Quantum magnets are known for their rich phase diagrams which depend on system geometry as well as physical parameters, such as applied field, pressure, and temperature. They can be generally described by the Heisenberg Hamiltonian,

,

where the first term is the exchange interaction between the atomic spins, and the second term is the interaction of the spins with an applied magnetic field along the z-direction. Since the exchange integral J_ij falls of rapidly with the spatial separation between the spins, it can often be truncated to nearest-neighbor interactions. However, sometimes longer-range contributions are important, and can lead to interesting frustration effects. Positive exchange integrals arise from antiferromagnetic interactions that favor a staggered alignment of the spins in the crystal, i.e. neighboring spins are anti-aligned and belong to different sublattices.

The dependence of the degree of antiferromagnetic ordering on system dimensionality and geometry can be illustrated by considering the nearest-neighbor antiferromagnetic Heisenberg Hamiltonian of chains,ladders, and planes with spin-1/2 atoms:

While in the absence of an applied field anti-alignment is favored in all three cases, the character of the zero-temperature ground state in these systems is quite different. In the chain, the spin-spin correlations fall off algebraically, indicating quasi-long-range order. In the ladder, their decay is exponential, corresponding to short-range antiferromagnetic order with a finite correlation length. In contrast, the plane has long-range antiferromagnetic order.

Using the recently developed stochastic series expansion quantum Monte Carlo algorithm, we have studied the phase diagrams of such low-dimensional quantum antiferromagnets as a function of applied magnetic field and temperature. Let us first focus on the properties of individual antiferromagnetic spin-1/2 Heisenberg ladders. At zero-temperature, the Heisenberg ladder in a magnetic field has three phases: it is a non-magnetic quantum spin liquid below a threshhold field h_c1, partially polarized at intermediate field strengths between h_c1 and h_c2, and fully polarized for fields larger than h_c2. When such ladders are embedded in a 3D crystal, these zero-temperature regimes of the quasi-1D ladder subsystems give rise to finite-temperature phase transitions in the crystal. The resulting temperature-field phase diagram is shown in the figure below. At high temperatures, the system is paramagnetic. As it is cooled down, it enters a quasi-1D Luttinger phase if the applied field is between h_c1 and h_c2 . Upon further cooling, the system then orders antiferromagnetically. At zero-temperature there are two quantum critical points, h_c1(T=0) and h_c2(T=0).

The scaling behavior in the vicinity of these quantum critical points is of interest to the experimental and theoretical communities involved with quantum magnets. The associated power-law dependence of the critical magnetic field on the temperature |h- h_c|~T^α can be extracted from fitting the data points obtained from our simulations, as shown in the insets of the above figure. For the case of weakly-coupled ladders embedded in a cubic crystal one finds significantly different scaling properties at the lower and the upper critical fields, indicating that quantum fluctuations that are strong at the lower quantum critical point are largely suppressed at h_c2 where the extracted exponent is close to the mean-field expectation α=2.

It was proposed that these types of phase transitions can be interpreted as Bose-Einstein condensations of magnons. However, it has been quite difficult to obtain the corresponding scaling behavior from simulations and experiments. More precisely, the scaling exponent associated with a Bose-Einstein condensation is expected to be α=3/2, but the bulk of the data of simulations as well as experimental measurements suggest larger values for α. We have recently proposed a resolution of this controversy, based on a large-scale simulation study of weakly-coupled dimers. We observed that the value of the extracted scaling exponent depends on the windows in the magnetic field (red and blue circles in the above figure) that are chosen to represent the scaling regimes. As these window are made smaller, it was found that the Bose-Einstein prediction is approached in the limiting case of infinitesimal windows around the quantum critical points. This observation also offers a natural explanation why the scaling exponents extracted from experimental data tend to be larger than 3/2. There are typically very few available data points at ultra-small temperatures close to h_c(T=0). However, the inclusion of data points that are further away from the scaling regimes in the fitting extraction of the critical exponents leads to overestimation of α.

Magnetic materials that are realizations of weakly-coupled antiferromagnetic dimer systems include the family of (Tl,K)CuCl₃  (crystal structure shown above) where the relevant spin-1/2 degrees of freedom are contributed by the Cu²⁺ ions. Interestingly, inelastic neutron scattering studies have shown that the triplet excitation spectra in KCuCl₃ are more dispersive than in TlCuCl_{3 ,} indicating that the former material has stronger inter-dimer couplings. It is conceivable that by applying external pressure or substituting the K-ion by an appropriate dopant, this system can be driven across a quantum critical point, beyond which there is long-range antiferromagnetic order even in the absence of an applied magnetic field. Based on our simulations, this quantum critical point should occur close to J’=0.25J in 2D, where J’ is the inter-dimer coupling and J is the intra-dimer coupling. Here it is assumed that the primary consequence of external presuure is an increase of the ration J’/J.

The pressure-induced quantum critical point shown in the above figure is determined by a finite-size scaling analysis of the staggered magnetization of systems with up to 400 spins. The antiferromagnetic order parameter is extracted from stochastic series quantum Monte Carlo simulations. This algorithm involves expansions of the partition function in inverse temperature, uses local and global system updates, and is significantly more efficient than conventional quantum Monte Carlo schemes. Recently proposed technical developments, using directed loops in the detailed balance to improve the sampling efficiency, have also been implemented.