Mechanisms of finite-temperature magnetism in the three-dimensional Hubbard model

We examine the nature of the transition to the antiferromagnetically ordered state in the half-filled three-dimensional Hubbard model using the dual-fermion multiscale approach. Consistent with analytics, in the weak-coupling regime we find that spin-flip excitations across the Fermi surface are important and that the strong-coupling regime is described by Heisenberg physics. In the intermediate-interaction, strong-correlation regime we find aspects of both local and nonlocal correlations. We analyze the critical exponents of the transition in the strong-coupling regime and find them to be consistent with Heisenberg physics down to an interaction of $U/t=10$.

Figure 1

Transition temperature $T_c$ as a function of $U$ (both in units of $t=1$) of the Hubbard model in three dimensions at particle-hole symmetry, as obtained by DF (filled dark circles) and DMFT [dash-dotted (blue) line)]. The comparison curves are obtained by the random phase approximation (RPA) from Refs. [10] and [24] with a factor of 1/3 in $T_c$ to account for quantum fluctuations [61] (dashed black line), dynamical cluster approximation [DCA; filled (red) hexagons] [39], lattice quantum Monte Carlo [QMC; filled (green) squares] [25], determinantal diagrammatic Monte Carlo [DDMC; filled (blue) triangles] [28], and dynamical vertex approximation [$\mathrm{D}\mathrm{\Gamma }\mathrm{A}$; filled (yellow) diamonds] [47]. See also Ref. [47]. The combination of RPA values at small $U$ with the QMC [25], the DCA [38], and large-$U$ asymptotics is plotted as the thick solid black line.

Figure 2

Estimate for the density of states at the Fermi level, $-\beta G(\beta /2)$, as a function of the temperature $T$ and interaction strength $U$. Inset: The cut at constant temperature $T=0.355$.

Figure 3

(a) Local, nearest-neighbor, and next-nearest neighbor spin-spin correlation function as a function of the interaction $U$ at $T\gtrsim T_c$. The formation of a well-defined magnetic moment occurs at large $U$. (b) Spatial dependence of the spin susceptibility along the real $x$ axis at different values of $U$ at $T\gtrsim T_c$.

Figure 4

(a) Inverse static magnetic susceptibility at $q=(\pi ,\pi ,\pi )$ and $U/t=0.25$, 5.0, 17.0 as obtained within the DF (filled symbols, dotted lines) and DMFT (open symbols, solid lines) methods and (b) corresponding correlation length. Dashed lines show the extrapolated low-temperature behavior, whereas dashed horizontal lines in (b) indicate the cutoff value for the inverse correlation length ${\xi }_c^{-1}=6/L$.

Figure 5

(a) Inverse magnetic susceptibility and (b) inverse correlation length as a function of $T-T_c$ at $U/t=11$, 14, and 17, plotted on a logarithmic scale. The thick solid line represents the power-law dependence with Heisenberg exponent (a) $\gamma \sim 1.4$ and (b) $\nu =0.7$, and the dashed line shows the power-law line with the Ising critical exponent (a) $\gamma \sim 1.24$ and (b) $\nu =0.63$.

Figure 6

Critical exponents $\gamma$ (susceptibility; top panel) and $\nu$ (correlation length; bottom panel) of the Hubbard model as obtained by the DF method at $U>U_H$ [filled (red) diamonds]. A comparison with the $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ result [47] and the DF method for the Falicov-Kimball model [50] is provided. The Heisenberg and Ising exponents are shown as dotted and dash-dotted lines, respectively.

Figure 7

Top: Inverse magnetic susceptibility at $\mathbf{q}=(\pi ,\pi ,\pi )$ for $U/t=11,14,17$ as a function of $T-T_c$. Circles, DF values; dashed line, linear high-temperature mean-field fit; bold solid line, fit of critical exponent in the critical region. Bottom: Residue values of the susceptibility fit plotted versus the temperature.

Figure 8

Inverse correlation length (top) and residue values of the power-law fit (bottom) at $U/t=11,14,17$ as a function of $T-T_c$. The critical region, obtained from the data in Fig. 7, is illustrated by the bold line.

Figure 9

Same as Fig. 7 for values of $U/t=11,12,13$ comparing the static approximation to the DF results at more than one bosonic frequency. Open symbols, static-approximation ladder DF approximation (LDFA) results; filled symbols, LDFA results using 120 bosonic frequencies.