The Bose–Einstein condensate (BEC) is a fascinating state of matter predicted to occur for particles obeying Bose statistics. Although the BEC has been observed with bosonic atoms in liquid helium and cold gases, the concept is much more general. We here review analogous states, where excitations in magnetic insulators form the BEC. In antiferromagnets, elementary excitations are magnons, quasiparticles with integer spin and Bose statistics. In certain experiments their density can be controlled by an applied magnetic field leading to the formation of a BEC. Furthermore, interactions between the excitations and the interplay with the crystalline lattice produce very rich physics compared with the canonical BEC. Studies of magnon condensation in a growing number of magnetic materials thus provide a unique window into an exciting world of quantum phase transitions and exotic quantum states, with striking parallels to phenomena studied in ultracold atomic gases in optical lattices

We investigate weakly coupled spin-1/2 ladders in a magnetic field. The work is motivated by recent experiments on the compound (C 5 H 12 N) 2 CuBr 4 (BPCB). We use a combination of numerical and analytical methods, in particular, the density-matrix renormalization group (DMRG) technique, to explore the phase diagram and the excitation spectra of such a system. We give detailed results on the temperature dependence of the magnetization and the specific heat, and the magnetic-field dependence of the nuclear-magnetic-resonance relaxation rate of single ladders. For coupled ladders, treating the weak interladder coupling within a mean-field or quantum Monte Carlo approach, we compute the transition temperature of triplet condensation and its corresponding antiferromagnetic order parameter. Existing experimental measurements are discussed and compared to our theoretical results. Furthermore, we compute, using time-dependent DMRG, the dynamical correlations of a single spin ladder. Our results allow to describe directly the inelastic neutron scattering cross section up to high energies. We focus on the evolution of the spectra with the magnetic field and compare their behavior for different couplings. The characteristic features of the spectra are interpreted using different analytical approaches such as the mapping onto a spin chain, a Luttinger liquid or onto a t-J model. For values of parameters for which such measurements exist, we compare our results to inelastic neutron scattering experiments on the compound BPCB and find excellent agreement. We make additional predictions for the high-energy part of the spectrum that are potentially testable in future experiments.

Spirals and helices are common motifs of long-range order in magnetic solids, and they may also be organized into more complex emergent structures such as magnetic skyrmions and vortices. A new type of spiral state, the spiral spin-liquid, in which spins fluctuate collectively as spirals, has recently been predicted to exist. Here, using neutron scattering techniques, we experimentally prove the existence of a spiral spin-liquid in MnSc2S4 by directly observing the 'spiral surface' -a continuous surface of spiral propagation vectors in reciprocal space. We elucidate the multi-step ordering behavior of the spiral spin-liquid, and discover a vortex-like triple-q phase on application of a magnetic field. Our results prove the effectiveness of the J1-J2 Hamiltonian on the diamond lattice as a model for the spiral spin-liquid state in MnSc2S4, and also demonstrate a new way to realize a magnetic vortex lattice.Magnetic frustration, where magnetic moments (spins) are coupled through competing interactions that cannot be simultaneously satisfied 1 , usually leads to highly cooperative spin fluctuations 2,3 and unconventional longrange magnetic order 4,5 . An archetypal ordering in the presence of frustration is the spin spiral. Competing interactions and spiral orders give rise to many phenomena in magnetism, including the multitudinous magnetic phases of rare earth metals 6 , domains with multiferroic properties 7,8 , and topologically non-trivial structures such as the emergent skyrmion lattice 9,10 .Recently, a new spiral state -a spiral spin-liquid in which the ground states are a massively degenerate set of coplanar spin spirals -was predicted to exist in the J 1 -J 2 model on the diamond lattice (see Fig. 1a) [11][12][13] . Although the diamond lattice is bipartite, and therefore unfrustrated at the near-neighbour (J 1 ) level, the second-neighbour coupling (J 2 ) can generate strong competition. For classical spins, mean-field calculations show that when |J 2 /J 1 | > 0.125 the spiral spin-liquid appears, and that it is signified by an unusual continuous surface of propagation vectors q in reciprocal space (see Fig. 1b for the spiral surface of |J 2 /J 1 | = 0.85). At finite temperature, thermal fluctuations might select some specific q-vectors on the spiral surface 11 , resulting in an orderby-disorder transition 14,15 .Until now, several series of A-site spinels, in which the magnetic A ions form a diamond lattice, have been investigated, including: the cobaltates Co 3 O 4 and CoRh 2 O 4 16 ; the aluminates M Al 2 O 4 with M = Fe, Co, Mn 17-20 ; and the scandium thiospinels M Sc 2 S 4 with M = Fe, Mn 21 . For the spinels with Fe 2+ at the A-site, the e g orbital angular momentum of Fe 2+ is active, making the pure spin J 1 -J 2 model inadequate 22 . Among the other compounds, CoAl 2 O 4 and MnSc 2 S 4 manifest the strongest frustration. For CoAl 2 O 4 , the ratio of |J 2 /J 1 | has been identified as 0.109 19 , which is near, but still lower than, the 0.125 threshold for the spiral spin-liquid state. Many expe...