New and Old Results in Resultant Theory

Resultants are getting increasingly important in modern theoretical physics: they appear whenever one deals with non-linear (polynomial) equations, with non-quadratic forms or with non-Gaussian integrals. Being a subject of more than three-hundred-year research, resultants are of course rather well studied: a lot of explicit formulas, beautiful properties and intriguing relationships are known in this field. We present a brief overview of these results, including both recent and already classical. Emphasis is made on explicit formulas for resultants, which could be practically useful in a future physics research.Comment: 50 pages, 15 figure

Ultraviolet Behavior of the Gluon Propagator in the Maximal Abelian Gauge

The ultraviolet asymptotic behavior of the gluon propagator is evaluated in the maximal Abelian gauge in the SU(2) gauge theory on the basis of the renormalization-group improved perturbation theory at the one-loop level. Square-root singularities obtained in the Euclidean domain are attributed to artifacts of the one-loop approximation in the maximal Abelian gauge and the standard normalization condition for the propagator used in our study. It is argued that this gauge is essentially nonperturbative.Comment: 15 pages, 2 figure

Bulk rheology and microrheology of active fluids

We simulate macroscopic shear experiments in active nematics and compare them with microrheology simulations where a spherical probe particle is dragged through an active fluid. In both cases we define an effective viscosity: in the case of bulk shear simulations this is the ratio between shear stress and shear rate, whereas in the microrheology case it involves the ratio between the friction coefficient and the particle size. We show that this effective viscosity, rather than being solely a property of the active fluid, is affected by the way chosen to measure it, and strongly depends on details such as the anchoring conditions at the probe surface and on both the system size and the size of the probe particle.Comment: 12 pages, 10 figure

Faces of matrix models

Partition functions of eigenvalue matrix models possess a number of very different descriptions: as matrix integrals, as solutions to linear and non-linear equations, as tau-functions of integrable hierarchies and as special-geometry prepotentials, as result of the action of W-operators and of various recursions on elementary input data, as gluing of certain elementary building blocks. All this explains the central role of such matrix models in modern mathematical physics: they provide the basic "special functions" to express the answers and relations between them, and they serve as a dream model of what one should try to achieve in any other field.Comment: 10 page

Black Hole Motion in Entropic Reformulation of General Relativity

We consider a system of black holes -- a simplest substitute of a system of point particles in the mechanics of general relativity -- and try to describe their motion with the help of entropic action: a sum of the areas of black hole horizons. We demonstrate that such description is indeed consistent with the Newton's laws of motion and gravity, modulo numerical coefficients, which coincide but seem different from unity. Since a large part of the modern discussion of entropic reformulation of general relativity is actually based on dimensional considerations, for making a next step it is crucially important to modify the argument, so that these dimensionless parameters acquire correct values.Comment: 6 page

Temperature-induced topological phase transition in HgTe quantum wells

We report a direct observation of temperature-induced topological phase transition between trivial and topological insulator in HgTe quantum well. By using a gated Hall bar device, we measure and represent Landau levels in fan charts at different temperatures and we follow the temperature evolution of a peculiar pair of "zero-mode" Landau levels, which split from the edge of electron-like and hole-like subbands. Their crossing at critical magnetic field $B_c$ is a characteristic of inverted band structure in the quantum well. By measuring the temperature dependence of $B_c$, we directly extract the critical temperature $T_c$, at which the bulk band-gap vanishes and the topological phase transition occurs. Above this critical temperature, the opening of a trivial gap is clearly observed.Comment: 5 pages + Supplemental Materials; Phys. Rev. Lett. (accepted

Interplay between lattice, orbital, and magnetic degrees of freedom in the chain-polymer Cu(II) breathing crystals

The chain-polymer Cu(II) breathing crystals C21H19CuF12N4O6 were studied using the x-ray diffraction and ab initio band structure calculations. We show that the crystal structure modification at T=146 K, associated with the spin crossover transition, induces the changes of the orbital order in half of the Cu sites. This in turn results in the switch of the magnetic interaction sign in accordance with the Goodenough-Kanamori-Andersen theory of the coupling between the orbital and spin degrees of freedom.Comment: 6 pages, 7 figure

Optical vector network analysis of ultra-narrow transitions in 166^{166}Er3+^{3+}:7^7LiYF4_4

We present optical vector network analysis (OVNA) of an isotopically purified $^{166}$Er$^{3+}$:$^7$LiYF$_4$ crystal. The OVNA method is based on generation and detection of modulated optical sideband by using a radio-frequency vector network analyzer. This technique is widely used in the field of microwave photonics for the characterization of optical responses of optical devices such as filters and high-Q resonators. However, dense solid-state atomic ensembles induce a large phase shift on one of the optical sidebands which results in the appearance of extra features on the measured transmission response. We present a simple theoretical model which accurately describes the observed spectra and helps to reconstruct the absorption profile of a solid-state atomic ensemble as well as corresponding change of the refractive index in the vicinity of atomic resonances.Comment: 4 pages, 5 figure