The anisotropic Heisenberg model close to the Ising limit: triangular lattice vs. effective models

TL;DR

This work analyzes the easy-axis anisotropic Heisenberg model on the triangular lattice and its effective bipartite reductions on the honeycomb and square lattices. Using DMRG and ED, it compares magnetization curves $m(h)$, transverse order $m_ opple$, and spin stiffness $ρ_s$, showing that reduced models reproduce many finite-field TL features near the correspondence regime ($m>1/2$), but TL at $h=0$ remains gapped for $α\lesssim α^*$ with a crossover to gapless behavior around $α^*\lesssim 0.5$. It demonstrates significant LSWT limitations in the small-$α$ and small-$h$ regime due to magnon repulsion and correlations, and identifies a gapped spin-solid phase for $h<h^*(α)$ with a supersolid-Y region for $h_*<h<h_c$. The results help interpret experiments on KSCO and highlight the need for beyond-LSWT theories to capture the full evolution of gaps, magnetization, and transverse order in frustrated quantum magnets.

Abstract

Stimulated by recent experiments on materials representing the realization of the anisotropic Heisenberg spin-$1/2$ model on the triangular lattice, we explore further properties of such a model in the easy-axis regime $α= J_\perp/J_z < 1$, as well as effective models that also capture such physics. We show that anisotropic Heisenberg models on the honeycomb lattice and even on the square lattice reveal similarities to the full triangular lattice in the magnetization curve as well as in the transverse magnetization (superfluid) order parameter $m_\perp$ at finite fields. Still, at $α\ll 1$, results reveal gapless excitations and small but finite $m_\perp >0 $ at effective fields corresponding to the triangular case without the field. In contrast, several additional numerical studies of the full model on the triangular lattice confirm the existence of the gap at $α\ll 1$. In particular, the magnetization curve $m(h)$ as well as the spin stiffness $ρ_s$ indicate (at zero field) a transition/crossover from gapped to gapless regime at $α\sim α^*$ with $α^* \lesssim 0.5$. We also show that deviations from the linear spin-wave theory and the emergence of the gap can be traced back to the strong effective repulsion between magnon excitations, having similarity to strongly correlated systems.

Figures (8)

• Figure 1: Magnetization curves $m(h)$ vs. renormalized fields $\tilde{h} = h/(zSJ)$ for (a) the honeycomb lattice (HcL) and ( b) the square lattice (SqL), for different anisotropies $\alpha = 0.1, 0.2, 0.5$, obtained with DMRG calculation on lattices with $N=72$ sites and $N=64$ sites, respectively. The gray dashed lines indicate $m=1/2$ and $\tilde{h} = 1$, corresponding to $m=0$ and $h=0$, respectively, within the TL model. The coloured dashed lines represent the augmented LSWT results for $m(\tilde{h})$, discussed in Sec. IV.
• Figure 2: Transverse order parameter $m_\perp^2$ vs. magnetization $m$, extracted from ED results for DSSP on systems with $N =36$ and $N=40$ sites on (a) HcL, and (b) SqL, for different $\alpha = 0.1 - 1.0$. Vertical $m=1/2$ line indicates the correspondence to the $h=m=0$ model on TL. The gray dashed curve represents the expected LSWT dependence $m^2_\perp = \zeta(1 - m^2)$ with $\zeta = 1/8$.
• Figure 3: Transverse order parameter $m_\perp^2$ at $m = 1/2$ for HcL : (a) finite-size $1/N$ scaling of results for different $\alpha$, (b) $m_\perp^2$, obtained on $N=40$ sites, and extrapolated $N \to \infty$ values vs. $\alpha$.
• Figure 4: Magnetization curve $m$ vs. normalized field $h/(\alpha J)$, for $\alpha =0.1$, obtained with ED for $N = 30, 36$ lattices and via DMRG for $N=48-72$ lattices, with the quadratic extrapolation (red dashed line). The gray dashed line represents the result of LSWT, explained and discussed in Sec. IV.
• Figure 5: Scaled gap $h^*/(\alpha J)$ vs. $\alpha$, as obtained by extrapolation of $m(h)$ results presented in Fig. \ref{['fig4']} and in Appendix \ref{['chap_append']}.