Roles of easy-plane and easy-axis XXZ anisotropy and bond alternation in a frustrated ferromagnetic spin-$\frac{1}{2}$ chain

The spin-$1/2$ Heisenberg chain with a ferromagnetic first-neighbor exchange coupling $J_1$ and an antiferromagnetic second-neighbor $J_2$ has a Haldane dimer ground state with an extremely small spin gap. Thus the ground state is readily altered by perturbations. Here, we investigate the effects of XXZ exchange magnetic anisotropy of both the easy-axis and easy-plane types and an alternation in $J_1$ on the ground state, the spin gap, and magnetic properties of the frustrated ferromagnetic spin-$1/2$ chain. It is found that there are two distinct dimerized spin-gap phases, in one of which the spin gap and the magnetic susceptibility are extremely small around the SU(2) symmetric case and in the other they are moderately large far away from the SU(2) symmetric case. A small alternation in the amplitude of $J_1$ rapidly shortens the pitch of spin correlations towards the four-spin periodicity, as in the limit of $J_1/J_2\to 0$. These effects are not sufficient to quantitatively explain overall experimentally observed magnetic properties in the quasi-one-dimensional spin-gapped magnetoelectric cuprate ${\mathrm{Rb}}_2{\mathrm{Cu}}_2{\mathrm{Mo}}_3{\mathrm{O}}_{12}$ that exhibits ferroelectricity stabilized by a magnetic field. Our results are also relevant to ${\mathrm{Cs}}_2{\mathrm{Cu}}_2{\mathrm{Mo}}_3{\mathrm{O}}_{12}$, where the ferromagnetic intrachain and antiferromagnetic interchain order has recently been found, in a single chain level. We also reveal the nature of symmetry-protected topological phase transitions in the model by mapping onto effective spin-1 chain models.

Figure 1

(a) Ground-state phase diagram, (b) spin gap ${\mathrm{\Delta }}_{\mathrm{G}}$, (c) wave number $q_{\max }$ of the maximum spin-spin correlation function, and (d) transverse magnetic susceptibility ${\chi }^x$ of the bond-alternated XXZ model, ${\mathcal{H}}_{\mathrm{XXZ}}+{\mathcal{H}}_{\delta \mathrm{XXZ}}$. Note that in the panel (a), some symbols are overlapped with each other, forming straight lines. Thin solid lines for phase boundaries in (a) are guides to the eye. Black, green, and brown lines represent second-order, first-order, and either second-order or weakly first-order phase transitions. (See Sec. 3.) In the hatched areas, our iTEBD calculations using up to the matrix dimensions 300 did not converge. In the white regions in the figure panel (b), the spin gap vanishes, i.e., ${\mathrm{\Delta }}_{\mathrm{G}}=0$. In the white line respecting the SU(2) symmetry, i.e., ${\mathrm{\Delta }}^{xy}={\mathrm{\Delta }}^z=1$, the magnetic susceptibility vanishes, i.e., ${\chi }^x=0$. In (d), ${\chi }^x{|}_{h^x\to 0}=\infty$ in the TLL phase is depicted in pink color.

Figure 2

Order parameters ($M:◯$, ${\mathcal{O}}_{\mathrm{uudd}}:\square$, $(D^x+D^y)D^z:\bullet$) as functions of $J_1/J_2$ for ${\mathrm{\Delta }}^z=1$. Parameter sets $({\mathrm{\Delta }}^{xy},\delta )$ for (a)–(h) are shown at the left top of each panels. The regions filled in dirk gray, light gray, red, blue, and yellow colors correspond to the FPF, PPF, ${\mathrm{D}}_{+}$, ${\mathrm{D}}_{-}$, and UUDD phases, respectively.

Figure 3

Order parameters ($M:◯$, ${\mathcal{O}}_{\mathrm{uudd}}:\square$, $(D^x+D^y)D^z:\bullet$) and correlation length ($\xi :△$) as functions of ${\mathrm{\Delta }}^{xy}$ for ${\mathrm{\Delta }}^z=1$. A magnified view of ${\left({\mathcal{O}}_{\mathrm{uudd}}\right)}^8$ around ${\mathrm{\Delta }}^{xy}=0.92$ is shown in the inset of (b).

Figure 4

Energy difference between the lowest energy $E_1$ specified with $S_{\mathrm{T}}^z=1$ and the energy $E_{\mathrm{uudd}}$ of up-up-down-down state for ${\mathrm{\Delta }}^{xy}=0$ and ${\mathrm{\Delta }}^z=1$ under the periodic boundary condition. Each vertical dotted line indicates the FPF-UUDD phase transition points for each $\delta$. The region A, B, and C are $J_1/J_2>-1/(1-\delta )$, $-2/(1-\delta )<J_1/J_2<-1/(1-\delta )$, and $J_1/J_2<-2/(1-\delta )$, respectively. The UUDD state is a ground state in the region A and B, and the FPF state is a ground state in the region C.

Figure 5

Two parameter scaling for $\varepsilon$ and $\delta \tau$ with respect to $\chi$ up to 40 for ${\mathcal{H}}_{\mathrm{XXZ}}$ with $J_1/J_2=-5/3$ and ${\mathrm{\Delta }}^{xy}={\mathrm{\Delta }}^z=1$. Scaling parameters $c_1=1.09\pm 0.03$ and $c_2=-1.87\pm 0.06$ are estimated by the Bayesian inference method [78].

Figure 6

Example of $\chi$ dependence of the order parameters for ${\mathrm{\Delta }}_{}^{xy}=0.45$, ${\mathrm{\Delta }}_{}^z=1$, and $\delta =0.02$ with $\chi =100$, 200, and 300. The vertical broken lines are phase boundary shown in Fig. 1  or 2.

Figure 7

Example of a finite size scaling of the spin gap for $J_1^{}/J_2^{}=-2.56$, ${\mathrm{\Delta }}_{}^{xy}=0.9$, ${\mathrm{\Delta }}_{}^z=1$, and $\delta =0.2$ with $\chi =400$, 600, and 800. The data of $\chi =\infty$ are estimated by the linear fitting with respect to $1/\chi$.