Fig. 2: Numerical observation of the spin-Peierls instability of the triangular-lattice AFM Heisenberg model.

Density matrix renormalisation group calculations of the next-nearest-neighbour frustrated triangular lattice model (J₂/J₁ = 1/8) are presented, where the undistorted lattice displays a gapless spin-liquid ground state. a Energy gain of the spin system under various simulated lattice distortions on the L = 6 circumference cylinder. Patterns considered are generated by momenta K₃, and M_1,2,3 (here the purple shaded region represents the range of responses for the three M_a, showing minimal dependence on cylinder orientation; see Supplementary Note 3), as well as the full 12-site distortion, defined by Eq. (5). b Nearest-neighbour exchanges under the 12-site unit cell distortion; red/blue bonds show enhanced/suppressed bonds. c System-size dependence of energy gain for L = 3, 6. We compare three distinct distortion patterns generated by the momenta K₃, M₃, and X₃ which are compatible with the cylinder geometry. Logarithmic axes show the quadratic energy gain for each pattern. d L-dependence of the energy gain for the data point δ = 0.002 for cylinders with circumference of up to L = 9. The colours indicate the same distortion patterns as in c. Only the distortion at X₃ shows strong L-dependence, which confirms that a weak-coupling instability of the U(1) Dirac spin liquid is realised in this numerical study of the J₁–J₂ model. e,f The spin-spin correlation function on dimers \(\langle {\overrightarrow{S}}_{i}\cdot {\overrightarrow{S}}_{j}\rangle\) in the presence of a 12-site lattice distortion b. Subfigure e shows the theoretical prediction from the leading order analysis of the symmetry of operators in the CFT spectrum (on an arbitrary scale); and f is the numerical calculation performed with DMRG, evaluated with distortion parameter δ  = 0.05 and L = 6 and averaged over orientations relative to the cylinder axis.