We prove that the entanglement entropy of the ground state of a locally gapped frustration-free 2D lattice spin system satisfies an area law with respect to a vertical bipartition of the lattice into left and right regions. We first establish that the ground state projector of any locally gapped frustration-free 1D spin system can be approximated to within error є by a degree O(√nlog(є−1)) multivariate polynomial in the interaction terms of the Hamiltonian. This generalizes the optimal bound on the approximate degree of the boolean AND function, which corresponds to the special case of commuting Hamiltonian terms. For 2D spin systems we then construct an approximate ground state projector (AGSP) that employs the optimal 1D approximation in the vicinity of the boundary of the bipartition of interest. This AGSP has sufficiently low entanglement and error to establish the area law using a known technique.We prove that the entanglement entropy of the ground state of a locally gapped frustration-free 2D lattice spin system satisfies an area law with respect to a vertical bipartition of the lattice into left and right regions. We first establish that the ground state projector of any locally gapped frustration-free 1D spin system can be approximated to within error є by a degree O(√nlog(є−1)) multivariate polynomial in the interaction terms of the Hamiltonian. This generalizes the optimal bound on the approximate degree of the boolean AND function, which corresponds to the special case of commuting Hamiltonian terms. For 2D spin systems we then construct an approximate ground state projector (AGSP) that employs the optimal 1D approximation in the vicinity of the boundary of the bipartition of interest. This AGSP has sufficiently low entanglement and error to establish the area law using a known technique.

arXiv:2103.02492ACM Symposium on Theory of Computing (STOC)Jun 9, 202240 citations

Two-dimensional spin-1 frustrated Heisenberg model with valence-bond ground states

Zi Cai, Shu Chen, Su-Peng Kou +1 more

In this paper, we propose a method to understand the nature of the quantum disorder phase of the two-dimensional (2D) high-spin frustrated model. The ground state and excitation properties of a fully frustrated 2D spin-1 model are studied based on a model whose ground state can be found exactly. By analogy to the pseudopotential approach in the fractional quantum Hall effect, we conclude that the ground states of the fully frustrated spin-1 model are doubly degenerate valence-bond solid (VBS) states along the horizontal or vertical direction of the square lattice. We also find that the ground state could be characterized by a nonzero string order, which rarely happened in the 2D case. The method that we used reveals the connection between the fractional quantum Hall effect and the frustrated 2D antiferromagnetism system. The VBS states capture the main character of the disordered phase in the 2D spin-1 frustrated system and can be verified by a numerical method.In this paper, we propose a method to understand the nature of the quantum disorder phase of the two-dimensional (2D) high-spin frustrated model. The ground state and excitation properties of a fully frustrated 2D spin-1 model are studied based on a model whose ground state can be found exactly. By analogy to the pseudopotential approach in the fractional quantum Hall effect, we conclude that the ground states of the fully frustrated spin-1 model are doubly degenerate valence-bond solid (VBS) states along the horizontal or vertical direction of the square lattice. We also find that the ground state could be characterized by a nonzero string order, which rarely happened in the 2D case. The method that we used reveals the connection between the fractional quantum Hall effect and the frustrated 2D antiferromagnetism system. The VBS states capture the main character of the disordered phase in the 2D spin-1 frustrated system and can be verified by a numerical method.

arXiv:cond-mat/0611170Physical Review BAug 24, 20076 citations

Simulations of Quantum Spin Models on 2D Frustrated Lattices

Roger G. Melko

Bulletin of the American Physical SocietyMar 15, 20060 citations

Ground state many-body quantum entanglement of frustrated transverse field models on square lattice

Vikas Vijigiri, K. G. Paulson, Saptarshi Mandal +1 more

We study the ground state (GS) many-body quantum entanglement of two different transverse field models on a quasi-2D square lattice relevant to a Hydrogen-bonded crystal, i.e, squaric acid. We measure the genuine multipartite qubit-entanglement ( $C_{\text{GME}}(ψ)$ ) of the ground state of very generic models with all the possible cases of exchange couplings considered under defect free and one lattice site defect conditions. Our results show that creation, decay of multipartite entanglement occur for different combinations of coupling strength. When frustration is maximum the system exhibits a peak in concurrence after a gradual increase from disentangled state at zero field followed by an asymptotic decay at large fields. In contrast, for a marginally frustrated (degenerate) case though the concurrence shows a peak, the entanglement is non-zero and large at zero fields. Our results discuss the sensitivity of the qubit-entanglement with underlying GS of varying degree of degeneracy. We conclude that despite of their similarities in ground state properties, yet we see a difference in the degree of entanglement between the two models. We conjecture this result could be due to the difference in the amount of degeneracy and the quantum ground states of both Hamiltonians that could dictate the results even in the thermodynamic limit.We study the ground state (GS) many-body quantum entanglement of two different transverse field models on a quasi-2D square lattice relevant to a Hydrogen-bonded crystal, i.e, squaric acid. We measure the genuine multipartite qubit-entanglement ($C_{\text{GME}}(ψ)$) of the ground state of very generic models with all the possible cases of exchange couplings considered under defect free and one lattice site defect conditions. Our results show that creation, decay of multipartite entanglement occur for different combinations of coupling strength. When frustration is maximum the system exhibits a peak in concurrence after a gradual increase from disentangled state at zero field followed by an asymptotic decay at large fields. In contrast, for a marginally frustrated (degenerate) case though the concurrence shows a peak, the entanglement is non-zero and large at zero fields. Our results discuss the sensitivity of the qubit-entanglement with underlying GS of varying degree of degeneracy. We conclude that despite of their similarities in ground state properties, yet we see a difference in the degree of entanglement between the two models. We conjecture this result could be due to the difference in the amount of degeneracy and the quantum ground states of both Hamiltonians that could dictate the results even in the thermodynamic limit.

arXiv:2101.12443Jan 29, 20210 citations

Entanglement and matrix product states in one-dimensional quantum lattice systems

Guifré Vidal

Jan 1, 20060 citations

C_{\text{GME}}(ψ)

) of the ground state of very generic models with all the possible cases of exchange couplings considered under defect free and one lattice site defect conditions. Our results show that creation, decay of multipartite entanglement occur for different combinations of coupling strength. When frustration is maximum the system exhibits a peak in concurrence after a gradual increase from disentangled state at zero field followed by an asymptotic decay at large fields. In contrast, for a marginally frustrated (degenerate) case though the concurrence shows a peak, the entanglement is non-zero and large at zero fields. Our results discuss the sensitivity of the qubit-entanglement with underlying GS of varying degree of degeneracy. We conclude that despite of their similarities in ground state properties, yet we see a difference in the degree of entanglement between the two models. We conjecture this result could be due to the difference in the amount of degeneracy and the quantum ground states of both Hamiltonians that could dictate the results even in the thermodynamic limit.