Entanglement in spin-1/2 dimerized Heisenberg systems

Anzi Hu, You‐Quan Li, Sun, Zhe, Xiaoguang Wang

Published Jan 1, 2023• 9 citations

Abstract

We study entanglement in dimerized Heisenberg systems. In particular, we give exact results of ground-state pairwise entanglement for the four-qubit model by identifying a Z2 symmetry. Although the entanglements cannot identify the critical point of the system, the mean entanglement of nearest-neighbor qubits really does, namely, it reaches a maximum at the critical point.

DOI: 10.57912/23888739.v1

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Entanglement in spin-1/2 dimerized Heisenberg systems

Anzi Hu, You‐Quan Li, Sun, Zhe, Xiaoguang Wang

Published Jan 1, 2023• 9 citations

Publisher

Abstract

DOI: 10.57912/23888739.v1

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arXiv:quant-ph/0501117Communications in Theoretical PhysicsJun 1, 200511 citations

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Finding an exact solution for a realistic interacting quantum many-body problem is often challenging. There are only a few problems where an exact solution can be found, usually in a narrow parameter space. Here, we propose a spin- $\frac{1}{2}$ Heisenberg model on a square lattice with spatial anisotropy and bond depletion for the nearest-neighbor antiferromagnetic interactions but not for the next-nearest-neighbor interactions. This model has an exact and unique dimer ground state at ${J}_{2}/{J}_{1}=\frac{1}{2}$ ; a dimer state is a product state of spin singlets on dimers (here, staggered nearest-neighbor bonds). We examine this model by employing the bond-operator mean-field theory and exact diagonalization. These analytical and numerical methods precisely affirm the correctness of the dimer ground state at the exact point $({J}_{2}/{J}_{1}=\frac{1}{2})$ . As one moves away from the exact point, the dimer order melts and vanishes when the spin gap becomes zero. The mean-field theory with harmonic approximation indicates that the dimer order persists for $\ensuremath{-}0.35\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}1.35$. However, in nonharmonic approximation, the upper critical point lowers by 0.28 to 1.07, but the lower critical point remains intact. The exact diagonalization results suggest that the latter approximation fares better. The model reveals N\'eel order below the lower critical point and stripe magnetic order above the upper critical point. It has a topologically equivalent model on a honeycomb lattice where the nearest-neighbor interactions are still spatial anisotropic, but the bond depletion shifts into the isotropic next-neighbor interactions. Moreover, these models can also be generalized in the three dimensions.Finding an exact solution for a realistic interacting quantum many-body problem is often challenging. There are only a few problems where an exact solution can be found, usually in a narrow parameter space. Here, we propose a spin-$\frac{1}{2}$ Heisenberg model on a square lattice with spatial anisotropy and bond depletion for the nearest-neighbor antiferromagnetic interactions but not for the next-nearest-neighbor interactions. This model has an exact and unique dimer ground state at ${J}_{2}/{J}_{1}=\frac{1}{2}$; a dimer state is a product state of spin singlets on dimers (here, staggered nearest-neighbor bonds). We examine this model by employing the bond-operator mean-field theory and exact diagonalization. These analytical and numerical methods precisely affirm the correctness of the dimer ground state at the exact point $({J}_{2}/{J}_{1}=\frac{1}{2})$. As one moves away from the exact point, the dimer order melts and vanishes when the spin gap becomes zero. The mean-field theory with harmonic approximation indicates that the dimer order persists for $\ensuremath{-}0.35\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}1.35$. However, in nonharmonic approximation, the upper critical point lowers by 0.28 to 1.07, but the lower critical point remains intact. The exact diagonalization results suggest that the latter approximation fares better. The model reveals N\'eel order below the lower critical point and stripe magnetic order above the upper critical point. It has a topologically equivalent model on a honeycomb lattice where the nearest-neighbor interactions are still spatial anisotropic, but the bond depletion shifts into the isotropic next-neighbor interactions. Moreover, these models can also be generalized in the three dimensions.

arXiv:2401.08157Physical review. B./Physical review. BSep 3, 20243 citations

We study the entanglement of thermal and ground states in the Heisenberg

\mathrm{XX}

qubit rings with a magnetic field. A general result is found that for even-number rings, pairwise entanglement between nearest-neighbor qubits is independent of both the sign of exchange interaction constants and the sign of magnetic fields. As an example we study the entanglement in the four-qubit model and find that the ground state of this model without magnetic fields is shown to be a four-body maximally entangled state measured by the square N-bit concurrence.

\frac{1}{2}

Heisenberg model on a square lattice with spatial anisotropy and bond depletion for the nearest-neighbor antiferromagnetic interactions but not for the next-nearest-neighbor interactions. This model has an exact and unique dimer ground state at

{J}_{2}/{J}_{1}=\frac{1}{2}

; a dimer state is a product state of spin singlets on dimers (here, staggered nearest-neighbor bonds). We examine this model by employing the bond-operator mean-field theory and exact diagonalization. These analytical and numerical methods precisely affirm the correctness of the dimer ground state at the exact point

({J}_{2}/{J}_{1}=\frac{1}{2})

. As one moves away from the exact point, the dimer order melts and vanishes when the spin gap becomes zero. The mean-field theory with harmonic approximation indicates that the dimer order persists for $\ensuremath{-}0.35\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}1.35$. However, in nonharmonic approximation, the upper critical point lowers by 0.28 to 1.07, but the lower critical point remains intact. The exact diagonalization results suggest that the latter approximation fares better. The model reveals N\'eel order below the lower critical point and stripe magnetic order above the upper critical point. It has a topologically equivalent model on a honeycomb lattice where the nearest-neighbor interactions are still spatial anisotropic, but the bond depletion shifts into the isotropic next-neighbor interactions. Moreover, these models can also be generalized in the three dimensions.