Finite-Temperature Free Fermions and the Kardar-Parisi-Zhang Equation at Finite Time
Langue
en
Article de revue
Ce document a été publié dans
Physical Review Letters. 2015-03-18, vol. 114, n° 11, p. 110402 (1-5)
American Physical Society
Résumé en anglais
We consider the system of $N$ one-dimensional free fermions confined by a harmonic well $V(x) = m\omega^2 {x^2}/{2}$ at finite inverse temperature $\beta = 1/T$. The average density of fermions $\rho_N(x,T)$ at position ...Lire la suite >
We consider the system of $N$ one-dimensional free fermions confined by a harmonic well $V(x) = m\omega^2 {x^2}/{2}$ at finite inverse temperature $\beta = 1/T$. The average density of fermions $\rho_N(x,T)$ at position $x$ is derived. For $N \gg 1$ and $\beta \sim {\cal O}(1/N)$, $\rho_N(x,T)$ is described by a scaling function interpolating between a Gaussian at high temperature, for $\beta \ll 1/N$, and the Wigner semi-circle law at low temperature, for $\beta \gg N^{-1}$. In the latter regime, we unveil a scaling limit, for $\beta {\hbar \omega}= b N^{-1/3}$, where the fluctuations close to the edge of the support, at $x \sim \pm \sqrt{2\hbar N/(m\omega)}$, are described by a limiting kernel $K^{\rm ff}_b(s,s')$ that depends continuously on $b$ and is a generalization of the Airy kernel, found in the Gaussian Unitary Ensemble of random matrices. Remarkably, exactly the same kernel $K^{\rm ff}_b(s,s')$ arises in the exact solution of the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimensions at finite time $t$, with the correspondence $t= b^3$.< Réduire
Mots clés en anglais
Fermion systems and Electron gas
Matrix theory
Probability theory Stochastic processes and Statistics
Project ANR
Marcheurs Browniens répulsifs et matrices aléatoires - ANR-11-BS04-0013
Paris Sciences et Lettres - ANR-10-IDEX-0001
Paris Sciences et Lettres - ANR-10-IDEX-0001
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