Emptiness Instanton in Quantum Polytropic Gas

TL;DR

The paper addresses the emptiness formation probability (EFP) in a one-dimensional quantum polytropic gas with equation of state $P \sim \rho^\gamma$, showing that for large empty intervals the probability is controlled by an Emptiness Instanton derived via imaginary-time hydrodynamics. It introduces an elegant integral representation for the emptiness-instantiation potential $\mathcal{V}_n(\lambda,\bar{\lambda})$, enabling analytic continuation from integer to arbitrary polytropic indices $n$ and linking the calculation to Dotsenko–Fateev integrals familiar from conformal field theory. The authors compute explicit spatiotemporal density profiles, revealing an astroid-shaped empty region with universal $3/2$-type cusp behavior at the center and $n$-dependent exponents at the region boundaries, thereby generalizing prior integer-$n$ results. This work extends limit-shape and instanton analyses to interacting 1D quantum fluids, and suggests connections to full counting statistics and non-integrable configurations, with potential applications to Chaplygin gas and beyond.

Abstract

The emptiness formation problem is addressed for a one-dimensional quantum polytropic gas characterized by an arbitrary polytropic index $γ$, which defines the equation of state $P \sim ρ^γ$, where $P$ is the pressure and $ρ$ is the density. The problem involves determining the probability of the spontaneous formation of an empty interval in the ground state of the gas. In the limit of a macroscopically large interval, this probability is dominated by an instanton configuration. By solving the hydrodynamic equations in imaginary time, we derive the analytic form of the emptiness instanton. This solution is expressed as an integral representation analogous to those used for correlation functions in Conformal Field Theory. Prominent features of the spatiotemporal profile of the instanton are obtained directly from this representation.

Figures (4)

• Figure 1: Emptiness instanton spatiotemporal density profiles for various polytropic indices, both integer and non-integer. From left to right: upper row, $n=0$, $n=0.5$, lower row, $n=1$, $n=\sqrt{2}$. Color scheme for the density $\rho(\!x\!,\!\tau\!)$ is shown on the right.
• Figure 2: Left: Temporal extension of the emptiness instanton, $\tau_c$, as a function of $n$, given by Eq. \ref{['eq:tauc']}. Right: Dimensionless action $f(n)$ of the emptiness instanton, Eq. \ref{['eq:fn']}. For free fermions, $f(0) = \pi/2$, and in the limit $n \to \infty$, it approaches the asymptotic value $f(\infty) = 2$. While our results are valid for $n > -1/2$ (solid curve), both $\tau_c$ and $f(n)$ can be analytically continued as shown by dashed curves to the case of the Chaplygin gas, $n = -1$, where $\tau_c=f(n) = 0$.
• Figure 3: Left: contours $\mathcal{C}(\lambda,\bar{\lambda})$ and $\mathcal{C}(\mathrm{i} , -\mathrm{i})$ in Eqs \ref{['eq:vn2']} and\ref{['eq:vnintn']}. Right: Pochhammer contour $\Pi o\!\chi$ in Eqs \ref{['eq:vnanyn']} and \ref{['eq:dvnanyn']}.
• Figure 4: Left: Density profile at $\tau=0$. Right: Density profile at $x=0$ given by Eq. (\ref{['eq:rhotau']})