Interface fluctuations associated with split Fermi seas

Abstract

We consider the asymptotic behaviour of a family of unidimensional lattice fermion models, which are in exact correspondence with certain probability laws on partitions and on unitary matrices. These models exhibit limit shapes, and in the case where the bulk of these shapes are described by analytic functions, the fluctuations around their interfaces have been shown to follow a universal Tracy-Widom distribution or its higher-order analogue. Non-differentiable bulk limit shape functions arise when a gap appears in some quantum numbers of the model, in other words when the Fermi sea is split. We show that split Fermi seas give rise to new interface fluctuations, governed by integer powers of universal distributions. This breakdown in universality is analogous to the behaviour of a random Hermitian matrix when the support of its limiting eigenvalue distribution has multiple cuts, with oscillations appearing in the limit of the two-point correlation function. We show that when the Fermi sea is split in the lattice fermion model, there are multiple cuts in the eigenvalue support of the corresponding unitary matrix model.

Figures (3)

• Figure 1: Plots of the function $D$ determining the Fermi seas (top), the limit fermion density $\varrho$ (middle) and corresponding limiting eigenvalue density $\rho$ in the $\theta \sim \ell/b$ regime for four sets of coefficients $\gamma$ with $\gamma_1 = 1$ and $\gamma_r = 0$ for $r >2$. For $\gamma_2 = \tfrac{1}{10}$ and $\gamma_2 = -\tfrac{1}{8}$ (illustrated in the central columns), the Fermi sea has one-cut throughout and the limit density is smooth in the bulk; the $\gamma_2 = -\tfrac{1}{8}$ model is order $m=2$ multicritical. For $\gamma_2 = \pm \tfrac{1}{3}$ (the leftmost and rightmost columns), the Fermi sea splits into two in the bulk, and there is a discontinuity in the derivative of the limit density where it splits. In the $\gamma_2=\frac{1}{3}$ case, there are two cuts in the Fermi sea $I_{b^-}$ before the right edge. Note that in this case the limiting eigenvalue density goes to $0$ at two points.
• Figure 2: Saddle points of the action $S(z;x)$ for $\gamma_1 = 1$, $\gamma_2 = -\tfrac{1}{3}$ and $\gamma_r = 0$ for $r>2$ (simple saddle points are correspond to single black dots, double saddle points correspond to encircled black dots), along with the appropriate integration contours. The frozen region corresponds to $x \leq-\tilde{b} = -\tfrac{10}{3}$ where we have $I_x = [-\pi,\pi]$. The appropriate integration contours are shown on the left, with the saddle points for $x= \tfrac{7}{2}$ above and $x=-\tfrac{10}{3}$ below. For $-\tilde{b}<x\leq \hat{b}=\tfrac{2}{3}$, $I_x$ has a single cut and contains $0$, and the contours have the form shown in the central column (with saddle points for $x=-\tfrac{2}{3}$ above and $x=\frac{2}{3}$ below, where there is a double saddle point at $1$). For $\hat{b}<x< b = \tfrac{41}{24}$, $I_x$ has two cuts, and we use contours of the kind show in the top right. The empty region corresponds to $x\geq b$, and we use the undeformed contours $c_+,c_-$ shown in the bottom right (with the saddle points for $x = \tfrac{7}{4}$). The corresponding $D$ and limit density $\varrho$ are shown on the right of Figure \ref{['fig:4casesDetc']}.
• Figure 3: Left, the order $2$ saddle points of the action $S(z;b)$ for $\gamma_1 = 1$, $\gamma_2 = -\tfrac{1}{3}$ and $\gamma_r = 0$ for $r>2$, at the right edge $b = \frac{41}{24}$. The central region $I$ of the integration contours $c_+\times c_-$ is shown in black. In the centre, plots of the real part of $S(z;x)$ for $z$ near the saddle points $e^{i\chi_b}$ above and $e^{-i\chi_b}$ below with $\chi_b = \arccos \frac{3}{8}$, with blues where $\mathrm{Re}(S(z;b)) <0$ and reds where $\mathrm{Re}(S(z;b)) >0$. Right, the corresponding plots in the case where $\gamma_1,\gamma_2, \gamma_3,\gamma_4$ are tuned so that the action $S(z;b)$ has order $4$ saddle points at $e^{\pm i \arccos \frac{3}{8} }$ on the right edge.