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arXiv:2608.09785

Superconductivity, pseudogap and marginal Fermi liquid in a relative-momentum-local theory

Jian-Jian Miao

Published Aug 10, 2026• 0 citations
arXivHTMLPDF

Abstract

Cuprate superconductors host \(d\)-wave superconductivity (SC), a pseudogap (PG) with Fermi arcs and a strange metal (SM) with incoherent excitations and temperature-linear resistivity. These regimes emerge from the same doped Mott system but are usually described separately. Here a relative-momentum-local (RML) theory is constructed, with the exact spin-singlet Cooper representation of antiferromagnetic superexchange in the \(t\)-\(J\) model serving as a common thread. Pair correlations organize the low-energy behaviour of this Cooper channel into phase-coherent static SC, phase-incoherent quasi-static PG and phase-incoherent dynamic SM. A static RML prescription yields an exactly solvable PG Hamiltonian with nodal electron poles, antinodal gaps, resolution-broadened Fermi arcs and gapless charge-\(2e\) Cooper surfaces. A Cooper-surface Lifshitz transition generates logarithmic enhancements in the specific-heat coefficient and charge response without a spin singularity. The particle-particle continuum at finite centre-of-mass momentum produces Ohmic damping, while a dynamic RML prescription yields a one-loop self-energy with marginal-Fermi-liquid scaling. Spectral-weight transfer in the physical pair spectrum diagnoses the PG--SM crossover, whereas nonzero phase stiffness and global phase coherence identify SC. The RML theory relates single-particle, pair-sensitive and thermodynamic probes to three infrared organizations of one Cooper channel. A decisive experimental signature would be zero-energy Cooper-surface ridges in two-electron angle-resolved photoemission spectroscopy (2e-ARPES).

DOI
10.48550/arxiv.2608.09785

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Cuprate superconductors host (d)-wave superconductivity (SC), a pseudogap (PG) with Fermi arcs and a strange metal (SM) with incoherent excitations and temperature-linear resistivity. These regimes emerge from the same doped Mott system but are usually described separately. Here a relative-momentum-local (RML) theory is constructed, with the exact spin-singlet Cooper representation of antiferromagnetic superexchange in the (t)-(J) model serving as a common thread. Pair correlations organize the low-energy behaviour of this Cooper channel into phase-coherent static SC, phase-incoherent quasi-static PG and phase-incoherent dynamic SM. A static RML prescription yields an exactly solvable PG Hamiltonian with nodal electron poles, antinodal gaps, resolution-broadened Fermi arcs and gapless charge-(2e) Cooper surfaces. A Cooper-surface Lifshitz transition generates logarithmic enhancements in the specific-heat coefficient and charge response without a spin singularity. The particle-particle continuum at finite centre-of-mass momentum produces Ohmic damping, while a dynamic RML prescription yields a one-loop self-energy with marginal-Fermi-liquid scaling. Spectral-weight transfer in the physical pair spectrum diagnoses the PG--SM crossover, whereas nonzero phase stiffness and global phase coherence identify SC. The RML theory relates single-particle, pair-sensitive and thermodynamic probes to three infrared organizations of one Cooper channel. A decisive experimental signature would be zero-energy Cooper-surface ridges in two-electron angle-resolved photoemission spectroscopy (2e-ARPES).
Despite the intense theoretical and experimental effort, an understanding of the superconducting pairing mechanism of the high-temperature superconductors leading to an unprecedented high transition temperature Tc{T}_{c}Tc​ is still lacking. An additional puzzle is the unknown connection between the superconducting gap and the so-called pseudogap which is a central property of the most unusual normal state. Angle-resolved photoemission spectroscopy (ARPES) measurements have revealed a gaplike behavior on parts of the Fermi surface, leaving a nongapped segment known as Fermi arc around the diagonal of the Brillouin zone. Two main interpretations of the origin of the pseudogap have been proposed: either the pseudogap is a precursor to superconductivity, or it arises from another order competing with superconductivity. Starting from the t-J model, in this paper we present a microscopic approach to investigate physical properties of the pseudogap phase as well as the superconducting phase in the framework of a renormalization scheme called projector-based renormalization method. This approach is based on a stepwise elimination of high-energy transitions using unitary transformations. We arrive at renormalized ``free'' Hamiltonians for correlated electrons for both phases. Our microscopic approach allows us to explain the experimental findings in the underdoped as well as in the optimal hole doping regime. The ARPES spectral function along the Fermi surface turns out to be in good agreement with experiment: For the pseudogap phase we find well-defined excitation peaks around $\ensuremath{\omega}=0$ near the nodal direction, which become strongly suppressed around the antinodal point. The origin of the pseudogap can be traced back to a suppression of spectral weight from incoherent excitations in a small $\ensuremath{\omega}$ range around the Fermi energy. In the superconducting phase, the order parameter turns out to have ddd-wave symmetry with a coherence length of a few lattice constants. In good agreement with experiments, we find no superconducting solutions for very small hole doping.
Despite the intense theoretical and experimental effort, an understanding of the superconducting pairing mechanism of the high-temperature superconductors is still lacking. An additional puzzle is the unknown connection between the superconducting gap and the so-called pseudogap which is a central property of the most unusual normal state. Angle-resolved photoemission spectroscopy (ARPES) measurements have revealed a gap-like behavior on parts of the Fermi surface, leaving a non-gapped segment known as Fermi arc around the diagonal of the Brillouin zone. Starting from the ttt-JJJ model, in this paper we present a microscopic approach to investigate physical properties of the pseudogap phase in the framework of a novel renormalization scheme called PRM. This approach is based on a stepwise elimination of high-energy transitions using unitary transformations. We arrive at a renormalized 'free' Hamiltonian for correlated electrons. The ARPES spectral function along the Fermi surface turns out to be in good agreement with experiment: We find well-defined excitation peaks around ω=0ω=0ω=0 near the nodal direction, which become strongly suppressed around the antinodal point. The origin of the pseudogap can be traced back to a suppression of spectral weight from incoherent excitations in a small ωωω-range around the Fermi energy. In a subsequent paper, also the supercunducting phase at moderate hole doping will be discussed within the PRM approach.
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.
Abstract Previous theoretical studies (Phillips et al 2020 Nat. Phys. 16 1175) based on the Hatsugai–Kohmoto model have examined the stability of s -wave superconductivity in strongly correlated systems, demonstrating that correlations alone can substantially modify superconducting (SC) behavior. Motivated by this perspective, but going beyond these studies, we perform self-consistent microscopic calculations of d -wave superconductivity in strongly correlated systems by employing an exactly solvable correlated model that hosts a pseudogap phase and a partially flat band (Worm et al 2024 Phys. Rev. Lett. 133 166501). We show that pseudogap correlations and SC order affect the low-energy spectrum in qualitatively different ways: the former leads to a momentum-localized suppression of spectral weight, whereas the latter induces a coherent reorganization of quasiparticle excitations. Moreover, we demonstrate that the interplay between SC order and pseudogap correlations naturally generates a SC dome in the temperature-doping phase diagram, with optimal doping located near the quantum critical point separating the pseudogap and metallic phases. Furthermore, d x 2 − y 2 -wave superconductivity is found to be remarkably robust, remaining energetically dominant over both d x y -wave and s -wave pairing channels across a wide doping range. Our results offer a potential route for a direct and controlled connection between pseudogap correlations and the emergence of the SC dome in cuprates.
We propose that the nonequilibrium current measured in the aaa-bbb plane of an underdoped cuprate (in either the strange metal or pseudogap regime) in contact with either an overdoped cuprate or a standard Fermi liquid can be used to diagnose how different the pseudogap and strange metals are from a Fermi liquid. Naively, one expects the strange metal to be more different from a Fermi liquid than the pseudogap is. We compute the expected nonequilibrium transport signal with the three Green's functions that are available in the literature: (1) marginal Fermi-liquid theory, (2) the phenomenological ansatz for the pseudogap regime, and (3) the Wilsonian reduction of the Hubbard model which contains both the strange metal and pseudogap. All three give linear III-VVV curves at low bias voltages. Significant deviations from linearity at higher voltages obtain only in the marginal Fermi-liquid approach. The key finding, however, is that III-VVV curves for the strange metal/Fermi-liquid contact exceed that of the pseudogap/Fermi-liquid system. If this is borne out experimentally, this implies that the strange metal is less orthogonal to a Fermi liquid than the pseudogap is. Within the Wilsonian reduction of the Hubbard model, this result is explained in terms of a composite-particle picture. Namely, the pseudogap corresponds to a confinement transition of the charge degrees of freedom present in the strange metal. In the strange metal the composite excitations break up and electron quasiparticles scatter off bosons. The bosons here, however, do not arise from phonons but from the charge degrees of freedom responsible for dynamical spectral weight transfer.
The pseudogap state of cuprate high-temperature superconductors has been often viewed as either a yet unknown competing order or a precursor state to superconductivity. While awaiting the resolution of the pseudogap problem in cuprates, we demonstrate that local pairing fluctuations, vortex liquid dynamics, and other precursor phenomena can emerge quite generally whenever fermionic excitations remain gapped across the superconducting transition, regardless of the gap origin. Our choice of a tractable model is a lattice band insulator with short-range attractive interactions between fermions in the sss-wave channel. An effective crossover between Bardeen-Cooper-Schrieffer (BCS) and Bose-Einstein condensate (BEC) regimes can be identified in any band insulator above two dimensions, while in two dimensions only the BEC regime exists. The superconducting transition is ``unconventional'' (non-pair-breaking) in the BEC regime, identified by either the bosonic mean field or XY\mathit{XY}XY universality class. The insulator adjacent to the superconductor in the BEC regime is a bosonic Mott insulator of Cooper pairs, which may be susceptible to charge density wave ordering. We construct a function of the many-body excitation spectrum whose nonanalytic changes define a sharp distinction between band and Mott insulators. The corresponding ``second-order transition'' can be observed out of equilibrium by driving a Cooper pair laser in the Mott insulator. We explicitly show that the gap for charged bosonic excitations lies below the threshold for Cooper pair breakup in any BEC regime, despite quantum fluctuations. Our discussion ends with a view of possible consequences for cuprates, where antinodal pair dynamics has certain features in common with our simple sss-wave picture.