{
  "id": "2207.13161",
  "version": "v1",
  "published": "2022-07-26T19:31:56.000Z",
  "updated": "2022-07-26T19:31:56.000Z",
  "title": "Projector formalism for kept and discarded spaces of matrix product states",
  "authors": [
    "Andreas Gleis",
    "Jheng-Wei Li",
    "Jan von Delft"
  ],
  "comment": "14 pages, 2 figures",
  "categories": [
    "quant-ph",
    "cond-mat.mes-hall",
    "cond-mat.str-el"
  ],
  "abstract": "Any matrix product state $|\\Psi\\rangle$ has a set of associated kept and discarded spaces, needed for the description of $|\\Psi\\rangle$, and changes thereof, respectively. These induce a partition of the full Hilbert space of the system into mutually orthogonal spaces of irreducible $n$-site variations of $|\\Psi\\rangle$. Here, we introduce a convenient projector formalism and diagrammatic notation to characterize these $n$-site spaces explicitly. This greatly facilitates the formulation of MPS algorithms that explicitly or implicitly employ discarded spaces. As an illustration, we derive an explicit expression for the $n$-site energy variance and evaluate it numerically for a model with long-range hopping. We also describe an efficient algorithm for computing low-lying $n$-site excitations above a finite MPS ground state.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-26T19:31:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "matrix product state",
      "discarded spaces",
      "finite mps ground state",
      "full hilbert space",
      "site energy variance"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}