Finite-Temperature Free Fermions and the Kardar-Parisi-Zhang Equation at Finite Time

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 $\beta \sim \mathcal{O}(1/N)$, ${\rho }_N(x,T)$ is given by a scaling function interpolating between a Gaussian at high temperature, for $\beta \ll 1/N$, and the Wigner semicircle law at low temperature, for $\beta \gg N^{-1}$. In the latter regime, we unveil a scaling limit, for $\beta \hslash \omega =b{N}^{-1/3}$, where the fluctuations close to the edge of the support, at $x\sim \pm \sqrt{2\hslash N/(m\omega )}$, are described by a limiting kernel $K_b^{ff}(s,s^{\prime })$ 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 $K_b^{ff}(s,s^{\prime })$ arises in the exact solution of the Kardar-Parisi-Zhang equation in $1+1$ dimensions at finite time $t$, with the correspondence $t=b^3$.

Figure 1

Plot of the scaling function $R(y,z)$ associated with the density ${\rho }_N(x,T)$ (7), given in Eq. (8).

Figure 2

Plot of ${\tilde{\rho }}_{\text{edge}}(s)$ given in Eq. (23) corresponding to two different (scaled) temperatures $b=0.5$ (dotted line) and $b=5$ (solid line). Inset: Plot of ${\tilde{\rho }}_{\text{edge}}(s)$ corresponding to $b\to \infty$ given in the text and shown here for comparison with the main plot.

Figure 3

Plot of the 2-point correlation function at the edge $g_b^{ff}(s,s^{\prime }=-4)$ as a function of $s$ and for scaled inverse temperatures $b=0.5$ and 5. Inset: Plot of the 2-point correlation function in the bulk $g_y^{\text{bulk}}(v)$ versus $v$ for scaled inverse temperatures $y=0.3,3$, and 10.