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

The ground-state phase diagram of a spin S = 1/2 XXZ Heisenberg chain with spatially modu-lated Dzyaloshinskii-Moriya interaction H = Sigma(n){J((SnSn+1x)-S-x + (SnSn+1y)-S-y) + J(z)S(n)(z)S(n+1)(z) + [D-0 + (-1)D-n(1)]((SnSn+1y)-S-x - (SnSn+1x)-S-y)} 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 frequencies, i.e., the scaling dimensions of the two competing cosine perturbation terms, depend on the effective anisotropy parameter gamma* = J(z)/root J(2) + D-0(2) + D-1(2). Exploring the ground-state properties of the DSG model we show 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/root 2; (iii) the gapped composite (C1) phase characterized by coexistence of the long-range-ordered (LRO) dimerization pattern epsilon similar to (-1)(n)(SnSn+1) with the LRO alternating spin chirality pattern kappa similar to (-1)(n)((SnSn+1y)-S-x - (SnSn+1x)-S-y) at gamma(C1)* < gamma* < gamma(C2)*; and (iv) at gamma* > gamma(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* = gamma(C2)* from C1 to C2 phase is of the Ising type.