Boundary versus bulk behavior of time-dependent correlation functions in one-dimensional quantum systems

TL;DR

This paper investigates how boundaries affect time-dependent correlation functions in one-dimensional quantum systems, revealing oscillation behaviors and decay patterns that differ between integrable and nonintegrable models, supported by numerical simulations.

Contribution

It extends mobile impurity models to open boundary conditions, providing new insights into boundary effects on correlation functions in quantum fluids.

Findings

Boundary autocorrelations oscillate with the same frequency as bulk in integrable models.

Power-law decay of oscillations at the boundary differs from the bulk, with boundary decay becoming exponential in nonintegrable models.

Numerical tDMRG results confirm theoretical predictions for various spin chains.

Abstract

We study the influence of reflective boundaries on time-dependent responses of one-dimensional quantum fluids at zero temperature beyond the low-energy approximation. Our analysis is based on an extension of effective mobile impurity models for nonlinear Luttinger liquids to the case of open boundary conditions. For integrable models, we show that boundary autocorrelations oscillate as a function of time with the same frequency as the corresponding bulk autocorrelations. This frequency can be identified as the band edge of elementary excitations. The amplitude of the oscillations decays as a power law with distinct exponents at the boundary and in the bulk, but boundary and bulk exponents are determined by the same coupling constant in the mobile impurity model. For nonintegrable models, we argue that the power-law decay of the oscillations is generic for autocorrelations in the bulk, but turns into an exponential decay at the boundary. Moreover, there is in general a nonuniversal shift of the boundary frequency in comparison with the band edge of bulk excitations. The predictions of our effective field theory are compared with numerical results obtained by time-dependent density matrix renormalization group (tDMRG) for both integrable and nonintegrable critical spin-$S$ chains with $S=1/2$, $1$ and $3/2$.