Optimizing linked-cluster expansions by white graphs

We introduce a white-graph expansion for the method of perturbative continuous unitary transformations when implemented as a linked-cluster expansion. The essential idea behind an expansion in white graphs is to perform an optimized bookkeeping during the calculation by exploiting the model-independent effective Hamiltonian in second quantization and the associated inherent cluster additivity. This approach is shown to be especially well suited for microscopic models with many coupling constants, since the total number of relevant graphs is drastically reduced. The white-graph expansion is exemplified for a two-dimensional quantum spin model of coupled two-leg $\mathit{XXZ}$ ladders.

Figure 1

All fluctuations involving the highlighted links are pooled together in this representation. The overall contribution of these fluctuations to the hopping element $t_{i,j}$ in the thermodynamic limit (left panel) can be identified with a reduced contribution of a disconnected cluster (right panel) vanishing due to the linked-cluster theorem.

Figure 2

Fluctuation pattern indicated by the highlighted links corresponds to fluctuations associated with the vacuum. These fluctuations are irrelevant for the local one-particle hopping element $t_{i,i}$ in the thermodynamic limit (left panel).

Figure 3

Three different labelings of the same graph are depicted with the corresponding adjacency matrices and the resulting keys. The first labeling corresponds to the canonical labeling and the resulting key is the graph key.

Figure 4

(a) Standard representation of intermediate states following Eq. (22) so that ${\alpha }_j$ corresponds to a rational number. (b) Standard calculation on white graphs using the generalized monomials as written in Eq. (24). (c) Improved calculation on white graphs following Eq. (25) which uses the product structure of the generalized monomials.

Figure 5

Illustration of the quantum spin model of coupled two-leg $\mathit{XXZ}$ ladders. Filled circles represent quantum spin 1/2, which are linked by various solid lines illustrating the different exchange couplings $J_{\perp },J_{\parallel }$, and $J_{\mathrm{int}}$. The black box shows the two-site unit cell of the system corresponding to the rungs of the ladder and $\delta x$ $\left(\delta y\right)$ refers to the distance between two neighboring rungs belonging to different (same) ladders.

Figure 6

Illustration of two different fluctuation patterns contributing to the local one-particle hopping element $t_{i,i}$ in thermodynamic limit (left panel) for the quantum spin model of coupled two-leg $\mathit{XXZ}$ ladders. The solid lines correspond to the different exchange couplings $J_{\perp },J_{\parallel }$, and $J_{\mathrm{int}}$. The occurrence of a link in Eq. (20) is visualized by an arc defining the fluctuation pattern. All fluctuations with this pattern are pooled together in this representation. Both fluctuations are associated with topologically distinct graphs ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ (right panel) solely because their colorings are different. Decomposing the results of the calculation with respect to the fluctuation pattern allows one to gain both contributions by embedding a single white graph ${\mathcal{G}}^{\mathrm{white}}$ and “evaluating” the resulting coloring. The evaluation yields polynomials in the different perturbation parameters $J_{\gamma }$.

Figure 7

Number of white (colored) graphs are shown as filled black (cyan-gray) circles as a function of the perturb-ative order $m$ for the model of coupled two-leg $\mathit{XXZ}$ Heisenberg ladders. The dashed cyan-gray curve represents a fit $aexp\left(bm\right)$ through the data points corresponding to colored graphs.

Figure 8

One-magnon gap $\mathrm{\Delta }/\tilde{J}$ as a function of $\tau$ for the isotropic square lattice $J_{\mathrm{leg}}=J_{\mathrm{rung}}=J_{\mathrm{int}}=\lambda =1$ for different bare series and different (biased) $d\text{log}$ Padé extrapolants. Inset: zoom close to $\tau =1$.