Non-perturbative effects of short-range spatial correlations at the two-particle level

TL;DR

The work tackles the non-perturbative effects of short-range spatial correlations on two-particle quantities in the 2D Hubbard model at half-filling by employing cellular dynamical mean-field theory (CDMFT) with a full Bethe-Salpeter equation treatment across charge, spin, and particle-particle channels. By enforcing Ward identities and conducting a thorough two-particle analysis, the authors show that short-range antiferromagnetic fluctuations drive the first divergence of the two-particle irreducible charge vertex to smaller interactions than in DMFT, signaling a breakdown of perturbation theory in two dimensions and linking this breakdown to the Mott transition. A key finding is that the sign change of the leading eigenvalue of the generalized charge susceptibility is a prerequisite for the Mott instability and related phase-separation tendencies, with non-local spin fluctuations playing a decisive role at intermediate temperatures. The results illuminate the intimate connection between perturbative breakdown, non-local correlations, and the Mott transition in 2D, and provide a framework to study thermodynamic instabilities and their momentum-space structure in more realistic, non-perturbative regimes relevant to cuprates and related materials.

Abstract

By means of cellular dynamical mean-field theory (CDMFT) we study how short-range correlations drive the breakdown of the self-consistent perturbation theory in two-dimensional systems and the most relevant physical consequences associated to it. To this aim, we first derive in a structured and consistent way the Bethe-Salpeter equation (BSE) formalism at the CDMFT level in all physical channels, explicitly addressing the important aspect of the related Ward identities. In this context, we perform systematic calculations of the BSE for the two-dimensional Hubbard model at half-filling at intermediate coupling. Our study illustrates how the divergence of a fundamental building block of the BSE in the charge channel, the two-particle irreducible vertex, systematically occurs at lower interactions than in the (purely local) DMFT case, due to short-range antiferromagnetic fluctuations. Further, the change of sign of the eigenvalues of the generalized charge susceptibility associated to the vertex divergences is identified as the essential prerequisite to drive, at larger interaction values, the physics of the Mott transition in two dimensions, as well as of the adjacent phase-separation instabilities.

Figures (12)

• Figure 1: The unit cell of the superlattice with a four-atomic basis on the left-hand side. The four sites are labelled from $0$ to $3$. Red indicates specific geometrically equivalent sites in different unit cells. On the right-hand side, the reciprocal superlattice basis is indicated by x-markers, where the shading gives the reduced Brillouin zones ($\mathrm{RBZ}$) in comparison to the Brillouin zone of the initial one-atomic basis Hubbard model Ayral2017. Any vector in the initial Brillouin zone can be decomposed by the closest reciprocal lattice vector $\mathbf{Q}$ and a vector $\tilde{\mathbf{q}} \in \mathrm{RBZ}$.
• Figure 2: Examples of lattice modulations encoded in the CDMFT scheme: The left-hand side plot displays commensurate [$\mathbf{q}=(0,0)$] anti-ferromagnetism [$\mathbf{Q}=(\pi,\pi)$], while the right-hand side displays different superlattice variations [$\mathbf{q}=(\pi/2,\pi/2)$], from the same cluster modulations [$\mathbf{Q}=(\pi,\pi)$].
• Figure 3: Representation of the ph-BSE Eq. (\ref{['Eq:BSE']}) and pp-BSE Eq. (\ref{['Eq:BSE_pp']}) for the generalized susceptibilities $\chi_{\mathrm{sp}/\mathrm{ch}}^{\mathrm{ijhl}}$ and $\tilde{\chi}_{\mathrm{pp},\uparrow\downarrow}^{\mathrm{ih|jl}}$, respectively.
• Figure 4: Benchmarking. Upper panel: Comparison of the anti-ferromagnetic response function $\chi_\mathrm{sp}^{\mathbf{q}=0,\mathbf{Q}=(\pi,\pi)}(\omega_m=0)$ of the $2\times2$-CDMFT from the BSE (black $+$-marker) and applied field calculations (red circle marker) to the DMFT Schaefer2021 (gray square marker) and $8\times8$-CDMFT Schaefer2021 (gray circle marker), at $U=2t$. The diagrammatic Monte-Carlo (DiagMC) calculations for the Hubbard model Schaefer2021 are given for comparison as dotted line. $\beta=1/T$ gives the inverse temperature. Lower panel: Exemplary Ward identity for the two particle irreducible vertex Eq. (\ref{['Eq:Ward-Gamma']}) in the charge channel for $U=5.9t$ and $T=t/15$ for the orbital combination $\mathrm{i=0,j=3,h=0}$ and $\omega=0$ where each marker represents a fermionic Matsubara frequency. Red solid lines refer to the left, dotted black lines to the right-hand side of the equation, respectively.
• Figure 5: Left: Location of the first vertex divergence line in the $T-U$ phase-diagram of the 2D Hubbard model, computed in CDMFT with a $N_c=2\times 2$ cluster (red continuous line), and DMFT Schaefer2013 (black continuous line), compared to the corresponding results for a $2\times 2$ isolated Hubbard cluster (red dashed line) and the $1\times 1$ Hubbard atom Schaefer2013 (black dashed line). Red shading indicates the incoherent quasiparticle regime (Inc. QP), turquoise the coherent quasiparticle (Coh. QP) one, and the white area the paramagnetically enforced (PM enf.) region, below the Néel-temperature of the cluster KlettPhdKlett2020Fratino2017Meixner2024. The gray shaded area gives the CDMFT metal-insulator coexistence region Park2008. As comparison, the black $\times$-marker gives the divergence for CDMFT given for a single temperature in Vucicevic2018. Middle: Zoom on the divergence line of $\Gamma_c$ with (red bold line) and without the non-local spin-transverse diagrams (orange dashed line). The black line indicates the corresponding DMFT divergence line Schaefer2013. The vertical gray line indicates a temperature cut at $U=4.8t$, for which thermodynamic quantities are given in the right-hand side plot. To the right of the divergence lines, we present the local spectral weight (bordeaux-colored $\times$ marker) and the double occupancy $D$ ($\circ$ marker) on the horizontal axis, while the vertical axis gives the temperature. The dashed green line for $D$ results form the Migdal formula Eq. (\ref{['Eq:Migdal']}), the grey, bold line from the generalized response function and the orange dash-dotted line excludes the non-local spin-transverse diagrams. This is contrasted to an area of incoherent quasiparticles in red shading, the red horizontal line the location of the first divergence line of $\Gamma_c$ and the gray, dotted line in the plot indicates the Kondo temperature $T_K$, see App. \ref{['App:Kondo']}.