Sec. IV.

Entanglement in Heisenberg chain only with nearest-neighbor interaction
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Analytical results of Entanglement in Heisenberg models
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We study entanglement of states of the system at thermal equilibrium described by the density operator $\rho(T)=\exp(-\beta
H)/Z$, where $\beta=1/k_BT$, $k_B$ is the Boltzmann’s constant, which is assumed to be one throughout the paper , and $Z=\text{Tr}\{\exp(-\beta H)\}$ is the partition function. The entanglement in the thermal state is called thermal entanglement.

We consider two kinds of spins, spin $\frac{1}{2}$ and $1$, alternating on a ring with antiferromagnetic exchange coupling. The Hamiltonian is given by $$\begin{aligned}
H_0&=&\sum_{i=1}^{N/2} ({\bf s}_i\cdot{\bf S}_{i}+{\bf S}_{i}\cdot
{\bf
s}_{i+1}), (N\in \text{even})\label{H01}\\
 H_0&=&\sum_{i=1}^{(N-1)/2} ({\bf s}_i\cdot{\bf S}_{i}+{\bf S}_{i}\cdot {\bf s}_{i+1})+
 {\bf s}_{\frac{(N+1)}{2}}\cdot{\bf s}_{1},\nonumber\\
 &&(N\in \text{odd}),\label{H02}\end{aligned}$$ where ${\bf s}_i$ and ${\bf S}_i$ are spin-1/2 and spin-1 operators, respectively. The exchange interactions exist only between nearest neighbors, and they are of the same strength which are set to one. We adopt the periodic boundary condition. In Fig. 1, we give the schematic representation of the above Hamiltonian. Next, we first consider the models with two and three spins, and aim at getting analytical results of entanglement.

### Two-spin case

For the two-spin case, the Hamiltonian (\[H01\]) reduces to $H_0={\bf s}_1\cdot{\bf S}_1$. 