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

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.Quantitative information very often comes in the form of a series expansion in a dimensionless variable [1]. Often one knows only the first few terms; but even if one knows many, the raw series is usually useful only for a limited range of values of the expansion parameter. One may therefore turn to various methods for improving the convergence properties. This is justified if one expects that the solution, in the region of interest of the expansion variable, is an analytic continuation of the solution in the region in which one is performing the expansion. The best-known method is the Padé approximation [2] and its various Hermite-Padé generalizations [3].On the other hand, when extrapolations from one parameter region to another were called for within particle physics, the solution was the Gell-Mann-Low renormalization group (GML-RG) [4,5]. This method is normally not comparable to the Padé approximation because it takes as input not only the known terms of the expansion, but also certain known nontrivial symmetries in the problem, an approach that was later formalized and extended to the study of differential equations using Lie group-theoretic methods [6]. We should also mention a related body of work in Ref. [7].We analyze the mechanism of operations of GML-RG in the single-variable (SV) case. We show that the SV GML-RG requires no special symmetry and is in fact an integral Hermite-Padé approximation [8], a connection that remains largely unexplored. The method is applicable to any series expansion, but is especially good for interpolation between a power-series expansion (for small values of a variable) and a scaling law (for large values of the variable), where the scaling law exponent is a known continuous function of a parameter, and one seeks, e.g., an approximation for the prefactor. As a case in point, we treat the density matrix of the Lieb-Liniger gas [9], an object of considerable interest to both cold gases [10,11] and integrable systems [12] communities.The GML-RG suggests itself very naturally in problems where a physical quantity at one value of a param-eter is given as an expansion in terms of the value of the same quantity at another value of the parameter, a situation usually caused by regularization and renormalization [5]. A simple single-parameter example is Twhich is the Born series f...

Low-Lying Energy Levels of a One-Dimensional Weakly Interacting Bose Gas under Zero Boundary Conditions

“…It is important that our solutions for E 0 and E(k) coincide with that ones obtained by the exactly solvable approach, based of the Bethe ansatz [7][8][9]. In addition, our solution for the density matrix F 1 (x 1 , x 2 )| T =0 coincides with the solution obtained within other methods for periodic BCs [39,[41][42][43][44][45][46][47][48]. This clearly shows that the Bogoliubov approximation is quite accurate for a finite 1D Bose system with weak coupling.…”

“…It is seen from Fig. 1 that F 1 (x 1 , x (87) is very close to the solution for a periodic system at T = 0, which reads [39,[41][42][43][44][45][46][47][48]]…”

Correlations in low-dimensional quantum gases

“…(III.66), with the connection (4,3), Eq. (III.96), yields the connection (4,0) first published in [302]: More generally, all correlations of the model are encoded in the connections of type (l, 0) and (0, k), as a consequence of integrability.…”

Connection between nonlocal one-body and local three-body correlations of the Lieb-Liniger model

“…given in [10] without proof. The right-hand sides of these last two equalities involve moments of the pseudomomentum distribution only, and it is a general fact that all correlations of the model are defined through connections of type (l, 0) and (0, k), as a consequence of integrability.…”

“…In this work, we focus on the link between non-local correlation functions of the Lieb-Liniger model and its integrals of motion, thus elucidating a special structure of the ground state for this integrable model. In particular, we derive a relation, first proposed in [10], that links the fourth coefficient of the Taylor expansion of the onebody correlation function at short distances with various moments of the quasi-momentum distribution and their derivatives with respect to the coupling constant. Then, * Maxim.Olchanyi@umb.edu we use a recently-developed method [11,12], and generalize recent conjectures [12][13][14], to evaluate these quantities with excellent accuracy in a wide range of interaction strengths.…”