Quasiparticle structure and coherent propagation in the $t-J_{z}-J_{\perp}$ model

TL;DR

This paper models the quasiparticle in a 2D quantum antiferromagnet using a generalized $t-J_{z}-J_{ot}$ model, deriving an effective Hamiltonian to understand its coherent propagation and comparing results with numerical data.

Contribution

It introduces a method to visualize and analyze the quasiparticle formation and propagation in the $t-J_{z}-J_{ot}$ model by diagonalizing the $t-J_{z}$ model and applying perturbation theory in $J_{ot}.

Findings

Quasiparticle properties are robust when extending from anisotropic to isotropic limits.

Derived quasiparticle dispersion matches well with numerical results for small clusters.

Effective Hamiltonian captures the interaction between quasiparticles and spin waves.

Abstract

Numerical studies, from variational calculation to exact diagonalization, all indicate that the quasiparticle generated by introducing one hole into a two-dimensional quantum antiferromagnet has the same nature as a string state in the $t-J_{z}$ model. Based on this observation, we attempt to visualize the quasiparticle formation and subsequent coherent propagation at low energy by studying the generalized $t-J_{z}-J_{\perp}$ model in which we first diagonalize the $t-J_{z}$ model and then perform a {\em degenerate} perturbation in $J_{\perp}$. We construct the quasiparticle state and derive an effective Hamiltonian describing the coherent propagation of the quasiparticle and its interaction with the spin wave excitations in the presence of the N\'{e}el order. We expect that qualitative properties of the quasiparticle remain intact when analytically continuing $J_{\perp}$ from the anisotropic $J_{\perp} < J_{z}$ to the isotropic $J_{\perp} = J_{z}$ limit, despite the fact that the spin wave excitations change from gapful to gapless. Extrapolating to $J_{\perp}=J_{z}$, our quasiparticle dispersion and spectral weight compare well with the exact numerical results for small clusters.