseudogap end point of hole-doped cuprate superconductors

In hole-doped cuprate superconductors, the enigmatic pseudogap phase terminates at a critical doping $p^*$, where the normal-state Fermi surface undergoes a documented transformation (the carrier density dropping from $n = 1 + p$ to $n = p$), where thermodynamic signatures of quantum criticality appear (a strongly peaked electronic specific heat with a $-T\log T$ dependence), and from which a fan of strange-metal behavior with $T$-linear resistivity emanates. Unlike heavy-fermion and iron-based superconductors, where quantum criticality is attributable to a quantum phase transition toward a symmetry-breaking phase, no symmetry-breaking order has been identified at $p^*$, and the pseudogap exists as a spectral gap without a corresponding order parameter. The open problem — whose beyond-Landau framing is the unknown origin of the criticality at $p^*$ and the missing low-energy effective theory — is to determine the nature of this putative beyond-Landau quantum criticality: whether $p^*$ is a genuine zero-temperature phase transition or a crossover (e.g., a Widom line above a hidden Mott-transition endpoint), and if a transition, whether it involves an as-yet-unidentified symmetry-breaking order, a topological Lifshitz/van Hove transition of the Fermi surface, or Fermi-volume-changing fractionalized physics; and to construct the low-energy effective theory governing the critical fan. The problem remains open: the competing scenarios are under active development and experimental test, and none is established.

Background

The parent compounds of the high-temperature superconducting (high-$T_c$) cuprates are antiferromagnetic Mott insulators (insulators due to strong electron-electron repulsion rather than band filling); doping the CuO$_2$ planes with holes $p$ yields a superconducting dome that peaks near $p \approx 0.16$ and, beyond it, a conventional Fermi liquid. Between the Mott insulator and the overdoped metal lies the pseudogap phase: below a doping-dependent temperature $T^*(p)$, the antinodal parts of the Fermi surface (parallel to the Cu-O bond directions) lose low-energy spectral weight, while nodal arcs remain. $T^*(p)$ decreases with doping and terminates at a critical doping $p^*$ (about 0.19 in YBa$_2$Cu$_3$O$_{6+x}$ and Bi2212, about 0.22-0.24 in La-based cuprates). No long-range order that breaks a lattice or spin symmetry has been unambiguously established as ending at $p^*$, so the pseudogap remains a 'spectral gap without an order parameter'.

A quantum critical point (QCP) is a continuous phase transition tuned to zero temperature by a non-thermal parameter such as doping; the finite-temperature 'quantum critical fan' above it produces anomalous metallic behavior, most famously resistivity linear in temperature. In the Landau paradigm, phase transitions are classified by symmetry breaking with a local order parameter, and this framework (with controlled extensions) accounts for the quantum criticality of heavy-fermion compounds and iron-based superconductors, where a magnetic or nematic order parameter is suppressed to zero temperature and its fluctuations drive the criticality. In the cuprates the situation is inverted: the strongest quantum-critical-like signatures occur at $p^*$, yet the critical degrees of freedom have no known symmetry-breaking identity. This is what makes the problem 'beyond Landau'.

The experimental facts an answer must confront are well established. In Nd-LSCO, the resistivity is $T$-linear down to the lowest temperatures exactly at $p^*$, and the Hall coefficient changes sharply across it (Daou et al. 2009). Pulsed-field Hall measurements in YBCO show the carrier density inferred from $n = 1/(eR_H)$ drop from $n = 1 + p$ above $p^*$ to $n = p$ below it, while charge-order-driven Fermi-surface reconstruction ends at a distinctly lower doping $p = 0.16$, so the pseudogap phenomenon is not charge order (Badoux et al. 2016); the same drop is seen in Nd-LSCO with $p^* = 0.235$ (Collignon et al. 2017). Calorimetry in fields large enough to suppress superconductivity shows the electronic specific heat $C_{\mathrm{el}}$ strongly peaked at $p^*$ with $C_{\mathrm{el}}/T$ exhibiting a $-T\log T$ dependence as $T$ approaches zero, a canonical quantum-critical signature (Michon et al. 2019; Girod et al. 2021). The Wiedemann-Franz law (the statement that in a Fermi liquid the electrical and thermal conductivities are locked by the Lorenz number) holds at $p^*$ in Bi2201 even though the conductivity changes abruptly there (Michon et al. 2018), constraining the critical scattering. Angle-dependent magnetoresistance in Nd-LSCO confirms a clear Fermi-surface transformation at $p^*$ most consistent with a reconstruction (Fang et al. 2022). The strange metal itself shows 'Planckian' scattering rates of order $k_B T/\hbar$.

Four broad classes of interpretation compete. (1) A Lifshitz transition: a topological change of the Fermi surface without symmetry breaking; here the transition from a hole-like to an electron-like Fermi surface as doping moves a saddle-point van Hove singularity (a divergence in the density of states where the Fermi surface changes topology) through the Fermi level. In La-based cuprates the van Hove doping lies near $p^*$, and ARPES-based analysis argues the specific-heat peak follows from the three-dimensional van Hove singularity without any pseudogap QCP (Zhong et al. 2022); cluster calculations show that strong correlations change the character of the Lifshitz transition precisely when it coincides with the pseudogap boundary (Aguiar et al. 2025). (2) Hidden symmetry-breaking order: the nematic susceptibility (response to a breaking of the four-fold rotational symmetry of the CuO$_2$ plane) diverges as $p^*$ is approached from above in Bi2201 (Ishida et al. 2020), and low-energy spin fluctuations can account for the $p^*$ specific-heat peak in LSCO (Zhu et al. 2023). (3) Beyond-Landau fractionalization: in a 'fractionalized Fermi liquid' (FL*), the pseudogap metal has small hole pockets of total area set by the doped holes rather than the full electron count, violating the Luttinger volume constraint; Sachdev (2025) argues the pseudogap 'fan' requires such a Fermi-volume-changing quantum phase transition without symmetry breaking, with a critical theory related to the Sachdev-Ye-Kitaev model. (4) No QCP at all: dynamical mean-field theory studies describe $T^*(p)$ as a Widom line (the line of maximal response above the endpoint of a first-order Mott transition), making the pseudogap boundary a sharp crossover (Sordi et al. 2012). Numerical work on the two-dimensional Hubbard model using cluster dynamical mean-field theory and its extensions (methods that embed a finite cluster of correlated sites in a self-consistent electronic bath) now directly exhibits a pseudogap-metal-to-Fermi-liquid transition with $\omega/T$ scaling of spin and current susceptibilities and a marginal-Fermi-liquid self-energy in the critical fan (Pelz et al. 2026; Tanaka 2026). The 2021 review by T. Li that motivates this problem frames the common thread: the electron in the cuprates carries a dual local-moment and itinerant-quasiparticle character, and the field's central difficulty is the lack of a universal low-energy effective theory of the high-$T_c$ phenomenology beyond the Landau paradigm. The same question has now been extended to superconducting infinite-layer nickelates, where $T$-linear resistivity also onsets at a critical doping at the end of a pseudogap-like regime (Iorio-Duval et al. 2025).

Problem Statement

Determine the nature of the quantum criticality at the pseudogap end point $p^*$ of hole-doped cuprate superconductors — the doping where the pseudogap line $T^*(p)$ terminates, the low-temperature normal-state Fermi surface undergoes its documented transformation (carrier density changing from $n = 1 + p$ to $n = p$; whether this is equivalently a hole-like-to-electron-like change of Fermi-surface topology, as proposed for La-based cuprates, is itself one of the contested readings within the problem), and a fan of strange-metal behavior with $T$-linear resistivity emanates — in a regime where, unlike heavy-fermion and iron-based superconductors, the criticality cannot be attributed to a quantum phase transition toward an identified symmetry-breaking phase. The problem comprises the following linked questions: (1) Is $p^*$ a genuine zero-temperature phase transition (quantum critical point), or a crossover such as the Widom line (the line of maximal response above the endpoint of a hidden first-order Mott transition)? (2) If it is a transition, what is its character — an as-yet-unidentified symmetry-breaking order (e.g., nematic, loop-current, or spin order), a topological Lifshitz transition (a change of Fermi-surface topology without symmetry breaking) associated with the van Hove singularity (a saddle-point divergence of the density of states, possibly modified by strong correlations), or a Fermi-volume-changing transition without symmetry breaking (e.g., fractionalized Fermi liquid or holon-metal physics, where the pseudogap metal has small Fermi pockets enclosing only the doped holes, violating the Luttinger volume constraint)? (3) What is the low-energy effective theory governing the quantum-critical fan, i.e., the strange metal, including how it reconciles the local-moment and itinerant-quasiparticle character of the doped holes, and how it reproduces the established constraints: the peaked $C_{\mathrm{el}}/T$ with $-T\log T$ at $p^*$, the abrupt carrier-density change, Wiedemann-Franz-law validity at $p^*$ (in a Fermi liquid, the electrical and thermal conductivities are locked by the Lorenz number), $T$-linear resistivity as $T$ approaches zero at $p^*$, and the absence of identified symmetry breaking. (4) What experiment (or class of experiments) decisively discriminates among these scenarios? All four questions serve one coherent scientific objective — identifying the critical degrees of freedom at the pseudogap end point and the low-energy effective theory that organizes the pseudogap, its termination, and the strange metal. A full resolution requires all four: the transition-versus-crossover determination; the character of the transition, or an explicit argument that the transition reading is excluded; a low-energy effective theory reproducing the listed constraints; and a named decisive experiment.

References

1. T. Li, 'A short review of the recent progresses in the study of the cuprate superconductivity', Chinese Physics B 30, 100508 (2021). DOI: 10.1088/1674-1056/abfa04; arXiv:2103.13595. https://arxiv.org/abs/2103.13595
2. H.-C. Kao, D. Li, B. Rosenstein, 'Unified intermediate coupling description of pseudogap and strange metal phases of cuprates', Physical Review B 107, 054508 (2023). DOI: 10.1103/PhysRevB.107.054508; arXiv:2208.10093. https://arxiv.org/abs/2208.10093
3. R. Daou, N. Doiron-Leyraud, D. LeBoeuf, S. Y. Li, F. Laliberté, O. Cyr-Choinière, Y. J. Jo, L. Balicas, J.-Q. Yan, J.-S. Zhou, J. B. Goodenough, L. Taillefer, 'Linear temperature dependence of resistivity and change in the Fermi surface at the pseudogap critical point of a high-Tc superconductor', Nature Physics 5, 31-34 (2009). DOI: 10.1038/nphys1109; arXiv:0806.2881. https://arxiv.org/abs/0806.2881
4. S. Badoux, W. Tabis, F. Laliberté, G. Grissonnanche, B. Vignolle, D. Vignolles, J. Béard, D. A. Bonn, W. N. Hardy, R. Liang, N. Doiron-Leyraud, L. Taillefer, C. Proust, 'Change of carrier density at the pseudogap critical point of a cuprate superconductor', Nature 531, 210-214 (2016). DOI: 10.1038/nature16983; arXiv:1511.08162. https://arxiv.org/abs/1511.08162
5. C. Collignon, S. Badoux, S. A. A. Afshar, B. Michon, F. Laliberté, O. Cyr-Choinière, J.-S. Zhou, S. Licciardello, S. Wiedmann, N. Doiron-Leyraud, L. Taillefer, 'Fermi-surface transformation across the pseudogap critical point of the cuprate superconductor La$_{1.6-x}$Nd$_{0.4}$Sr$_x$CuO$_4$', Physical Review B 95, 224517 (2017). DOI: 10.1103/PhysRevB.95.224517; arXiv:1607.05693. https://arxiv.org/abs/1607.05693
6. G. Sordi, P. Sémon, K. Haule, A.-M. S. Tremblay, 'Pseudogap temperature as a Widom line in doped Mott insulators', Scientific Reports 2, 547 (2012). DOI: 10.1038/srep00547; arXiv:1110.1392. https://arxiv.org/abs/1110.1392
7. B. Michon, C. Girod, S. Badoux, J. Kačmarčík, Q. Ma, M. Dragomir, H. A. Dabkowska, B. D. Gaulin, J.-S. Zhou, S. Pyon, T. Takayama, H. Takagi, S. Verret, N. Doiron-Leyraud, C. Marcenat, L. Taillefer, T. Klein, 'Thermodynamic signatures of quantum criticality in cuprate superconductors', Nature 567, 218-222 (2019). DOI: 10.1038/s41586-019-0932-x; arXiv:1804.08502. https://arxiv.org/abs/1804.08502
8. B. Michon, A. Ataei, P. Bourgeois-Hope, C. Collignon, S. Y. Li, S. Badoux, A. Gourgout, F. Laliberté, J.-S. Zhou, N. Doiron-Leyraud, L. Taillefer, 'Wiedemann-Franz law and abrupt change in conductivity across the pseudogap critical point of a cuprate superconductor', Physical Review X 8, 041010 (2018). DOI: 10.1103/PhysRevX.8.041010; arXiv:1805.04589. https://arxiv.org/abs/1805.04589
9. K. Ishida, S. Hosoi, Y. Teramoto, T. Usui, Y. Mizukami, K. Itaka, Y. Matsuda, T. Watanabe, T. Shibauchi, 'Divergent nematic susceptibility near the pseudogap critical point in a cuprate superconductor', Journal of the Physical Society of Japan 89, 064707 (2020). DOI: 10.7566/JPSJ.89.064707; arXiv:1908.07167. https://arxiv.org/abs/1908.07167
10. Y. Fang, G. Grissonnanche, A. Legros, S. Verret, F. Laliberté, C. Collignon, A. Ataei, M. Dion, J. Zhou, D. Graf, M. J. Lawler, P. Goddard, L. Taillefer, B. J. Ramshaw, 'Fermi surface transformation at the pseudogap critical point of a cuprate superconductor', Nature Physics 18, 558-564 (2022). DOI: 10.1038/s41567-022-01514-1; arXiv:2004.01725. https://arxiv.org/abs/2004.01725
11. C. Girod, D. LeBoeuf, A. Demuer, G. Seyfarth, S. Imajo, K. Kindo, Y. Kohama, M. Lizaire, A. Legros, A. Gourgout, H. Takagi, T. Kurosawa, M. Oda, N. Momono, J. Chang, S. Ono, G.-q. Zheng, C. Marcenat, L. Taillefer, T. Klein, 'Normal state specific heat in the cuprates La$_{2-x}$Sr$_x$CuO$_4$ and Bi$_{2+y}$Sr$_{2-x-y}$La$_x$CuO$_{6+\delta}$ near the critical point of the pseudogap phase', Physical Review B 103, 214506 (2021). DOI: 10.1103/PhysRevB.103.214506; arXiv:2101.09221. https://arxiv.org/abs/2101.09221
12. M. Zhu, D. J. Voneshen, S. Raymond, O. J. Lipscombe, C. C. Tam, S. M. Hayden, 'Spin fluctuations associated with the collapse of the pseudogap in a cuprate superconductor', Nature Physics 19, 99-105 (2023). DOI: 10.1038/s41567-022-01825-3; arXiv:2201.11719. https://arxiv.org/abs/2201.11719
13. Y. Zhong, Z. Chen, S.-D. Chen, K.-J. Xu, M. Hashimoto, Y. He, S.-I. Uchida, D. Lu, S.-K. Mo, Z.-X. Shen, 'Differentiated roles of Lifshitz transition on thermodynamics and superconductivity in La$_{2-x}$Sr$_x$CuO$_4$', Proceedings of the National Academy of Sciences USA 119, e2204630119 (2022). DOI: 10.1073/pnas.2204630119; arXiv:2303.11622. https://arxiv.org/abs/2303.11622
14. S. Jung, B. Seok, C. J. Roh, D. Kim, Y. Lee, S. Kang, S. Ishida, S. Shin, H. Eisaki, T. W. Noh, D. Song, C. Kim, 'Spontaneous breaking of mirror symmetry beyond critical doping in Pb-Bi2212' (2023). arXiv:2306.02621. https://arxiv.org/abs/2306.02621
15. H. Park, J. H. Park, J. Hwang, 'Planckian behavior of cuprates at the pseudogap critical point simulated via flat electron-boson spectral density', Heliyon 10, e28056 (2024). DOI: 10.1016/j.heliyon.2024.e28056; arXiv:2308.05290. https://arxiv.org/abs/2308.05290
16. M. Berben, J. Ayres, C. Duffy, R. D. H. Hinlopen, Y.-T. Hsu, M. Leroux, I. Gilmutdinov, M. Massoudzadegan, D. Vignolles, Y. Huang, T. Kondo, T. Takeuchi, J. R. Cooper, S. Friedemann, A. Carrington, C. Proust, N. E. Hussey, 'Compartmentalizing the cuprate strange metal' (2022). arXiv:2203.04867. https://arxiv.org/abs/2203.04867
17. S. Sachdev, 'The foot, the fan, and the cuprate phase diagram: Fermi-volume-changing quantum phase transitions', Physica C: Superconductivity and its Applications 633, 1354707 (2025). DOI: 10.1016/j.physc.2025.1354707; arXiv:2501.16417. https://arxiv.org/abs/2501.16417
18. Y.-H. Xing, W.-M. Liu, X.-T. Zhang, 'Strange metal at the Lifshitz transition' (2024). arXiv:2407.02270. https://arxiv.org/abs/2407.02270
19. M. C. O. Aguiar, H. Bragança, I. Paul, M. Civelli, 'Pseudogap-induced change in the nature of the Lifshitz transition in the two-dimensional Hubbard model', Brazilian Journal of Physics 56, 48 (2025). DOI: 10.1007/s13538-025-01958-9; arXiv:2509.09783. https://arxiv.org/abs/2509.09783
20. P. M. Bonetti, M. Christos, A. Nikolaenko, A. A. Patel, S. Sachdev, 'Fractionalized Fermi liquids and the cuprate phase diagram', Reports on Progress in Physics 89, 044501 (2026). DOI: 10.1088/1361-6633/ae530d; arXiv:2508.20164. https://arxiv.org/abs/2508.20164
21. A. Tanaka, 'Pseudogap and strange metal states in the square-lattice Hubbard model: A comprehensive study', Physical Review B 114, 045114 (2026). DOI: 10.1103/4kld-1x5d; arXiv:2511.07726. https://arxiv.org/abs/2511.07726
22. C. Iorio-Duval, E. Beauchesne-Blanchet, F. Perreault, J. L. Santana González, S. Üstün Kaykusuz, W. Sun, D. Graf, Y. F. Nie, A. Gourgout, G. Grissonnanche, 'Quantum critical origin of strange-metals at the end of a pseudogap phase in infinite-layer nickelates' (2025). arXiv:2510.12786. https://arxiv.org/abs/2510.12786
23. T. P. Singar, I. Maggio-Aprile, G. Gu, C. Renner, 'Vortex core spectroscopy links pseudogap and Lifshitz critical point in a cuprate superconductor' (2025). arXiv:2512.13588. https://arxiv.org/abs/2512.13588
24. M. Pelz, G. Kotliar, J. von Delft, A. Gleis, 'Dynamical scaling near the pseudogap quantum critical point of the two-dimensional Hubbard model' (2026). arXiv:2605.15060. https://arxiv.org/abs/2605.15060
25. J.-J. Miao, 'Superconductivity, pseudogap and marginal Fermi liquid in a relative-momentum-local theory' (2026). arXiv:2608.09785. https://arxiv.org/abs/2608.09785