Scaling of entanglement during the quantum phase transition for Ising spin systems on triangular and Sierpiński fractal lattices

Adopting concurrence as entanglement measure, we study entanglement and quantum phase transition of the Ising spin systems on the triangular lattice and Sierpiński fractal lattices by using the quantum renormalization-group method. It is found that the ground-state entanglement between two spins (or spin blocks) depends on the following factors: the size of system, the magnetic field, the exchange coupling, and the structure of lattice. As the size of the system becomes large, (a) the range of the magnetic field, in which the entanglement exists, contracts gradually and focuses on the critical point; and (b) the first derivative of entanglement shows singular behavior, and its maximum or minimum is approaching to the critical point gradually. The scaling behaviors of entanglement on the different lattices are similar but the scaling relations are diverse. For the triangular lattice, the space dimensionality determines the scaling relationship between the critical exponent of the entanglement and the critical exponent correlation length. However, for fractal lattices, it is the fractal dimensionality but not the space one to determine this relationship.

Figure 1

The procedure of quantum RG transformation for the triangular (a) and fractal lattices (b). The basic cluster of the lattices are shown as the $n$ structure.

Figure 2

The evolution of concurrence C vs the effective field ($g=h/J$) in terms of the RG iteration on the triangular lattice.

Figure 3

The evolution of the first derivative of the concurrence under different RG steps on the triangular lattice.

Figure 4

The scaling behavior of the minimum of the first derivative of concurrence for various system sizes on the triangular lattice. The RG steps show that the minimum diverges as $|\left(\frac{dC}{dg}\right){|}_{g_m}|\sim N^{\theta } (\theta =0.797)$.

Figure 5

The evolution of concurrence C vs the effective field ($g=h/J$) in terms of the RG iteration on the Sierpiński triangular lattice (a) and pyramid lattice (b).

Figure 6

The evolution of the first derivative of the concurrence under different RG steps on the Sierpiński triangular lattice (a) and pyramid lattice (b).

Figure 7

The scaling behavior of the minimum of the first derivative of concurrence for various system sizes on the Sierpiński triangular lattice (a) and pyramid lattice (b). The RG steps show that the minimum diverges as $|\left(\frac{dC}{dg}\right){|}_{g_m}|\sim N^{\theta }$.