# Counting statistics for noninteracting fermions in a d -dimensional potential – Archive ouverte HAL

### Naftali R. Smith 1 Pierre Le Doussal 2 Satya N. Majumdar 3 Grégory Schehr 3

#### Naftali R. Smith, Pierre Le Doussal, Satya N. Majumdar, Grégory Schehr. Counting statistics for noninteracting fermions in a d -dimensional potential. Physical Review E , American Physical Society (APS), 2021, 103 (3), ⟨10.1103/PhysRevE.103.L030105⟩. ⟨hal-03179783⟩

We develop a first-principle approach to compute the counting statistics in the ground-state of $N$ noninteracting spinless fermions in a general potential in arbitrary dimensions $d$ (central for $d>1$). In a confining potential, the Fermi gas is supported over a bounded domain. In $d=1$, for specific potentials, this system is related to standard random matrix ensembles. We study the quantum fluctuations of the number of fermions ${\cal N}_{\cal D}$ in a domain $\cal{D}$ of macroscopic size in the bulk of the support. We show that the variance of ${\cal N}_{\cal D}$ grows as $N^{(d-1)/d} (A_d \log N + B_d)$ for large $N$, and obtain the explicit dependence of $A_d, B_d$ on the potential and on the size of ${\cal D}$ (for a spherical domain in $d>1$). This generalizes the free-fermion results for microscopic domains, given in $d=1$ by the Dyson-Mehta asymptotics from random matrix theory. This leads us to conjecture similar asymptotics for the entanglement entropy of the subsystem $\cal{D}$, in any dimension, supported by exact results for $d=1$.

• 1. LPMS - Laboratoire de Physique des Matériaux et des Surfaces
• 2. Champs Aléatoires et Systèmes hors d'Équilibre
• 3. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques