Riemann-Hilbert approach to the time-dependent generalized sine kernel

Abstract

We derive the leading asymptotic behavior and build a new series representation for the Fredholm determinant of integrable integral operators appearing in the representation of the time and distance dependent correlation functions of integrable models described by a six-vertex R-matrix. This series representation opens a systematic way for the computation of the long-time, long-distance asymptotic expansion for the correlation functions of the aforementioned integrable models away from their free fermion point. Our method builds on a Riemann-Hilbert based analysis.

Fredholm determinants, full counting statistics and Loschmidt echo for domain wall profiles in one-dimensional free fermionic chains

“…First, we make an approximation for the kernel (11), which allows us to split it into the generalized sine-kernel and regular corrections, which can be disregarded in the leading order. Second, we obtain the leading asymptotics of log χ(λ) at t → ∞ using known results for the generalized sine-kernel [31,67,68]. The thus obtained leading asymptotics coincides with the Levitov-Lesovik formula for FCS [21][22][23] obtained within what can be called the hydrodynamic or semiclassical approach.…”

Time and temperature-dependent correlation function of an impurity in one-dimensional Fermi and Tonks–Girardeau gases as a Fredholm determinant

“…Let us discuss how our results can be used to investigate the mobile impurity dynamics. Various asymptotic formulas for a Fredholm determinant can be obtained by formulating the matrix Riemann-Hilbert problem and solving it asymptotically [1,[22][23][24][25]. In model (2.12) Fredholm determinant representations are known in the  ¥ g limit [12,13], and corresponding asymptotic solutions of the matrix Riemann-Hilbert problem have been discussed in [17,[26][27][28].…”

Correlation functions of one-dimensional strongly interacting two-component gases

“…The Fredholm determinants appearing in the representations for the dynamic correlators are also integrable which means that their asymptotics can be derived by solving an associated RHP. Their kernels are particular cases of the time dependent generalization of the sine-kernel for which a comprehensive asymptotic analysis was performed in [108] by Kozlowski. We are interested in the large x behavior of the dynamic correlators for a fixed value of the saddle point k 0 = x/2t.…”

“…The last relation shows that lim k→∞ χ(k) = I 2 + 1 k The asymptotic solution of the RHP in both space-like and time-like regime can be found in [108] and is briefly presented in Appendices D and E. In this section we assume x > 0 and t > 0. The asymptotic behavior of the time dependent Fredholm determinant is given by the following theorem:…”

“…This constitutes the anyonic generalization of the fermionic and bosonic result obtained by Izergin and Pronko [106]. The infrared asymptotics of the correlators at zero temperature and zero magnetic field are computed using the solutions of the associated Riemann-Hilbert problems to the static and time generalization of the sine-kernel [107,108]. The spin part of the asymptotics is exponentially decaying (the correlation length is independent on statistics) coupled with an oscillatory component with frequency depending on the statistics parameter while the charge part presents scaling with anomalous exponents.…”