Vector-spin-chirality order in a dimerized frustrated spin-$\frac{1}{2}$ chain

A frustrated spin-$1/2$ XXZ chain model comprising a ferromagnetic nearest-neighbor coupling with the bond alternation, $J_1(1\pm \delta )<0$, and an antiferromagnetic second-neighbor exchange coupling $J_2>0$ is studied at zero and weak magnetic fields by means of density-matrix renormalization-group calculations of order parameters, correlation functions, and the entanglement entropy, as well as an Abelian bosonization analysis. At zero magnetic field, the bond alternation $\delta >0$ suppresses the gapless phase characterized by a vector-chiral (VC) long-range order (LRO) and a quasi-LRO of an incommensurate spin spiral, whereas this phase occupies a large region in the space of $J_1/J_2$ and the easy-plane exchange anisotropy for $\delta =0$ [S. Furukawa et al., Phys. Rev. Lett. 105, 257205 (2010)]. Then, four gapped phases are found to appear as the exchange anisotropy varies from the SU(2)-symmetric case to the U(1)-symmetric case: the Haldane dimer (D${}_{+}$) phase with the same sign of the $x,y$- and $z$-component dimer order parameters, two VC dimer (VCD${}_{+}$/VCD${}_{-}$) phases with the sign of the $z$-component dimer order parameter being unaltered/reversed, and the even-parity dimer (D${}_{-}$) phase. At small magnetic fields, a field-induced ring-exchange interaction, which is proportional to a staggered scalar chirality and a magnetic flux penetrating the associated triangle, drives a transition from the D${}_{-}$ phase into a VC-Neel-dimer (VCND) phase, but not from the D${}_{+}$ phase. This VCND phase is stable up to the large magnetic field at which the Zeeman term closes the spin gap. A possible relevance to Rb${}_2$Cu${}_2$Mo${}_3$O${}_{12}$ is discussed.

Figure 1

(a) The lattice structure and (b)–(d) the phase diagrams of ${\widehat{\mathcal{H}}}_{\delta \mathrm{xxz}}$ [Eq. (1)] for (b) $\delta =0$, (c) $\delta =0.02$, and (d) $\delta =0.04$. The phase diagrams are obtained with DMRG calculations up to the system size $N=320$ and the 300 renormalized basis states ($m=300$). We have also checked a convergence of the results by increasing $m$ up to 500 in the case of $J_1/J_2=-2.5$ and $\delta =0.02$. A possible VC Haldane dimer (VCD${}_{+}$) phase is restricted inside the red shaded area at most in (b). A possible gapless VC dimer (VCD${}_0$) phase with only the $x,y$-component dimer order is restricted inside the black shaded area at most in (c) and (d).

Figure 2

Correlation functions $C_N^{\left(\alpha \right)}\left(r\right)$ for (a) $\Delta =1.00$ in the Haldane dimer (D${}_{+}$) phase, (b) $\Delta =0.92$ near the boundary between the D${}_{+}$ phase and VCD${}_{+}$ phase, (c) $\Delta =0.914$ in the VCD${}_{+}$ phase, (d) $\Delta =0.88$ near the boundary between the VCD${}_{+}$ phase and VCD${}_{-}$ phase or in a very narrow gapless VCD${}_0$ phase, (e) $\Delta =0.80$ in the VCD${}_{-}$ phase, and (f) $\Delta =0.70$ in the even-parity dimer (D${}_{-}$) phase in the case of $J_1/J_2=-2.5$ and $\delta =0.02$. The left and middle subpanels of (a)–(e) show the spatial decays of $C_{N_0}^{\left(\alpha \right)}\left(r\right)$ with $N_0=320$ and the $N$ dependence of $C_N^{\left(\alpha \right)}\left(r_N\right)$ with $r_N=N/4$ for $N=80,112,160,224$ and 320, respectively [$C_N^{\left(O_1^z\right)}\left(r\right)$: $cmykblue◯$; $C_N^{\left(O_2^z\right)}\left(r\right)$: $cmykpurple\diamond$; $C_N^{\left({\kappa }^z\right)}\left(r\right)$: $\square$; $C_N^{\left(D^z\right)}\left(r\right)$: $cmykgreen△$; $C_N^{\left(D^x\right)}\left(r\right)$: $cmykred▽$]. The right subpanels show the spatial decay of spin correlation functions ($C_N^{\left(S^z\right)}\left(r\right):cmykblue+$, $C_N^{\left(S^x\right)}\left(r\right):cmykred\times$). The results are obtained with $m=500$ and are almost unchanged within the symbol size by taking $m=300$. For a guide to the eyes, the cubic-spline interpolated curve is plotted for each correlation function.

Figure 3

(a) Bulk order parameters ${\mathrm{Ave}}_j\langle {\widehat{\kappa }}_{j+1/2}^z\rangle$, ${\mathrm{Ave}}_j\langle {\widehat{D}}_j^x\rangle$, and ${\mathrm{Ave}}_j\langle {\widehat{D}}_j^z\rangle$ extrapolated to $N\to \infty$ for $J_1/J_2=-2.5$ and $\delta =0.02$. See the text for the broken lines and the gray line. (b)–(e) Order parameters for $N=80$, 112, 160, 224, and 320. The order parameters in the bulk limit are estimated by a linear extrapolation of the data for $N=224$ and 320. The error bars in (a) are chosen to be the difference between the data for $N=320$ and the extrapolated value. All of the results are obtained with $m=500$ and are unchanged by taking $m=400$ within the symbol size.

Figure 4

(a) Finite-size dependence of $\tilde{c}$ around the D${}_{-}$-VCD${}_{-}$ boundary (${\Delta }_{{\mathrm{D}}_{-}{\text{-VCD}}_{-}}$) for $J_1/J_2=-2.5$ and $\delta =0.02$ ($m=500$). (b) The peak position ${\tilde{\Delta }}_{{\mathrm{D}}_{-}{\text{-VCD}}_{-}}$ of $\tilde{c}$ vs $1/N$ are open squares. The filled symbol represents ${\Delta }_{{\mathrm{D}}_{-}{\text{-VCD}}_{-}}$ determined from Fig. 3. (c) Finite-size dependence of $\tilde{c}$ at the criticality ${\Delta }_{{\mathrm{D}}_{-}{\text{-VCD}}_{-}}$. The gray lines are a guide to the eyes.

Figure 5

(a) Schematic picture of the staggered scalar spin chirality and (b) phase diagram of ${\widehat{\mathcal{H}}}_{\delta \mathrm{xxz}}+{\widehat{\mathcal{H}}}_{\mathrm{ssc}}$ around $\Delta =0.7$ for $J_1/J_2=-2.5$ and $\delta =0.02$. The number $m$ of the renormalized basis state in the DMRG method is 300.

Figure 6

Correlation functions $C_N^{\left(\alpha \right)}\left(r\right)$ for (a) $\Delta =0.712$, $gh=0.00$ in the VCD${}_{-}$ phase, (b) $\Delta =0.712$, $gh=0.30$ in the VCND phase, and (c) $\Delta =0.702$, $gh=0.06$ in the D${}_{-}$ phase. The left and middle subpanels of (a)–(c) show the spatial decays of $C_{N_0}^{\left(\alpha \right)}\left(r\right)$ with $N_0=320$ and the $N$ dependence of $C_N^{\left(\alpha \right)}\left(r_N\right)$ with $r_N=N/4$ for $N=80,112,160,224$, and 320, respectively [$C_N^{\left(O_1^z\right)}\left(r\right)$: $cmykblue◯$; $C_N^{\left(O_2^z\right)}\left(r\right)$: $cmykpurple\diamond$; $C_N^{\left({\kappa }^z\right)}\left(r\right)$: $\square$; $C_N^{\left(D^z\right)}\left(r\right)$: $cmykgreen△$; $C_N^{\left(D^x\right)}\left(r\right)$: $cmykred▽$]. The right subpanels show the spatial decay of spin-correlation functions ($C_N^{\left(S^z\right)}\left(r\right)$: $cmykblue+$; $C_N^{\left(S^x\right)}\left(r\right)$: $cmykred\times$). The results are obtained by taking $m=300$.

Figure 7

(a) Energy difference between lowest energy of $S_{\mathrm{tot}}^z=1$ ($E_1$) and $S_{\mathrm{tot}}^z=0$ ($E_0$). Interpolation curves of $E_1-E_0$ as a function of $h$ are the cubic spline. The level cross point, namely, $E_1-E_0=0$, is represented by ${\tilde{h}}_{\mathrm{c}}$. (b) Finite-size dependence of ${\tilde{h}}_{\mathrm{c}}$. The finite-size data are naively extrapolated to $h_c\simeq 0.0459$ as $N\to \infty$ using the polynomial function.

Figure 8

Phase diagram of low magnetic field region for ${\widehat{\mathcal{H}}}_{\delta \mathrm{xxz}}+{\widehat{\mathcal{H}}}_{\mathrm{ssc}}+{\widehat{\mathcal{H}}}_Z$ at $J_1/J_2=-2.5$, $\delta =0.02$: (a) $\Delta =0.71$, (b) $\Delta =0.708$, and (c) $\Delta =0.706$. The broken lines between VCD${}_{-}$ and D${}_{-}$ in (b) and (c) represent $gh=0.16$ and $0.22$, respectively, which are the transition values as shown in Fig. 5. The number of retain state $m$ in the DMRG method is 300.

Figure 9

Spatially averaged order parameters for (a) $\Delta =0.8$ and (b) $\Delta =0.91$, where $J_1/J_2=-2.5$, $\delta =0.02$, $N=320$, and $m=500$. The parameter $L$ represents the number of sites over which the spatial average is taken.

Figure 10

(a) Vector-chirality order parameter ${\mathrm{Ave}}_j\langle {\widehat{\kappa }}_{j+1/2}^z\rangle$ for $N$ up to 320 ($J_1/J_2=-2.5$ and $\delta =0.02$). (b) The $N$ dependence of ${\tilde{\Delta }}_{{\mathrm{D}}_{-}{\text{-VCD}}_{-}}$. The broken line represents a linear extrapolation of the two largest-$N$ results. (c) ${\left({\mathrm{Ave}}_j\langle {\widehat{\kappa }}_{j+1/2}^z\rangle \right)}^8$ vs $\Delta$ near the ${\Delta }_{{\mathrm{D}}_{+}{-\mathrm{VCD}}_{+}}$ phase boundary. The broken line represents the least-squares fit. All of the results are obtained with $m=500$ and are unchanged within the symbol size by taking $m=400$.