There is a growing suspicion that thermodynamic properties of many novel materials at finite temperatures may be a consequence of their vicinity to quantum critical points, i.e. changes of the ground-state symmetry as a function of some intrinsic parameter p. Such quantum phase transitions (QPTs) may play a key role in the high-temperature cuprate superconductors, in heavy fermion materials, and in low-dimensional quantum magnets. Regarding the latter, antiferromagnetically (AFM) coupled spin-1/2 dimer magnets are of particular interest. Many molecular magnetic materials can be understood in terms of one-, two- and three-dimensional networks of spin dimers [1]. In dimer magnets, p can be identified with the ratio J/j of the intra-dimer exchange to an effective inter-dimer coupling. For |p|≫1 and J > 0, the ground state can be understood in terms of a product state of weakly interacting spin-singlets which display no magnetic long-range order (MLRO). However, as p is lowered to |p|∼1 and in spacial dimensions D ≥ 2, a QPT to MLRO may arise with a commensuration vector Q depending on the details of the inter-dimer coupling. In this context, the impact of disorder has become a recent topic. Here we focus on the effects of substitutional disorder arising from non-magnetic defects. In molecular dimer magnets, such defects can be introduced in a controlled way, as e.g. in the coupled ladder compound [Ph(NH₃)](18C₆)[Ni(dmit)₂]_1–x[Au(dmit)₂]_x, where the [Ni(dmit)₂]⁻ ion leads to spin-1/2, while for Ni → Au spin-0 results [2]. To be specific, we study the 2D version of the so-called Kondo-necklace model of coupled spin-1/2 dimers:

The disorder we consider amounts to a random removal of spins from sites l_d of the upper layer ‘I’ of Fig. 1, thereby introducing non-magnetic defects at a concentration c. It has been conjectured that generically such defects in AFM quantum spin systems will enhance local AFM spin correlations and therefore stabilize or trigger MLRO on unfrustrated lattices [6,7]. To test this conjecture in the present case, we have evaluated the AFM order parameter, i.e. the longitudinal staggered structure factor:

Fig. 2 summarizes our results. It shows the low-temperature squared staggered moment MQ2=3Sn(Q) vs. J for various defect concentrations and for fixed inter-dimer exchange j ≡ 1 at a system size of N = 2 × N_x × N_y = 2 × 24 × 24. Extensive finite-size scaling analysis has been performed at various J and c to ensure that the systematic finite-size corrections to MQ2 are on the order of, or less than ∼10% and have no impact on the results discussed here [10]. Moreover, the inverse temperature β = 1/T has been chosen such as to represent the zero-temperature limit. At c = 0, MQ2 is finite below a critical value of J = J_c and drops to approximately zero for J > J_c. Extrapolation of MQ2(J<Jc) by a power law leads to J_c ≈ 1.41(2). We identify J_c with the QPT and expect AFMLRO for J < J_c in the thermodynamic limit at T = 0. For J > J_c we find no other transitions, i.e. the systems connect adiabatically to the limit of J = ∞. Therefore, it is in a dimerized state with no MLRO. For a finite defect density, c > 0, the situation changes dramatically. For all values of J investigated, we find that the staggered moment, i.e. the order parameter of the AFM state remains finite, also for J > J_c. Our results do not rule out a large upward renormalization of the QPT as a function of c, i.e. beyond the range of J considered here. However, it is rather likely that the QPT is suppressed by the disorder and Jc(c≠0)=∞ for all c. In any case Fig. 2 demonstrates the main point of this short note, i.e. that non-magnetic disorder can induce MLRO in a state which is non-magnetic otherwise in the clean limit, i.e. we find order-from-disorder.