Rotationally invariant slave-boson and density matrix embedding theory: Unified framework and comparative study on the one-dimensional and two-dimensional Hubbard model

We present detailed benchmark ground-state calculations of the one- and two-dimensional Hubbard model utilizing the cluster extensions of the rotationally invariant slave-boson mean-field theory and the density matrix embedding theory. Our analysis shows that the overall accuracy and the performance of these two methods are very similar. Furthermore, we propose a unified computational framework that allows us to implement both of these techniques on the same footing. This provides us with a different line of interpretation and paves the ways for developing systematically distinct generalizations of these complementary approaches.

Figure 1

Schematic representation of the RISB and DMET algorithm.

Figure 2

Energy $E/t$ for (a) DMET and (b) RISB as a function of occupancy $n$ in the 1D Hubbard model with the nearest-neighbor hopping at $U=1t,4t,8t$ for cluster size $N_c=1,2,4$, indicated by the blue solid, green dashed, and red dotted lines, respectively. The solid black lines denote the results from BA.

Figure 3

Occupancy $n$ for (a) DMET and (b) RISB as a function of chemical potential $\mu$ in the 1D Hubbard model with the nearest-neighbor hopping at $U=1t,4t,8t$ for cluster size $N_c=1,2,4$, indicated by the blue solid, green dashed, and red dotted lines, respectively. The solid black lines denote the results from BA.

Figure 4

Double occupancy $\langle n_{\uparrow }{n}_{\downarrow }\rangle$ for (a) DMET and (b) RISB as a function of interaction $U$ in the half-filled 1D Hubbard with the nearest-neighbor hopping for cluster size $N_c=1,2,4$, indicated by the blue solid, green dashed, and red dotted lines, respectively. The solid black lines denote the results from BA.

Figure 5

Energy $E/t$ as a function of inverse cluster size $1/N_c$ in the 1D Hubbard model with the nearest-neighbor hopping for (a) $U=4t$ and $n=1$, (b) $U=8t$ and $n=1$, (c) $U=4t$ and $n=0.75$, and (d) $U=8t$ and $n=0.75$. The blue circles correspond to the DMET values in our simulation. The red squares are our RISB results. The green triangles are the data from Zheng et al. with antiferromagnetic order [13]. The black solid lines are the results from BA.

Figure 6

Clusters with sizes (a) $N_c=1$, (b) $N_c=2$, (c) $N_c=4$, and (d) $N_c=6$, used in our simulation. The red arrows indicate the lattice vectors. The blue lines delimit the unit cells.

Figure 7

Energy $E/t$ for (a) DMET and (b) RISB as a function of interaction $U$ in the half-filled 2D Hubbard model on a square lattice with the nearest-neighbor hopping at cluster size, $N_c=1,2,4$, indicated by the blue, green, and red lines, respectively. The solid, dashed, and dotted lines represent the PM metal, PM insulator, and AFM solutions, respectively. The critical interaction $U_c$ is indicated by the vertical line. The black solid circles indicate the results in the TL from Refs. [13, 15]. The gray arrow indicates the $U_c$ from cellular-DMFT (CDMFT) with $N_c=4$ in Ref. [45]. The inset of (a) shows the magnified plot around $U_c$.

Figure 8

Occupancy $n$ as a function of chemical potential $\mu$ in the PM phase of the 2D Hubbard model on a square lattice with the nearest-neighbor hopping at $U=12t$ for cluster sizes $N_c=2$ and 4, indicated by the green dashed and red dotted lines, respectively.