The paper by M. Fannes, B. Nachtergaele, and R. F. Werner explores a class of translation-invariant states on quantum spin chains, characterized by finite-dimensional correlations across bonds. These states are generalized valence bond states and are dense in the set of all translation-invariant states. The authors develop a comprehensive theory of ergodic decomposition, including the decomposition into periodic "Néel ordered" states, with exponential decay of correlations. They show that all these states can be obtained as "local functions" of "purely generated states," which are ground states for finite-range VBS interactions. The theory does not require symmetry with respect to a local gauge group. The paper also discusses a one-parameter family of examples that are not isotropic except for a special case, which coincides with the one-dimensional antiferromagnet studied by Affleck, Kennedy, Lieb, and Tasaki. The introduction highlights the complexity of determining ground state properties in quantum spin systems compared to classical statistical mechanics, and the challenges in establishing properties like uniqueness, degeneracy, and spectral gaps. The authors aim to provide explicit answers to these questions for a specific class of states and Hamiltonians.The paper by M. Fannes, B. Nachtergaele, and R. F. Werner explores a class of translation-invariant states on quantum spin chains, characterized by finite-dimensional correlations across bonds. These states are generalized valence bond states and are dense in the set of all translation-invariant states. The authors develop a comprehensive theory of ergodic decomposition, including the decomposition into periodic "Néel ordered" states, with exponential decay of correlations. They show that all these states can be obtained as "local functions" of "purely generated states," which are ground states for finite-range VBS interactions. The theory does not require symmetry with respect to a local gauge group. The paper also discusses a one-parameter family of examples that are not isotropic except for a special case, which coincides with the one-dimensional antiferromagnet studied by Affleck, Kennedy, Lieb, and Tasaki. The introduction highlights the complexity of determining ground state properties in quantum spin systems compared to classical statistical mechanics, and the challenges in establishing properties like uniqueness, degeneracy, and spectral gaps. The authors aim to provide explicit answers to these questions for a specific class of states and Hamiltonians.