arXiv:2412.11686 · SciPost Physics

Emptiness instanton in quantum polytropic gas

Alexander G. Abanov, Dimitri M. Gangardt

Published Apr 9, 2025• 0 citations

Abstract

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

DOI: 10.21468/scipostphys.18.4.122

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arXiv:2412.11686 · SciPost Physics

Emptiness instanton in quantum polytropic gas

Alexander G. Abanov, Dimitri M. Gangardt

Published Apr 9, 2025• 0 citations

arXiv HTML PDF

Abstract

DOI: 10.21468/scipostphys.18.4.122

Related papers

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We study the emptiness formation probability (EFP) in the ground state of the repulsive one-dimensional Lieb-Liniger Bose gas. For a macroscopic empty interval of length

2R

, its leading asymptotic behavior is described by a rate function

f(γ_0)

, defined by

-\log P(R)\sim(ρ_0R)^2f(γ_0)

, where

ρ_0

is the mean density and

γ_0

is the dimensionless interaction strength. We propose a parameter-free integral equation for

f(γ_0)

. Starting from the exact dual-field Fredholm-determinant representation of the EFP, we show how the conjectured kernel arises formally and identify the uniform asymptotic statement that remains to be proven for a rigorous derivation. The conjecture reproduces the Tonks-Girardeau and weak-coupling limits, as well as the first correction obtained independently in both limits. It also agrees at the few-percent level with numerical minimization of the Lieb-Liniger hydrodynamic action over more than four orders of magnitude in coupling. The numerical calculation yields the corresponding emptiness instantons and their astroid-like vacuum regions.

|x|<R

. For a weakly interacting liquid there is parametrically wide regime ${n}^{\ensuremath{-}1}<R<\ensuremath{\xi}$ (here

n

is the average density and $\ensuremath{\xi}$ is the healing length) where EFP exhibits a nontrivial crossover from the Poisson to the Gaussian behavior. We employ the instanton technique [NATO Science Series II: Mathematics, Physics and Chemistry (2004), Vol. 221] to study quantitative details of these regime and compare it with previously reported special cases.

We consider the quantum Non-linear Schrödinger equation

i\partial_tΨ=-\partial_x^2Ψ+2cΨ^\daggerΨ^2

with positive coupling constant

c

varying from zero to infinity. We study quantum correlation functions of this model using the determinant representation of these correlation functions. We consider the case of a finite density ground state and evaluate, using the Riemann Hilbert problem, the asymptotics of the probability that no particles are present in the space interval

[0,x]

in the large

x

limit. We call this the probability of phase separation or alternately the emptiness formation probability.

We study a correlation function for the one-dimensional isotropic

{XY}

model (

{XX0}

model), which is called the Emptiness Formation Probability (EFP). It is the probability of the formation of a ferromagnetic string in the anti-ferromagnetic ground state. Using the expression of the EFP as a Toeplitz determinant, we discuss its asymptotic behaviors. We also compare the analytical results with numerical calculations as the density-matrix renormalization group and the quantum Monte-Carlo method.