Fig. 4: Simulations of annealing processes.
a, b Transverse field dependence of staggered magnetization \({m}_{{{{{{{{\rm{s}}}}}}}}}^{z}=\langle | {\sum }_{j}{S}_{j}^{z}{(-1)}^{{j}_{a}+{j}_{c}}| \rangle /(NS)\) and 3\(\langle | {\sum }_{j}{S}_{j}^{z} \exp [i2\pi ({j}_{a}+ {j}_{c})/3]| \rangle /(2NS)\) calculated at μ0Hz = 0 and 2 T, respectively. Here, N is the site number, S = 1/2, and ja and jc are site indexes along the a- and c-axis, respectively. The critical transverse fields Γc are marked. c Density matrix ratios, W↓/W↑. Here, \({W}_{\uparrow }=\langle \uparrow | \exp (-\beta {{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{s}}}}}}}}})| \uparrow \rangle\) and \({W}_{\downarrow }=\langle \downarrow | \exp (-\beta {{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{s}}}}}}}}})| \downarrow \rangle\) are exactly calculated using the single-site molecular-field Hamiltonian \({{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{s}}}}}}}}}\) = − hSz − ΓSx. d, e, g, h Monte Carlo step (MCS) dependence of the energy per site (E) calculated at T = 1, 1.8 K in μ0Hz = 0, 2 T. f, i MCS dependence of longitudinal magnetization calculated at μ0H z = 2 T. The original spin Hamiltonian of α-CoV2O6 is used in (a, b, d–i), and the datasets in (d–i) are averaged over 64 independent samples. To mimic the experimental procedures, the simulations (d–i) initiated with random states at μ0H z = − 4.2 T. We gradually raised μ0Hz until reaching the target field and then simulated the relaxation processes in subsequent MCS. All the SSE-QMC calculations (a, b, d–i) were performed on a 6 × 6 × 12 (432-site) cluster with periodic boundary conditions.
