Mutual relationship and common origin of the cuprate pseudogap and strange metal: testing the Mott-Slater localization / van-Hove-singularity coincidence scenario

The pseudogap and the strange metal are the two central unexplained normal-state phenomena of cuprate superconductors, and how they are related to each other - two regimes of one underlying physics or two distinct phenomena - remains an open problem. A comprehensive ladder dual-fermion study of the square-lattice Hubbard model (Arata Tanaka, Phys. Rev. B 114, 045114 (2026)) found that both states appear only when the hole concentration $p_{\mathrm{MS}}$ below which Mott-Heisenberg and Slater electron-localization mechanisms operate nearly coincides with the doping at which the van Hove singularity at the zone-boundary point $X=(\pi,0)$ sits at the Fermi level: antiferromagnetic fluctuations then open a pseudogap near $X$ at low doping, while pinning of the van Hove point keeps the antiferromagnetic nesting condition fulfilled almost everywhere on the Fermi surface, generating a marginal-Fermi-liquid strange metal with nearly quantum-critical, Planckian-limit spin fluctuations whose range extends up to the pseudogap endpoint $p^*$. The open problem is to determine, for the real materials, the mutual relationship of the two states and whether this common-origin scenario - or a competing quantum-critical-point account - is correct, by confrontation with experiment across cuprate families and with controlled many-body theory.

Background

Cuprate high-temperature superconductors are layered copper oxides whose CuO2 planes, when doped away from an antiferromagnetic Mott-insulating parent compound by a hole concentration $p$ per copper site, pass through a pseudogap regime and a superconducting dome. Two normal-state (non-superconducting) anomalies have resisted explanation for four decades. First, below a crossover temperature $T^*$ (above the superconducting transition temperature $T_c$), low-energy electronic spectral weight is partially suppressed - the pseudogap - strongest near the antinodal points $X=(\pi,0)$ and $(0,\pi)$ of the Brillouin zone, so that angle-resolved photoemission spectroscopy (ARPES) sees only disconnected 'Fermi arcs' near the nodal diagonal; the pseudogap phase ends at a hole concentration $p^*$. Second, at higher temperatures and dopings the strange metal shows electrical resistivity linear in $T$ over hundreds of kelvin, with a transport scattering rate of order the Planckian scale $\hbar/k_B T$ and marginal-Fermi-liquid phenomenology in which the electron self-energy obeys $\mathrm{Im}\,\Sigma_k(\omega)=T f(\omega/T)$ with weak momentum dependence. Experiments in strong magnetic fields that suppress superconductivity find $T$-linear resistivity persisting toward $T=0$ near $p^*$, together with thermodynamic anomalies such as $C_e/T\sim\ln(1/T)$, suggesting a hidden quantum critical point (QCP) at $p^*$; in several cuprate families $p^*$ lies close to the Lifshitz transition at which the Fermi surface, passing through the van Hove singularity (VHS) - the saddle point of the band at $X$ where the density of states diverges and the Fermi velocity vanishes - changes topology.

The one-band Hubbard model on the square lattice, with nearest-neighbor hopping $t$, next-nearest-neighbor hopping $t'$ (which moves the VHS relative to the Fermi level as a function of doping), and on-site repulsion $U$, is the minimal model of these planes. Antiferromagnetic (AFM) spin fluctuations with wave vector near $Q=(\pi,\pi)$ scatter electrons between Fermi-surface portions related by $Q$ (the 'nesting' condition); when the renormalized VHS point sits at the Fermi level, the flat antinodal dispersion makes nesting nearly fulfilled everywhere on the Fermi surface. Electron localization at weak coupling by AFM band folding is called the Slater mechanism, to be distinguished from Mott-Heisenberg localization driven by local-moment (Mottness) physics; the ladder dual-fermion approximation - a dynamical mean-field theory based diagrammatic extension that sums particle-hole ladder diagrams built from a local impurity vertex, here with a Moriya-lambda-like self-energy correction - captures both tendencies together with long-range AFM correlations.

In such a calculation, Tanaka (Phys. Rev. B 114, 045114 (2026)) found that the pseudogap and the strange metal both appear in the square-lattice Hubbard model only when the characteristic hole concentration $p_{\mathrm{MS}}$ below which Mott-Heisenberg and Slater localization tendencies set in (operationally, where $(\partial\mu/\partial T)_p=0$; $p_{\mathrm{MS}}\approx 0.18$ for $U=6t$ and $\approx 0.2$ for $U=8t$) nearly coincides with the hole concentration at which the VHS sits at the Fermi level. Comparing parameter sets with the VHS far below ($t'=0$), near ($t'=-0.2t$), and far above ($t'=-0.3t$) $p_{\mathrm{MS}}$, only the near-coincident case produces both states. When the coincidence holds, pinning of the VHS near the Fermi level keeps the AFM nesting condition fulfilled almost everywhere on the Fermi surface over a wide doping range, producing a strange metal with nearly quantum-critical spin fluctuations, $\omega/T$ scaling of an overdamped spin mode with Planckian relaxation $\hbar\Gamma\approx 2k_B T$, and marginal-Fermi-liquid $T$-linear transport; the strange metal then extends from the AFM QCP up to $p^*$, which lies near $p_{\mathrm{MS}}$. The pseudogap, by contrast, forms near $X$ in the renormalized-classical (quasi-static) magnetic regime at $p<p^*$, and the model's predictions are claimed to be consistent with La$_{2-x}$Sr$_x$CuO$_4$ (LSCO) resistivity and neutron data and with ARPES on (Bi,Pb)$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ ((Bi,Pb)2212). Competing accounts exist: dynamical cluster approximation studies with a numerical-renormalization-group solver find instead a continuous quantum phase transition at $p^*$ between a pseudogap metal and a Fermi liquid whose quantum-critical fan is itself the strange metal, and cluster dynamical mean-field theory shows that van Hove singularities control pseudogap onset in doped Mott insulators without invoking the coincidence. What no study settles is the question for the real materials.

Problem Statement

For cuprate superconductors, in their natural variation of hole doping, momentum structure, and material families: determine the mutual relationship between the pseudogap state and the strange metal state - whether they are two regimes of a single underlying physics and, if so, what that physics is, or two essentially distinct phenomena - and decide whether their common origin is the mechanism proposed by the ladder dual-fermion study of the square-lattice Hubbard model, namely the near-coincidence between the hole concentration $p_{\mathrm{MS}}$ below which Mott-Heisenberg and Slater electron-localization mechanisms operate and the hole concentration at which the van Hove singularity at the zone-boundary point $X=(\pi,0)$ sits at the Fermi level, with antiferromagnetic fluctuations opening the pseudogap near $X$ in the renormalized-classical regime while pinning of the van Hove point keeps the antiferromagnetic nesting condition fulfilled almost everywhere on the Fermi surface, extending quantum-critical, Planckian-limit, marginal-Fermi-liquid behavior from the antiferromagnetic quantum critical point up to the pseudogap endpoint $p^*$. An answer must establish or refute this unified scenario for the real materials - for example through decisive measurements across cuprate families testing whether $p^*$, the van Hove/Lifshitz doping, and the localization scale coincide exactly where both states exist and fail to coincide where they do not, or through controlled many-body treatments of the Hubbard and Emery (three-band Cu-O) models that confirm or break the $p_{\mathrm{MS}}$-van Hove coincidence requirement - and must identify what relationship holds if this scenario is wrong.

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