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Noninteracting fermions at finite temperature in a d -dimensional trap: Universal correlations

hal.structure.identifierLaboratoire Ondes et Matière d'Aquitaine [LOMA]
dc.contributor.authorDEAN, David S.
hal.structure.identifierLaboratoire de Physique Théorique de l'ENS [École Normale Supérieure] [LPTENS]
dc.contributor.authorLE DOUSSAL, Pierre
hal.structure.identifierLaboratoire de Physique Théorique et Modèles Statistiques [LPTMS]
dc.contributor.authorMAJUMDAR, Satya N.
hal.structure.identifierLaboratoire de Physique Théorique et Modèles Statistiques [LPTMS]
dc.contributor.authorSCHEHR, Grégory
dc.date.created2016-09-17
dc.date.issued2016-12
dc.identifier.issn1050-2947
dc.description.abstractEnWe study a system of $N$ non-interacting spin-less fermions trapped in a confining potential, in arbitrary dimensions $d$ and arbitrary temperature $T$. The presence of the trap introduces an edge where the average density of fermions vanishes. Far from the edge, near the center of the trap (the so called "bulk regime"), physical properties of the fermions have traditionally been understood using the Local Density Approximation. However, this approximation drastically fails near the edge where the density vanishes. In this paper we show that, even near the edge, novel universal properties emerge, independently of the details of the confining potential. We show that for large $N$, these fermions in a confining trap, in arbitrary dimensions and at finite temperature, form a determinantal point process. As a result, any $n$-point correlation function can be expressed as an $n \times n$ determinant whose entry is called the kernel. Near the edge, we derive the large $N$ scaling form of the kernels. In $d=1$ and $T=0$, this reduces to the so called Airy kernel, that appears in the Gaussian Unitary Ensemble (GUE) of random matrix theory. In $d=1$ and $T>0$ we show a remarkable connection between our kernel and the one appearing in the $1+1$-dimensional Kardar-Parisi-Zhang equation at finite time. Consequently our result provides a finite $T$ generalization of the Tracy-Widom distribution, that describes the fluctuations of the rightmost fermion at $T=0$. In $d>1$ and $T \geq 0$, while the connection to GUE no longer holds, the process is still determinantal whose analysis provides a new class of kernels, generalizing the $1d$ Airy kernel at $T=0$ obtained in random matrix theory. Some of our finite temperature results should be testable in present-day cold atom experiments, most notably our detailed predictions for the temperature dependence of the fluctuations near the edge.
dc.language.isoen
dc.publisherAmerican Physical Society
dc.rights.urihttp://creativecommons.org/licenses/by-sa/
dc.title.enNoninteracting fermions at finite temperature in a d -dimensional trap: Universal correlations
dc.typeArticle de revue
dc.identifier.doi10.1103/PhysRevA.94.063622
dc.subject.halPhysique [physics]/Matière Condensée [cond-mat]/Mécanique statistique [cond-mat.stat-mech]
dc.subject.halPhysique [physics]/Matière Condensée [cond-mat]/Gaz Quantiques [cond-mat.quant-gas]
dc.subject.halPhysique [physics]/Physique mathématique [math-ph]
dc.subject.halMathématiques [math]/Probabilités [math.PR]
dc.identifier.arxiv1609.04366
bordeaux.journalPhysical Review A : Atomic, molecular, and optical physics [1990-2015]
bordeaux.page063622 (1-41)
bordeaux.volume94
bordeaux.issue6
bordeaux.peerReviewedoui
hal.identifierhal-01421984
hal.version1
hal.popularnon
hal.audienceInternationale
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-01421984v1
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