Spin Gap and String Order Parameter in the Ferromagnetic Spiral Staircase Heisenberg Ladder: A Quantum Monte Carlo Study

We consider a spin-$1/2$ ladder with a ferromagnetic rung coupling $J_{\perp }$ and inequivalent chains. This model is obtained by a twist ($\theta$) deformation of the ladder and interpolates between the isotropic ladder ($\theta =0$) and the $SU(2)$ ferromagnetic Kondo necklace model ($\theta =\pi$). We show that the ground state in the ($\theta$, $J_{\perp }$) plane has a finite string order parameter characterizing the Haldane phase. Twisting the chain introduces a new energy scale, which we interpret in terms of a Suhl-Nakamura interaction. As a consequence we observe a crossover in the scaling of the spin gap at weak coupling from $\Delta /J_{\parallel }\propto J_{\perp }/J_{\parallel }$ for $\theta <{\theta }_c\simeq 8\pi /9$ to $\Delta /J_{\parallel }\propto (J_{\perp }/J_{\parallel }{)}^2$ for $\theta >{\theta }_c$. Those results are obtained on the basis of large scale quantum Monte Carlo calculations.

Figure 1

(a) Sketch of Spiral Staircase Heisenberg Ladder. (b) View of the model from the top. (c) For $\theta =\pi$, the model maps to the 1D $SU(2)$ ferromagnetic Kondo necklace model [9].

Figure 2

(a) Spin gap $\Delta (J_{\perp })$ as a function of $|J_{\perp }/J_{\parallel }|$ for different twist angles $\theta$. The gap is rescaled by $J_{\mathrm{eff}}=\frac{J_{\parallel }}{4}[1+{cos}^2(\theta /2)]$ such that in the large-$|J_{\perp }|$ limit, it converges asymptotically toward the Haldane gap of a spin-1 chain. At weak couplings, we have carried out QMC simulations up to $\beta J_{\parallel }=2500$ and $2\times 512$ spins to ensure size and temperature convergence. Inset: zoom on the weak-coupling region. (b) Results for spin gap on a semilogarithmic scale.

Figure 3

Spin-spin correlation function (a) on the first leg and (b) on the second leg for the Kondo necklace model ($\theta =\pi$) at different couplings $J_{\perp }/J_{\parallel }$ on a $2\times 800$ lattice. Simulations are carried out at $\beta J_{\parallel }=7000$ ($J_{\perp }/J_{\parallel }=-0.3$, $-0.4$), $\beta J_{\parallel }=5000$ ($J_{\perp }/J_{\parallel }=-0.5$), and $\beta J_{\parallel }=2000$ ($J_{\perp }/J_{\parallel }=-0.6$).

Figure 4

Dynamical spin-spin correlations at $J_{\perp }/J_{\parallel }=1$ for the ladder system ($\theta =0$) and the Kondo necklace model ($\theta =\pi$). Here we consider a bonding combination of the spins across the rungs ($\beta J_{\parallel }=200$, $L=100$).

Figure 5

(a) String order parameter ${\mathcal{O}}_s$ and ${\mathcal{O}}_H$ as a function coupling $J_{\perp }/J_{\parallel }$ and several twist angles. For $\theta =8\pi /9$, $\pi$ finite-size effects are still present for the considered $L=800$ lattice in the parameter range $|J_{\perp }/J_{\parallel }|<0.5$. For $|J_{\perp }/J_{\parallel }|>1.0$ the system size $L=400$ is sufficiently large enough to guarantee convergence. Simulations are carried out up to $\beta J_{\parallel }=7000$. (b) Finite-size scaling of the order parameters for the parameter sets $J_{\perp }/J_{\parallel }=-0.2$, $\theta =8\pi /9$ (blue) and $J_{\perp }/J_{\parallel }=-0.3$, $\theta =\pi$ (red). The data for ${\mathcal{O}}_H$ are fitted to the form: ${\mathcal{O}}_H\propto L^{-\alpha }\exp (-L/\xi )$.