Combined analytical and numerical approach to magnetization plateaux in one-dimensional spin tube antiferromagnets

In this paper, we investigate the properties of frustrated three-leg spin-$S$ tubes under a magnetic field. We concentrate on two kind of geometries for these tubes, one of which is relevant for the compound $\left[{\left({\mathrm{CuCl}}_2\mathrm{tachH}\right)}_3\mathrm{Cl}\right]{\mathrm{Cl}}_2$. We perform an analytical large-$S$ path-integral approach to determine the phase diagrams, which exhibit several magnetization plateaux. Moreover, by combining the field-theory approach with a strong-coupling one, we also investigate the issue of gapless nonmagnetic excitations on some plateaux, dubbed chirality degrees of freedom for both tubes: on increasing the intertriangle exchange couplings, a gapless chiral phase transforms into a gapped “ferrochiral” state, possibly through intermediate “spin-imbalanced” states. All these predictions are confirmed numerically using large-scale density-matrix renormalization-group simulations for the $S=3/2$ case.

Figure 1

Lattice structure of the simple spin tube.

Figure 2

Scaling dimension $\delta$ of the cosine operator in the action (13) as a function of $J_{\parallel }/J_{\perp }$ for $S=3/2$ at the values of $m$ fulfilling the condition (14). The perturbation is relevant and opens a gap in the spectrum if $\delta <2$, thus the plateaux should disappear for $J_{\parallel }/J_{\perp }\gtrsim 0.1$.

Figure 3

Values of the effective $XXZ$ Hamiltonian parameters $J_{xy}$, $J_z$, and $\Delta =J_z/J_{xy}$ describing the chirality on the $m=7/6$ plateau for a spin tube $S=3/2$. The inset shows that at a critical coupling $J_{\parallel ,c}^h/J_{\perp }=0.042$, the chirality excitations become gapped and the model enters the ferrochiral phase.

Figure 4

Magnetization per spin curve ($m$) vs magnetic field $h$ for the simple spin tube in the case $S=3/2$. DMRG simulations were performed with $L=64$ and $J_{\parallel }/J_{\perp }=0.1$. Finite-size steps are almost not visible on this scale.

Figure 5

Finite-size scaling of the plateaux widths for different couplings $J_{\parallel }/J_{\perp }$.

Figure 6

Phase diagram of the three-leg spin tube with $S=3/2$ as a function of the coupling $J_{\parallel }$ and magnetic field $h$. Several magnetization plateaux can be observed (filled areas) and an additional $m=0$ plateau is found (bold line). Data correspond to numerical simulations on $3\times 32$ lattice with DMRG. Inside the extreme plateaux $m=1/6$ and $7/6$, hashed red regions correspond to critical chirality phases (see Sec. 2d2).

Figure 7

Block entropy $S_{vN}\left(\ell \right)$ vs block length $d\left(\ell \right|L)$ (starting at one end of the tube) for several magnetization plateaux on $L=32$ spin tube. Coupling constants are fixed to $J_{\parallel }/J_{\perp }=0.02$.

Figure 8

Block entropy $S_{vN}\left(\ell \right)$ vs block length $d\left(\ell \right|L)$ (starting at one end of the tube) for $L=16$ (open symbols) and $L=32$ (filled symbols). Coupling constants are fixed to $J_{\parallel }/J_{\perp }=0.1$. (a) Magnetizations corresponding to plateaux; (b) intermediate magnetizations.

Figure 9

Local magnetizations obtained by DMRG simulations on a $3\times 32$ spin tube with spin 3/2 on the $m=7/6$ plateau. Upper and lower panels correspond, respectively, to (a) $J_{\parallel }/J_{\perp }=0.04$ and (b) $J_{\parallel }/J_{\perp }=0.06$.

Figure 10

Lattice structure of the twisted spin tube.

Figure 11

Magnetization curve ($m$) vs magnetic field $h$ for the twisted three-leg spin tube in the case $S=3/2$. DMRG simulations were performed with $L=32$ and $J_d/J_{\perp }=0.1$.

Figure 12

Phase diagram of the twisted spin tube with $S=3/2$ as a function of the couplings $J_d$ and magnetic field $h$. Several magnetization plateaux can be observed (filled regions) and an additional $m=0$ plateau is also found (but is much smaller and not shown on the figure). Data correspond to numerical simulations on $3\times 32$ lattice with DMRG. Red hashed areas correspond to critical chirality regions (see text).

Figure 13

Local magnetizations obtained by DMRG simulations on a $3\times 32$ twisted spin tube with spin 3/2 on the $m=1/6$ plateau. Upper and lower panels correspond, respectively, to (a) $J_d/J_{\perp }=0.2$ and (b) $J_d/J_{\perp }=0.3$.