Effect of the hole doping into the Haldane-gap state on the one-dimensional $S=1$ $t$-$J$ model

TL;DR

This paper investigates how hole doping affects the ground state and excitations of the one-dimensional S=1 t-J model, revealing phase separation, Tomonaga-Luttinger liquid behavior, and persistent string correlations despite doping.

Contribution

It provides a detailed phase diagram of the doped S=1 chain, highlighting the coexistence of phase separation and Luttinger liquid phases, and explores the robustness of string correlations under doping.

Findings

Phase separation in weak-hopping region.

Tomonaga-Luttinger liquid phase in strong-hopping region.

String correlation persists with hole doping.

Abstract

The ground state and excitation spectra of the $S=1$ Heisenberg spin chain with hole hopping are investigated by means of the numerical-diagonalization method and by the help of the Zhang-Arovas picture. As for the charge sector, two phases are found. It is shown that in the weak-hopping region, a phase separation occurs, while in the strong-hopping region, the system is in the Tomonaga-Luttinger liquid phase. Irrespective of the presence of the spin gap, the present phase diagram is similar to that reported for the $S=1/2$ $t$-$J$ chain with gapless spin excitation rather than to that for the magnetically-frustrated $t$-$J$ chain with spin gap. For the spin sector, the string correlation is found to develop withstanding the hole doping except in a region, whose region is characterized by a generalized string correlation. This remains an open question to be studied in future.