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Valence bond states on quantum spin chains as ground states with spectral gap

M. Fannes, Bruno Nachtergaele, Reinhard F. Werner

The authors show that every translation invariant valence bond state on a one-dimensional quantum spin chain arises as the unique ground state of a certain family of finite-range interactions. For each interaction in this family they show the existence of a nonzero spectral gap above the ground state energy. A special example of this structure is a state recently studied by Affleck et al. (1987). For the Hamiltonian studied by these authors they estimate the gap, and prove that it lies between 1/3 and 10/27.The authors show that every translation invariant valence bond state on a one-dimensional quantum spin chain arises as the unique ground state of a certain family of finite-range interactions. For each interaction in this family they show the existence of a nonzero spectral gap above the ground state energy. A special example of this structure is a state recently studied by Affleck et al. (1987). For the Hamiltonian studied by these authors they estimate the gap, and prove that it lies between 1/3 and 10/27.

Journal of Physics A Mathematical and GeneralFeb 21, 199157 citations

Multipartite entanglement in quantum spin chains (4 pages)

Dagmar Bruß, Nilanjana Datta, Chiara Macchiavello +2 more

Physical Review AJan 1, 20053 citations

Multipartite entanglement in quantum phase transitions

Marco Gabbrielli

This dissertation collects results about witnessing multipartite entanglement in critical spin systems and free-fermion models both at zero and finite temperature.

arXiv:1810.10537Oct 24, 20181 citation

The Spectral Gap for Some Spin Chains With Discrete Symmetry Breaking

Bruno Nachtergaele

We prove that for any finite set of generalized valence bond solid (GVBS) states of a quantum spin chain there exists a translation invariant finite-range Hamiltonian for which this set is the set of ground states. This result implies that there are GVBS models with arbitrary broken discrete symmetries that are described as combinations of lattice translations, lattice reflections, and local unitary or anti-unitary transformations. We also show that all GVBS models that satisfy some natural conditions have a spectral gap. The existence of a spectral gap is obtained by applying a simple and quite general strategy for proving lower bounds on the spectral gap of the generator of a classical or quantum spin dynamics. This general scheme is interesting in its own right and therefore, although the basic idea is not new, we present it in a system-independent setting. The results are illustrated with an number of examples. Keywords: quantum spin chains, Heisenberg model, spectral gap, symmetr...We prove that for any finite set of generalized valence bond solid (GVBS) states of a quantum spin chain there exists a translation invariant finite-range Hamiltonian for which this set is the set of ground states. This result implies that there are GVBS models with arbitrary broken discrete symmetries that are described as combinations of lattice translations, lattice reflections, and local unitary or anti-unitary transformations. We also show that all GVBS models that satisfy some natural conditions have a spectral gap. The existence of a spectral gap is obtained by applying a simple and quite general strategy for proving lower bounds on the spectral gap of the generator of a classical or quantum spin dynamics. This general scheme is interesting in its own right and therefore, although the basic idea is not new, we present it in a system-independent setting. The results are illustrated with an number of examples. Keywords: quantum spin chains, Heisenberg model, spectral gap, symmetr...

Jan 1, 199480 citations

Entanglement Hamiltonians for quantum spin chains

Ronny Thomale, Stephan Rachel, Daniel P. Arovas +1 more

Bulletin of the American Physical SocietyMar 24, 20110 citations

Entanglement in Finitely Correlated Spin States

Spyridon Michalakis, Bruno Nachtergaele

We consider entanglement properties of pure finitely correlated states (FCS). We derive bounds for the entanglement of a spin with an interval of spins in an arbitrary pure FCS. Finitely correlated states are also known as matrix product states or generalized valence-bond states. The bounds become exact in the case where one considers the entanglement of a single spin with a half-infinite chain to the right of it. Our bounds provide a proof of the recent conjecture by Benatti, Hiesmayr, and Narnhofer that their necessary condition for nonvanishing entanglement in terms of a single spin and the memory of the FCS is also sufficient. We also generalize the study of entanglement in the Affleck-Kennedy-Lieb-Tasaki model by Fan, Korepin, and Roychowdhury. Our result permits a more efficient calculation, numerically and in some cases analytically, of the entanglement of arbitrary finitely correlated quantum spin chains.We consider entanglement properties of pure finitely correlated states (FCS). We derive bounds for the entanglement of a spin with an interval of spins in an arbitrary pure FCS. Finitely correlated states are also known as matrix product states or generalized valence-bond states. The bounds become exact in the case where one considers the entanglement of a single spin with a half-infinite chain to the right of it. Our bounds provide a proof of the recent conjecture by Benatti, Hiesmayr, and Narnhofer that their necessary condition for nonvanishing entanglement in terms of a single spin and the memory of the FCS is also sufficient. We also generalize the study of entanglement in the Affleck-Kennedy-Lieb-Tasaki model by Fan, Korepin, and Roychowdhury. Our result permits a more efficient calculation, numerically and in some cases analytically, of the entanglement of arbitrary finitely correlated quantum spin chains.

arXiv:math-ph/0606018Physical Review LettersOct 4, 200611 citations

Entangled finitely correlated quantum spin chains

Abstract

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Entangled finitely correlated quantum spin chains

Abstract

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