Lanczos-boosted numerical linked-cluster expansion for quantum lattice models

Numerical linked-cluster expansions allow one to calculate finite-temperature properties of quantum lattice models directly in the thermodynamic limit through exact solutions of small clusters. However, full diagonalization is often the limiting factor for these calculations. Here we show that a partial diagonalization of the largest clusters in the expansion using the Lanczos algorithm can be as useful as full diagonalization for the method while mitigating some of the time and memory issues. As test cases, we consider the frustrated Heisenberg model on the checkerboard lattice and the Fermi-Hubbard model on the square lattice. We find that our approach can surpass state of the art in performance in a parallel environment.

Figure 1

A $3\times 3$ section of the checkerboard lattice Heisenberg model. $J$ is the strength of the exchange interaction on nearest-neighbor bonds and $J^{\prime }$ is the interaction on next-nearest-neighbor bonds in every other $2\times 2$ plaquette.

Figure 2

Square expansion for the checkerboard lattice Heisenberg model, where clusters at higher orders are generated by adding corner-sharing $2\times 2$ blocks with next-nearest-neighbor bonds to smaller clusters in lower orders. Shown are the topologically distinct clusters up to the third order of the expansion.

Figure 3

Sorted first 400 eigenvalues of the Heisenberg Hamiltonian for a 16-site cluster in the four spin-up sector with $n_{\mathrm{Hilbert}}=1820$. Results from the Lanczos algorithm deviate from the exact ones very early in the spectrum due to the existence of degeneracies and the loss of orthogonalization between vectors in the Krylov space. A full reorthogonalization step in the algorithm restores orthogonality of the Lanczos vector at every iteration and captures a larger percentage of the exact spectrum as the number of iterations increases.

Figure 4

Energy per site of the Heisenberg model on a checkerboard lattice with $J=0.5$ and $J^{\prime }=1.0$ as a function of temperature. The results within the convergence region are valid in the thermodynamic limit. Thin dotted lines are bare sums in the fifth and sixth orders. The latter is taken from Ref. [5]. Thick black lines are results in the fourth, fifth, and sixth orders from Euler resummations after regular sum of the first three terms in the series. These results are obtained through full diagonalization using a Lapack routine. Color lines show the results from our Lanczos algorithm for the fifth or sixth order with different number of iterations. As the latter increases, a larger percentage of the low-lying eigenvalues for every cluster in that order approach their exact values, and the corresponding curves match the exact curve up to higher temperatures. The blue dashed line represents results from ED for the case where 90% of the eigenvalues are used in the thermal averages.

Figure 5

(a) The heat capacity and (b) entropy of the Heisenberg model on a checkerboard lattice with $J=0.5$ and $J^{\prime }=1.0$ as a function of temperature. Lines are the same as Fig. 4. In case of the heat capacity, partial diagonalization using the Lanczos algorithm barely reaches the minimum convergence temperature of the series with six orders. However, a $n_{\mathrm{iter}}$ of 50% of the size of the Hilbert space is enough to reach the convergence temperature for the entropy with five or six orders.

Figure 6

The average energy per site of the half-filled Fermi-Hubbard model with $U=8t$ as a function of temperature. Thick color lines are the bare results of the square expansion with two and three squares. The two clusters in the third order, shown in Fig. 2, have 10 sites each and are diagonalized using the Lanczos algorithm with $n_{\mathrm{iter}}=$ 50% of the size of the Hilbert space. Thin black lines are Euler resummations after regular sum of the first six terms in the site expansion for the same model up to the 10th order. Both orders 9 and 10 are obtained using the Lanczos partial diagonalization algorithm with $n_{\mathrm{iter}}$ equal to 50% and 30% of $n_{\mathrm{Hilbert}}$, respectively. Results from order 3 of the square expansion and the 10th order of the site expansion agree down to $T\sim 0.15t$.

Figure 7

Scaling of the Lanczos partial diagonalization algorithm with full reorthogonalization for up to the fifth order. For comparison, we also show the same scaling for ED for which we use the Intel's MKL to diagonalize every cluster on a core. The performance of MKL improves rapidly by increasing the number of cores until we have as many cores as the number of clusters in the last order of the series. No speedup can be achieved beyond that. The Lanczos algorithm with $n_{\mathrm{iter}}=n_{\mathrm{Hilbert}}$ quickly surpasses MKL in performance and continues to offer some speedup by increasing the number of cores. Choosing $n_{\mathrm{iter}}=0.5n_{\mathrm{Hilbert}}$ reduces the computational time by roughly a factor of 3.