Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0910.0565

A Hermite-Padé perspective on Gell-Mann--Low renormalization group: an application to the correlation function of Lieb-Liniger gas

Vanja Dunjko, Maxim Olshanii

Published Oct 3, 2009• 0 citations
arXivHTMLPDF

Abstract

While Padé approximation is a general method for improving convergence of series expansions, Gell-Mann--Low renormalization group normally relies on the presence of special symmetries. We show that in the single-variable case, the latter becomes an integral Hermite-Padé approximation, needing no special symmetries. It is especially useful for interpolating between expansions for small values of a variable and a scaling law of known exponent for large values. As an example, we extract the scaling-law prefactor for the one-body density matrix of the Lieb-Liniger gas. Using a new result for the 4th-order term in the short-distance expansion, we find a remarkable agreement with known ab initio numerical results.

DOI
10.48550/arxiv.0910.0565

Related papers

A Hermite–Padé perspective on the renormalization group, with an application to the correlation function of Lieb–Liniger gas

Vanja Dunjko, Maxim Olshanii

Renormalization group (RG) methods are based on symmetry principles that are peculiar to quantum field theory and certain related settings. However, in cases when the β-function depends only on the coupling, the RG technology does include a part—which is related to the equation for the running coupling and which we will call the 'single variable RG' (SVRG)—that turns out to be a purely formal method for improving the convergence properties of series expansions. In fact, we show that SVRG is an integral Hermite–Padé approximation preceded by a series reversion. A complementary viewpoint on what SVRG does to an expansion is that it performs a 'minimal upgrade to a self-similar form', the minimality being crucial. In applications, SVRG is especially useful in problems in which one knows not only the first few terms of an expansion valid for small values of a variable, but also the exponent (but not the prefactor) of the scaling law for large values of the variable. As a practical example, we extract the scaling-law prefactor for the one-body density matrix of the Lieb–Liniger gas. Using a new result for the fourth-order term in the short-distance expansion, we find remarkable agreement with known quantum diffusion Monte Carlo numerical results.

Journal of Physics A Mathematical and Theoretical•Jan 5, 2011•11 citations

Applications of Padé approximants to correlation functions

R. Paul

In this paper we have presented a general scheme whereby the method of Padé approximants may be applied to the power series expansion of time dependent correlation functions. It has been demonstrated by actual numerical calculations, using the computer simulated results of Berne and Harp [5], that even a few terms in the power series expansion are capable of reproducing the correlation curves to a fairly large extent. It is pointed out that the method can be extended to higher orders without the introduction of fitting parameters, and without violating the time parity requirements of the Liouville equation. In the final analysis the method is a truly microscopic method since it relies on the microscopic hamiltonian, and the underlying principles are very straightforward. The basic requirements are the spectral moments, but we do not need many of these as the Padé approximants are highly convergent.

Molecular Physics•Sep 1, 1980•11 citations

The short-distance first-order correlation function of the interacting one-dimensional Bose gas

Maxim Olshanii, Vanja Dunjko

We derive exact closed-form expressions for the first few terms of the short-distance Taylor expansion of the one-body correlation function of the Lieb–Liniger gas.

New Journal of Physics•Jul 28, 2003•9 citations

Correlation functions of the Lieb-Liniger gas and the LeClair-Mussardo formula

Garry Goldstein, Natan Andrei

In this letter we derive formulas for multi point correlation functions, in the thermodynamic limit, for the Lieb Liniger gas taken with respect to arbitrary eigenstates. These results apply for the ground state, thermal states and GGE states. We obtain these correlation functions as a series of multiple integrals of progressively higher dimensions. These integrals converge rapidly for short distance correlation functions and low densities of particles. The series derived matches exactly the LeClair Mussardo formula for correlation functions of relativistic integrable models.

arXiv:1507.00077•Jul 1, 2015•2 citations

Correlation functions of a Lieb–Liniger Bose gas

G. E. Astrakharchik, S. Giorgini

The ground-state correlation functions of a one-dimensional homogeneous Bose system described by the Lieb-Liniger Hamiltonian are investigated by using exact quantum Monte Carlo techniques. This article is an extension of a previous study published in Phys. Rev. A {\bf 68}, 031602 (2003). New results on the local three-body correlator as a function of the interaction strength are included and compared with the measured value from three-body loss experiments. We also carry out a thorough study of the short- and long-range behavior of the one-body density matrix.

arXiv:cond-mat/0510463•Journal of Physics B Atomic Molecular and Optical Physics•May 2, 2006•45 citations

Ground-state energy and excitation spectrum of the Lieb-Liniger model : accurate analytical results and conjectures about the exact solution

Guillaume Lang, F. W. J. Hekking, Anna Minguzzi

We study the ground-state properties and excitation spectrum of the Lieb-Liniger model, i.e. the one-dimensional Bose gas with repulsive contact interactions. We solve the Bethe-Ansatz equations in the thermodynamic limit by using an analytic method based on a series expansion on orthogonal polynomials developed in and push the expansion to an unprecedented order. By a careful analysis of the mathematical structure of the series expansion, we make a conjecture for the analytic exact result at zero temperature and show that the partially resummed expressions thereby obtained compete with accurate numerical calculations. This allows us to evaluate the density of quasi-momenta, the ground-state energy, the local two-body correlation function and Tan’s contact. Then, we study the two branches of the excitation spectrum. Using a general analysis of their properties and symmetries, we obtain novel analytical expressions at arbitrary interaction strength which are found to be extremely accurate in a wide range of intermediate to strong interactions.

arXiv:1609.08865•SciPost Physics•Jul 7, 2017•59 citations
While Padé approximation is a general method for improving convergence of series expansions, Gell-Mann--Low renormalization group normally relies on the presence of special symmetries. We show that in the single-variable case, the latter becomes an integral Hermite-Padé approximation, needing no special symmetries. It is especially useful for interpolating between expansions for small values of a variable and a scaling law of known exponent for large values. As an example, we extract the scaling-law prefactor for the one-body density matrix of the Lieb-Liniger gas. Using a new result for the 4th-order term in the short-distance expansion, we find a remarkable agreement with known ab initio numerical results.
Renormalization group (RG) methods are based on symmetry principles that are peculiar to quantum field theory and certain related settings. However, in cases when the β-function depends only on the coupling, the RG technology does include a part—which is related to the equation for the running coupling and which we will call the 'single variable RG' (SVRG)—that turns out to be a purely formal method for improving the convergence properties of series expansions. In fact, we show that SVRG is an integral Hermite–Padé approximation preceded by a series reversion. A complementary viewpoint on what SVRG does to an expansion is that it performs a 'minimal upgrade to a self-similar form', the minimality being crucial. In applications, SVRG is especially useful in problems in which one knows not only the first few terms of an expansion valid for small values of a variable, but also the exponent (but not the prefactor) of the scaling law for large values of the variable. As a practical example, we extract the scaling-law prefactor for the one-body density matrix of the Lieb–Liniger gas. Using a new result for the fourth-order term in the short-distance expansion, we find remarkable agreement with known quantum diffusion Monte Carlo numerical results.
In this paper we have presented a general scheme whereby the method of Padé approximants may be applied to the power series expansion of time dependent correlation functions. It has been demonstrated by actual numerical calculations, using the computer simulated results of Berne and Harp [5], that even a few terms in the power series expansion are capable of reproducing the correlation curves to a fairly large extent. It is pointed out that the method can be extended to higher orders without the introduction of fitting parameters, and without violating the time parity requirements of the Liouville equation. In the final analysis the method is a truly microscopic method since it relies on the microscopic hamiltonian, and the underlying principles are very straightforward. The basic requirements are the spectral moments, but we do not need many of these as the Padé approximants are highly convergent.
In this letter we derive formulas for multi point correlation functions, in the thermodynamic limit, for the Lieb Liniger gas taken with respect to arbitrary eigenstates. These results apply for the ground state, thermal states and GGE states. We obtain these correlation functions as a series of multiple integrals of progressively higher dimensions. These integrals converge rapidly for short distance correlation functions and low densities of particles. The series derived matches exactly the LeClair Mussardo formula for correlation functions of relativistic integrable models.
The ground-state correlation functions of a one-dimensional homogeneous Bose system described by the Lieb-Liniger Hamiltonian are investigated by using exact quantum Monte Carlo techniques. This article is an extension of a previous study published in Phys. Rev. A {\bf 68}, 031602 (2003). New results on the local three-body correlator as a function of the interaction strength are included and compared with the measured value from three-body loss experiments. We also carry out a thorough study of the short- and long-range behavior of the one-body density matrix.
We study the ground-state properties and excitation spectrum of the Lieb-Liniger model, i.e. the one-dimensional Bose gas with repulsive contact interactions. We solve the Bethe-Ansatz equations in the thermodynamic limit by using an analytic method based on a series expansion on orthogonal polynomials developed in and push the expansion to an unprecedented order. By a careful analysis of the mathematical structure of the series expansion, we make a conjecture for the analytic exact result at zero temperature and show that the partially resummed expressions thereby obtained compete with accurate numerical calculations. This allows us to evaluate the density of quasi-momenta, the ground-state energy, the local two-body correlation function and Tan’s contact. Then, we study the two branches of the excitation spectrum. Using a general analysis of their properties and symmetries, we obtain novel analytical expressions at arbitrary interaction strength which are found to be extremely accurate in a wide range of intermediate to strong interactions.