Abstract:
Two decades ago, Sadovskii found an exact solution of a model describing a pseudogap in electron energy spectrum (first introduced by Lee, Rice and Anderson). The discovery of a pseudogap in high-Tc superconductors has revived the interest to his exact solution. I review the model with the emphasis on physical content, point out an error in the original Sadovskii's solution and explain which problem he actually solved. A recent incorporation of Sadovskii's ideas into a description of "hot spots" on the Fermi surface in cuprate superconductors (Schmalian, Pines and Stojkovic) is briefly discussed.

Abstract:
This paper explores the large-S route to quantum disorder in the Heisenberg antiferromagnet on the pyrochlore lattice and its homologues in lower dimensions. It is shown that zero-point fluctuations of spins shape up a valence-bond solid at low temperatures for one two-dimensional lattice and a liquid with very short-range valence-bond correlations for another. A one-dimensional model demonstrates potential significance of quantum interference effects (as in Haldane's gap): the quantum melting of a valence-bond order yields different valence-bond liquids for integer and half-integer values of S.

Abstract:
I present an exactly solvable model of a pseudogap with two zero-energy fermion modes coupled to each other by a classical source of frequency omega_0 and strength |Delta|. A suitably defined fermion propagator has an infinite number of poles at frequencies that are multiple integers of omega_0. In the adiabatic limit, omega_0 << |\Delta|, the situation is qualitatively different from the static case omega_0=0: the residue of the pole at omega=0 (a remnant of the bare fermion) vanishes linearly with omega_0, a result that could not be anticipated by perturbation theory; the multiple poles of the propagator coalesce into a continuum instead of forming two single poles at +-|Delta|, which should be interpreted as inhomogeneous broadening of the Bogoliubov quasiparticles.

Abstract:
Linear and angular momenta of a soliton in a ferromagnet are commonly derived through the application of Noether's theorem. We show that these quantities exhibit unphysical behavior: they depend on the choice of a gauge potential in the spin Lagrangian and can be made arbitrary. To resolve this problem, we exploit a similarity between the dynamics of a ferromagnetic soliton and that of a charged particle in a magnetic field. For the latter, canonical momentum is also gauge-dependent and thus unphysical; the physical momentum is the generator of magnetic translations, a symmetry combining physical translations with gauge transformations. We use this analogy to unambiguously define conserved momenta for ferromagnetic solitons. General considerations are illustrated on simple models of a domain wall in a ferromagnetic chain and of a vortex in a thin film.

Abstract:
The physics of a parity anomaly, potentially observable in a narrow-gap semiconductor, is revisited. Fradkin, Dagotto, and Boyanovsky have suggested that a Hall current of anomalous parity can be induced by a Peierls distortion on a domain wall. I argue that a perturbation inducing the parity anomaly must break the time reversal symmetry, which rules out the Peierls distortion as a potential cause. I list all possible perturbations that can generate the anomaly.

Abstract:
Spin excitations in an ordered Heisenberg magnet are magnons--bosons with spin 1. That may change when frustration and quantum fluctuations suppress the order and restore the spin-rotation symmetry. We show that spin excitations in the $S=1/2$ Heisenberg antiferromagnet on kagome are spinons--fermions with spin 1/2. In the ground state the system can be described as a collection of small, heavy pairs of spinons with spin 0. A magnetic excitation of lowest energy amounts to breaking up a pair into two spinons at a cost of $0.06 J$.

Abstract:
The ground state of the S=1/2 Heisenberg antiferomagnet on kagome can be viewed as a collection of fermionic spinons bound into small, heavy singlet pairs. Low-energy magnetic excitations in this system correspond to breaking the pairs into individual spinons. We calculate the structure factor for inelastic neutron scattering from independent spinon pairs.

Abstract:
We derive the spectrum of low-frequency spin waves in skyrmion crystals observed recently in noncentrosymmetric ferromagnets. We treat the skyrmion crystal as a superposition of three helices whose wavevectors form an equilateral triangle. The low-frequency spin waves are Goldstone modes associated with displacements of skyrmions. Their dispersion is determined by the elastic properties of the skyrmion crystal and by the kinetic terms of the effective Lagrangian, which include both kinetic energy and a Berry-phase term reflecting a nontrivial topology of magnetization. The Berry-phase term acts like an effective magnetic field, mixing longitudinal and transverse vibrations into a gapped cyclotron mode and a twist wave with a quadratic dispersion.

Abstract:
We investigate charged Bose liquid immersed in uniform background charge at zero temperature. Novel phenomena, such as oscillatory shielding of external localized electric charge, rotons and charge density waves (charge stripes in two dimensions), occur in any dimensions. Oscillatory shielding is caused by mixing between scalar boson exchange and Coulomb interactions, which mediate opposite forces. On the other hand, rotons and charge density waves are due to attractive local self-interaction of bosons. Rotons can be regarded as a finite size charge density wave packet without any back flow. We also comment on charge stripes observed recently in cuprates and nickelates.

Abstract:
We study quantum spin ice in an external magnetic field applied along a $<100>$ direction. When quantum spin fluctuations are weak, elementary excitations are quantum strings with monopoles at their ends manifested as multiple spin-wave branches in the dynamical structure factor. Strong quantum fluctuations make the string tension negative and give rise to the deconfinement of monopoles. We discuss our results in the light of recent neutron scattering experiments in $\mathrm{Yb_2Ti_2O_7}$