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We prove that translationally invariant Hamiltonians of a chain of $n$ qubitswith nearest-neighbour interactions have two seemingly contradictory features.Firstly in the limit $n\rightarrow\infty$ we show that any translationallyinvariant Hamiltonian of a chain of $n$ qubits has an eigenbasis such thatalmost all eigenstates have maximal entanglement between fixed-size sub-blocksof qubits and the rest of the system; in this sense these eigenstates are likethose of completely general Hamiltonians (i.e. Hamiltonians with interactionsof all orders between arbitrary groups of qubits). Secondly in the limit$n\rightarrow\infty$ we show that any nearest-neighbour Hamiltonian of a chainof $n$ qubits has a Gaussian density of states; thus as far as the eigenvaluesare concerned the system is like a non-interacting one. The comparison appliesto chains of qubits with translationally invariant nearest-neighbourinteractions, but we show that it is extendible to much more general systems(both in terms of the local dimension and the geometry of interaction).Numerical evidence is also presented which suggests that the translationalinvariance condition may be dropped in the case of nearest-neighbour chains.

Spectra and eigenstates of spin chain Hamiltonians