{
  "id": "2308.16179",
  "version": "v1",
  "published": "2023-08-30T17:57:06.000Z",
  "updated": "2023-08-30T17:57:06.000Z",
  "title": "Out-of-time-order correlator, many-body quantum chaos, light-like generators, and singular values",
  "authors": [
    "Ke Huang",
    "Xiao Li",
    "David A. Huse",
    "Amos Chan"
  ],
  "comment": "6 + 15 pages, 3 + 11 figures. Comments are welcome",
  "categories": [
    "quant-ph",
    "cond-mat.stat-mech",
    "hep-th",
    "nlin.CD"
  ],
  "abstract": "We study out-of-time-order correlators (OTOCs) of local operators in spatial-temporal invariant or random quantum circuits using light-like generators (LLG) -- many-body operators that exist in and act along the light-like directions. We demonstrate that the OTOC can be approximated by the leading singular value of the LLG, which, for the case of generic many-body chaotic circuits, is increasingly accurate as the size of the LLG, $w$, increases. We analytically show that the OTOC has a decay with a universal form in the light-like direction near the causal light cone, as dictated by the sub-leading eigenvalues of LLG, $z_2$, and their degeneracies. Further, we analytically derive and numerically verify that the sub-leading eigenvalues of LLG of any size can be accessibly extracted from those of LLG of the smallest size, i.e., $z_2(w)= z_2(w=1)$. Using symmetries and recursive structures of LLG, we propose two conjectures on the universal aspects of generic many-body quantum chaotic circuits, one on the algebraic degeneracy of eigenvalues of LLG, and another on the geometric degeneracy of the sub-leading eigenvalues of LLG. As corollaries of the conjectures, we analytically derive the asymptotic form of the leading singular state, which in turn allows us to postulate and efficiently compute a product-state variational ansatz away from the asymptotic limit. We numerically test the claims with four generic circuit models of many-body quantum chaos, and contrast these statements against the cases of a dual unitary system and an integrable system.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-30T17:57:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "many-body quantum chaos",
      "out-of-time-order correlator",
      "singular value",
      "light-like generators",
      "sub-leading eigenvalues"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}