We revisit the phase transition from the Néel ordered to a valence bond solid (VBS) state in the two-dimensional $J_1\text{−}{J}_2$ antiferromagnetic Heisenberg model. In the first part we address the question whether or not this transition could be an example of a second-order phase transition due to a deconfinement of spinons. We give arguments based on series expansion and spin-wave theory that this is not the case and the transition is most likely first order. The method proposed here to detect first-order phase transitions seems to be very sensitive and might be useful in other models as well. In the second part we analyze possible VBS patterns in the magnetically disordered phase based on numerical data for different susceptibilities, obtained in the ordered phase, which test the breaking of lattice symmetries. We conclude that a columnar dimerization pattern is the most likely candidate.

We analyze the phase diagram of the frustrated Heisenberg antiferromagnet, the $J_1{-J}_2$ model, in two dimensions. Two quantum phase transitions in the model are already known: the second-order transition from the Néel state to the spin liquid state at ${(J}_2{/J}_1{)}_{c2\mathrm{}}\approx 0.38,$ and the first-order transition from the spin liquid state to the collinear state at ${(J}_2{/J}_1{)}_{c4\mathrm{}}\approx \mathrm{0.60.}$ We have found evidence for two second-order phase transitions: the transition from the spin columnar dimerized state to the state with plaquette-type modulation at ${(J}_2{/J}_1{)}_{c3\mathrm{}}=0.50\mathrm{}\pm 0.02,$ and the transition from the simple Néel state to the Néel state with spin columnar dimerization at ${(J}_2{/J}_1{)}_{c1\mathrm{}}=0.34\mathrm{}\pm \mathrm{0.04.}$ We also present an independent calculation of ${(J}_2{/J}_1{)}_{c2\mathrm{}}\approx 0.38$ using another approach.

We study topological defects in anisotropic ferromagnets with competing interactions near the Lifshitz point. We show that Skyrmions and bimerons are stable in a large part of the phase diagram. We calculate Skyrmion-Skyrmion and meron-meron interactions and show that Skyrmions attract each other and form ring-shaped bound states in a zero magnetic field. At the Lifshitz point merons carrying a fractional topological charge become deconfined. These results imply that unusual topological excitations may exist in weakly frustrated magnets with conventional crystal lattices.

We consider quantum spin systems with dimerization, which at strong coupling have singlet ground states. To account for strong correlations, the $S=1\mathrm{}$ elementary excitations are described as a dilute Bose gas with infinite on-site repulsion. This approach is applied to the two-layer Heisenberg antiferromagnet at $T=0\mathrm{}$ with general couplings. Our analytic results for the triplet gap, the excitation spectrum, and the location of the quantum critical point are in excellent agreement with numerical results obtained by dimer series expansions.

We calculate exactly the vacuum polarization charge density in the field of a subcritical Coulomb impurity, $Z|e|/r$, in graphene. Our analysis is based on the exact electron Green’s function, obtained by using the operator method, and leads to results that are exact in the parameter $Z\alpha$, where $\alpha$ is the “fine-structure constant” of graphene. Taking into account also electron-electron interactions in the Hartree approximation, we solve the problem self-consistently in the subcritical regime, where the impurity has an effective charge $Z_{\mathrm{eff}}$, determined by the localized induced charge. We find that an impurity with bare charge $Z=1$ remains subcritical, $Z_{\mathrm{eff}}\alpha <1/2$, for any $\alpha$, while impurities with $Z=2$, 3 and higher can become supercritical at certain values of $\alpha$.

The phonon Hall effect has been observed in the paramagnetic insulator ${\mathrm{Tb}}_3{\mathrm{Gd}}_5{\mathrm{O}}_{12}$. A magnetic field applied perpendicularly to a heat current induces a temperature gradient that is perpendicular to both the field and the current. We show that this effect is due to resonant skew scattering of phonons from the crystal field states of superstoichiometric ${\mathrm{Tb}}^{3+}$ ions. This scattering originates from the coupling between the quadrupole moment of ${\mathrm{Tb}}^{3+}$ ions and the lattice strain. The estimated magnitude of the effect is consistent with experimental observations at $T\sim 5\mathrm{K}$ and can be significantly enhanced by increasing temperature.