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Journal of Physics Condensed Matter

Quantum entanglement and criticality of the antiferromagnetic Heisenberg model in an external field

Guanghua Liu, Ruoyan Li, Guang-Shan Tian

Published May 25, 2012• 4 citations
Publisher

Abstract

By Lanczos exact diagonalization and the infinite time-evolving block decimation (iTEBD) technique, the two-site entanglement as well as the bipartite entanglement, the ground state energy, the nearest-neighbor correlations, and the magnetization in the antiferromagnetic Heisenberg (AFH) model under an external field are investigated. With increasing external field, the small size system shows some distinct upward magnetization stairsteps, accompanied synchronously with some downward two-site entanglement stairsteps. In the thermodynamic limit, the two-site entanglement, as well as the bipartite entanglement, the ground state energy, the nearest-neighbor correlations, and the magnetization are calculated, and the critical magnetic field h(c) = 2.0 is determined exactly. Our numerical results show that the quantum entanglement is sensitive to the subtle changing of the ground state, and can be used to describe the magnetization and quantum phase transition. Based on the discontinuous behavior of the first-order derivative of the entanglement entropy and fidelity per site, we think that the quantum phase transition in this model should belong to the second-order category. Furthermore, in the magnon existence region (h < 2.0), a logarithmically divergent behavior of block entanglement which can be described by a free bosonic field theory is observed, and the central charge c is determined to be 1.

DOI
10.1088/0953-8984/24/25/256002

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By Lanczos exact diagonalization and the infinite time-evolving block decimation (iTEBD) technique, the two-site entanglement as well as the bipartite entanglement, the ground state energy, the nearest-neighbor correlations, and the magnetization in the antiferromagnetic Heisenberg (AFH) model under an external field are investigated. With increasing external field, the small size system shows some distinct upward magnetization stairsteps, accompanied synchronously with some downward two-site entanglement stairsteps. In the thermodynamic limit, the two-site entanglement, as well as the bipartite entanglement, the ground state energy, the nearest-neighbor correlations, and the magnetization are calculated, and the critical magnetic field h(c) = 2.0 is determined exactly. Our numerical results show that the quantum entanglement is sensitive to the subtle changing of the ground state, and can be used to describe the magnetization and quantum phase transition. Based on the discontinuous behavior of the first-order derivative of the entanglement entropy and fidelity per site, we think that the quantum phase transition in this model should belong to the second-order category. Furthermore, in the magnon existence region (h < 2.0), a logarithmically divergent behavior of block entanglement which can be described by a free bosonic field theory is observed, and the central charge c is determined to be 1.
The matrix product state (MPS) is utilized to investigate the ground state properties and quantum phase transitions (QPTs) of the dimerized antiferromagnetic Heisenberg (DAH) model. The ground state MPS wavefunctions determined by the infinite time-evolving block decimation (iTEBD) algorithm are shown to be very efficient descriptions of DAH model. In the thermodynamic limit, the quantum entanglement, the bond energy, and the nearest-neighbor correlations are calculated. It is revealed that the singular behavior of the bipartite entanglement can detect the QPTs directly. The critical point J c 2 = 1.0 is determined evidently, and the quantum phase transition is argued to belong to the second-order category. At the critical point, logarithmic divergent character of the block entanglement is observed, and the system can be described by a free bosonic field theory.
We consider a two-chain, spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg spin ladder in an external magnetic field HHH. The spin ladder is known to undergo second-order quantum phase transitions (QPTs) at two critical values Hc1{H}_{c1}Hc1​ and Hc2{H}_{c2}Hc2​ of the magnetic field. There are now known examples of strongly coupled (where the rung exchange interaction is much stronger than the nearest-neighbor intrachain exchange interaction) organic ladder compounds in which QPTs have been experimentally observed. In this paper, we investigate whether well-known bipartite entanglement meaures such as one-site von Neumann entropy, two-site von Neumann entropy, and concurrence develop special features close to the quantum critical points. As suggested by an earlier theorem, the first derivatives of the measures with respect to magnetic field are expected to diverge as $H\ensuremath{\rightarrow}{H}_{c1}$ and $H\ensuremath{\rightarrow}{H}_{c2}$. Based on numerical diagonalization data and a mapping of the strongly coupled ladder Hamiltonian onto the XXZXXZXXZ chain Hamiltonian, for which several analytical results are known, we find that the derivatives of the entanglement measures diverge as $H\ensuremath{\rightarrow}{H}_{c2}$ but remain finite as $H\ensuremath{\rightarrow}{H}_{c1}$. The reason for this discrepancy is analyzed. We further calculate two recently proposed quantum information theoretic measures, the reduced fidelity and reduced fidelity susceptibility, and show that these measures provide appropriate signatures of the QPTs occuring at the critical points H=Hc1H={H}_{c1}H=Hc1​ and H=Hc2H={H}_{c2}H=Hc2​.
Bipartite entanglement, entanglement spectrum, and Schmidt gap in S=1 bond-alternative antiferromagnetic Heisenberg chain are investigated by the infinite time-evolving block decimation (iTEBD) method. The quantum phase transition (QPT) from the singlet-dimer phase to the Haldane phase can be detected by the singular behavior of bipartite entanglement, the sudden change of the entanglement spectrum, and the completely vanishing of the Schmidt gap. The critical point is determined to be around r c ≃ 0.587, and the second-order character of the QPT is verified. Doubly degenerate entanglement spectra of both even and odd bonds are observed in the Haldane phase, by which one can distinguish the Haldane phase from the singlet-dimer phase easily. Nearest-neighbor antiferromagnetic correlations and next-nearest-neighbor ferromagnetic correlations are found in the whole parameter region. At the critical massless point, although exponentially decaying antiferromagnetic correlation is observed, it approaches to a constant value finally. Therefore, long-range correlations exist and the correlation length becomes divergent at the critical point.
We investigate the ground-state phase diagram for a spin-one quantum Heisenberg antiferromagnetic chain with exchange and single-ion anisotropies in an external magnetic field by using the infinite time-evolving block decimation algorithm to compute the ground-state fidelity per lattice site. We detect all phase boundaries solely by computing the ground-state fidelity per lattice site, with the prescription that a phase transition point is attributed to a pinch point on the ground-state fidelity surface. Furthermore, the results indicate that a magnetization plateau corresponds to a fidelity plateau on the ground-state fidelity surface, thus offering an alternative route for investigating the magnetization processes of quantum many-body spin systems. We characterize all phases by using the local-order parameter, the spin correlation, the momentum distribution of the spin correlation structure factor, and mutual information as a function of the lattice distance. The commensurate and incommensurate phases are distinguished by the mutual information. In addition, the central charges at criticalities are identified by performing a finite-entanglement scaling analysis. The results show that all phase transitions between spin liquids and magnetization plateaus belong to the Pokrovsky-Talapov universality class.
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.
Abstract By using the infinite time-evolving block decimation, we study quantum fidelity and entanglement entropy in the spin-1/2 Heisenberg alternating chain under an external magnetic field. The effects of the magnetic field on the fidelity are investigated, and its relation with the quantum phase transition (QPT) is analyzed. The phase diagram of the model is given accordingly, which supports the Haldane phase, the singlet-dimer phase, the Luttinger liquid phase and the paramagnetic phase. The scaling of entanglement entropy in the gapless Luttinger liquid phase is studied, and the central charge c = 1 is obtained. We also study the relationship between the quantum coherence, string order parameter and QPTs. Results obtained from these quantum information observations are consistent with the previous reports.