Vector chiral and multipolar orders in the spin-$\frac{1}{2}$ frustrated ferromagnetic chain in magnetic field

We study the one-dimensional spin-1/2 Heisenberg chain with competing ferromagnetic nearest-neighbor $J_1$ and antiferromagnetic next-nearest-neighbor $J_2$ exchange couplings in the presence of magnetic field. We use both numerical approaches (the density-matrix renormalization-group method and exact diagonalization) and effective-field-theory approach and obtain the ground-state phase diagram for wide parameter range of the coupling ratio $J_1/J_2$. The phase diagram is rich and has a variety of phases, including the vector chiral phase, the nematic phase, and other multipolar phases. In the vector chiral phase, which appears in relatively weak magnetic field, the ground state exhibits long-range order (LRO) of vector chirality which spontaneously breaks a parity symmetry. The nematic phase shows a quasi-LRO of antiferronematic spin correlation and arises as a result of formation of two-magnon bound states in high magnetic fields. Similarly, the higher multipolar phases, such as triatic $\left(p=3\right)$ and quartic $\left(p=4\right)$ phases, are formed through binding of $p$ magnons near the saturation fields, showing quasi-LRO of antiferromultipolar spin correlations. The multipolar phases cross over to spin-density-wave phases as the magnetic field is decreased before encountering a phase transition to the vector chiral phase at a lower field. The implications of our results to quasi-one-dimensional frustrated magnets (e.g., ${\text{LiCuVO}}_4$) are discussed.

Figure 1

(Color online) Magnetic phase diagram of the spin-1/2 zigzag chain with ferromagnetic $J_1$ and antiferromagnetic $J_2$ (a) in the $J_1/J_2$ versus $h/J_2$ plane and (b) in the $J_1/J_2$ versus $M$ plane. Crosses show the transition and crossover points obtained from the magnetization curves and correlation functions. In (a), symbols VC, N, IN, T, Q, and F indicate the vector chiral $\left(\Delta S_{\text{tot}}^z=1\right)$, nematic $\left(\Delta S_{\text{tot}}^z=2\right)$, incommensurate nematic $\left(\Delta S_{\text{tot}}^z=2\right)$, triatic $\left(\Delta S_{\text{tot}}^z=3\right)$, quartic $\left(\Delta S_{\text{tot}}^z=4\right)$, and ferromagnetic (fully polarized) phases, respectively. Here $\Delta S_{\text{tot}}^z$ is the unit of changes in the total $S_{\text{tot}}^z={\sum }_l{s}_l^z$ when the magnetic field $h$ is swept. There are also two kinds of spin-density-wave phases: ${\text{SDW}}_2$ $\left(\Delta S_{\text{tot}}^z=2\right)$ and ${\text{SDW}}_3$ $\left(\Delta S_{\text{tot}}^z=3\right)$, which are related to the nematic and triatic phases, respectively. The solid curve shows the saturation field $h_s$ and dotted lines are the guide for the eyes. In (b), symbols indicate parameter points for which their ground-state phase is identified by analysis of correlation functions. Shaded regions in (b) correspond to the magnetization jump at the first-order transition (see Sec. 4).

Figure 2

(Color online) Typical behaviors of various correlation functions in (a) the vector chiral (VC) phase, (b) the nematic (N) phase, (c) the ${\text{SDW}}_2$ phase, (d) the incommensurate nematic (IN) phase, (e) the triatic (T) phase, and (f) the ${\text{SDW}}_3$ phase. Absolute values of spatially averaged correlation functions are plotted. (For the averaging procedure, see Sec. 5B.) In (b) and (d), the triatic correlation function ${⟨s_0^{+}{s}_1^{+}{s}_2^{+}{s}_r^{-}{s}_{r+1}^{-}{s}_{r+2}^{-}⟩}_{\text{av}}$ is smaller than ${10}^{-5}$.

Figure 3

Schematic picture of the vector chiral order. The arrows on bonds indicate breaking of the parity symmetry by the vector chiral order ${\kappa }_l^{\left(n\right)}={\left({\mathit{s}}_l\times {\mathit{s}}_{l+n}\right)}^z$, which obeys the relation $J_1⟨{\kappa }^{\left(1\right)}⟩+2J_2⟨{\kappa }^{\left(2\right)}⟩=0$. The circulation of the $s^z$ spin current, shown by the dashed arrows, is alternating, and there is no net spin current flow.

Figure 4

Schematic picture of antiferronematic quasi-long-range order in the nematic phase. Ellipses represent directors of the nematic order on each bond.

Figure 5

Schematic pictures of incommensurate nematic quasi-long-range order in the incommensurate chiral nematic phase. Ellipses represent directors of the nematic order on each bond. The numerical results in Sec. 7 indicate that this chiral symmetry is not broken in the incommensurate nematic phase; the ground state is given by equal superposition of the upper and lower configurations.

Figure 6

Schematic picture of antiferrotriatic quasi-long-range order in the triatic phase. Solid triangles represent spin structure of the triatic order formed by three $s=1/2$ spins on each plaquette.

Figure 7

Dependence on $J_1/J_2$ of the center-of-mass momentum $k$ for the two-magnon bound state. The momentum deviates continuously from $\pi$ at $J_1/J_2\simeq -2.669$. The incommensurate momentum $k$ is fitted well to $k/\pi =1-0.67\sqrt{-2.669-J_1/J_2}$ as shown by the solid curve. The dashed line is the classical estimate $k=2\text{arccos}\left(-J_1/4J_2\right)$ for two scattering magnons.

Figure 8

(Color online) Dispersion curves of multimagnon bands at the saturation field for (a) $J_1/J_2=-3.0$ and (b) $J_1/J_2=-3.6$. The solid curves are the dispersions of bound states and the dashed curves show the lower edges of the continuum of scattering states. For clarity, only the states of up to six magnons are shown in (a) and up to eight magnons in (b). The numbers printed beside the curves denote the number of magnons. Bound states inside the scattering continuum are not shown.

Figure 9

Magnetization curves for (a) $J_1/J_2=-2.0$, (b) $J_1/J_2=-2.4$, (c) $J_1/J_2=-2.5$, (d) $J_1/J_2=-3.0$, (e) $J_1/J_2=-3.4$, and (f) $J_1/J_2=-3.6$. The dotted lines represent the boundaries of the regions of $\Delta S_{\text{tot}}^z=1$ and $\Delta S_{\text{tot}}^z\ge 2$.

Figure 10

(Color online) Averaged correlation functions for $L=120$ spins with $J_1/J_2=-2.7$ and $M=0.1$ in the vector chiral phase: (a) vector chiral correlation functions ${⟨{\kappa }_0^{\left(1\right)}{\kappa }_r^{\left(1\right)}⟩}_{\text{av}}$, ${⟨{\kappa }_0^{\left(2\right)}{\kappa }_r^{\left(2\right)}⟩}_{\text{av}}$, and ${⟨{\kappa }_0^{\left(1\right)}{\kappa }_r^{\left(2\right)}⟩}_{\text{av}}$, where ${\kappa }_r^{\left(n\right)}={\left({\mathit{s}}_r\times {\mathit{s}}_{r+n}\right)}^z$ [Eq. (5)], (b) transverse-spin-correlation function ${⟨s_0^x{s}_r^x⟩}_{\text{av}}$, and (c) nematic correlation function ${⟨s_0^{+}{s}_1^{+}{s}_r^{-}{s}_{r+1}^{-}⟩}_{\text{av}}$. Open symbols represent the DMRG data. Truncation errors are smaller than the size of the symbols. The solid lines and solid circles in (b) and (c) are fits to Eqs. (21, 23), respectively.

Figure 11

(Color online) Averaged, normalized vector chiral correlations at the distance $r=L/2$ for $L=120$ spin zigzag chain. Circles, squares, and triangles represent $⟨{\kappa }_0^{\left(1\right)}{\kappa }_{L/2}^{\left(1\right)}⟩$, ${\left(2J_2/\left|J_1\right|\right)}^2⟨{\kappa }_0^{\left(2\right)}{\kappa }_{L/2}^{\left(2\right)}⟩$, and $\left(2J_2/\left|J_1\right|\right)⟨{\kappa }_0^{\left(1\right)}{\kappa }_{L/2}^{\left(2\right)}⟩$, respectively. (a) $J_1/J_2=-2.5$, (b) $J_1/J_2=-2.6$, (c) $J_1/J_2=-2.7$, and (d) $J_1/J_2=-2.8$. Solid lines are the guide for the eyes. Vertical dashed lines represent the boundaries of the chiral ordered phase in low magnetization regime and the chiral disordered phase in the high magnetization regime. The shaded region corresponds to the magnetization jump at the first-order transition.

Figure 12

(Color online) $J_1/J_2$ dependence of (a) the exponent $K_{+}$ and (b) the incommensurate wave number $Q$ for $L=120$ spin zigzag chain in the vector chiral phase. Solid and open symbols in (a) represent the estimates obtained from the fitting of ${⟨s_0^x{s}_r^x⟩}_{\text{av}}$ and ${⟨s_0^{+}{s}_1^{+}{s}_r^{-}{s}_{r+1}^{-}⟩}_{\text{av}}$, respectively. The error bars represent the difference of the estimates obtained from the fitting of the data of different ranges. In (b), only the results from ${⟨s_0^x{s}_r^x⟩}_{\text{av}}$ are shown since the estimates from ${⟨s_0^x{s}_r^x⟩}_{\text{av}}$ and ${⟨s_0^{+}{s}_1^{+}{s}_r^{-}{s}_{r+1}^{-}⟩}_{\text{av}}$ are identical to each other within their error bars. Dashed line in (b) represents the classical pitch angle, ${\varphi }^c=\text{arccos}\left(-J_1/4J_2\right)$.

Figure 13

(Color online) Correlation functions for $L=120$ spin zigzag chain with $J_1/J_2=-2.0$, $M=0.2$ and 0.4, and their fits to the theory for the nematic and ${\text{SDW}}_2$ phases: (a) Friedel oscillations in the local spin polarization $⟨s_l^z⟩$, (b) longitudinal-spin fluctuation $⟨s_l^z{s}_{l^{\prime }}^z⟩-⟨s_l^z⟩⟨s_{l^{\prime }}^z⟩$, and (c) nematic correlation function $⟨s_l^{+}{s}_{l+1}^{+}{s}_{l^{\prime }}^{-}{s}_{l^{\prime }+1}^{-}⟩$. The open symbols represent the DMRG data. Truncation errors are smaller than the size of the symbols. In (b) and (c), the data for $l=L/2-\left[r/2\right]$ and $l^{\prime }=L/2+\left[\left(r+1\right)/2\right]$ are plotted as a function of $r=\left|l-l^{\prime }\right|$. The results of the fitting are shown by solid symbols in (a) and (b) and by dashed curves in (c). The data for $M=0.4$ are multiplied by a factor 2 in (a) and shifted by 0.01 in (b). (d) Absolute values of the averaged transverse-spin-correlation function ${⟨s_0^x{s}_r^x⟩}_{\text{av}}$.

Figure 14

(Color online) $M$ dependence of the exponent $\eta$ in the nematic phase $\left(\eta >1\right)$ and ${\text{SDW}}_2$ phase $\left(\eta <1\right)$. The estimates are obtained from the fitting of $⟨s_l^z{s}_{l^{\prime }}^z⟩$. The error bars represent the difference of the estimates obtained from the fitting using the data of different ranges.

Figure 15

(Color online) Averaged correlation functions for $L=120$ spin zigzag chain with $J_1/J_2=-2.7$ and $M=0.4$ in the incommensurate nematic phase: (a) longitudinal-spin fluctuation ${\left[⟨s_0^z{s}_r^z⟩-⟨s_0^z⟩⟨s_r^z⟩\right]}_{\text{av}}$, (b) nematic correlation function ${⟨s_0^{+}{s}_1^{+}{s}_r^{-}{s}_{r+1}^{-}⟩}_{\text{av}}$, (c) absolute values of the transverse-spin-correlation function ${⟨s_0^x{s}_r^x⟩}_{\text{av}}$, and (d) vector chiral correlation functions ${⟨{\kappa }_0^{\left(1\right)}{\kappa }_r^{\left(1\right)}⟩}_{\text{av}}$, ${⟨{\kappa }_0^{\left(2\right)}{\kappa }_r^{\left(2\right)}⟩}_{\text{av}}$, and ${⟨{\kappa }_0^{\left(1\right)}{\kappa }_r^{\left(2\right)}⟩}_{\text{av}}$. Truncation errors are smaller than the size of the symbols. Insets in (a), (b), and (d) show the absolute values of the data in a log-log scale.

Figure 16

(Color online) Squared modulus of the Fourier transform, ${\left|s^z\left(k\right)\right|}^2$, which is an even function of $k$ and where $s^z\left(k\right)=\left(1/\sqrt{L}\right){\sum }_l{e}^{ikl}\left(⟨s_l^z⟩-M\right)$, for $J_1/J_2=-2.7$ and $M=0.4$ in the incommensurate nematic phase. The solid, dashed, and dotted lines represents the data for $L=160$, 120, and 80, respectively. Inset: the local spin polarization $⟨s_l^z⟩$ for $J_1/J_2=-2.7$, $M=0.4$, and $L=160$.

Figure 17

(Color online) Correlation functions for $L=120$ spin zigzag chain with $J_1/J_2=-3.0$, $M=0.2$ and 0.4 and their fits to the theory for the triatic phase: (a) Friedel oscillations in the local spin polarization $⟨s_l^z⟩$, (b) longitudinal-spin fluctuation $⟨s_l^z{s}_{l^{\prime }}^z⟩-⟨s_l^z⟩⟨s_{l^{\prime }}^z⟩$, and (c) triatic correlation function $⟨s_l^{+}{s}_{l+1}^{+}{s}_{l+2}^{+}{s}_{l^{\prime }}^{-}{s}_{l^{\prime }+1}^{-}{s}_{l^{\prime }+2}^{-}⟩$. The open symbols represent the DMRG data. Truncation errors are smaller than the size of the symbols. In (b) and (c), the data for $l=L/2-\left[r/2\right]$ and $l^{\prime }=L/2+\left[\left(r+1\right)/2\right]$ are plotted as a function of $r=\left|l-l^{\prime }\right|$. The results of the fitting are shown by solid symbols in (a) and (b) and by dashed curves in (c). The data for $M=0.4$ are multiplied by a factor 2 in (a) and shifted by 0.01 in (b). (d) Absolute values of the averaged transverse-spin and nematic correlation functions, ${⟨s_0^x{s}_r^x⟩}_{\text{av}}$ and ${⟨s_0^{+}{s}_1^{+}{s}_r^{-}{s}_{r+1}^{-}⟩}_{\text{av}}$.

Figure 18

(Color online) $M$ dependence of the exponent $\eta$ in the triatic phase $\left(\eta >1\right)$ and ${\text{SDW}}_3$ phase $\left(\eta <1\right)$. The estimates are obtained from the fitting of $⟨s_l^z{s}_{l^{\prime }}^z⟩$ for $J_1/J_2\ge -3.0$ and $⟨s_l^z⟩$ for $J_1/J_2=-3.2$ and $-3.4$. The error bars represent the difference of the estimates obtained from the fitting using the data of different ranges.

Figure 19

(Color online) Friedel oscillations in the local spin polarization $⟨s_l^z⟩$ for $L=120$ spin zigzag chain with $J_1/J_2=-3.6$ and $M=0.2$ and 0.4, and their fits to the theory for the quartic phase. The open symbols represent the DMRG data and solid symbols show the results of the fitting. Truncation errors of the DMRG data are smaller than the size of the symbols. The data for $M=0.4$ are multiplied by a factor 2.