In the quest of Bose-Einstein condensation, an extraordinary state of matter in a very low temperature, people suffer from many obstacles for the reason that it can be stabilized only under some extreme conditions. Thanks to the recent development of cooling techniques, its realization in a laboratory is no longer a dream. However, once it cannot survive with normal temperature or pressure, it is hard to make use of its desirable properties such as the zero viscosity. On the other hand, a more extraordinary state of matter, combining the characteristics of superfluid and solid, has been proposed for long and named after the "supersolid" state. Despite a huge amount of effort, the realization of supersolid in bosonic systems is difficult even in a low-temperature environment, synthesized within a laboratory. Such fact also hinders us from investigating its further properties and mechanism.
Luckily, we have an another option.
Considering the similarity between bosons and spins, magnetic materials might be able to provide a better playground for us to investigate the bosonic condensation. When bosons condensate after cooled down, it costs zero energy to introduce or remove a particle from it. The above fact suggests that the number of total boson is no longer a "good" quantum number, meaning that a "symmetry" has been broken. Scientists refer to this symmetry group as the U(1) group and its group elements compose a continuous circle in the complex plane. In the spin language, when U(1) symmetry is broken the off-diagonal spin moment (lying in the x-y plane) appears, which corresponds to the bosonic condensation. The merit of studying the spin systems for searching and realizing the Bose-Einstein condensation lies on the fact that we do not need to push the system under the extreme conditions, as long as the Curie temperature is high enough for our target material of interest.
If the above statement sounds reasonable to you, our next step is to figure out what kind of magnetic material could be a good candidate. Despite some artificial systems made of optical lattices with laser cooling cold-atom techniques, a possible scenario living in magnetic materials arises from the idea of localized singlets. Imagine that within a lattice, we have two spin degrees of freedom both equal to 1/2 on every site, and then the local Hilbert space on each lattice site is equal to four: One singlet state and three triplet states. If a magnetic material is composed of localized singlets, which can be detected by a constant magnetic susceptibility, then we can easily picture that on each site, the singlet state is gapped from the rest three triplet states.
Now, imagine what will happen once we turn on the magnetic field. As we all know, the three triplet states possess Sz=+1,0,-1 separately. Suppose the field aligns along the plus direction, the gap between singlet and Sz=+1 triplet decreases as we enhance the field strength, and eventually disappears. What would happen after gap closing is that the singlet and Sz=+1 triplet states hybridize, leading to the off-diagonal magnetic moment and thus condensation.
Despite a quite simple picture, where to look for an appropriate candidate of material becomes an issue. The key ingredient that we might need is a strong enough on-site coupling that couples the two spin degrees of freedom and makes them form a spin singlet. For such purpose, we will need a strong enough antiferromagnetic Heisenberg interaction on each site, and thus it becomes a Kondo lattice. A recent work by Jin et al. utilizing first-principle calculation proposed that such Kondo-lattice model is likely to be responsible for a peculiar magnetic behavior of a nickelate compound, Ba2NiO2(AgSe)2, hosting the spin singlet state as its ground state.
Given all the above-mentioned facts and discoveries, we study the microscopic two-dimensional Kondo-necklace model in the square lattice using the numerical technique called infinite projected entangled-pair state (iPEPS), a powerful tensor network ansatz in two dimensions. When the magnetic field is zero, we can locate the region in the parameter space where the spin singlet state becomes the ground state. As we turn on the magnetic field, as explained earlier the hybridization of two distinct states will take place after hc, leading to the spin condensation. For a further fruitful scenario, we introduce the XXZ anisotropy to the inter-site Heisenberg coupling. We discover that with a strong enough anisotropy strength, a new state of condensation along with a real-space solid order can be realized, suggesting the existence of an exotic spin supersolid state.
Last but not least, studying various nickelate compounds has been a hot topic for investigation due to their many interesting and potential features, such as the superconductivity. For Ba2NiO2(AgSe)2, we have also proposed an effective many-body Hamiltonian that we believe can be used to describe its behavior under doping. If this model can be later shown possessing the superconducting state as its ground state under a certain doping level, it might suggest the existence of superconductivity in Ba2NiO2(AgSe)2, which has yet been discovered.
In sum, our work reveals many aspects of the intriguing properties for a Kondo material such as Ba2NiO2(AgSe)2, and we hope that it can help trigger some further research interest in related materials and topics, in both theoretical and experimental sides.