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arXiv:1910.08054 · Physical Review A

Emptiness formation probability in one-dimensional Bose liquids

Hsiu-Chung Yeh, Alex Kamenev

Published Feb 28, 2020• 4 citations
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Abstract

We study the emptiness formation probability (EFP) in interacting one-dimensional Bose liquids, which is the probability that a snapshot of its ground state reveals exactly zero particles within the interval $|x|<R$. 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.

DOI
10.1103/physreva.101.023623

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We study the emptiness formation probability (EFP) in interacting one-dimensional Bose liquids, which is the probability that a snapshot of its ground state reveals exactly zero particles within the interval ∣x∣<R|x|<R∣x∣<R. For a weakly interacting liquid there is parametrically wide regime n\ensuremath−1<R<\ensuremathξ{n}^{\ensuremath{-}1}<R<\ensuremath{\xi}n\ensuremath−1<R<\ensuremathξ (here nnn is the average density and \ensuremathξ\ensuremath{\xi}\ensuremathξ 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 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 2R2R2R, its leading asymptotic behavior is described by a rate function f(γ0)f(γ_0)f(γ0​), defined by −log⁡P(R)∼(ρ0R)2f(γ0)-\log P(R)\sim(ρ_0R)^2f(γ_0)−logP(R)∼(ρ0​R)2f(γ0​), where ρ0ρ_0ρ0​ is the mean density and γ0γ_0γ0​ is the dimensionless interaction strength. We propose a parameter-free integral equation for f(γ0)f(γ_0)f(γ0​). 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.
Abstract We study large deviations in interacting quantum liquids with the polytropic equation of state P ( ρ ) ∼ ρ γ , where ρ is density and P is pressure. By solving hydrodynamic equations in imaginary time we evaluate the instanton action and calculate the emptiness formation probability (EFP), the probability that no particle resides in a macroscopic interval of a given size. Analytic solutions are found for a certain infinite sequence of rational polytropic indexes γ and the result can be analytically continued to any value of γ ⩾ 1. Our findings agree with (and significantly expand on) previously known analytical and numerical results for EFP in quantum liquids. We also discuss interesting universal spacetime features of the instanton solution.
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.
We consider the quantum Non-linear Schrödinger equation i∂tΨ=−∂x2Ψ+2cΨ†Ψ2i\partial_tΨ=-\partial_x^2Ψ+2cΨ^\daggerΨ^2i∂t​Ψ=−∂x2​Ψ+2cΨ†Ψ2 with positive coupling constant ccc 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][0,x][0,x] in the large xxx limit. We call this the probability of phase separation or alternately the emptiness formation probability.
The competition between particle and pair condensation in Bose liquids is studied using a mean field variational ground state that comprises both the Bogoliubov approximation for weakly interacting Bose gases and the Valatin-Butler pair state. We first consider structureless bosons, disregarding the fact that an attractive gas will necessarily collapse. We show that a pair state occurs only if the attraction is strong enough to bind two bosons together. The limit of low densities is treated exactly. At higher density, we use a separable interaction model and we show that the pairs dissociate at a critical density Nc (where their ionization gap vanishes). Past that threshold, ordinary particle condensation sets in. In order to cure collapse, we consider bosons with an internal spin structure : the interaction is binding in some states, repulsive on the average. Taking spin 1 particles as an example, we repeat the previous analysis, and we substantiate its conclusions while retaining a positive compressibility. In the pair regime the ground state is isotropic (singlet pairs). Past the dissociation threshold, an anisotropic one particle condensate appears. We show that it is linearly polarized (in an arbitrary direction), and non fragmented. We discuss briefly its physical properties, and we suggest that the transition from pair to particle condensation might be first order.
We consider an impurity in a one-dimensional weakly interacting Bose gas and analytically calculate the density profile of the Bose gas. Within the mean-field approximation, by increasing the distance from the impurity, the Bose gas density saturates exponentially fast to its mean thermodynamic-limit value at distances beyond the healing length. The effect of quantum fluctuations drastically changes this behavior, leading to a power-law decay of the density deviation from the mean density. At distances longer than the healing length and shorter than a new length scale proportional to the impurity coupling strength, the power-law exponent is 2, while at the longest distances the corresponding exponent becomes 3. The latter crossover does not exist in two special cases. The first one is realized for infinitely strongly coupled impurity; then the density deviation always decays with the exponent 2. The second special case occurs when the new length scale is smaller than the healing length, i.e., at weak impurity coupling; then the density deviation always decays with the exponent 3. The obtained results are exact in the impurity coupling strength and account for the leading order in the interaction between the particles of the Bose gas.