Numerical Linked-Cluster Approach to Quantum Lattice Models

We present a novel algorithm that allows one to obtain temperature dependent properties of quantum lattice models in the thermodynamic limit from exact diagonalization of small clusters. Our numerical linked-cluster approach provides a systematic framework to assess finite-size effects and is valid for any quantum lattice model. Unlike high temperature expansions, which have a finite radius of convergence in inverse temperature, these calculations are accurate at all temperatures provided the range of correlations is finite. We illustrate the power of our approach studying spin models on kagomé, triangular, and square lattices.

Figure 1

NLC results up to 5 (thin lines) and 6 (thick lines) triangles for entropy (a) and specific heat (b) of the transverse Ising model on the kagomé lattice as a function of temperature ($T$) for five different values of the transverse field ($h_x$). Except for very low but nonzero transverse field, the direct sum converges at all temperature.

Figure 2

NLC results up to 7 (thin lines) and 8 (thick lines) triangles for the entropy (a) and specific heat (b) of the $XXZ$ models on the kagomé lattice.

Figure 3

Specific heat of kagomé lattice Heisenberg model as a function of temperature $T$. The direct sum of HTE to order 13–16 is shown to diverge around $T=1$. NLC up to 7 and 8 triangle clusters converges below $T=0.4$. Two Padé approximants are also shown (see Ref. 16 for details), one of which is close to NLC result down to $T=0.3$. The NLC results indicate that there may be a second peak below $T=0.3$.

Figure 4

Specific heat of kagomé lattice Ising model in a transverse field $h_x=0.5$ and a longitudinal field $h_z=0.25$. The $T=0$ phase diagram of the model is shown in the inset (the line $h_x=0$ is critical). The parameters correspond to large but finite correlation length, where bare sums up to 3, 4, 5, and 6 triangle clusters diverge below $T=1$ but the sequence extrapolation methods converge at all temperatures. The subscript “2” in Wynn means that two cycles were applied to accelerate convergence [19].

Figure 5

The entropy of square (SQ) and triangular (TR) lattice Heisenberg models. Various extrapolations of NLC are compared with results obtained by BM [18]. Subscripts in Wynn’s and Brezinski’s (Brez) extrapolation results show the number of cycles applied to accelerate convergence [5].