Ground-state phase diagram of a spin-$\frac{1}{2}$ frustrated ferromagnetic XXZ chain: Haldane dimer phase and gapped/gapless chiral phases

The ground-state phase diagram of a spin-$\frac{1}{2}$ XXZ chain with competing ferromagnetic nearest-neighbor ($J_1<0$) and antiferromagnetic second-neighbor ($J_2>0$) exchange couplings is studied by means of the infinite time evolving block decimation algorithm and effective field theories. For the SU(2)-symmetric (Heisenberg) case, we show that the nonmagnetic phase in the range $-4<J_1/J_2<0$ has a small but finite ferromagnetic dimer order. We argue that this spontaneous dimer order is associated with effective spin-1 degrees of freedom on dimerized bonds, which collectively form a valence bond solid state as in the spin-1 antiferromagnetic Heisenberg chain (the Haldane spin chain). We thus call this phase the Haldane dimer phase. With easy-plane anisotropy, the model exhibits a variety of phases including the vector chiral phase with gapless excitations and the even-parity dimer and Néel phases with gapped excitations, in addition to the Haldane dimer phase. Furthermore, we show the existence of gapped phases with coexisting orders in narrow regions that intervene between the gapless chiral phase and any one of Haldane dimer, even-parity dimer, and Néel phases. Possible implications for quasi-one-dimensional edge-sharing cuprates are discussed.

Figure 1

The ground-state phase diagram of the model (1) with $J_1<0$ and $J_2>0$. The vector chiral phase, which has a nonvanishing vector chirality (2), extends between the two boundaries with the “$\square$” symbols. These boundaries are determined as in Figs. 2 and 3 (see the vertical solid lines in these figures). Around the highly degenerate point $(J_1/J_2,\Delta )=(-4,1)$, these two boundaries could not be determined accurately, but we expect both of them to continue to this point. It was also difficult to draw the boundary around the right top corner of the phase diagram. The onsets of (Haldane and even-parity) dimer and Néel orders occur inside the vector chiral phase, as indicated by the “$\times$” symbols (determined as in Figs. 14 and 15). Thus there are narrow intermediate phases (between the “$\square$” and “$\times$” symbols) where two kinds of orders coexist. The phase boundaries among the TLL, even-parity dimer, and Néel phases are determined in a previous work.[24] On the right of the “$\circ$” symbols, the even-parity dimer and Néel phases alternately appear when approaching the point $(J_1/J_2,\Delta )=(-4,1)$: the first transition occurs at $\Delta \approx 0.7$ (“$\diamond$” symbols) and the second at $\Delta \approx 0.93$ (not shown).[24] On the line where $\Delta =1$ and $J_1/J_2<-4$, the ground state is ferromagnetic (FM). The “+” symbols inside the dimer and Néel phases indicate the Lifshitz line on which the short-range spin correlation changes its character from incommensurate (IC) to commensurate (C); see Fig. 16.

Figure 2

(a) Chiral and (b), (c) dimer order parameters and (d) half-chain entanglement entropy as functions of $\Delta$ for fixed $J_1/J_2=-2$. These are calculated by the iTEBD with Schmidt ranks $\chi =200$ and 300. (c) is a zoom of (b). In (d), $S_{1:2}^{\mathrm{vN}}$ and $S_{2:3}^{\mathrm{vN}}$ are defined for the bipartitions of the system at the bonds $(1,2)$ and $(2,3)$, respectively. Solid vertical lines indicate the boundaries of the vector chiral phase, and are determined from the onsets of the vector chiral order parameter in (a) or more accurately from the peaks in the entanglement entropy in (d). Broken vertical lines indicate the transition points on which dimer orders set in. These points are difficult to locate within the analysis of dimer order parameters in (b) and (c) and are instead determined by the analysis of spin correlations in Fig. 14. Narrow intermediate phases exist between solid and broken vertical lines, where the vector chiral and dimer orders coexist. In the intermediate phase in $0.61\lesssim \Delta \lesssim 0.63$, a dip in the entanglement entropy is seen in (d), as zoomed in the inset.

Figure 3

(a) Chiral and (b) Néel order parameters and (c) half-chain entanglement entropy as a function of $J_1/J_2$ for fixed $\Delta =0.8$, calculated by the iTEBD. The solid and broken vertical lines indicate the onsets of the vector chiral and Néel orders, respectively. The former is determined by the peak position in the entanglement entropy in (c), while the latter is determined in Fig. 15. In the narrow intermediate phase in $-3.225\lesssim J_1/J_2\lesssim -3.200$, the vector chiral and Néel orders coexist.

Figure 4

Zigzag chain picture for the $J_1$-$J_2$ chain model (1). The $x$ axis indicates the coordinate for the continuum description.

Figure 5

Numerical solutions to the one-loop RG equations (16) for (a) $J_1/J_2=0.2$ and (b) $J_1/J_2=-0.2$. We set $\lambda =1$ [see Eq. (14)]. It is found that the three coupling constants $G_1$, $G_{\mathrm{tw}}$, and $G_{\mathrm{dtw}}$ are most relevant and grow under the RG, with asymptotically a simple ratio $G_1:G_{\mathrm{tw}}:G_{\mathrm{dtw}}=2:1:(-1)$ or $2:(-1):1$ for $J_1>0$ and $J_1<0$, respectively. In the plots, factors $\pm 2$ are multiplied to $G_{\mathrm{tw}}$ and $G_{\mathrm{dtw}}$ so that these ratios can be visually confirmed.

Figure 6

(a) Nearest-neighbor spin correlations $\langle {\mathit{S}}_j·{\mathit{S}}_{j+1}\rangle$ and the dimer order parameter $D_{123}$, (b) the spin correlation length $\xi$, and (c) the string correlation (34) with $\ell =2$ and $r=50$, as a function of $J_1/J_2$ in the isotropic case $\Delta =1$. In (a), $D_{123}$ multiplied by 40 is also plotted for $J_1/J_2<0$ (filled circular symbols). In (b), $\xi$ is too small to determine around $J_1/J_2=2$.

Figure 7

Sketches of (a) the singlet dimer state and (b) the Haldane dimer state. The thick lines indicate valence bonds. In (b), the encircled bonds indicate emergent spin-1's. From each of them, a valence bond emanate to each left and right; the wave function is given by a superposition of such valence bond covering states. Vertical cuts (dashed lines) are introduced to probe a hidden order; the number of valence bonds crossing with each cut is shown in the square. The alternation of odd and even numbers is found in both the states.

Figure 8

Determination of the spin correlation length $\xi$, illustrated for $J_1/J_2=-1.8$. Assuming the asymptotic behavior (33), we plot the function $r^{1/2}{e}^{r/\xi }\langle {\mathit{S}}_1·{\mathit{S}}_{1+r}\rangle$ and tune $\xi$ such that the oscillation width of this function becomes as constant as possible as a function of $r$.[5] While the oscillations arise from the cosine factor in Eq. (33), it is difficult to extract the pitch angle $Q$ from this figure; instead, calculations in Sec. 4b3 give $Q/\left(2\pi \right)\approx 0.235$.

Figure 9

String correlation function (34) for $J_1/J_2=1.4$ and $-2.0$ in the isotropic case $\Delta =1$. For both values of $J_1/J_2$, $O^z(2,2+2r)$ remains finite in the long-distance limit, while $O^z(1,1+2r)$ decays to zero.

Figure 10

Entanglement spectra $\{-2\ln {\lambda }_{m\alpha }\}$ for (a) $J_1/J_2=1.4$ and (b) $J_1/J_2=-2$. The left and right panels are for the bipartition of the system at the bonds $(1,2)$ and $(2,3)$, respectively. $m$ refers to the magnetization in the right half of the system. The lower entanglement level corresponds to the more important weight in the total state. We note that as an example, the exact singlet dimer ground state of Ref. 2 shows $-2\ln {\lambda }_{\pm \frac{1}{2},1}=\ln 2$ and $-2\ln {\lambda }_{0,1}=0$ for the two types of bipartition (with all the other levels at infinity).

Figure 11

Expected phase diagram of the zigzag ladder model with alternating nearest-neighbor couplings $J_1$ and $J_1^{\prime }$. $|J_1|/J_2$ and $|J_1^{\prime }|/J_2$ are assumed to be small. The solid diagonal line $J_1=J_1^{\prime }$ corresponds to the original model (1) (with $\Delta =1$), and represents the first-order phase transition line in the current model. The vertical and horizontal dashed lines correspond to a usual ladder model (no phase transition on these lines). In four insets of zigzag ladders, thick lines indicate valence bonds, and ovals indicate the formation of effective spin-1's.

Figure 12

In-plane spin correlation function $|\langle S_1^{+}{S}_{1+r}^{-}\rangle |$ for fixed $\Delta =0.8$ and various values of $J_1/J_2$ in the gapless chiral phase. Logarithmic scales are used in both axes.

Figure 13

Equal-time spin structure factors [see Eq. (66)], (a) $S^{+-}\left(q\right)$ and (b) $S^{zz}\left(q\right)$, in the gapless chiral phase. Calculations were done for the same parameter points as in Fig. 12, and we set $L=100$.

Figure 14

In-plane spin correlation function, calculated for fixed $J_1/J_2=-2$ and various values of $\Delta$ around the transition points shown in Fig. 2. The symbols “C” and “D” indicate our estimates of the transition points (with a precision of 0.005) for the onsets of the vector chiral and dimer orders, respectively. Logarithmic scale is used for the horizontal axis. At the BKT transition related to the onset of the dimer order, the plotted function is expected to become linear in the long-distance limit, which we use to determine the “D” points. In (b), all the curves are slightly bent downward around $r=70$ (broken vertical line) due to the finiteness of the Schmidt rank $\chi (=300)$ in iTEBD, so we use the range $r\lesssim 70$ for our analysis.

Figure 15

In-plane spin correlation, calculated for fixed $\Delta =0.8$ and various values of $J_1/J_2$ around the transition points shown in Fig. 3. The symbols “C” and “N” indicate our estimates of the transition points (with a precision of 0.005) for the onsets of the vector chiral and Néel orders, respectively.

Figure 16

Pitch angle $Q$ as a function of $J_1/J_2$ for different values of $\Delta$. This angle is determined by finding the peak in the in-plane structure factor $S^{+-}\left(q\right)$ as shown in Fig. 13a. The classical value $Q=\arccos (-\frac{J_1}{4J_2})$, which is independent of $\Delta$, is plotted together for comparison.

Figure 17

Plots of $r^2{\left|\langle S_1^x{S}_{1+r}^x\rangle \right|}^2$ for fixed $\Delta =0$ and various values of $J_1/J_2$ around the TLL-dimer transition point studied in Ref. 4. A logarithmic scale is used for the horizontal axis. The symbol “D” indicates the estimate of the transition point within the current analysis (with a precision of $0.05$), which agrees reasonably well with the previous accurate estimate[4] $J_2/J_1\approx 3.0893$.