Low Dimensional Physics and Gauge Principles

The following sections are included:

• FOREWORD

• CONTENTS

We consider integrable models, i.e. models defined by R-matrices, on random Manhattan lattices (RML). The set of random Manhattan lattices is defined as the set dual to the lattice random surfaces embedded on a regular d-dimensional lattice. As an example we formulate a random matrix model where the partition function reproduces annealed average of the XXZ Heisenberg chain over all RML. A technique is presented which reduces the random matrix integration in partition function to an integration over their eigenvalues.

We present the results of magnetic properties and entanglement of the distorted diamond chain model for azurite using pure quantum exchange interactions. The magnetic properties and concurrence as a measure of pairwise thermal entanglement have been studied by means of variational mean-field like treatment based on Gibbs-Bogoliubov inequality. Such a system can be considered as an approximation of the natural material azurite, Cu₃(CO₃)₂(OH)₂. For values of exchange parameters, which are taken from experimental results, we study the thermodynamic properties, such as specific heat and magnetic susceptibility. We also have studied the thermal entanglement properties and magnetization plateau of the distorted diamond chain model for azurite.

We also consider entangled diamond chain with exactly solvable Ising and anisotropic Heisenberg (Ising-XXZ) model. The thermal entanglement (concurrence) is constructed for different values of anisotropic Heisenberg parameter, magnetic field and temperature.

We calculate the set of corrections to the branching ratio and to the photon energy spectrum of the decay process originating from the interference of diagrams involving the electromagnetic dipole operator O₇ with diagrams involving the chromomagnetic dipole operator O₈. The corrections evaluated here are one of the elements needed to complete the calculations of the branching ratio at next-to-next-to-leading order in QCD. We conclude that this set of corrections does not change the central value of the Standard Model prediction for by more than 1%.

The progress on calorons (finite temperature instantons) is sketched. In particular there is some interest for confining temperatures, where the holonomy (the asymptotic value of the Polyakov loop) is non-trivial. In the last section I give more recent results by others.

The purpose of the “bootstrap program” for integrable quantum field theories in 1+1 dimensions is to construct explicitly a model in terms of its Wightman functions. In this article, the “bootstrap program” is briefly reviewed and mainly illustrated in terms of the SU(N) Gross-Neveu model and the O(N) σ- model. Systems of SU(N)- and O(N)-matrix difference equations are solved by means of the off-shell version of the nested algebraic Bethe ansatz. In the nesting process for the O(N) case a new object, the Π-matrix, is introduced to overcome the complexities of the O(N) group structure. Some explicit examples are discussed.

We consider some aspects of spontaneous breaking of Lorentz Invariance in field theories, discussing the possibility that the certain tensor operators may condensate in the ground state in which case the tensor Goldstone particles would appear. We analyze their dynamics and discuss to which extent such a theory could imitate the gravity. We are also interested if the universality of coupling of such ‘gravitons’ with other particles can be achieved in the infrared limit.

We present some general approach to emergent gauge theories and consider in significant detail the emergent tensor field gravity case. In essence, an arbitrary local theory of a symmetric two-tensor field H_μν in Minkowski spacetime is considered, in which the equations of motion are required to be compatible with a nonlinear σ model type length-fixing constraint leading to spontaneous Lorentz invariance violation, SLIV (M is the proposed scale for SLIV). Allowing the parameters in the Lagrangian to be adjusted so as to be consistent with this constraint, the theory turns out to correspond to linearized general relativity in the weak field approximation, while some of the massless tensor Goldstone modes appearing through SLIV are naturally collected in the physical graviton. The underlying diffeomophism invariance emerges as a necessary condition for the tensor field H_μν not to be superfluously restricted in degrees of freedom, apart from the constraint due to which the true vacuum in the theory is chosen by SLIV. The emergent theory appears essentially nonlinear, when expressed in terms of the pure Goldstone tensor modes and contains a plethora of new Lorentz and CPT violating couplings. However, these couplings do not lead to physical Lorentz violation once this tensor field gravity is properly extended to conventional general relativity.

According to General Relativity, as distinct from Newtonian gravity, motion under gravity is treated by a theory that deals, initially, only with test particles. At the same time, satellite measurements deal with extended bodies. We discuss the correspondence between geodesic motion in General Relativity and the motion of an extended body by means of the Ehlers-Geroch theorem, and in the context of the recently launched LAser RElativity Satellite (LARES). Being possibly the highest mean density orbiting body in the Solar system, this satellite provides the best realization of a test particle ever reached experimentally and provides a unique possibility for testing the predictions of General Relativity.

We consider the several phenomena which are taking place in Quantum Dots (QD) and Quantum Rings (QR): The connection of the Quantum Chaos (QC) with the reflection symmetry of the QD, Disappearance of the QC in the tunnel coupled chaotic QD, electron localization and transition between Double Concentric QR in the transverse magnetic field, transition of electron from QR to the QD located in the center of QR. Basis of this consideration is the effective Schrödinger equation for the corresponding systems.

We study an exactly solvable model of non-Abelian anyons symmetric under the quantum double of the dihedral group D₃ on a one-dimensional lattice. Bethe ansatz methods are employed to compute the ground states of this model in different regions of the parameter space. The finite size spectrum is studied and the corresponding low energy field theories are identified.

We derive an effective spin Hamiltonian for the one-dimensional half-filled spin asymmetric ionic-Hubbard model in the limit of strong on-site repulsion. We have shown that the effective Hamiltonian, which describes spin degrees of freedom of the initial lattice electron system is given by the Hamiltonian of the frustrated XXZ Heisenberg chain with an additional alternating three spin coupling term in the presence of alternating magnetic field.

We discuss the hyperbolicity in perturbed Friedmann-Robertson-Walker Universe and the cummulative role of inhomogeneities and their link to observations. The degree of the randomness defined by the Kolmogorov stochasticity parameter, acts as an efficient descriptor for astrophysical datasets such as those of the cosmic microwave background radiation and X-ray clusters of galaxies.

We have calculated the electromagnetic properties of neutrinos at an interface. Particulary, we have measured the transition radiation process by neutrinos ν → ν_γ at an interface of two media. The medium fulfills the dual purpose of inducing an effective neutrino-photon vertex and of modifying the photon dispersion relation. The transition radiation occurs when at least one of those two quantities have different values in different media. The neutrinos are taken to be with only standard-model couplings and neutrino mass is ignored due to its negligible contribution. We present a result for the probability of the transition radiation which is both accurate and analytic. For example, E_ν = 1MeV neutrino crossing polyethylene-vacuum interface the transition radiation probability is about 10⁻³⁹ and the energy intensity is about 10⁻³⁴eV. At the surface of the neutron stars the transition radiation probability may be ∼ 10⁻²⁰.

We analyze the correlations in the cosmological cosmic microwave background radiation and those of point sources, i.e. galaxies and quasars. The Wilkinson Microwave Anisotropy Probe 7-year data are used along with the method developed by Tegmark et al. for obtaining of foreground reduced maps, combined with the Kolmogorov parameter. We obtain the cross correlations in the temperature and K-maps, and report the detection of still unidentified point sources, along with the previously known ones, and hence, showing the power of the applied technique.

The four-fermion gravitational interaction is induced by torsion. It gets dominating below the Planck scale. The regular, axial-axial part of this interaction by itself does not stop the gravitational compression. However, the anomalous, vector-vector interaction results in a natural way both in big bounce and in inflation.

We consider the capture and ejection of dark matter by the Solar System. Both processes are due to the gravitational three-body interaction of the Sun, a planet, and a dark matter particle. Simple estimates are presented for the capture cross-section, as well as for density and velocity distribution of captured dark matter particles close to the Earth.

Electrons in a two-dimensional periodic potential with a magnetic field is considered. The Hall conductances are shown to be integers, i.e Chen numbers. Thus these electronic bands give one of the first examples of topologically ordered state of matter.

There are two cases where QCD string is described by an effective theory of long strings: the static potential and meson scattering amplitudes in the Regge regime. I show how the former can be solved in the mean-field approximation, justified by the large number of space-time dimensions, and argue that it turns out to be exact for the Nambu–Goto string. By adding extrinsic curvature I demonstrate how the tachyonic instability of the ground-state energy can be cured by operators less relevant in the infrared.

Based on the talks given at the NBI Summer Institute “Strings, gauge theory and the LHC”, Copenhagen August, 22 – September, 02 2011 and the Workshop “Low dimensional physics and gauge principles”, Yerevan September 21 – 26, 2011.

This note is an attempt to explain in simple words why the famous relation E = mc² misrepresents the essence of Einstein's relativity theory. The first part of the note is addressed to high-school teachers, the rest – to those university professors of Physics who allow themselves to say that the mass of a body increases with its velocity.

After reviewing briefly the dimensional reduction of Chern–Pontryagin densitie, we define new Chern–Simons densities expressed in terms of Yang-Mills and Higgs fields. These are defined in all dimensions, including in even dimensional spacetimes. They are constructed by subjecting the dimensionally reduced Chern–Pontryagin densites to further descent by two steps.

Considered is spherical shell as a model for visible universe and parameters that such model must have to comply with the observable data. The topology of the model requires that motion of all galaxies and light must be confined inside a spherical shell. Consequently the observable universe cannot be defined as a sphere centered on the observer, rather it is an arc length within the volume of the spherical shell. The radius of the shell is 4.46 ± 0.06 Gpc, which is for factor π smaller than radius of a corresponding 3-sphere. However the event horizon, defined as the arc length inside the shell, has the size of 14.0 ± 0.2 Gpc, which is in agreement with the observable data. The model predicts, without inflation theory, the isotropy and uniformity of the CMB. It predicts the correct value for the Hubble constant H₀ = 67.26 ± 0.90 km/s/Mpc, the cosmic expansion rate H(z), and the speed of the event horizon in agreement with observations. The theoretical suport for shell model comes from general relativity, curvature of space by mass, and from holographic principle. The model explains the reason for the established discrepancy between the non-covariant version of the holographic principle and the calculated dimensionless entropy (S/k) for the visible universe, which exceeds the entropy of a black hole. The model is in accordance with the distribution of radio sources in space, type Ia data, and data from the Hubble Ultra Deep Field optical and near-infrared survey.

The structure of atomic levels originating from the lowest Landau level in a superstrong magnetic field is analyzed. The influence of the screening of the Coulomb potential on the values of critical nuclear charge is studied.

We consider shadows around black holes as a tool to evaluate their parameters. Kerr and Reissner – Nordström cases are analyzed. Different ways to find signatures of extra dimensions are discussing in the literature. If the Randall–Sundrum II braneworld scenario is adopted then a metric of black holes may be different from a standard one and the Schwarzschild metric has to be changed with the Reissner – Nordström metric with a tidal charge. The model may be applied for the black hole at the Galactic Center. Since the Schwarzschild and Reissner – Nordström metrics are different for a significant charge one can expect that geodesics of bright stars near the black hole and their observed fluxes may be different for these metrics due to a difference in gravitational lensing. Therefore, a deviation from the Schwarzschild case flux may be measured for a significant charge and the signature of extra dimensions may found. However, as a theoretical analysis shows that a tidal charge has to be very close to zero and a suggested charge value which may lead to measurable flux deviations for S2 like stars (or their essential displacements or astrometrical lensing) is not consistent with observational constraints on a shadow size.

We study the scattering dynamics of a spinor wavefunction in a random environment on a two-dimensional lattice. In the presence of particle-hole symmetry we find diffusion on large scales. The latter is described by a non-interacting Grassmann field, indicating a special kind of asymptotic freedom on large length scales in d = 2.

The following sections are included:

• AUTHOR INDEX