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 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 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:
(1) |
![](/chimie/article/CRCHIM_2007__10_1-2_79_0/jpg/src/tex/figures/gr1.jpg)
2D Kondo-necklace with 2 × Nx × Ny sites (periodic boundary conditions assumed). Spin-1/2 moments are located on the solid bullets. Solid (dashed) lines refer to inter(intra)-dimer exchange J(j ≡ 1).
The disorder we consider amounts to a random removal of spins from sites ld 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:
(2) |
Fig. 2 summarizes our results. It shows the low-temperature squared staggered moment vs. J for various defect concentrations and for fixed inter-dimer exchange j ≡ 1 at a system size of N = 2 × Nx × Ny = 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 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, is finite below a critical value of J = Jc and drops to approximately zero for J > Jc. Extrapolation of by a power law leads to Jc ≈ 1.41(2). We identify Jc with the QPT and expect AFMLRO for J < Jc in the thermodynamic limit at T = 0. For J > Jc 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 > Jc. 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 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.
![](/chimie/article/CRCHIM_2007__10_1-2_79_0/jpg/src/tex/figures/gr2.jpg)
Staggered structure factor SA(Q) vs. J close to the QPT for various defect concentrations c. Inverse temperatures β = 1/T are β = 100 at c = 0 and β = 1024 at c = 0.03, 0.1, and 0.2. Disorder results include averages over ∼900 system realizations each. Statistical errors are less than the solid-circle marker size.
Acknowledgement
Part of this work has been funded by the German Science Foundation, DFG, under Grant Nos. BR 1084/2-2 and BR 1084/4-1.