Cluster density matrix embedding theory for quantum spin systems

We applied cluster density matrix embedding theory, with some modifications, to a spin lattice system. The reduced density matrix of the impurity cluster is embedded in the bath states, which are a set of block-product states. The ground state of the impurity model is formulated using a variational wave function. We tested this theory in a two-dimensional spin-1/2 $J_1\text{−}{J}_2$ model for a square lattice. The ground-state energy (GSE) and the location of the phase boundaries agree well with the most accurate previous results obtained using the quantum Monte Carlo and coupled cluster methods. Moreover, this cluster density matrix embedding theory is cost effective and convenient for calculating the von Neumann entropy, which is related to the quantum phase transition.

Figure 1

A square lattice under periodic boundary conditions is divided into $2\times 2$ spin clusters, where A is treated as an impurity cluster and the remaining spin clusters are treated as bath states.

Figure 2

Energy curves for various values of $J_2$; the impurity schemes are a $2\times 2$ spin cluster (empty squares), a $2\times 4$ spin cluster (empty circles), and only the central $\left(2\times 2\right)$ half of the $2\times 4$ spin cluster (full circles); the reference data are extrapolated results obtained using the CCM (empty triangles).

Figure 3

The position distributions of site energies for $J_2=0$ (a) and $J_2=0.5$ (b) obtained using cluster-DMET with $2\times 4$ (squares) and $2\times 2$ (circles) impurity clusters. $R_i$ is the position of site $i$; the lowest values of the curves correspond to the impurity sites; the surrounding spins produce an embedding effect on the impurity cluster; and the remaining bath spins produce a $2\times 2$ cluster mean field.

Figure 4

The entanglement entropy curves for various $J_2$; all curves exhibit a discontinuity at $J_2=0.62$, indicating a first-order transition from the QP phase to the collinear phase.

Figure 5

The derivative of the von Neumann entanglement entropy $dS\left(\rho \right)/d{J}_2$ plotted versus $J_2$; all curves peak at $J_2\approx 0.42$, corresponding to a second-order transition from the Néel phase to the QP phase.