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Rotating trapped fermions in two dimensions and the complex Ginibre ensemble: Exact results for the entanglement entropy and number variance

Bertrand Lacroix-A-Chez-Toine, Satya N. Majumdar, Grégory Schehr

2019
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Physical Review A
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We consider N non-interacting fermions in a 2d harmonic potential of trapping frequency ω and in a rotating frame at angular frequency Ω, with 0<ω - Ω≪ω. At zero temperature, the fermions are in the non-degenerate lowest Landau level and their positions are in one to one correspondence with the eigenvalues of an N× N complex Ginibre matrix. For large N, the fermion density is uniform over the disk of radius √(N) centered at the origin and vanishes outside this disk. We compute exactly, for any
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... inite N, the Rényi entanglement entropy of order q, S_q(N,r), as well as the cumulants of order p, 〈N_r^p〉_c, of the number of fermions N_r in a disk of radius r centered at the origin. For N ≫ 1, in the (extended) bulk, i.e., for 0 < r/√(N) < 1, we show that S_q(N,r) is proportional to the number variance Var (N_r), despite the non-Gaussian fluctuations of N_r. This relation breaks down at the edge of the fermion density, for r ≈√(N), where we show analytically that S_q(N,r) and Var (N_r) have a different r-dependence.

doi:10.1103/physreva.99.021602
fatcat:oz4m6yx3abbczff2t2ju5uucxi