There is a long time gap between the formulation of the basic theory of low-dimensional (low-D) magnetism as advanced by Ising, Heisenberg and Bethe and its experimental verification. The latter started not long before the discovery of high-T_C superconductivity in cuprates and has been boosted by this discovery result in an impressive succession of newly observed physical phenomena. Milestones on this road were the compounds which reached their quantum ground states upon lowering the temperature either gradually or through different instabilities. The gapless and gapped ground states for spin excitations in these compounds are inherent for isolated half-integer spin and integer spin chains, respectively. The same is true for the compounds hosting odd and even leg spin ladders. Some complex oxides of transition metals reach gapped ground state by means of spin-Peierls transition, charge ordering or orbital ordering mechanisms. However, the overwhelming majority of low-dimensional systems arrive to a long-range ordered magnetic state, albeit quite exotic realizations. Under a magnetic field some frustrated magnets stabilize multipolar order, e.g., showing a spin-nematic state in the simplest quadropolar case. Finally, numerous square, triangular, kagome and honeycomb layered lattices, along with Shastry–Sutherland and Nersesyan–Tsvelik patterns constitute the playground to check the basic concepts of two-dimensional magnetism, including resonating valence bond state, Berezinskii–Kosterlitz–Thouless transition and Kitaev model.

From thermodynamics, local spin density $\mathrm{\text{approximation}}+\mathrm{\text{Hubbard}}$ $U$ studies and exact diagonalizations of a five-band Hubbard model on ${\mathrm{CuO}}_2$ stripes we find that ${\mathrm{Li}}_2{\mathrm{ZrCuO}}_4$ (${\mathrm{Li}}_2{\mathrm{CuZrO}}_4$ in traditional notation) is close to a ferromagnetic critical point. Analyzing its susceptibility $\chi (T)$ and specific heat $c_p(T,H)$ within a Heisenberg model, we show that the ratio of the 2nd to the 1st neighbor exchange integrals $\alpha =-J_2/J_1\sim 0.3$ is close to the critical value ${\alpha }_c=\frac{1}{4}$. Comparing with related chain cuprates we explain the rather strong field dependence of $c_p$, the monotonic downshift of the peak of $\chi (T)$, and its increase for $\alpha \to {\alpha }_c+0$.

The plate-like single crystals of tetragonal (P4/nmm) FeSe_1−x superconductors were grown using the KCl–AlCl₃ flux technique which produced single crystalline tetragonal samples of about 4 × 4 × 0.1 mm³ dimensions. The energy dispersive X-ray spectroscopy established a ratio of Fe : Se = 1 : 0.96 ± 0.02. The resistivity and magnetization measurements revealed a sharp superconducting transition at T_c = 9.4 K. Multiple Andreev reflections spectroscopy pointed to the existence of two-gap superconductivity with the gap values Δ_L = 2.4 ± 0.2 meV and Δ_S = 0.75 ± 0.1 meV at 4.2 K.

The physical properties of perovskites of the type $\mathrm{A}{\mathrm{C}}_3{\mathrm{B}}_4{\mathrm{O}}_{12}$ , whose structure derives from simple perovskites $\mathrm{A}\mathrm{B}{\mathrm{O}}_3$ , are reviewed. The A position is subject to strong structural distortions and splits into two new positions A and C. In the structure of $\mathrm{A}{\mathrm{C}}_3{\mathrm{B}}_4{\mathrm{O}}_{12}$ vacancies and any cations with a large radius, irrespective of their charge state, can be present in the icosahedral environment of A: ${\mathrm{Na}}^{+}$ , ${\mathrm{Cd}}^{2+}$ , ${\mathrm{Ca}}^{2+}$ , ${\mathrm{Sr}}^{2+}$ , ${\mathrm{Y}}^{3+}$ , ${\mathrm{Ln}}^{3+}$ , and ${\mathrm{Nd}}^{4+}$ . The C position in the square environment of oxygen can be occupied only by the Jahn–Teller cations ${\mathrm{Cu}}^{2+}$ and ${\mathrm{Mn}}^{3+}$ . Transition and nontransition metal ions—${\mathrm{Mn}}^{3+}$ , ${\mathrm{Fe}}^{3+}$ , ${\mathrm{Al}}^{3+}$ , ${\mathrm{Cr}}^{3+}$ , ${\mathrm{Ti}}^{4+}$ , ${\mathrm{Mn}}^{4+}$ , ${\mathrm{Ge}}^{4+}$ , ${\mathrm{Ru}}^{4+}$ , ${\mathrm{Ir}}^{4+}$ , ${\mathrm{Ta}}^{5+}$ , ${\mathrm{Nb}}^{5+}$ , ${\mathrm{Ta}}^{5+}$ , ${\mathrm{Sb}}^{5+}$ —can occupy the B position in an octahedral environment. Some members of the family of complex perovskites possess properties which are characteristic for systems with heavy fermions; collinear and noncollinear magnetic structures with high ordering temperatures occur in these materials; tunneling magnetoresistance and high permittivity are observed. The diversity and unique properties make these materials attractive for practical applications.