Emptiness Instanton in Quantum Polytropic Gas

TL;DR

This paper derives an analytic instanton solution for the probability of empty interval formation in a one-dimensional quantum polytropic gas, revealing detailed spatiotemporal instanton features.

Contribution

It provides the first explicit analytic form of the emptiness instanton for arbitrary polytropic index in quantum gases, using hydrodynamic equations in imaginary time.

Findings

Analytic instanton solution expressed as an integral representation.

Explicit spatiotemporal profile of the emptiness instanton.

Applicable to quantum gases with arbitrary polytropic index.

Abstract

The emptiness formation problem is addressed for a one-dimensional quantum polytropic gas characterized by an arbitrary polytropic index $\gamma$, which defines the equation of state $P \sim \rho^\gamma$, where $P$ is the pressure and $\rho$ 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.