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. A collection of bosonic particles, such as liquid helium or ultracold gases, can condense into a ground state in which the atoms flow as a ‘superfluid’ without scattering. Magnetic materials further illustrate the generality of the effect, as described in this review.

Bose–Einstein condensation in magnetic insulators

We review ground states and excitations of a quantum antiferromagnet on triangular and other frustrated lattices. We pay special attention to the combined effects of magnetic field h, spatial anisotropy R and spin magnitude S. The focus of the review is on the novel collinear spin density wave and spin nematic states, which are characterized by fully gapped transverse spin excitations with Sz = ± 1. We discuss extensively the R − h phase diagram of the antiferromagnet, both in the large-S semiclassical limit and the quantum S = 1/2 limit. When possible, we point out connections with experimental findings.

https://iopscience.iop.org/article/10.1088/0034-4885/78/5/052502/pdf

Unusual ordered phases of highly frustrated magnets: a review

It has long been noticed that special lattices contain single-electron flat bands (FB) without any dispersion. Since the kinetic energy of electrons is quenched in the FB, this highly degenerate energy level becomes an ideal platform to achieve strongly correlated electronic states, such as magnetism, superconductivity, and Wigner crystal. Recently, the FB has attracted increasing interest because of the possibility to go beyond the conventional symmetry-breaking phases towards topologically ordered phases, such as lattice versions of fractional quantum Hall states. This article reviews different aspects of FBs in a nutshell. Starting from the standard band theory, we aim to bridge the frontier of FBs with the textbook solid-state physics. Then, based on concrete examples, we show the common origin of FBs in terms of destructive interference, and discuss various many-body phases associated with such a singular band structure. In the end, we demonstrate real FBs in quantum frustrated materials and organometallic frameworks.

Exotic electronic states in the world of flat bands: From theory to material

The energy dispersion of fermions or bosons vanishes in momentum space if destructive quantum interference occurs in a frustrated Kagome lattice with only nearest-neighbor hopping. A discrete flat band (FB) without any dispersion is consequently formed, promising the emergence of fractional quantum Hall states at high temperatures. Here, we report the experimental realization of an FB with possible nontrivial topology in an electronic Kagome lattice on twisted multilayer silicene. Because of the unique low-buckled two-dimensional structure of silicene, a robust electronic Kagome lattice has been successfully induced by moire patterns after twisting the silicene multilayers. The electrons are localized in the Kagome lattice because of quantum destructive interference, and thus, their kinetic energy is quenched, which gives rise to an FB peak in the density of states. A robust and pronounced one-dimensional edge state has been revealed at the Kagome edge, which resides at higher energy than the FB. Our observations of the FB and the exotic edge state in electronic Kagome lattice open up the possibility that fractional Chern insulators could be realized in two-dimensional materials.

https://ro.uow.edu.au/cgi/viewcontent.cgi?article=4419&context=aiimpapers

Realization of flat band with possible nontrivial topology in electronic Kagome lattice

https://cyberleninka.org/article/n/859500.pdf

Quantum kagome frustrated antiferromagnets: One route to quantum spin liquids

It is important to study the many-body effects of the electrons on the flat band lattice, since the flatness of the dispersion may lead to instabilities in the presence of interactions. To clarify the effects of flat dispersion for superconductivity, we investigate a simple two-band BCS Hamiltonian, where one of the bands has a flat dispersion, in the mean field approximation. A superconducting gap Δ and a critical temperature Tc strongly depend on the pairing coupling constant V. Especially, for small coupling constant, Δ and Tc are proportional to V, which are quite different from the single band BCS cases. Higher Tc, compared to the normal BCS system, are realized due to the existence of the flat band.

BCS theory on a flat band lattice

We investigate the interplay of Dzyaloshinskii-Moriya interactions and an external field in spin-$1∕2$ dimers. For isolated dimers and at low field, we derive simple expressions for the staggered and uniform magnetizations which show that the orientation of the uniform magnetization can deviate significantly from that of the external field. In fact, in the limit where the $\mathbf{D}$ vector of the Dzyaloshinskii-Moriya interaction is parallel to the external field, the uniform magnetization actually becomes perpendicular to the field. For larger fields, we show that the staggered magnetization of an isolated dimer has a maximum close to one-half the polarization, with a large maximal value of $0.35g{\ensuremath{\mu}}_{B}$ in the limit of very small Dzyaloshinskii-Moriya interaction. We investigate the effect of interdimer coupling in the context of ladders with density-matrix renormalization-group (DMRG) calculations and show that, as long as the values of the Dzyaloshinskii-Moriya interaction and of the exchange interaction are compatible with respect to the development of a staggered magnetization, the simple picture that emerges for isolated dimers is also valid for weakly coupled dimers with minor modifications. The results are compared with torque measurements on ${\mathrm{Cu}}_{2}{({\mathrm{C}}_{5}{\mathrm{H}}_{12}{\mathrm{N}}_{2})}_{2}{\mathrm{Cl}}_{4}$.

/pdf/uniform-and-staggered-magnetizations-induced-by-39pccx33df.pdf

Uniform and staggered magnetizations induced by Dzyaloshinskii-Moriya interactions in isolated and coupled spin-1/2 dimers in a magnetic field

Cs2CuBr4 is a new two-dimensional spin- 1 2 system, where 1 3 -plateau has been observed in external magnetic fields for H ∥ b and H ∥ c . It is expected that magnetic behaviors of the material are well explained by a two-dimensional antiferromagnetic Heisenberg model on a distorted triangular lattice. In the model, there are two types of interactions, J 1 and J 2 , where J 1 chains are coupled with interchain interactions J 2 . Using an exact diagonalization method, we investigate magnetic properties, especially magnetization curve. In the magnetization, 1 3 -plateau appears for 0.7 ≲ J 2 / J 1 ≲ 1.3 . In the plateau phase, three-fold degenerate translational symmetry broken ground state is realized. Our results indicate that the material has a more frustrated character than what has been expected from a classical theory.

Magnetization plateaux of Cs2CuBr4

We investigate a detail of a dodecamer cluster ordering in a double-exchange spin ice model on a kagome lattice. In frustrated systems, ordinary spin orderings are suppressed and macroscopic degeneracy remains down to low temperatures. In some frustrated systems, the degeneracy is lifted due to residual interactions and cluster orderings are stabilized. In the present model, the spin ice state is first formed at intermediate temperatures, and further entropies are released at lower temperatures as the dodecamer phase emerges. Since the spin symmetry is not broken in the dodecamer phase, there still exists macroscopic degeneracy. At further low temperatures, a possible spin ordering due to inter-dodecamer interactions is proposed. We discuss that such a multiple-site clustering larger than a bond-pair might be generic to frustrated systems where macroscopic degeneracy is lifted by residual interactions.

Frustration-induced Dodecamer Ordering in the Double-Exchange Spin Ice Model on the Kagomé Lattice

We show that Dzyaloshinskii-Moriya (DM) interactions can substantially modify the phase diagram of spin-1/2 Heisenberg ladders in a magnetic field provided they compete with exchange. For nonfrustrated ladders, they induce a local magnetization along the DM vector that turns the gapless intermediate phase into an Ising phase with broken translational symmetry, while for frustrated ladders, they extend the Ising order of the half-integer plateau to the surrounding gapless phases of the purely Heisenberg case. Implications for experimental ladder and dimer systems are discussed.

/pdf/ising-phases-of-heisenberg-ladders-in-a-magnetic-field-2xviyhdahr.pdf

Shin Miyahara

Papers

BCS theory on a flat band lattice

Uniform and staggered magnetizations induced by Dzyaloshinskii-Moriya interactions in isolated and coupled spin-1/2 dimers in a magnetic field

Magnetization plateaux of Cs2CuBr4

Frustration-induced Dodecamer Ordering in the Double-Exchange Spin Ice Model on the Kagomé Lattice

Ising phases of Heisenberg ladders in a magnetic field