Long-range spin chirality dimer order in the Heisenberg chain with modulated Dzyaloshinskii-Moriya interactions

Abstract: The ground state phase diagram of a spin $S=1/2$ $XXZ$ Heisenberg chain with spatially modulated Dzyaloshinskii-Moriya (DM) interaction $ {\cal H}= \sum_n J\left[\left(S^x_n S^x_{n+1} +S^y_n S^y_{n+1}+\Delta S^z_n S^z_{n+1}\right)+(D_0+(-1)^n D_1)\left(S^x_n S^y_{n+1} -S^y_n S^{x}_{n+1} \right) \right] $ is studied using the continuum-limit bosonization approach and extensive density matrix renormalization group computations. It is shown that the effective continuum-limit bosonized theory of the model is given by the double frequency sine-Gordon model (DSG) where the frequences i.e. the scaling dimensions of the two competing cosine perturbation terms depend on the effective anisotropy parameter $\gamma^*=J\Delta /\sqrt{J^2+D_0^2+D_1^2}$. Exploring the ground state properties of the DSG model we have shown that the zero-temperature phase diagram contains the following four phases: (i) the ferromagnetic phase at $\gamma^*<-1$; (ii) the gapless Luttinger-liquid (LL) phase at $-1<\gamma^*< \gamma^*_{c1}=-1/\sqrt{2}$; (iii) the gapped composite (C1) phase characterized by coexistence of the long-range-ordered (LRO) dimerization pattern $\epsilon \sim (-1)^n (S_n S_{n+1})$ with the LRO alternating spin chirality pattern $\kappa \sim (-1)^{n}\left(S^{x}{n}S^{y}{n+1} -S^{y}{n}S^{x}{n+1} \right)$ at $\gamma^{\ast}_{c1}<\gamma^{\ast} <\gamma^{\ast}_{c2}$; and (iv) at $\gamma^{\ast} >\gamma^{\ast}_{c2}>1$ the gapped composite (C2) phase characterized in addition to the coexisting spin dimerization and alternating chirality patterns, by the presence of LRO antiferromagnetic order. The transition from the LL to the C1 phase at $\gamma^*_{c1}$ belongs to the Berezinskii-Kosterlitz-Thouless universality class, while the transition at $\gamma^*_{c2}$ from C1 to C2 phase is of the Ising type.