Spectra and Eigenstates of Spin Chain Hamiltonians

J. P. Keating, N. Linden, H. J. Wells*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

34 Citations (Scopus)

Abstract

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 $${n \rightarrow \infty}$$n→∞ 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 $${n \rightarrow \infty}$$n→∞ 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. The comparison applies to chains of qubits with translationally invariant nearest-neighbour interactions, 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 that suggests that the translational invariance condition may be dropped in the case of nearest-neighbour chains.

Original languageEnglish
Pages (from-to)81-102
Number of pages22
JournalCommunications in Mathematical Physics
Volume338
Issue number1
Early online date1 May 2015
DOIs
Publication statusPublished - 1 Aug 2015

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