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arXiv:2511.07726 · Physical review. B./Physical review. B

Pseudogap and strange metal states in the square-lattice Hubbard model: A comprehensive study

Arata Tanaka

Published Jun 24, 2026• 1 citation
arXivHTMLPDF

Abstract

To clarify the origin of the pseudogap and strange metal states as well as their mutual relationship in cuprate superconductors, a comprehensive study on the spectral function, Fermi surface, resistivity and dynamical spin susceptivity of the Hubbard model on the square lattice has been conducted by means of the ladder dual-fermion approximation with an electron self-energy correction. It is found that the appearance of these two states requires that the characteristic hole concentration below which the Mott-Heisenberg and Slater mechanisms of electron localization occurs $p_{\rm MS}$ nearly coincides with the hole concentration where the Van Hove singularity (VHS) point is placed at the vicinity of the Fermi level. When this condition is met the VHS point is pinned at which the nesting condition of the antiferromagnetic (AFM) fluctuation is fulfilled almost everywhere on the Fermi surface in wide range of the hole concentration in a metallic state, i.e., the strange metal state. The spin fluctuation of the strange metal state is nearly quantum critical and the dynamical spin susceptivity is well described by overdamped spin wave having the $ω/T$ scaling with the relaxation rate at the Planckian limit. Because of these distinctive features of the strange metal state, the $k$ dependence of scattering rate of electrons is small and electrons behave as the marginal Fermi liquid, resulting in $T$-linear resistivity. In contrast, the pseudogap state is magnetically in the renormalized classical regime and the pseudogap is formed near the X point where the nesting condition of the short-range AFM order is fulfilled.

DOI
10.1103/4kld-1x5d

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To clarify the origin of the pseudogap and strange metal states as well as their mutual relationship in cuprate superconductors, a comprehensive study on the spectral function, Fermi surface, resistivity and dynamical spin susceptivity of the Hubbard model on the square lattice has been conducted by means of the ladder dual-fermion approximation with an electron self-energy correction. It is found that the appearance of these two states requires that the characteristic hole concentration below which the Mott-Heisenberg and Slater mechanisms of electron localization occurs pMSp_{\rm MS}pMS​ nearly coincides with the hole concentration where the Van Hove singularity (VHS) point is placed at the vicinity of the Fermi level. When this condition is met the VHS point is pinned at which the nesting condition of the antiferromagnetic (AFM) fluctuation is fulfilled almost everywhere on the Fermi surface in wide range of the hole concentration in a metallic state, i.e., the strange metal state. The spin fluctuation of the strange metal state is nearly quantum critical and the dynamical spin susceptivity is well described by overdamped spin wave having the ω/Tω/Tω/T scaling with the relaxation rate at the Planckian limit. Because of these distinctive features of the strange metal state, the kkk dependence of scattering rate of electrons is small and electrons behave as the marginal Fermi liquid, resulting in TTT-linear resistivity. In contrast, the pseudogap state is magnetically in the renormalized classical regime and the pseudogap is formed near the X point where the nesting condition of the short-range AFM order is fulfilled.
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 mysterious metallic phase showing perfect TTT-linear resistivity and a universal scattering rate 1/τ=αPkBT/ℏ1/τ= α_P k_B T /\hbar1/τ=αP​kB​T/ℏ with a universal prefactor αP∼1α_P \sim 1αP​∼1 and logarithmic-in-temperature singular specific heat coefficient, so-called Planckian metal phase was observed in various overdoped high-TcT_cTc​ cuprate superconductors over a finite range in doping. Here, we propose a microscopic mechanism for this exotic state based on quantum-critical bosonic charge Kondo fluctuations coupled to both spinon and a heavy conduction-electron Fermi surfaces within the heavy-fermion formulation of the slave-boson ttt-JJJ model. Using a controlled perturbative renormalization group (RG) analysis, we examine the competition between the pseudogap phase, characterized by Anderson's Resonating-Valence-Bond spin-liquid, and the Fermi-liquid state, characterized by the electron hoping (effective charge Kondo effect). We find a quantum-critical metallic phase with a universal Planckian ℏω/kBT\hbar ω/k_B Tℏω/kB​T scaling in scattering rate near a localized-delocalized (pseudogap-to-Fermi liquid) charge Kondo breakdown transition. Our results are in excellent agreement with the recent experimental observations on optical conductivity (without fine-tuning) in Nat. Commun. 14, 3033 (2023), universal doping-independent field-to-temperature scaling in magnetoresistance in Nature 595, 661 (2021), and the marginal Fermi-liquid spectral function observed in ARPES (Science 366, 1099 (2019)) as well as Hall coefficient in various overdoped cuprates in Nature 595, 661 (2021) and Annu. Rev. Condens. Matter Phys. 10, 409 (2019). Our mechanism offers a microscopic understanding of the quantum-critical Planckian metal phase observed in cuprates d-wave superconducting, and Fermi liquid phases.
In this study, the metal-insulator transition in the square-lattice Hubbard model at half-filling is revisited in relation to the DOS and spectral functions by means of the ladder dual fermion approximation (LDFA). For this purpose, a new expression of the two-body Green's function in the form of resolvents is proposed, which provides tractable and efficient means to calculate the local vertex function with the Lanczos exact diagonalization (ED) method. This makes it possible to use the Lanczos ED method as a solver of the effective impurity Anderson model for LDFA, opening up the way to access low temperatures for these perturbative extensions of the dynamical mean-field theory and to obtain accurate DOS and spectral functions on the real frequency axis by a new variant of the maximum entropy method. It is found that for $U\ensuremath{\le}3.5t$, as temperature decreases, the pseudogap formation due to antiferromagnetic correlations in the quasiparticle peak of the spectral function occurs at the XXX point $[\mathbit{k}=(\ensuremath{\pi},0)]$, spreads through the Fermi surface and ends at the M2{M}_{2}M2​ point $[\mathbit{k}=(\ensuremath{\pi}/2,\ensuremath{\pi}/2)]$. The almost simultaneous creation of the pseudogap and the loss of the Fermi liquid feature is consistent with that expected in the Slater regime. Although the pseudogap still appears in the quasi-particle-like single peak for $U\ensuremath{\ge}4.0t$, the Fermi-liquid feature is partially lost on the Fermi surface already at higher temperatures as expected in the Mott-Heisenberg regime, in which local spins are preformed at high temperatures. A sharp crossover from a pseudogap phase to a Mott insulator at finite ${U}^{*}\ensuremath{\approx}4.7t$ is found to occur below the temperature of the pseudogap formation similar to a previous study with the nonlinear $\ensuremath{\sigma}$ model approach.
This article is based on a talk by S.S. at the Nambu Memorial Symposium at the University of Chicago. We review ideas on the nature of the metallic states of the hole-doped cuprate high temperature superconductors, with an emphasis on the connections between the Luttinger theorem for the size of the Fermi surface, topological quantum field theories (TQFTs), and critical theories involving changes in the size of the Fermi surface. We begin with the derivation of the Luttinger theorem for a Fermi liquid, using momentum balance during a process of flux-insertion in a lattice electronic model with toroidal boundary conditions. We then review the TQFT of the Z2 spin liquid, and demonstrate its compatibility with the toroidal momentum balance argument. This discussion leads naturally to a simple construction of `topological' Fermi liquid states: the fractionalized Fermi liquid (FL*) and the algebraic charge liquid (ACL). We present arguments for a description of the pseudogap metal of the cuprates using Z2-FL* or Z2-ACL states with Ising-nematic order. These pseudogap metal states are also described as Higgs phases of a SU(2) gauge theory. The Higgs field represents local antiferromagnetism, but the Higgs-condensed phase does not have long-range antiferromagnetic order: the magnitude of the Higgs field determines the pseudogap, the reconstruction of the Fermi surface, and the Ising-nematic order. Finally, we discuss the route to the large Fermi surface Fermi liquid via the critical point where the Higgs condensate and Ising nematic order vanish, and the application of Higgs criticality to the strange metal.
Many strongly correlated systems exhibit strange metallic behavior in certain parameter regimes characterized by anomalous transport properties that are irreconcilable with a Fermi-liquid-like description in terms of quasiparticles. The Hubbard model is a standard theoretical starting point to examine the properties of such systems and also exhibits non-Fermi-liquid behavior in simulations. Here we analytically study the two-dimensional hole-doped Hubbard model in the large UUU limit, first identifying a doping-induced percolation transition in the low-energy sector occurring at a definite critical doping pc{p}_{c}pc​ depending on lattice structure. Using the critical properties near this transition we rewrite the Hubbard Hamiltonian and motivate a low-energy lattice-independent large-NNN model with distinct non-Fermi-liquid properties. We show that this model has linear-in-TTT resistivity with doping-dependent slope maximized at p=pcp={p}_{c}p=pc​ and power-law optical conductivity ${\ensuremath{\sim}|\ensuremath{\omega}|}^{\ensuremath{-}2/3}$. Though the parameters used in developing this theory mean it cannot be directly applied to the cuprate superconductors, we nevertheless reproduce several important phenomena observed in their strange metal phase and also predict these same qualitative behaviors to manifest in other lattices near concrete hole dopings.
A one band Hubbard model with intermediate coupling is shown to describe the two most important unusual features of a normal state: linear resistivity strange metal and the pseudogap. Both the spectroscopic and transport properties of the cuprates are considered on the same footing by employing a relatively simple post-Gaussian approximation valid for the intermediate couplings U/t=1.5−−4U/t=1.5--4U/t=1.5−−4 in relevant temperatures T>100K.T>100\phantom{\rule{3.33333pt}{0ex}}\mathrm{K}.T>100K. In the doping range p=0.1−−0.3\phantom{\rule{4pt}{0ex}}p=0.1--0.3p=0.1−−0.3, the value of UUU is smaller than that in the parent material. For a smaller doping, especially in the Mott insulator phase, the coupling is large compared to the effective tight binding scale and a different method is required. This scenario provides an alternative to the paradigm that the coupling should be strong, say U/t>6U/t>6U/t>6, in order to describe the strange metal. We argue that, to obtain phenomenologically acceptable underdoped normal state characteristics like T∗{T}^{*}T∗, pseudogap values, and spectral weight distribution, a large value of UUU is detrimental. Surprisingly the resistivity in the above temperature range is linear, $\ensuremath{\rho}={\ensuremath{\rho}}_{0}+\ensuremath{\alpha}\frac{{m}^{*}}{{e}^{2}n\ensuremath{\hbar}}T$, with the ``Planckian'' coefficient $\ensuremath{\alpha}$ of order 1.