{
  "id": "2112.13355",
  "version": "v2",
  "published": "2021-12-26T10:57:23.000Z",
  "updated": "2022-04-24T12:27:43.000Z",
  "title": "Counting statistics for non-interacting fermions in a rotating trap",
  "authors": [
    "Naftali R. Smith",
    "Pierre Le Doussal",
    "Satya N. Majumdar",
    "Gregory Schehr"
  ],
  "comment": "21 pages, 7 figures",
  "journal": "Phys. Rev. A 105, 043315 (2022)",
  "doi": "10.1103/PhysRevA.105.043315",
  "categories": [
    "cond-mat.stat-mech",
    "cond-mat.quant-gas",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study the ground state of $N \\gg 1$ noninteracting fermions in a two-dimensional harmonic trap rotating at angular frequency $\\Omega>0$. The support of the density of the Fermi gas is a disk of radius $R_e$. We calculate the variance of the number of fermions ${\\cal N}_R$ inside a disk of radius $R$ centered at the origin for $R$ in the bulk of the Fermi gas. We find rich and interesting behaviours in two different scaling regimes: (i) $\\Omega / \\omega <1 $ and (ii) $1 - \\Omega / \\omega = O(1/N)$, where $\\omega$ is the angular frequency of the oscillator. In the first regime (i) we find that ${\\rm Var}\\,{\\cal N}_{R}\\simeq\\left(A\\log N+B\\right)\\sqrt{N}$ and we calculate $A$ and $B$ as functions of $R/R_e$, $\\Omega$ and $\\omega$. We also predict the higher cumulants of ${\\cal N}_{R}$ and the bipartite entanglement entropy of the disk with the rest of the system. In the second regime (ii), the mean fermion density exhibits a staircase form, with discrete plateaus corresponding to filling $k$ successive Landau levels, as found in previous studies. Here, we show that ${\\rm Var}\\,{\\cal N}_{R}$ is a discontinuous piecewise linear function of $\\sim (R/R_e) \\sqrt{N}$ within each plateau, with coefficients that we calculate exactly, and with steps whose precise shape we obtain for any $k$. We argue that a similar piecewise linear behavior extends to all the cumulants of ${\\cal N}_{R}$ and to the entanglement entropy. We show that these results match smoothly at large $k$ with the above results for $\\Omega/\\omega=O(1)$. These findings are nicely confirmed by numerical simulations. Finally, we uncover a universal behavior of ${\\rm Var}\\,{\\cal N}_{R}$ near the fermionic edge. We extend our results to a three-dimensional geometry, where an additional confining potential is applied in the $z$ direction.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-04-24T12:27:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "non-interacting fermions",
      "rotating trap",
      "counting statistics",
      "angular frequency",
      "fermi gas"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "APS",
      "journal": "Phys. Rev. A"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}