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Journal of Applied Mathematics and Physics

Ground-State Energy and Entropy for One-Dimensional Heisenberg Chain with Alternating D-Term

Chunhuan Xiang, Honglei Wang

Published Jan 1, 2019• 1 citation
Publisher

Abstract

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.

DOI
10.4236/jamp.2019.75082

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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.
Abstract The quantum Gaussian phase transition is investigated for the infinite spin-1 bond-alternative Heisenberg model in one spatial dimension. By using a tensor network representation with an infinite matrix product state approach, the ground state energy, bipartite entanglement entropy, non-local string order, and fidelity per lattice site are calculated to characterize the phase transition. At the quantum phase transition point, the scaling behavior of various physical observables with respect to the finite truncation dimension are discussed for the ground state wavefunctions. In addition, the central charge is extracted from the finite entanglement entropies and the finite correlation lengths. Furthermore, the various critical exponents of the string order are calculated. The characteristic critical exponents and the central charge determine the universality class of the phase transition.
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Quantum entanglement and non-local string order are investigated in the one-dimensional spin-1 bond-alternating Heisenberg model for quantum phase transitions. Employing the infinite matrix product state representation, ground state wavefunctions are numerically obtained by using the infinite time evolving block decimation approach in the infinite lattice system. From a bipartite entanglement measure, i.e., von Neumann entropy, a quantum phase transition point corresponding to the singular point of the von Neumann entropy between the Haldane phase and the dimerised phase is found to be δc=0.26. In addition, based on the infinite matrix product state algorithm, the non-local string order is calculated. The result shows that the non-local string order is able to characterize the topological phase transitions.
By the infinite time-evolving block decimation method, the ground state properties of the S=1 Heisenberg–Ising alternating chain have been investigated in the thermodynamic limit. A ground-state phase diagram, which includes three different phases, i.e., an antiferromagnetic strip phase, a dimer phase, and an antiferromagnetic phase, is obtained. A modified definition of the string order is adopted to describe the dimer phase, and it is found to have nonzero antiferromagnetic topological string order. The quantum phase transitions (QPTs) in such a model can be well described by the singularities of bipartite entanglement and fidelity bifurcations, and two second-order QPTs points are determined to be at JI≃±0.66983. The central charges at both QPTs points are determined to be ∼1/2. Therefore, both QPTs belong to the Ising universality class with central charges c=c‾=1/2.
The quantum renormalization group method is applied to study the quantum criticality and entanglement entropy of the ground state of the Ising chain in the presence of antisymmetric anisotropic couplings and alternating exchange interactions. The quantum phase transitions can be characterized by the discontinuity in the second derivative of the energy of the renormalized ground state. A phase diagram is obtained by the critical boundary line. The first derivative of entanglement entropy also diverges at the same critical points after enough iterations of the renormalization of the coupling constants. The antisymmetric anisotropy and alternating interaction can enhance the renormalized entanglement via the creation of quantum fluctuations. The scaling behavior of the derivative of the entropy around the critical points manifests the logarithm dependence on the size of the spin system.
We study the fidelity susceptibility in an antiferromagnetic spin-1 XXZ chain numerically. By using the density-matrix renormalization group method, the effects of the alternating single-site anisotropy D on fidelity susceptibility are investigated. Its relation with the quantum phase transition is analyzed. It is found that the quantum phase transition from the Haldane spin liquid to periodic Néel spin solid can be well characterized by the fidelity. Finite size scaling of fidelity susceptibility shows a power-law divergence at criticality, which indicates the quantum phase transition is of second order. The results are confirmed by the second derivative of the ground-state energy. We also study the relationship between the entanglement entropy, the Schmidt gap and quantum phase transitions. Conclusions drawn from these quantum information observables agree well with each other.