Finite temperature free fermions and the Kardar-Parisi-Zhang equation at finite time

الأصل · EN

We consider the system of N one-dimensional free fermions confined by a harmonic well V(x) = mω² x²/2 at finite inverse temperature β= 1/T. The average density of fermions ρₙ(x,T) at position x is derived. For N ≫ 1 and β O(1/N), ρₙ(x,T) is given by a scaling function interpolating between a Gaussian at high temperature, for β≪ 1/N, and the Wigner semi-circle law at low temperature, for β≫ N⁻¹. In the latter regime, we unveil a scaling limit, for βℏ ω= b N⁻¹/³, where the fluctuations close to the edge of the support, at x ± √2ℏ N/(mω), are described by a limiting kernel K ffb(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 ffb(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= b³.