{
  "id": "1403.1121",
  "version": "v3",
  "published": "2014-03-05T13:37:11.000Z",
  "updated": "2015-01-07T10:04:29.000Z",
  "title": "Spectra and eigenstates of spin chain Hamiltonians",
  "authors": [
    "J. P. Keating",
    "N. Linden",
    "H. J. Wells"
  ],
  "comment": "Updated figures, as accepted in 'Communications in Mathematical Physics' on 5 January 2015",
  "categories": [
    "math-ph",
    "math.MP",
    "quant-ph"
  ],
  "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 $n\\rightarrow\\infty$ 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. 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 which suggests that the translational invariance condition may be dropped in the case of nearest-neighbour chains.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-07-14T12:48:18.000Z",
      "abstract": "We prove that translationally invariant Hamiltonians of $n$ qubits with nearest-neighbour interactions have two seemingly contradictory features. Firstly (in the limit $n\\rightarrow\\infty$) 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 we will show that the density of states of such systems is Gaussian; thus as far as the eigenvalues are concerned the system is like a non-interacting one. The results apply to chains of qubits with translation invariant nearest-neighbour interactions, but we will show that many of the results are extendible to much more general systems (both in terms of the local dimension, the geometry of interaction and, in the case of the density of states, the requirement of translation invariance).",
      "comment": "21 pages, 2 figures, typos corrected",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-01-07T10:04:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "spin chain hamiltonians",
      "eigenstates",
      "translation invariant nearest-neighbour interactions",
      "translationally invariant hamiltonians",
      "seemingly contradictory features"
    ],
    "publication": {
      "doi": "10.1007/s00220-015-2366-0",
      "journal": "Communications in Mathematical Physics",
      "year": 2015,
      "month": "Aug",
      "volume": 338,
      "number": 1,
      "pages": 81
    },
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015CMaPh.338...81K"
    }
  }
}