Impact of nonlocal spatial correlations for different lattice geometries

TL;DR

This work analyzes how lattice geometry modulates magnetic ordering in the Hubbard model by comparing DMFT (local correlations) with ladder D$\Gamma$A (nonlocal correlations) across 3d-sc, bcc, fcc, and 4d-sc lattices. Nonlocal fluctuations markedly reduce the Néel temperature $T_N$, with the reduction increasing as coordination decreases, and can even suppress order entirely on the frustrated fcc lattice; in four dimensions, mean-field critical behavior emerges and the DMFT-D$\Gamma$A discrepancy narrows as expected from high connectivity. The study also characterizes critical exponents near $T_N$, showing larger values for 3d-sc and bcc consistent with Berlin-Kac-like universality in 3D, while 4d-sc remains near the MF value with logarithmic corrections. Additionally, the authors quantify sum-rule violations in DMFT and demonstrate their mitigation by nonlocal correlations, providing practical guidance for estimating DMFT overestimation and guiding future extensions that include charge fluctuations.

Abstract

We analyze the impact of the lattice geometry on the thermodynamic transition to magnetically ordered phases in strongly interacting electron systems for various Bravais lattices in three and four dimensions, including both local and nonlocal correlation effects. In a first step we use the dynamical mean field theory (DMFT), which takes into account purely local correlations, to calculate the magnetic susceptibilities of the Hubbard model on three (3d-sc) and four dimensional (4d-sc) simple cubic/hypercubic, as well as on three dimensional body- (bcc) and face-centered (fcc) cubic lattices, and determine the transition temperature to the corresponding magnetically-ordered state. In a second step, we exploit the dynamical vertex approximation (D$Γ$A), a diagrammatic extension of DMFT, to include the effect of nonlocal correlations which are particularly important in the vicinity of the corresponding phase transition. For the bipartite 3d-sc, 4d-sc and bcc lattices nonlocal fluctuations lead to a substantial reduction of the DMFT transition temperature consistent to the overall tendency of mean-field approaches to overestimate the stability of ordered phases. As expected, the magnitude of the difference between the DMFT, being exact in the limit of large connectivity/dimensions, and D$Γ$A transition temperatures decreases with increasing coordination number. On a more practical perspective, these results also provide a reasonable guidance to evaluate the expected overestimation of the DMFT ordering temperature for different material geometries. For the fcc lattice, on the other hand, the ordered phase observed in DMFT vanishes completely within D$Γ$A which is consistent with the existence of strong geometric frustration in this lattice.

Figures (11)

• Figure 1: Three dimensional simple cubic (3d-sc), body-centered cubic (bcc), face centered cubic (fcc) and four dimensional simple cubic (4d-sc, tesseract) lattices considered in this study ordered by increasing coordination number $Z$ (from left to right) and increasing spatial dimension $d$ (from bottom to top). Arrows display the spin arrangement for the predominant magnetic order.
• Figure 2: Local spin susceptibilities $\chi_{m,\text{loc}}^\omega=\sum_\mathbf{q}\chi_{m,\mathbf{q}}^\omega$ for the three bipartite lattices at $U=2.0$ and $n=1.0$ for the three temperatures $\beta = 5.0$ (left panels), $\beta=10.0$ (middle panels) and $\beta=12.0$ (right panels) for the 3d-sc (upper row), bcc (middle row) and 4d-sc (lower row) lattices, respectively. Data are shown for DMFT (blue crosses) and lD$\Gamma$A (orange circles).
• Figure 3: Momentum dependence of the static DMFT susceptibilities $\chi_{m,\mathbf{q}}^{\omega=0}$ along a high-symmetry path for the 3d-sc (first panel), the bcc (second panel) and the 4d-sc (third panel) lattice, respectively, at $U=2.0$, $n=1.0$ and $\beta = 5.0$. The fourth (rightmost) panel shows the corresponding results for the the fcc lattice at $U=3.0$, $n=0.75$ and $\beta = 80.0$. Since there is no established convention for naming high symmetry points in the 4d-sc lattice, we choose a notation similar to the 3d-sc lattice: $\Gamma = (0,0,0,0)$, $X = (0,0,0,\pi)$, $M = (0,0,\pi,\pi)$, $R = (0,\pi,\pi,\pi)$, $S = (\pi,\pi,\pi,\pi)$. Insets: geometrical structure of the lattices replotted from Fig. \ref{['fig:lattice_overview']} where the arrows represents the geometric structure of the predominant spin order. In the fcc lattice, the state presented here is given by the ordering vector $\mathbf{q} = (0,0,2\pi)$.
• Figure 4: Local spin susceptibility for the fcc lattice at $U=3.0$ and $n=0.75$ for $\beta=5.0$ (inset of left panel), $\beta = 20.0$ (left panel), and $\beta = 150.0$ (right panel) at the ordering vector $\mathbf{q}_N = (0 ,0,2\pi)$.
• Figure 5: Inverse antiferromagnetic susceptibility $1/\chi_\text{AF}(T)=1/\chi_{m,\mathbf{q}=\mathbf{q}_N}^{\omega=0}(T)$ as function of the temperature $T$ at $U=2.0$ for the 3d-sc [$\mathbf{q}_N=(\pi,\pi,\pi)$, left panel], bcc [$\mathbf{q}_N=(2\pi,2\pi,2\pi)$, middle panel] and 4d-sc [$\mathbf{q}_N=(\pi,\pi,\pi,\pi)$, right panel] lattice, respectively. DMFT and D$\Gamma$A results are indicated by blue circles and orange triangle. Blue lines represent linear fits to the DMFT data. For the 3d-sc and the bcc lattices, the inset shows fits to the numerical lD$\Gamma$A data for the two fitting functions in Eqs. \ref{['equ:scalingfunction']} and \ref{['equ:fitsubleading']}. For the 4d-sc the orange line is a linear fit to the lD$\Gamma$A data.