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arXiv:2608.26640

Emptiness formation in the Lieb-Liniger gas: hydrodynamic instantons and a conjectured rate function

Boris A. Khanikati, Alexander G. Abanov

Published Aug 27, 2026• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2608.26640

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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 ${n}^{\ensuremath{-}1}<R<\ensuremath{\xi}$ (here nnn 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.
A hydrodynamic approach is used to calculate an asymptotics of the Emptiness Formation Probability - the probability of a formation of an empty space in the ground state of a quantum one-dimensional many body system. Quantum hydrodynamics of a system is represented as a Euclidian path integral over configurations of hydrodynamic variables. In the limit of a large size of the empty space, the probability is dominated by an instanton configuration, and the problem is reduced to the finding of an instanton solution of classical hydrodynamic equations. After establishing a general formalism, we carry out this calculation for several simple systems -- free fermions with an arbitrary dispersion and Calogero-Sutherland model. For these systems we confirm the obtained results by comparison with exact results known in Random Matrix theory. We argue that the nonlinear hydrodynamic approach might be useful even in cases where the linearized hydrodynamics fails.
We study a correlation function for the one-dimensional isotropic XY{XY}XY model (XX0{XX0}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.
Motivated by recent experiments we derive an exact expression for the correlation function entering the three-body recombination rate for a one-dimensional gas of interacting bosons. The answer, given in terms of two thermodynamic parameters of the Lieb-Liniger model, is valid for all values of the dimensionless coupling $\ensuremath{\gamma}$ and contains the previously known results for the Bogoliubov and Tonks-Girardeau regimes as limiting cases. We also investigate finite-size effects by calculating the correlation function for small systems of three, four, five, and six particles.
This PhD thesis is devoted to the study of the long-time and large-distance asymptotic behaviour of the dynamical field--field correlation function of the Lieb--Liniger model in the limit of infinite repulsion (the one-dimensional impenetrable Bose gas). Extending previous works we consider a large class of non-thermal equilibrium conditions in addition to the case of thermal equilibrium that had been studied before. Our starting point is an exact representation of the field--field correlation function as the Fredholm determinant of an integrable integral operators. The integrable operator depends parametrically on time t, distance x and on the filling fraction, which is a function that characterizes the thermal or non-thermal equilibrium conditions. Our work is based on the rigorous asymptotic analysis, using Riemann--Hilbert techniques, of a more general integral operator that depends on two more functional parameters. The long-time and large-distance behaviour of the field--field correlation function is derived for two classes of filling fractions. These classes are characterized by the number of poles on the real axis (a generalization of Fermi points) that, together with the unique saddle point, contribute to the asymptotic expansion. For each class, we derive the long-time and large-distance asymptotic behaviour as a series in x^(-1/2) as x and t go to infinity for a fixed ratio x / t. We provide explicit closed-form expressions for the leading and sub-leading terms, logarithmic corrections, and overall constants in terms of special functions and simple integrals. For the impenetrable Bose gas in thermal equilibrium, we verify the derived asymptotic expansions by comparing them with the existing results in the literature and with numerical data. Our analysis is based solely on the analytic properties of the functions parametrizing the kernel of the integrable integral operator. This generalized approach will enable future applications to the long-time and large-distance asymptotic analysis of the Lieb--Liniger model with finite coupling constant.
We consider the one-dimensional XXX spin 1/2 Heisenberg antiferromagnet at zero temperature and zero magnetic field. We are interested in a probability of a formation of a ferromagnetic string P(n) in the antiferromagnetic ground-state. We call it emptiness formation probability [EFP]. We suggest a new technique for computation of the EFP in the inhomogeneous case. It is based on the quantum Knizhnik-Zamolodchikov equation [qKZ]. We calculate EFP for n≤6 for the inhomogeneous case. The homogeneous limit confirms our hypothesis about the relation of quantum correlations and number theory. We also make a conjecture about a structure of EFP for arbrary n.