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arXiv:1411.6947 · Physical Review Letters

Fluctuation Diagnostics of the Electron Self-Energy: Origin of the Pseudogap Physics

O. Gunnarsson, Thomas Schäfer, J. P. F. LeBlanc, Emanuel Gull, Jaime Merino, Giorgio Sangiovanni, G. Rohringer, A. Toschi

Published Jun 10, 2015• 145 citations
arXivHTMLPDF

Abstract

We demonstrate how to identify which physical processes dominate the low-energy spectral functions of correlated electron systems. We obtain an unambiguous classification through an analysis of the equation of motion for the electron self-energy in its charge, spin, and particle-particle representations. Our procedure is then employed to clarify the controversial physics responsible for the appearance of the pseudogap in correlated systems. We illustrate our method by examining the attractive and repulsive Hubbard model in two dimensions. In the latter, spin fluctuations are identified as the origin of the pseudogap, and we also explain why d-wave pairing fluctuations play a marginal role in suppressing the low-energy spectral weight, independent of their actual strength.

DOI
10.1103/physrevlett.114.236402

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cond-mat.str-elCross-listedarXiv:2603.15790•Mar 18, 2026•0 citations
We demonstrate how to identify which physical processes dominate the low-energy spectral functions of correlated electron systems. We obtain an unambiguous classification through an analysis of the equation of motion for the electron self-energy in its charge, spin, and particle-particle representations. Our procedure is then employed to clarify the controversial physics responsible for the appearance of the pseudogap in correlated systems. We illustrate our method by examining the attractive and repulsive Hubbard model in two dimensions. In the latter, spin fluctuations are identified as the origin of the pseudogap, and we also explain why d-wave pairing fluctuations play a marginal role in suppressing the low-energy spectral weight, independent of their actual strength.
Correlated electron systems may give rise to multiple effective interactions whose combined impact on quasiparticle properties can be difficult to disentangle. We introduce an unambiguous decomposition of the electronic self-energy which allows us to quantify the contributions of various effective interactions simultaneously. We use this tool to revisit the hole-doped Hubbard model within the dynamical cluster approximation, where commonly spin fluctuations are considered to be the origin of the pseudogap. While our fluctuation decomposition confirms that spin fluctuations indeed suppress antinodal electronic spectral weight, we show that they alone cannot capture the pseudogap self-energy quantitatively. Nonlocal multiboson Feynman diagrams yield substantial contributions and are needed for a quantitative description of the pseudogap.
The two-dimensional attractive Hubbard model is studied in the weak-to-intermediate-coupling regime by employing a nonperturbative approach. It is shown that this approach is in quantitative agreement with Monte Carlo calculations for both single-particle and two-particle quantities. Both the density of states and the single-particle spectral weight show a pseudogap at the Fermi energy below some characteristic temperature T∗,{T}^{*},T∗, also in good agreement with quantum Monte Carlo calculations. The pseudogap is caused by critical pairing fluctuations in the low-temperature renormalized classical regime $(\ensuremath{\Elzxh}\ensuremath{\omega}<{k}_{B}T)$ of the two-dimensional system. With increasing temperature the spectral weight fills in the pseudogap instead of closing it, and the pseudogap appears earlier in the density of states than in the spectral function. Small temperature changes around T∗{T}^{*}T∗ can modify the spectral weight over frequency scales much larger than temperature. Several qualitative results for the s-wave case should remain true for d-wave superconductors.
While calculations and measurements of single-particle spectral properties often offer the most direct route to study correlated electron systems, the underlying physics may remain quite elusive, if information at higher particle levels is not explicitly included. Here, we present a comprehensive overview of the different approaches which have been recently developed and applied to identify the dominant two-particle scattering processes controlling the shape of the one-particle spectral functions and, in some cases, of the physical response of the system. In particular, we will discuss the underlying general idea, the common threads and the specific peculiarities of all the proposed approaches. While all of them rely on a selective analysis of the Schwinger-Dyson (or the Bethe-Salpeter) equation, the methodological differences originate from the specific two-particle vertex functions to be computed and decomposed. Finally, we illustrate the potential strength of these methodologies by means of their applications the two-dimensional Hubbard model, and we provide an outlook over the future perspective and developments of this route for understanding the physics of correlated electrons.
We have studied in terms of the effective T-matrix the self-energy part for both the attractive (U<0) and repulsive (U>0) Hubbard models. It is shown that the fluctuation pairing for strong coupling at T>T c for the attractive Hubbard model does not cause any pseudogap behavior in the single-particle density of states. We have also identified the origin of the false pseudogap for that case. Meanwhile, the fluctuation pairing significantly affect the density–density and the current–current correlation functions. As a result, there exists a pseudogap in the charge (and current) response for the attractive Hubbard model. In contrast, the pseudogap in the spin response appears to exist for the repulsive Hubbard model only. It is also shown that the contribution from the Hartree–"Cooper" channel gives an anomalous Fermi liquid term, Im χ c ≈-ω/T, to the charge response of the nearly antiferromagnetic Fermi liquid.
The theoretical investigation of spectral functions and pseudogap in systems with strongly correlated electrons is discussed, with the emphasis on the single-band t-J model as relevant for superconducting cuprates. The evidence for the pseudogap features from numerical studies of the model is presented. One of the promising methods to study spectral functions is the method of equations of motion. The latter can deal systematically with the local constraints and projected fermion operators inherent for strongly correlated electrons. In the evaluation of the self energy the decoupling of spin and single-particle fluctuations is performed. In an undoped antiferromagnet the method reproduces the selfconsistent Born approximation (SCBA). For finite doping the approximation evolves into a paramagnon contribution which retains large incoherent contribution in the hole part. On the other hand, the contribution of longer-range spin fluctuations is essential for the emergence of the pseudogap. The latter shows up at low doping in the effective truncation of the large Fermi surface, reduced electron density of states and at the same time reduced quasiparticle density of states at the Fermi level.
We have developed a semi-analytical framework formulated in the canonical fermion representation to investigate strongly correlated electron systems. We consider the U=∞\infty∞ Hubbard model and used the equation of motion method to calculate the fermion self-energy which has two parts: single and two-boson exchange processes. The emergent bosons here are self-generated local charge and spin-density fluctuations which become strongly time-dependent due to extreme correlations. The computed boson spectral density is a diffusive damped mode with a long tail. The electron self-energy at d=∞d=\inftyd=∞ is computed self-consistently. The corresponding fermionic spectral density displays a pronounced coherence peak at ω=0ω=0ω=0, while its frequency derivative develops a two-peak structure at finite ωωω. The resistivity shows a linear temperature dependence over a broad range, crossing over to coherent Fermi-liquid behavior at extremely low temperatures.