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arXiv:2505.20116 · Proceedings of the National Academy of Sciences

The finite-difference parquet method: Enhanced electron–paramagnon scattering opens a pseudogap

Jae-Mo Lihm, Dominik Kiese, Seung‐Sup B. Lee, Fabian B. Kugler

Published Mar 3, 2026• 3 citations
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Abstract

We present the finite-difference parquet method that greatly improves the applicability and accuracy of two-particle correlation approaches to interacting electron systems. This method incorporates the nonperturbative local physics from a reference solution and builds all parquet diagrams while circumventing potentially divergent irreducible vertices. Its unbiased treatment of different fluctuations is crucial for reproducing the strong-coupling pseudogap in the underdoped Hubbard model, consistent with diagrammatic Monte Carlo calculations. We reveal a strong-coupling spin-fluctuation mechanism of the pseudogap with decisive vertex corrections that encode the enhanced, energy-dependent scattering amplitude between electrons and antiferromagnetic spin fluctuations.

DOI
10.1073/pnas.2525308123

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We present the finite-difference parquet method that greatly improves the applicability and accuracy of two-particle correlation approaches to interacting electron systems. This method incorporates the nonperturbative local physics from a reference solution and builds all parquet diagrams while circumventing potentially divergent irreducible vertices. Its unbiased treatment of different fluctuations is crucial for reproducing the strong-coupling pseudogap in the underdoped Hubbard model, consistent with diagrammatic Monte Carlo calculations. We reveal a strong-coupling spin-fluctuation mechanism of the pseudogap with decisive vertex corrections that encode the enhanced, energy-dependent scattering amplitude between electrons and antiferromagnetic spin fluctuations.
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This thesis presents the development of new numerical methods for the treatment of strongly correlated electron systems based on self-consistent approaches at both the one and the two-particle level such as the parquet formalism. The parquet formalism was solved for the first time on a two-dimensional cluster. When the fully irreducible vertex is approximated by the bare vertex, we obtain the parquet approximation. Its validity was investigated by comparing results that it produces to those of other conserving approximations such as the FLuctuation EXchange (FLEX) approximation or the Second Order Perturbation Theory (SOPT). We found that it provides a significant improvement of FLEX or SOPT and a satisfactory agreement with Quantum Monte Carlo results despite instabilities in the self-consistency at low temperatures and for strong Coulomb interaction. We use the parquet formalism to study the Quantum Critical Point at finite doping in the Hubbard model by decomposing the vertex into its contributions from different channels. We apply this decomposition to the pairing channel and we find that the dominant contribution to the vertex originates in the spin channel even at the quantum critical doping. Furthermore, we explore the divergence of the two parts of the pairing matrix at optimal doping and observe that the irreducible vertex decreases monotonically as the doping is increased while the bare susceptibility exhibits an algebraic divergence at the quantum critical doping supporting the Quantum Critical BCS scenario proposed by She and Zaanen. To circumvent the instabilities in the iteration of the parquet formalism, we explored the dual fermion approach introduced by Rubtsov et al. Here, we extended the formalism to the Dynamical Cluster Approximation, in the process introducing a small parameter in the dual fermions perturbation theory. We demonstrate the quality of the resulting Dual Fermion DCA through a systematic study of the cluster size dependence and of the different perturbative approximations. These efforts represent the initial steps in the development of the Multi-Scale Many Body approach that appropriately treats correlations at different length scales.
We present a formalism for strongly correlated electron systems which consists in a local approximation of the dynamical three-leg interaction vertex. This vertex is self-consistently computed using a quantum impurity model with dynamical interactions in the charge and spin channels, similar to dynamical mean field theory approaches. The electronic self-energy and the polarization are both frequency and momentum dependent. The method interpolates between the spin-fluctuation or GW approximations at weak coupling and the atomic limit at strong coupling. We apply the formalism to the Hubbard model on a two-dimensional square lattice and show that as interactions are increased towards the Mott insulating state, the local vertex acquires a strong frequency dependence, driving the system to a Mott transition, while at low enough temperatures the momentum dependence of the self-energy is enhanced due to large spin fluctuations. Upon doping, we find a Fermi arc in the one-particle spectral function, which is one signature of the pseudogap state.
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We study a model of quasiparticles on a two-dimensional square lattice coupled to Gaussian distributed dynamical molecular fields. We consider two types of such fields, a vector molecular field that couples to the quasiparticle spin density and a scalar field coupled to the quasiparticle number density. The model describes quasiparticles coupled to spin or charge fluctuations, and is solved by a Monte Carlo sampling of the molecular field distributions. The nonperturbative solution is compared to various approximations based on diagrammatic perturbation theory. When the molecular field correlations are sufficiently weak, the diagrammatic calculations capture the qualitative aspects of the quasiparticle spectrum. For a range of model parameters near the magnetic boundary, we find that the quasiparticle spectrum is qualitatively different from that of a Fermi liquid, in that it shows a double peak structure, and that the diagrammatic approximations we consider fail to reproduce, even qualitatively, the nonperturbative results of the Monte Carlo calculations. This suggests that the magnetic pseudogap induced by a coupling to antiferromagnetic spin fluctuations and the spin splitting of the quasiparticle peak induced by a coupling to ferromagnetic spin fluctuations lie beyond diagrammatic perturbation theory. While a pseudogap opens when quasiparticles are coupled to antiferromagnetic fluctuations, such a pseudogap is not observed in the corresponding charge-fluctuation case for the range of parameters studied, where vertex corrections are found to effectively reduce the strength of the interaction. This suggests that one has to be closer to the border of long-range order to observe pseudogap effects in the charge-fluctuation case than for a spin-fluctuation-induced interaction under otherwise similar conditions. The diagrammatic approximations that contain first-order vertex corrections show the enhancement of the spin-fluctuation-induced interaction and the suppression of the effective interaction in the charge-fluctuation case. However, for the range of model parameters considered here, the multiple spin or charge-fluctuation exchange processes not included in the diagrammatic approximations considered are found to be important, especially for quasiparticles coupled to charge fluctuations.
We describe a controllable and unbiased strong-coupling diagrammatic Monte Carlo technique that is applicable to a wide range of fermionic systems and spin models. Unlike previous strong coupling methods that generally rely on the Grassmannian Hubbard-Stratonovich transformation, our construction is based on Wick's theorem and a recursive procedure to group contractions into effective connected vertices that are nonperturbative in all local physics and can be calculated exactly. The resulting expansion is described by simple diagrammatic rules that make it suitable for systematic treatment via stochastic sampling. Benchmarks against numerical linked cluster expansion display excellent agreement.