Variational Diagrammatic Monte-Carlo Built on Dynamical Mean-Field Theory

Abstract

We develop a variational perturbation expansion around dynamical mean-field theory (DMFT) that systematically incorporates nonlocal correlations beyond the local correlations treated by DMFT. We apply this approach to investigate how the DMFT critical temperature is suppressed from its mean-field value and how the critical behavior near the finite-temperature phase transition evolves from the mean-field to the Heisenberg universality class. By identifying the symmetry breaking of paramagnetic diagrammatic expansions as a signature of the Néel transition, we accurately predict the Néel temperature of the three-dimensional cubic Hubbard model across all interaction strengths with low computational cost. Introducing a variational order parameter, our method can be applied to both paramagnetic and long-range ordered states, such as antiferromagnetic order. We compute magnetization and antiferromagnetic susceptibility, demonstrating minor corrections to DMFT solutions in the weak-coupling regime while revealing significant modifications to these properties in the intermediate correlation regime. From the analysis of critical exponents, we establish the emergence of Heisenberg critical behavior beyond the mean-field nature of DMFT.

Figures (12)

• Figure 1: Comparison of the Néel temperature $T_N(U)$ obtained using VDMC with results from other numerical methods: the truncated-unity functional renormalization group (TUFRG) TUFRGTN; diagrammatic Monte Carlo methods, including CDet(PM) CDETPMTN and CDet(AF) VarPertHF; diagrammatic determinant Monte Carlo (DDMC) DDMCTN; auxiliary-field quantum Monte Carlo (AFQMC) AFQMCTN; quantum Monte Carlo (QMC) QMCTN; the dynamical vertex approximation (D$\Gamma$A) DGATN; the dynamical cluster approximation (DCA) DCATN; and the dual fermion method (DF) DFandDMFTTN. VDMC(PM) results are calculated using a small splitting starting point, whereas VDMC(AF) results are obtained by extrapolating the magnetization $m(T)$ to zero. Error bars for our results are smaller than the marker size.
• Figure 2: Magnetization $m(T)$ and critical behavior at different interaction strengths. The dashed lines indicate the critical behavior fitting: $m(T)=A(T_N-T)^\beta$.
• Figure 3: Antiferromagnetic susceptibility $\chi_{AF}=\chi(\pi,\pi,\pi)$ at $U=11$ for DMFT, VDMC, and dual Fermion DFandDMFTTN. (a) DMFT, VDMC and dual Fermion $\chi_{AF}$ plotted on linear scale. The dashed lines show the fit to the form $\chi_{AF}=A(T-T_N)^{-\gamma}$. (b) The residuals of the VDMC $\chi_{AF}$ fit close to $T_N$, which shows details of fitting around the phase transition. (c) DMFT and VDMC $\chi_{AF}$ plotted on log-log scale, compared with dual Fermion results DFandDMFTTN and several power-laws: mean field $\gamma_{\mathrm{MF}}=1$, Ising $\gamma_{\mathrm{Ising}}=1.24$, Heisenberg $\gamma_{\mathrm{Heisenberg}}=1.4$, and $\gamma=1.9$ proposed in Ref. DGAattractive.
• Figure 4: Magnetization of 3D cubic Hubbard model at half-filling under different values of interactions, temperatures, orders, and variational parameters. The blue dashed horizontal lines indicate the magnetization calculated from DMFT. The result of perturbation strongly relies on the starting point of perturbation. The points in the first column are all antiferromagnetic, while the points in the second column are all paramagnetic.
• Figure 5: Relative error of $\chi_{AF}(L)$ at $U=11$ and various temperatures. As the temperature decreases, larger system sizes are required to reach equally high precision.