A Unified Approach to Strong Local Correlations and Collective Fluctuations: Eliminating Divergence in the Spin Channel

Abstract

Dynamical mean-field theory (DMFT) provides an optimal local approximation for correlated lattice systems by mapping the lattice onto a self-consistent effective impurity model. To account for the missing long-range correlations, we propose a novel extended approach, which we term fluctuating dynamical mean-field theory (fDMFT). It incorporates collective fluctuations of auxiliary impurity models across different sites via functional integration. Technically, this method involves obtaining a family of DMFT solutions on a grid for a self-consistent auxiliary classical field applied to the lattice. While the result can, in principle, be improved diagrammatically, we find that the minimal version of the theory already yields accurate results, with lowest-order diagrammatic corrections offering only minor improvements. This consistent framework, based on our fluctuating local field concept, demonstrates superior performance for the nearly half-filled Hubbard model compared to other known diagrammatic extensions of DMFT.

Figures (9)

• Figure 1: A non-local Dual Fermion (DF) leading self-energy diagram $\Sigma_{12}^{\cal DF} (i \omega)$ built from two local impurity irreducible vertices $\gamma^{\cal I}$ connected by three dual propagators $\tilde{G}$. The dual propagator is defined as $\tilde{G}_{k, l}=G^{\cal D}_{k, l}-G^{\cal I}_{k, l}$, where $k$ and $l$ are generalized indices collecting the node label (or momentum), Matsubara frequency (or imaginary time), and spin, e.g. $k\equiv(j,i\omega_n,\sigma)$. Repeated internal indices are summed over, while the external indices are fixed.
• Figure 2: The Fluctuating Local Field (FLF) free energy $\tilde{F}_{\bf{h}}$ as a function of the order parameter $\bar{s}_{\bf{h}}$ for the square cluster with size $L = 8$ calculated via fDF approach. Left panel: $\mu=0.00$; right panel: $\mu=-0.68$. Colors indicate different inverse temperatures $\beta$.
• Figure 3: Schematic DMFT+FLF calculation procedure. For each value of the fluctuating field $h_j=j\,\delta h$ we run an inner DMFT self-consistency loop until convergence. At each grid point we then compute the effective field $h_j^*$, the free-energy increment $\tilde{F}_j$, and the averaged Green's function $\tilde{G}_j$; the statistical weights $\delta z_j$ are accumulated to obtain $Z$ and $G$, followed by the final normalization $G=G/Z$.
• Figure 4: Self-energy at the first Matsubara frequency, $\Sigma(\mathbf{k},i\omega_0)$ with $\omega_0=\pi/\beta$, along the $\Gamma\!-\!X\!-\!M\!-\!\Gamma$ path. Lines: $fDF$, crosses: DQMC. Left: half-filling, $\mu=0.00$; Right: doped case, $\mu=-0.68$. Data are shown for $\beta=2$--$6$ (as indicated in the legend).
• Figure 5: Curie constant $C=\chi/\beta$ in the antiferromagnetic channel as a function of inverse temperature $\beta$. Lines show DMFT, fDMFT, and fDF results; crosses indicate the DQMC reference data. Left panel: half filling, $\mu=0.00$; right panel: doped case, $\mu=-0.68$. The vertical dashed line marks the Néel point (see text).