Diagrammatic approach in the variational coupled-cluster method

We present an alternative approach based on a diagrammatic technique to the recently proposed variational coupled-cluster method. We apply this method to quantum antiferromagnetic bipartite lattices. In our analysis, infinite resummations of diagrams can be done in a straightforward manner. One such resummation reproduces spin-wave theory (SWT). Approximations beyond SWT can also be easily made. Interestingly, one such approximation produces a convergent, precise number for the order parameter of the one-dimensional isotropic model, in contrast to the well-known divergence of SWT; it also produces improved results for the two-dimensional square-lattice system. We also discuss relations and a possible combination of our approach with other established many-body theories.

Figure 1

Three basic elements for construction of diagrams, where simplified index notations $1\equiv (i_1,j_1)$, etc., are used.

Figure 2

First- and second-order contributions in the expansion of Eq. (18).

Figure 3

Diagrams of third-order contributions in the expansion of Eq. (18).

Figure 4

Similar to Fig. 3 but for the fourth-order contributions.

Figure 5

Similar to Fig. 3 but for the fifth-order contributions.

Figure 6

Diagrams of up to third-order contributions to the generating functional $W$ of Eq. (26) except ring diagrams $R_1,R_2,R_3$. The corresponding symmetry factors are, in the same order as the list of diagrams, $(1∕2,1∕2,1,1,1∕2,1,1∕2,1,1∕3\text{!},1∕2,1∕3)$.

Figure 7

The ring expansion of the one-body bare distribution function ${\tilde{g}}_1$ of Eq. (31).

Figure 8

First few contributions to the full one-body distribution function ${\rho }_1$ of Eq. (35), where open dots indicate no summations over its indices while solid dots indicate such summations as before.

Figure 9

Super-ring diagram expansion, similar to the ring diagram expansion of Fig. 7 but now involving Pauli lines and with resummations of ring diagrams already carried out in all exchange lines. See text for more details.