Kernels for non interacting fermions via a Green’s function approach with applications to step potentials
hal.structure.identifier | Laboratoire Ondes et Matière d'Aquitaine [LOMA] | |
dc.contributor.author | DEAN, David | |
hal.structure.identifier | Champs Aléatoires et Systèmes hors d'Équilibre | |
dc.contributor.author | DOUSSAL, Pierre Le | |
hal.structure.identifier | Laboratoire de Physique Théorique et Modèles Statistiques [LPTMS] | |
dc.contributor.author | MAJUMDAR, Satya N. | |
hal.structure.identifier | Laboratoire de Physique Théorique et Modèles Statistiques [LPTMS] | |
dc.contributor.author | SCHEHR, Grégory | |
hal.structure.identifier | Laboratoire de Physique Théorique et Modèles Statistiques [LPTMS] | |
dc.contributor.author | SMITH, Naftali R. | |
dc.date.issued | 2021-02-05 | |
dc.identifier.issn | 1751-8113 | |
dc.description.abstractEn | The quantum correlations of $N$ noninteracting spinless fermions in their ground state can be expressed in terms of a two-point function called the kernel. Here we develop a general and compact method for computing the kernel in a general trapping potential in terms of the Green's function for the corresponding single particle Schr\"odinger equation. For smooth potentials the method allows a simple alternative derivation of the local density approximation for the density and of the sine kernel in the bulk part of the trap in the large $N$ limit. It also recovers the density and the kernel of the so-called {\em Airy gas} at the edge. This method allows to analyse the quantum correlations in the ground state when the potential has a singular part with a fast variation in space. For the square step barrier of height $V_0$, we derive explicit expressions for the density and for the kernel. For large Fermi energy $\mu>V_0$ it describes the interpolation between two regions of different densities in a Fermi gas, each described by a different sine kernel. Of particular interest is the {\em critical point} of the square well potential when $\mu=V_0$. In this critical case, while there is a macroscopic number of fermions in the lower part of the step potential, there is only a finite $O(1)$ number of fermions on the shoulder, and moreover this number is independent of $\mu$. In particular, the density exhibits an algebraic decay $\sim 1/x^2$, where $x$ is the distance from the jump. Furthermore, we show that the critical behaviour around $\mu = V_0$ exhibits universality with respect with the shape of the barrier. This is established (i) by an exact solution for a smooth barrier (the Woods-Saxon potential) and (ii) by establishing a general relation between the large distance behavior of the kernel and the scattering amplitudes of the single-particle wave-function. | |
dc.description.sponsorship | Matrices aléatoires et fermions piégés - ANR-17-CE30-0027 | |
dc.language.iso | en | |
dc.publisher | IOP Publishing | |
dc.title.en | Kernels for non interacting fermions via a Green’s function approach with applications to step potentials | |
dc.type | Article de revue | |
dc.identifier.doi | 10.1088/1751-8121/abd9ef | |
dc.subject.hal | Physique [physics] | |
dc.identifier.arxiv | 2009.12882 | |
bordeaux.journal | Journal of Physics A: Mathematical and Theoretical | |
bordeaux.page | 084001 | |
bordeaux.volume | 54 | |
bordeaux.issue | 8 | |
bordeaux.peerReviewed | oui | |
hal.identifier | hal-03177657 | |
hal.version | 1 | |
hal.popular | non | |
hal.audience | Internationale | |
hal.origin.link | https://hal.archives-ouvertes.fr//hal-03177657v1 | |
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