Scaling Property of the F-AF Spin Chain Near the Exactly Solvable Point

TL;DR

This paper analyzes the ground state properties of a frustrated spin chain near an exactly solvable point, revealing a scaling relation for the phase boundary between spin fluid and dimer phases through perturbation and numerical methods.

Contribution

It introduces a perturbative approach combined with numerical analysis to determine the phase boundary scaling near the solvable point of the $J_1$-$J_2$ spin chain.

Findings

Derived the phase boundary equation $ ext{alpha}_c = 14 ext{lambda}_c^{4/3}$.

Identified the scaling property of the ground state energy.

Mapped the phase transition between spin fluid and dimer phases.

Abstract

We investigate the ground state of the $J_1$-$J_2$ spin-1/2 chain with $J_1<0$ and $J_2>0$ in the case that the nearest-neighbor $J_1$ interaction in the $z$-direction has a weak anisotropy as $J_1 (1-\alpha)$. We perform a perturbational analysis for small $\alpha$ and $\lambda \equiv J_2- |J_1|/4$ with the exact solution of the unperturbed ground state for $\alpha = \lambda = 0$. The scaling property of the ground state energy is examined in detail. By the numerical diagonalization analysis of finite size systems, we found the phase boundary equation between the spin fluid and dimer phases as $\alpha_{\rm c} = 14 \lambda_{\rm c}^{4/3}$.