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We study the ground-state information of one-dimensional Heisenberg chain with alternating D-term. Given the ground-state phase diagram, the ground-state energy and the entanglement entropy are obtained by tensor-net work algorithm. The phase transition points are shown in the entanglement entropy figure. The results are agreed with the phase diagram.

Recent improvements in experimental techniques [

This paper is organized as follows: in the second section, the model Hamiltonian and the ground-state phase diagram are presented. The matrix product state algorithm is simply introduced. The figure for ground-state energy and entanglement entropy for left and right section are shown. The final section is devoted to a summary and discussion.

The anti-ferromagnetic Heisenberg chain with alternating D-term for spin-1 [

H = ∑ l = 1 N J S l S l + 1 + D + ∑ l = 1 N / 2 S 2 l − 1 z 2 + D − ∑ l = 1 N / 2 S 2 l z 2 , ( J > 0 ) (1)

where J is the exchange coupling, D + = D 0 + δ D , D − = D 0 + δ D , and S is the spin-1 operator.

S x = 1 2 ( 0 1 0 1 0 1 0 1 0 ) , S y = 1 2 ( 0 i 0 i 0 − i 0 − i 0 ) , S z = ( 1 0 0 0 0 0 0 0 − 1 )

The parameters D_{0} and dD represent uniform and alternating components of uniaxial single-ion anisotropy, respectively. In what follows we set J = 1 to fix the energy scale. The ground-state phase diagram is given in Ref. [_{0} = 2, and dD as the control parameter.

The conformal central charge is an important content in field theory, which gives the type of the phase transition in theory. With conformal central charge c = 1, the transition line between the Larged-D and Haldane is expected to be described by the conformal field theory. The phases u0d0 and udud are expected as gapless system.

The matrix product state algorithm is given in Ref [

| Ψ t 〉 = exp ( − i H t ) | Ψ 0 〉 (2)

the Schmidt decomposition of | Ψ t 〉 is written as

| Ψ 〉 = ∑ α = 1 χ λ α [ r ] | Φ α [ ⊲ r ] 〉 ⊗ | Φ α [ r + 1 ⊳ ] 〉 (3)

The Equation (3) can be rewritten as

| Ψ 〉 = ∑ α , β = 1 χ ∑ i = 1 d λ α [ r ] Γ i α β [ r + 1 ] λ β [ r + 1 ] | Φ α [ ⊲ r ] 〉 | i [ r ] 〉 | Φ α [ r + 1 ⊳ ] 〉 (4)

where χ is the truncation dimension, d is the Hilbert space, λ is diagonal matrix, the G is three index tensor. By using the two-site Hamiltonian, which is expanded through a Suzuki-Trotter decomposition, the imaginary time evolution for the given wave-function is shown. The approximation ground-state wave-function is obtained until the approximation ground-state energy is lower enough.

The simulation results of the approximation ground-state energy for Equation (1) are shown in

The approximation ground-state energies with different control parameter dD agree with each other.

During the numerical simulation, two phase transitions are obtained, which are shown in entanglement entropy. The figure for entanglement entropy of the left one and the right one are given in

The anti-ferromagnetic Heisenberg chain with alternating D-term for spin-1 is investigated by using matrix product states. The approximation ground-state

energy and the entanglement entropy are shown in this paper. We simply analyzed the results, however, as we have been unable to determine the local order parameter. This is the next direction of research. Besides, we will use alternative techniques beyond the MPS paradigm to yield the scaling behavior of physical observable, which may be more suitable.

The authors declare no conflicts of interest regarding the publication of this paper.

Xiang, C.H. and Wang, H.L. (2019) Ground-State Energy and Entropy for One-Dimensional Heisenberg Chain with Alternating D-Term. Journal of Applied Mathematics and Physics, 7, 1220-1225. https://doi.org/10.4236/jamp.2019.75082