Multicritical Edge Statistics for the Momenta of Fermions in Nonharmonic Traps

We compute the joint statistics of the momenta $p_i$ of $N$ noninteracting fermions in a trap, near the Fermi edge, with a particular focus on the largest one $p_{\max }$. For a 1D harmonic trap, momenta and positions play a symmetric role, and hence the joint statistics of momenta are identical to that of the positions. In particular, $p_{\max }$, as $x_{\max }$, is distributed according to the Tracy-Widom distribution. Here we show that novel “momentum edge statistics” emerge when the curvature of the potential vanishes, i.e., for “flat traps" near their minimum, with $V(x)\sim x^{2n}$ and $n>1$. These are based on generalizations of the Airy kernel that we obtain explicitly. The fluctuations of $p_{\max }$ are governed by new universal distributions determined from the $n$th member of the second Painlevé hierarchy of nonlinear differential equations, with connections to multicritical random matrix models. Finite temperature extensions and possible experimental signatures in cold atoms are discussed.

Figure 1

(Left) (solid line) Scaling function $F_{2n=4}(s)$, Eq. (4), of the density at the edge, for $V(x)\sim x^4$. (Dotted line) $s\to -\infty$ asymptotics $F_4(s)\simeq |s{|}^{1/4}/\pi$, matching the bulk density, see above (4). (Inset) Same in log-linear scale showing the large $s>0$ oscillations. (Dotted line) Asymptotics in (6). (Right) Scaling function ${\mathcal{F}}_{2n}^{\prime }(s)$ of the PDF of $p_{\max }$ in (8)–(10). (Red) $n=2$. (Blue) The TW distribution ($n=1$) for comparison. (Dotted line) Large $s<0$ behavior, ${\mathcal{F}}_4^{\prime }(s)\sim \exp \{-[(\sqrt{8/3})/15]|s{|}^{5/2}\}$.