Nearest-neighbor approximation in one-excitation state evolution along spin-1/2 chain governed by XX-Hamiltonian

TL;DR

This paper investigates the limits of nearest-neighbor interaction approximation in long-time spin dynamics of spin-1/2 chains with power-law decaying interactions, identifying a critical decay exponent for approximation validity.

Contribution

It determines the critical decay exponent for the applicability of NNI in long-time evolution of spin chains governed by the XX-Hamiltonian.

Findings

Found the logarithmic dependence of the critical decay exponent on chain length.

Established the boundary for NNI applicability in systems with power-law interactions.

Analyzed the evolution of one-excitation states under different interaction decay rates.

Abstract

The approximation of nearest neighbor interaction (NNI) is widely used in short-time spin dynamics with dipole-dipole interactions (DDI) when the intensity of spin-spin interaction is $\sim 1/r^3$, where $r$ is a distance between those spins. However, NNI can not approximate the long time evolution in such systems. We consider the system with the intensity of the spin-spin interaction $\sim 1/r^{\alpha}$, $\alpha\ge 3$, and find the low boundary $\alpha_c$ of applicability of the NNI to the evolution of an arbitrary one-excitation initial quantum state in the homogeneous spin chain governed by the $XX$-Hamiltonian. We obtain the logarithmic dependence of $\alpha_c$ on the chain length.