Spectra and eigenstates of spin chain Hamiltonians

Understand

We prove that translationally invariant Hamiltonians of a chain of $n$ qubits with nearest-neighbour interactions have two seemingly contradictory features.

• Firstly in the limit $n\rightarrow\infty$ we show that any translationally invariant Hamiltonian of a chain of $n$ qubits has an eigenbasis such that almost all eigenstates have maximal entanglement between fixed-size sub-blocks of qubits and the rest of the system; in this sense these eigenstates are like those of completely general Hamiltonians (i.e.
• Hamiltonians with interactions of all orders between arbitrary groups of qubits).
• Secondly in the limit $n\rightarrow\infty$ we show that any nearest-neighbour Hamiltonian of a chain of $n$ qubits has a Gaussian density of states; thus as far as the eigenvalues are concerned the system is like a non-interacting one.