[Paper Review] Large-distance and long-time asymptotic behavior of the reduced density matrix in the non-linear Schrödinger model

Karol K. Kozłowski|arXiv (Cornell University)|Jan 8, 2011

Cold Atom Physics and Bose-Einstein Condensates36 references7 citations

TL;DR

This paper derives a multidimensional generalization of the Natte series for the time- and distance-dependent reduced density matrix in the non-linear Schrödinger model using finite-volume form factor expansions. It enables exact asymptotic analysis of correlation functions in interacting integrable models by mapping their long-distance/long-time behavior to equivalent free-fermion systems, revealing contributions from excitations beyond the Fermi surface that go beyond CFT/Luttinger liquid predictions.

ABSTRACT

Starting from the form factor expansion in finite volume, we derive the multidimensional generalization of the so-called Natte series for the zero-temperature, time and distance dependent reduced density matrix in the non-linear Schrödinger model. This representation allows one to extit{read-off} straightforwardly the long-time/large-distance asymptotic behavior of this correlator. This method of analysis reduces the complexity of the computation of the asymptotic behavior of correlation functions in the so-called interacting integrable models, to the one appearing in free fermion equivalent models. We compute explicitly the first few terms appearing in the asymptotic expansion. Part of these terms stems from excitations lying away from the Fermi boundary, and hence go beyond what can be obtained by using the CFT/Luttinger liquid based predictions.

Motivation & Objective

• To derive an exact asymptotic representation for the time- and distance-dependent reduced density matrix in the non-linear Schrödinger model at zero temperature.
• To extend the Natte series formalism to multidimensional integrals, enabling systematic asymptotic analysis of correlation functions in interacting integrable models.
• To identify contributions to the asymptotic expansion from excitations away from the Fermi boundary, which are inaccessible via standard CFT or Luttinger liquid approaches.
• To establish a direct correspondence between the asymptotic behavior of interacting models and their free-fermion equivalents, simplifying complex correlation function computations.
• To compute the first few terms in the asymptotic expansion explicitly, providing quantitative insight into long-distance and long-time decay properties.

Proposed method

• Utilizes the form factor expansion in finite volume to construct a multidimensional generalization of the Natte series for the reduced density matrix.
• Applies the non-linear steepest-descent method to analyze Riemann–Hilbert problems associated with Fredholm determinants representing the correlators.
• Employs a contour integral representation for the Fredholm determinant, with the contour shrinking to the interval $[-q, q]$ in the large-$L$ limit.
• Derives the leading-order behavior of the density $ ho^{(L)}$ by solving a non-linear integral equation and showing convergence to a solution $ ho$ satisfying $(I + rac{ u}{2 au} R) ho = h$.
• Uses the Euler–Maclaurin formula to approximate Riemann sums arising from discrete sums in the finite-volume form factor expansion.
• Establishes uniform convergence of the functional $ ilde{ ho}^{(L)}$ to $ ho$ in the large-$L$ limit, ensuring the validity of the asymptotic expansion.

Experimental results

Research questions

• RQ1 How can the long-time and large-distance asymptotic behavior of the reduced density matrix be systematically extracted in the non-linear Schrödinger model?
• RQ2 To what extent do contributions from excitations away from the Fermi surface affect the asymptotic decay of correlation functions?
• RQ3 Can the asymptotic behavior of correlation functions in interacting integrable models be reduced to that of free-fermion models via a systematic analytical framework?
• RQ4 What is the structure of the asymptotic expansion for the reduced density matrix beyond the leading-order power-law decay predicted by CFT?
• RQ5 How does the finite-volume form factor expansion facilitate the derivation of exact asymptotic results in the thermodynamic limit?

Key findings

• The paper derives a multidimensional generalization of the Natte series, enabling exact asymptotic analysis of the time- and distance-dependent reduced density matrix in the non-linear Schrödinger model.
• The asymptotic expansion includes contributions from excitations lying away from the Fermi boundary, which are not captured by standard CFT or Luttinger liquid theories.
• The leading-order behavior of the reduced density matrix is shown to be equivalent to that of a free-fermion model, with the correspondence established through the resolvent of the Lieb kernel.
• The solution $ ho^{(L)}$ to the finite-volume non-linear integral equation converges uniformly to $ ho$ as $L o ty$, with an $O(L^{-1})$ error term.
• The functional $ ilde{ ho}^{(L)}$ converges to $ ho$ in the space of holomorphic functions on $U_ ho$, ensuring the validity of the asymptotic expansion in the large-$L$ limit.
• The first few terms in the asymptotic expansion are computed explicitly, providing a quantitative description of the long-distance and long-time decay of the reduced density matrix.