Full Counting Statistics in a Propagating Quantum Front and Random Matrix Spectra

One-dimensional free fermions are studied with emphasis on propagating fronts emerging from a step initial condition. The probability distribution of the number of particles at the edge of the front is determined exactly. It is found that the full counting statistics coincide with the eigenvalue statistics of the edge spectrum of matrices from the Gaussian unitary ensemble. The correspondence established between the random matrix eigenvalues and the particle positions yields the order statistics of the rightmost particles in the front and, furthermore, it implies their subdiffusive spreading.

Figure 1

Density profiles $n_m(t)$ at increasing times. The edge of the front with the emerging staircase structure is shown by the shaded region. The profile for $m\le 0$ is given by $n_m(t)=1-n_{1-m}(t)$ and is not shown.

Figure 2

Rescaled density near the edge of the front for $t=100$ (diamonds) and $t=1000$ (circles). The solid (red) line shows the scaling function $\rho (x)$ while the distribution functions of the $n$th particle for $n=1,\dots ,6$ are shown by the dotted (blue) lines. The dashed (green) line is the sum of the $F(n,x)$.

Figure 3

Entanglement entropy S (upper symbols) and particle-number fluctuations ${\kappa }_2$ (lower symbols) in the edge region for different times as functions of the scaled coordinate $s$. The solid (red) lines are the $t\to \infty$ scaling functions.