Dual fermion dynamical cluster approach for strongly correlated systems

We have designed a multiscale approach for strongly correlated systems by combining the dynamical cluster approximation (DCA) and the recently introduced dual fermion formalism. This approach employs an exact mapping from a real lattice to a DCA cluster of linear size $L_c$ embedded in a dual fermion lattice. Short-length-scale physics is addressed by the DCA cluster calculation, while longer-length-scale physics is addressed diagrammatically using dual fermions. The bare and dressed dual fermionic Green functions scale as $\mathcal{O}(1/L_c)$, so perturbation theory on the dual lattice converges very quickly, e.g., the dual Fermion self-energy calculated with simple second-order perturbation theory is of order $\mathcal{O}(1/L_c^3)$ with third-order and three-body corrections down by an additional factor of $\mathcal{O}(1/L_c)$.

Figure 1

Scaling plot for the bare dual Green function. Here, we have used the 1D Hubbard model to analyze its scaling behavior. Except for very small $L_c$ values, the two ratios scale linearly according to Eq. (28).

Figure 2

Lowest-order contributions to the dual fermion self-energy from the two-body interaction (left) and to the two-body interaction from the three-body term (right). Since the bare $n$-body vertices depend only upon the small cluster $K$, the dual Green function line may be coarse grained, and they are therefore zero according to Eq. (22).

Figure 3

Leading nontrivial $n$-body vertex contribution to the self-energy. It is contructed by one $n$-body vertex and one ($n-1$)-body vertex. Since there are ($2n-2$) internal legs, this contribution scales as $\mathcal{O}(1/L_c^{2n-2})$.

Figure 4

(a) Equation for $T_c$. Transition temperatures on the dual fermion lattice are identical to those calculated on the real lattice. (b) The low-order corrections to the dual fermion irreducible vertex ${\Gamma }_d$. The second-order terms are of the order $\mathcal{O}(1/L_c^2)$ with corrections $\mathcal{O}(1/L_c^3)$. (c) Contributions to the dual fermion self-energy. It is dominated by the second-order term, which is of the order $\mathcal{O}(1/L_c^3)$ with corrections $\mathcal{O}(1/L_c^4)$. The self-energy adds relative corrections to ${\chi }_d^0$ of order $\mathcal{O}(1/L_c^4)$ (see the text for the detail), so the most important contributions to the equation for $T_c$ come from the second-order cross channel contributions to ${\Gamma }_d$.

Figure 5

Plots of leading eigenvalues for different cluster sizes for the anti-ferromagnetic channel with $U=6t$ and filling $\langle n\rangle =0.95$. Lines without symbols are results from the DCA calculation for clusters with sizes $N_c=8$, 12, and 16, while lines with symbols are results from the DFDCA calculation without self-energy correction. For the latter, we have used a linear size of the dual fermion lattice as large as several hundreds ($N=L\times L$, $L\sim 200$). The inset is an enlarged view around the transition point. Note that both calculations produce the same transition temperatures as expected.

Figure 6

Plots of leading eigenvalues for the antiferromagnetic channel. They are calculated with different approximate methods in the dual fermion lattice. The parameters used are $U=4t$, $\langle n\rangle =1$, the DCA cluster of size $N_c=1$, and the dual fermion lattice of a size $N=4\times 4$.

Figure 7

Plots of the inverse antiferromagnetic and $d$-wave pairing susceptibilities calculated with different approximate methods in the dual fermion calculation. The parameters used are $U=8t$ and $N_c=4$. The linear dependence of the results with $1/L^2$ (see Fig. 8) is used to extrapolate the $L=\infty$ limit results.

Figure 8

The $L$ dependence of the leading eigenvalue for different DCA clusters. The parameters used are $U=8t$, $\langle n\rangle =1$, ${\Sigma }_{\text{dual}}=0,$ and $\beta =4t/T$. The nice linear dependence of $1/L^2$ can be readily observed, which is due to the periodic boundary conditions used in the dual fermion calculation.