Entanglement witnesses in spin models

We construct entanglement witnesses using fundamental quantum operators of spin models which contain two-particle interactions and have a certain symmetry. By choosing the Hamiltonian as such an operator, our method can be used for detecting entanglement by energy measurement. We apply this method to the Heisenberg model in a cubic lattice with a magnetic field, the $XY$ model, and other familiar spin systems. Our method provides a temperature bound for separable states for systems in thermal equilibrium. We also study the Bose-Hubbard model and relate its energy minimum for separable states to the minimum obtained from the Gutzwiller ansatz.

Figure 1

Some of the most often considered lattice models: (a) Chain, (b) two-dimensional cubic, (c) hexagonal, and (d) triangular. Different symbols at the vertices indicate a possible partitioning into sublattices.

Figure 2

(a) Heisenberg chain of eight spins. Nearest-neighbor entanglement as a function of magnetic field $B$ and temperature $T$. (b) The same for an Ising spin chain. Here $k_B$ is the Boltzmann constant, $J$ and $J_x$ are coupling constants. Light color indicates the region where entanglement is detected by our method.