Universality for free fermions and the local Weyl law for semiclassical Schrödinger operators

Deleporte, Alix; Lambert, Gaultier

doi:10.48550/arxiv.2109.02121

Cited by 5 publications

(8 citation statements)

References 79 publications

(124 reference statements)

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“…1.1]. For confining potentials, this asymptotic fails close to the transition region {V = E} and a pointwise Weyl law was proved for V (x) = |x| 2 in [19] and more recently for general potentials in [7].…”

Section: Introductionmentioning

confidence: 94%

Fermionic semiclassical Lp estimates

Nguyen¹

2022

Preprint

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We generalize the semiclassical L p estimates of Koch, Tataru and Zworski in the setting of Schrödinger operators with confining potentials to density matrices. This is motivated by the problem of the concentration of free fermionic particles in a trapping potential. Our proof relies on semiclassical and many-body tools. As an application, we provide bounds on spectral clusters. We also discuss the optimality of the one-body and many-body bounds

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Section: Introductionmentioning

confidence: 94%

Fermionic semiclassical Lp estimates

Nguyen¹

2022

Preprint

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“…In this potential we consider m = O(µ) fermions, and the 1d fermion density vanishes outside the interval [r − ( ), r + ( )], where r ± ( ) are the two roots of V 0 (r) = µ. As shown in Figure 5, for a fixed µ and R, the integrand in (15)…”

Section: Rotating Harmonic Oscillator In 2d: General Frameworkmentioning

confidence: 99%

“…These results are valid when the argument of V 2 in ( 23) is of order unity. We now wish to use (23) in (15) in the regime R R e . The dominant contribution to Var N R is from e where r + ( e ) = R e and therefore, plugging (23) directly into (15) gives the correct leading-order result.…”

Section: B Edge Regime Formentioning

confidence: 99%

“…The noninteracting limit is experimentally relevant since the interactions between the particles can be tuned in experiments. Studies on the theoretical side have focused on the number density, its correlations and related properties, in real space and in momentum space [8][9][10][11][12][13][14][15]. Systems of noninteracting fermions display rich and interesting behaviour and nontrivial fluctuations even at zero temperature due to the Pauli exclusion principle.…”

Section: Introductionmentioning

confidence: 99%

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Counting statistics for non-interacting fermions in a rotating trap

Smith,

Doussal,

Majumdar

et al. 2021

Preprint

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We study the ground state of N 1 noninteracting fermions in a two-dimensional harmonic trap rotating at angular frequency Ω > 0. The support of the density of the Fermi gas is a disk of radius Re. We calculate the variance of the number of fermions NR 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) Ω/ω < 1 and (ii) 1 − Ω/ω = O(1/N ), where ω is the angular frequency of the oscillator. In the first regime (i) we find that Var NR (A log N + B)√ N and we calculate A and B as functions of R/Re, Ω and ω. We also predict the higher cumulants of NR 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 Var NR is a discontinuous piecewise linear function of ∼ (R/Re) √ 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 NR and to the entanglement entropy. We show that these results match smoothly at large k with the above results for Ω/ω = O(1). These findings are nicely confirmed by numerical simulations. Finally, we uncover a universal behavior of Var NR near the fermionic edge. We extend our results to a three-dimensional geometry, where an additional confining potential is applied in the z direction.

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“…This extends to models even beyond Wigner or covariance matrices, such as beta ensembles [6,40], discrete beta ensembles [8], as well as various determinantal point processes whose kernels are given by or approximate a spectral projection [21,53,54].…”

Section: Introductionmentioning

confidence: 99%

Almost-optimal bulk regularity conditions in the CLT for Wigner matrices

Landon¹,

Sosoe²

2022

Preprint

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We consider linear spectral statistics of the form trpϕpHqq for test functions ϕ of low regularity and Wigner matrices H with smooth entry distribution. We show that for functions ϕ in the Sobolev space H 1{2`ε or the space C 1{2`ε , that are supported within the spectral bulk of the semicircle distribution, these linear spectral statistics have asymptotic Gaussian fluctuations with the same variance as in the CLT for functions of higher regularity, for any ε ą 0.

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Universality for free fermions and the local Weyl law for semiclassical Schrödinger operators

Cited by 5 publications

References 79 publications

Fermionic semiclassical Lp estimates

Fermionic semiclassical Lp estimates

Counting statistics for non-interacting fermions in a rotating trap

Almost-optimal bulk regularity conditions in the CLT for Wigner matrices

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