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Unified Description of the Strange Metal, d-Wave Pairing, and Superconducting Dome in Cuprates from a Coherence Purity Principle

Kai Huang, Hongkui Liu, Ziwei Huang

Published May 12, 2026• 0 citations
Publisher

Abstract

The phase diagram of hole-doped cuprate superconductors contains three central puzzles: a dome-shaped superconducting phase, a strange metal normal state with Planckian dissipation, and robust d-wave pairing (dx2-y2 symmetry). We propose a unified framework based on a single material-dependent parameter, the coherence purity p, which measures the fraction of phase-coherent condensate relative to dissipative channels. Geometric frustration of the antiferromagnetic background on the CuO2 square lattice drives p to a maximum at optimal doping. The superconducting transition temperature follows T_c proportional to p/(1-p). The d-wave symmetry is derived as the leading pairing instability under on-site repulsion and nearest-neighbor superexchange, without being assumed. In the strange metal, the Planckian dissipation coefficient alpha = hbar/(tau kB T) is linked to the purity deficit 1-p through a universal amplification factor C ≈ 31, which we derive from the nodal Dirac structure of the d-wave state. Fitting the superconducting domes of LSCO and YBCO with the peak purity p_max = 0.97 (fixed from independent transport and high-energy experiments) yields excellent agreement using only two material-dependent scale parameters. The framework produces falsifiable predictions for the doping dependence of the superfluid density and for the product of maximum T_c and the Planckian coefficient. Keywords: high-temperature superconductivity, cuprates, strange metal, Planckian dissipation, d-wave pairing, coherence purity, superconducting dome, LSCO, YBCO

DOI
10.5281/zenodo.20139964

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The phase diagram of hole-doped cuprate superconductors contains three central puzzles: a dome-shaped superconducting phase, a strange metal normal state with Planckian dissipation, and robust d-wave pairing (dx2-y2 symmetry). We propose a unified framework based on a single material-dependent parameter, the coherence purity p, which measures the fraction of phase-coherent condensate relative to dissipative channels. Geometric frustration of the antiferromagnetic background on the CuO2 square lattice drives p to a maximum at optimal doping. The superconducting transition temperature follows T_c proportional to p/(1-p). The d-wave symmetry is derived as the leading pairing instability under on-site repulsion and nearest-neighbor superexchange, without being assumed. In the strange metal, the Planckian dissipation coefficient alpha = hbar/(tau kB T) is linked to the purity deficit 1-p through a universal amplification factor C ≈ 31, which we derive from the nodal Dirac structure of the d-wave state. Fitting the superconducting domes of LSCO and YBCO with the peak purity p_max = 0.97 (fixed from independent transport and high-energy experiments) yields excellent agreement using only two material-dependent scale parameters. The framework produces falsifiable predictions for the doping dependence of the superfluid density and for the product of maximum T_c and the Planckian coefficient. Keywords: high-temperature superconductivity, cuprates, strange metal, Planckian dissipation, d-wave pairing, coherence purity, superconducting dome, LSCO, YBCO
Abstract The mysterious metallic phase showing T-linear resistivity and a universal scattering rate 1 / τ = α P k B T / ℏ with a universal prefactor α P ∼ 1 and logarithmic-in-temperature singular specific heat coefficient, the so-called ‘Planckian metal phase’ was observed in various overdoped high- T c cuprate superconductors over a finite range in doping. Revealing the mystery of the Planckian metal state is believed to be the key to understanding the mechanism for high- T c superconductivity. Here, we propose a generic microscopic mechanism for this state based on quantum-critical local bosonic charge Kondo fluctuations coupled to both spinon and a heavy conduction-electron Fermi surface within the heavy-fermion formulation of the slave-boson t–J model. By a controlled perturbative renormalization group analysis, we examine the competition between the pseudogap phase, characterized by Anderson’s Resonating-Valence-Bond spin-liquid, and the Fermi-liquid state, modeled by the electron hopping (effective charge Kondo effect). We find a quantum-critical metallic phase with a universal Planckian ℏ ω / k B T scaling in scattering rate near an extended localized-delocalized (pseudogap-to-Fermi liquid) charge-Kondo breakdown transition. The d-wave superconducting ground state emerges near the transition. Unprecedented qualitative and quantitative agreements are reached between our theoretical predictions and various experiments, including optical conductivity, universal doping-independent field-to-temperature scaling in magnetoresistance, specific heat coefficient, marginal Fermi-liquid spectral function observed in ARPES, and Fermi surface reconstruction observed in Hall coefficients in various overdoped cuprates. Our mechanism offers a microscopic understanding of the quantum-critical Planckian metal phase observed in cuprates and its link to the pseudogap, d-wave superconducting, and Fermi liquid phases. It offers a promising route for understanding how d-wave superconductivity emerges from such a strange metal phase in cuprates–one of the long-standing open problems in condensed matter physics since 1990s–as well as shows a broader implication for the Planckian strange metal states observed in other correlated unconventional superconductors.
The mechanism controlling the exotic behavior of the transport properties in the strange metallic phase of high temperature superconductors is one of the main unresolved problems in condensed matter physics. Recent experimental observations have proven that space-time symmetry breaking due to charge density wave orderings is an ubiquitous feature of the phase diagram of high temperature superconductors and plays a dominant role in determining the transport properties of these materials. In this paper we show that in the strange metal phase of the cuprates, a whole set of transport coefficients are described by a universal hydrodynamic framework once one accounts for the effects of quantum critical charge density waves. We corroborate our theoretical prediction by measuring the DC transport properties of Bi-2201 close to optimal doping, proving the validity of our approach. Our argument does not assume any specific microscopic mechanism, relying only on the symmetries of the system, and can be used as a consistency check to understand the universality class governing the behavior of cuprate high temperature superconductors.
We use a variational approach to gain insight into the strongly correlated ddd-wave superconducting state of the high Tc{T}_{c}Tc​ cuprates at T=0T=0T=0. We show that strong correlations lead to qualitatively different trends in pairing and phase coherence: the pairing scale decreases monotonically with hole doping while the superconducting order parameter shows a nonmonotonic dome. We obtain detailed results for the doping dependence of a large number of experimentally observable quantities, including the chemical potential, coherence length, momentum distribution, nodal quasiparticle weight and dispersion, incoherent features in photoemission spectra, optical spectral weight, and superfluid density. Most of our results are in remarkable quantitative agreement with existing data and some of our predictions, first reported in Phys. Rev. Lett. 87, 217002 (2001), have been recently verified.
The Hubbard and closely related t-J models are exciting platforms for unconventional superconductivity (SC). Through state-of-the-art density matrix renormalization group calculations using the grand canonical ensemble, we address open issues regarding the ground-state phase diagram of the extended t-J model on a square lattice in the parameter regime relevant to cuprate superconductors. On large 8-leg cylinders, we demonstrate that the pure t-J model with only nearest-neighbor hoppings and superexchange interactions, for a wide range of doping ( δ = 0.1 to 0.2 ), hosts robust d-wave superconductivity possibly coexisting with weak unidirectional pair density wave. Furthermore, a small next nearest neighbor hopping t 2 suppresses pair and charge density waves, resulting in a uniform d-wave SC phase in both electron- and hole-doped cuprate model systems. Our work validates the t-J model as a proper minimum model for the emergence of superconductivity in cuprate superconductors.
We generalize the condition under which a quantum material exhibiting a pseudogap phase is a precursor to a superconducting (SC) dome. The result reveals the non-analytic T* as a function of doping. A well-known example is the high-Tc cuprates. Essentially, the SC dome is generated under two conditions: (1) that the pseudogap T* is a decreasing function of doping, due to the decrease in size of extended pairing of doped holes with doping, and most importantly, (2) that the rate of configurational-ordering parameter is an increasing function of doping as a result of the decrease in extended length of the disordered pairs. The theory behind these hinges on a novel strong entanglement and confinement hole pairing (ECHP) mechanism that unravels the microscopic features of the entire phase diagram of both electron and hole-doped high-Tc cuprates, recently discussed in the by Buot et al. The ECHP configurational ordering (CO) at the underdoped and overdoped regions qualitatively agrees with spin texture experiments, and strongly suggests that strange metal phase can only occur in the overdoped region, in agreement with experiments.
In a recent publication [Chen et al., Phys. Rev. B 86, 165136 (2012)], we identified a line of Lifshitz transition points separating the Fermi liquid and pseudogap regions in the hole-doped two-dimensional Hubbard model. Here, we extend the study to further determine the superconducting transition temperature in the phase diagram. By means of large-scale dynamical cluster quantum Monte Carlo simulations, we are able to identify the evolution of the ddd-wave superconducting dome in the hole-dope side of the phase diagram, with next-nearest-neighbor hopping (${t}^{\ensuremath{'}}$), chemical potential, and temperature as control parameters. To obtain the superconducting transition temperature Tc{T}_{c}Tc​, we employ two-particle measurements of the pairing susceptibilities. As ${t}^{\ensuremath{'}}$ goes from positive to negative values, we find the ddd-wave projected irreducible pairing vertex function is enhanced, and the curvature of its doping dependence changes from convex to concave, which fixes the position of the maximum superconducting temperature at the same filling ($n\ensuremath{\approx}0.85$) and constraints the dome from precisely following the Lifshitz line. We furthermore decompose the irreducible vertex function into fully irreducible, charge and spin components via the parquet equations, and consistently find that the spin component dominates the pairing vertex function in the doping range where the dome is located. Our investigations deepen the understanding of the phase diagram of the two-dimensional Hubbard model and, more importantly, pose new questions to the field. For example, we found as ${t}^{\ensuremath{'}}$ goes from positive to negative values, the curvature of the pairing strength as a function of doping changes from convex to concave, and the nature of the dominant fluctuations changes from charge degree of freedom to spin degree of freedom. The study of these issues will lead to further understanding of the phase diagram of the two-dimensional Hubbard model and also the physics of the hole-doped cuprate high-temperature superconductors.