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arXiv:2610.12339

True vs false Fermi surfaces in the Pseudogap regime and their transformation with doping and temperature in the Hubbard Model

Yu. M. Vil’k

Published Oct 8, 2026• 0 citations
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Abstract

Exact diagrammatic quantum Monte Carlo (DiagMC) results for the nearest-neighbor Hubbard model motivate a closer study of the pseudogap. Using the improved two-particle self-consistent approach (TPSC+), we analyze Energy (EDC) and Momentum Distribution Curves (MDC) simultaneously. We show that Fermi-liquid terminology breaks down in the pseudogap regime, requiring a distinction between true and false Fermi surfaces and zero-energy quasiparticle (ZEQ) lines. A false Fermi surface has a momentum-space spectral maximum but a frequency-space depression at zero energy, while a false ZEQ line violates the standard quasiparticle condition $\partial Σ'(\mathbf{k},ω)/\partial ω|_{ω=0}<0$.The pseudogap is driven by critical thermal spin fluctuations, which occur in two dimensions because of the Mermin-Wagner theorem. Commensurate fluctuations first open an antinodal pseudogap, leaving true Fermi arcs. With decreasing temperature, the Fermi surface evolves into hole- and electron-like false Fermi surfaces. Incommensurate fluctuations generate hot spots near the diagonal, where the pseudogap persists to the quantum critical point (QCP), whereas at $\mathbf{k}_{AN}$ it disappears before the QCP doping. There, the spectrum has two precursor antiferromagnetic (AFM) bands, both in the unoccupied ($ω>0$) region. We benchmark TPSC+ against DiagMC results for the Matsubara spectral proxy $-\mathrm{Im}[\mathcal{G}(\mathbf{k},iπT)]/π$. TPSC+ underestimates pseudogap suppression in the strong-interaction regime, reproducing DiagMC behavior at lower temperatures or doping. This proxy is equivalent to the spectral function with thermal broadening $η=πT$, which obscures features when $πT$ is not the smallest energy scale. Using $A(\mathbf{k},0)$, we find that hole-like Fermi surfaces emerge at any interaction strength at low temperature, even at weak coupling.

DOI
10.48550/arxiv.2610.12339

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We investigate the doped two-dimensional Hubbard model at finite temperature using controlled diagrammatic Monte Carlo calculations allowing for the computation of spectral properties in the infinite-size limit and, crucially, with arbitrary momentum resolution. We show that three distinct regimes are found as a function of doping and interaction strength, corresponding to a weakly correlated metal with properties close to those of the non-interacting system, a correlated metal with strong interaction effects including a reshaping of the Fermi surface, and a pseudogap regime at low doping in which quasiparticle excitations are selectively destroyed near the antinodal regions of momentum space. We study the physical mechanism leading to the pseudogap and show that it forms both at weak coupling when the magnetic correlation length is large and at strong coupling when it is shorter. In both cases, we show that spin-fluctuation theory can be modified in order to account for the behavior of the non-local component of the self-energy. We discuss the fate of the pseudogap as temperature goes to zero and show that, remarkably, this regime extrapolates precisely to the ordered stripe phase found by ground-state methods. This handshake between finite temperature and ground-state results significantly advances the elaboration of a comprehensive picture of the physics of the doped Hubbard model.
Long-wavelength spin fluctuations prohibit antiferromagnetic long-range order at finite temperature in two dimensions. Nevertheless, the correlation length starts to grow rapidly at a crossover temperature, leading to critical slowing down and to a renormalized-classical regime over a wide range of temperature, between a fraction of the mean-field transition temperature and the zero-temperature ordered state. This leads to a single-particle pseudogap of the kind observed in electron-doped cuprates. Very few theoretical methods can claim an accurate description of this regime. The challenge is that in this regime Fermi-liquid quasiparticles are already destroyed and new quasiparticles of the ordered state are not fully formed yet. Here, we study this problem for the two-dimensional Hubbard model by first generalizing the two-particle self-consistent approach. Using a special algorithm, spin fluctuations are treated in the thermodynamic limit even for large correlation lengths. The effects of Kanamori-Br\"uckner screening, of classical and of quantum fluctuations are taken into account. Results are presented at half filling for the one-band Hubbard model with nearest-neighbor hopping. They agree well with available benchmark diagrammatic quantum Monte Carlo at high temperature where the pseudogap opens up. In addition to temperature-dependent spectral properties, we find quantum corrections to the zero-temperature renormalized mean-field antiferromagnetic gap. Finally, analytic continuation of the Matsubara results for spectral functions show that the pseudogap opens up at significantly higher temperature than was previously identified based on the Matsubara data only.
We compute high-resolution angle-resolved photoemission spectroscopy of the Hubbard model using the unbiased determinant quantum Monte Carlo algorithm, revealing an asymmetry between electron and hole doping. Electron doping exhibits more coherent quasiparticles and stronger antiferromagnetic correlations compared to hole doping. At low doping, a nodal-antinodal dichotomy on the Fermi surface is observed, similar to cuprate experiments. The dichotomy reflects the momentum dependence of the Mott gap, as manifested in both the spectral function and the self-energy. For hole doping, we observe a transition towards the pseudogap, without signature of pocket formation. The simulated nuclear magnetic resonance pseudogap temperatures do not necessarily agree with the temperature determined by spectroscopy. These findings collectively suggest the pseudogap is a smooth crossover driven by strong correlations.
The effect of doping in the two-dimensional Hubbard model is studied within finite-temperature exact diagonalization combined with cluster dynamical mean-field theory. By employing a mixed basis involving cluster sites and bath molecular orbitals for the projection of the lattice Green's function onto $2\ifmmode\times\else\texttimes\fi{}2$ clusters, a considerably more accurate description of the low-frequency properties of the self-energy is achieved than in a pure site picture. To evaluate the phase diagram, the transition from Fermi-liquid to non-Fermi-liquid behavior for decreasing hole doping is studied as a function of Coulomb energy, next-nearest-neighbor hopping, and temperature. The self-energy component ${\ensuremath{\Sigma}}_{X}$ associated with $X=(\ensuremath{\pi},0)$ is shown to develop a collective mode above EF{E}_{F}EF​, whose energy and strength exhibits a distinct dispersion with doping. This low-energy excitation gives rise to non-Fermi-liquid behavior as the hole doping decreases below a critical value ${\ensuremath{\delta}}_{c}$, and to an increasing particle-hole asymmetry, in agreement with recent photoemission data. This behavior is consistent with the removal of spectral weight from electron states above EF{E}_{F}EF​ and the opening of a pseudogap, which increases with decreasing doping. The phase diagram reveals that ${\ensuremath{\delta}}_{c}\ensuremath{\approx}0.15\dots{}0.20$ for various system parameters. For electron doping, the collective mode of ${\ensuremath{\Sigma}}_{X}(\ensuremath{\omega})$ and the concomitant pseudogap are located below the Fermi energy, which is consistent with the removal of spectral weight from the hole states just below EF{E}_{F}EF​. The critical doping, which marks the onset of non-Fermi-liquid behavior, is systematically smaller than for hole doping.
Qualitative changes in thermodynamic and single-particle properties characterize the transition between the pseudogapped electronic liquid and the Fermi liquid. Recent cold-atom experiments on a simulator of the Hubbard model with nearest-neighbor hoppings [L. H. Kendrick , .] showed that the isothermal compressibility κ ( δ ) has a maximum as a function of doping δ . Here, we use the two-particle self-consistent plus approach to explain these experiments and connect the maximum in κ ( δ ) to the transformation of the single-particle spectrum from the pseudogapped to the metallic regime. This elucidates the nature of the pseudogap. Specifically, the maximum in κ ( δ ) practically coincides with the doping at which the precursor of the lower ( π , π ) spin-density-wave (SDW) band at the antinodal point crosses the zero-frequency ω = 0 . The Knight shift χ sp ( 0 , 0 ) ( δ ) , as a function of doping, should also have a maximum. In addition, we demonstrate that two-particle self-consistent plus correctly predicts a maximum in the temperature dependence of the Knight shift χ sp ( 0 , 0 ) ( T ) , consistent with recent ultracold-atom experiments [T. Chalopin , .]. The maxima in both quantities should exist, at sufficiently low temperatures ( T ), in both the intermediate U ≈ U Mott and weak U < U Mott interaction limits. In both limits, the mechanism is critical thermal SDW fluctuations. At the antinodal pseudogap, the correlation length at δ max ( T ) can be small, controlled not by static but by dynamic critical thermal fluctuations. We also find that the SDW fluctuations are incommensurate at δ = δ max . We predict that, at low T , the multiple peaks in the spin susceptibility in the incommensurate case lead to more than two SDW precursor peaks in the spectral function and density of states. By allowing access to parameter regimes relevant to cuprates—including further-neighbor hopping ( t ′ , t ′′ ) and low temperatures, our work provides a high-impact tool for further studies by the broader community.
The spin-fermion (SF) model postulates that the dominant coupling between low-energy fermions in near critical metals is mediated by collective spin fluctuations (paramagnons) peaked at the Néel wave vector, Q N , connecting hot spots on opposite sides of the Fermi surface. It has been argued that strong correlations at hot spots lead to a Fermi surface deformation (FSD) featuring flat regions and increased nesting. This conjecture was confirmed in the perturbative self-consistent calculations when the paramagnon propagator dependence on momentum deviation from Q N is given by χ − 1 ∝ | Δ q | . Using diagrammatic Monte Carlo (diagMC) technique we show that such a dependence holds only at temperatures orders of magnitude smaller than any other energy scale in the problem, indicating that a different mechanism may be at play. Instead, we find that a χ − 1 ∝ | Δ q | 2 dependence yields a robust finite- T scenario for achieving FSD. To link phenomenological and microscopic descriptions, we applied the connected determinant diagMC method to the ( t − t ′ ) Hubbard model and found that at large U / t > 5.5 before the formation of electron and hole pockets (i) the FSD defined as a maximum of the spectral function is not very pronounced; instead, it is the lines of zeros of the renormalized dispersion relation that deforms toward nesting, and (ii) the static spin susceptibility is well described by χ − 1 ∝ | Δ q | 2 . Flat FS regions yield a nontrivial scenario for realizing a non-Fermi liquid state. Published by the American Physical Society 2024