A recent experimental realization of the anisotropic J₁-J₂ spin-1/2 model [1] provides an opportunity to probe the quantum critical point, where theory predicts a spin-liquid behavior. The data indicate the presence of a partial spin-liquid contribution, possibly in coexistence with a gapped state driven by spin-Peierls instabilities. Motivated by these observations, we study the effect of spin-Peierls instability on the phase-diagram of a quasi one-dimensional (1D) version of the model - a two-leg ladder. We introduce a spin-Peierls instability along the legs of the ladder as well as XXZ anisotropy of the exchange constants. Two distinct forms of spin-Peierls instabilities on the ladder are considered: columnar and staggered dimerization on the two legs. Applying Bosonization to derive the low-energy theory, we find that the effective field theory describing the system is a self-dual sine-Gordon model, which favors ordering and the opening of a gap to excitations. The order parameter represents a superposition of longitudinal and transverse dimers on a four-spin plaquette. We found that depending on the spin-Peierls instability mode, closed (for the columnar mode) or open (for the staggered mode) rectangular plaquettes may form. While the closed plaquette order is stable, the open plaquettes are less stable and may break into a 1D spin-liquid at a fine-tuned point. This suggests the possibility of forming a mixed valence bond crystal-liquid state in a higher-dimensional, multi-leg realization of the model.

[1] O. Volkova,  I. Morozov, V. Shutov, E. Lapsheva, P. Sindzingre, O. Cépas, M. Yehia, V. Kataev, R. Klingeler, B. Büchner, and A. Vasiliev, Phys. Rev. B 82 054413 (2010);

V. Gnezdilov, P. Lemmens, Yu. G. Pashkevich, D. Wulferding, I. V. Morozov, O. S. Volkova, and A. Vasiliev, Phys Rev B 85 214403 (2012).