Geometry, topology, and physics of non-Abelian lattices

Conclusion  After reviewing in some detail the notion of non-Euclidean lattices, whose domain of physical realization lies mostly in the novel carbon structures of the family offullerenes, we have discussed a number of physical problems denned over such lattices. We have shown that the group-theoretical definition of these lattices leads to “designing” new tubular regular structures, endowed with symmetries unheard of in the frame of customary crystallography, which combine features of extreme complexity and, at the same time, of great regularity. We have compared the role of the non-Abelian symmetries which these super-lattices are characterized by, with that of (discrete) harmonic (Fourier) lattice symmetry typical of customary crystallographic lattices. Many novel features enter into play, due to thenon-flatness of the related lattice geometry, which led us to a novel—sometimes unexpected—insight into the dynamical and/or thermodynamical properties of various physical systems which have these lattices as ambient space. We have analyzed how lattice topology bears on the complex combinatorics (related to loop-counting) of the classical Ising model. These lattices, even though finite, are, of course, much closer to being three-dimensional than regular 2D lattices simply equipped with periodic boundary conditions. We have shown, on the other hand, how the relation between the lattice symmetry (for example, in the case of fullerene, the discrete subgroup ofSU(2) that we have denotedg ₆₀ and the symmetry proper to the Hamiltonian of quantum systems of many itinerant interacting electrons (Hubbard-like models) allows us to reduce the calculation of the system spectral properties to a “size” that can be dealt with numerically with present-day numerical exact diagonalization techniques much more easily than a regular 3D cluster with a quite smaller number of sites.     

Effect of a photonic band gap in the optical range on solid-state SiO2 cluster lattices — opals

It is shown that synthetic opals — cubic face-centered lattices of SiO₂ clusters — are systems which exhibit a number of properties of photonic crystals in the visible-light range. By filling the voids (pores) in such lattices with different materials it is possible to vary the optical contrast of the medium and to obtain crystals of both the lattice of spheres type and its three-dimensional replica. It is shown that under conditions of identical optical contrast and in the presence of an additional optical inhomogeneity of the spheres, the transparency of the lattice of spheres is lower than that of its replica based on homogeneous media. A refractive index modulation of 1.266 was achieved in the lattice of spheres. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 7, 496–501 (10 April 1996)     

On Classical and Quantum Objectivity

We propose a conceptual framework for understanding the relationship between observables and operators in mechanics. To do so, we introduce a postulate that establishes a correspondence between the objective properties permitting to identify physical states and the symmetry transformations that modify their gauge dependant properties. We show that the uncertainty principle results from a faithful—or equivariant—realization of this correspondence. It is a consequence of the proposed postulate that the quantum notion of objective physical states is not incomplete, but rather that the classical notion is overdetermined.     

Quantum cryptography based on homodyne detection (vacuum-state cryptosystem)

A new protocol is proposed for quantum cryptography. The protocol is based on the use of a set of measurements which make it possible to reconstruct completely the density matrix — the information carrier — of a physical system. Such a protocol can be implemented by means of homodyne detection (well known in quantum optics) of an electromagnetic field. An example is given of a quantum cryptosystem in which the vacuum state of the photon field is used as one of two information states. Pis’ma Zh. éksp. Teor. Fiz. 66, No. 1, 64–68 (10 July 1997)     

Electronic spectrum of a two-dimensional Fibonacci lattice

The electronic spectrum and wave functions of a new quasicrystal structure—a two-dimensional Fibonacci lattice—are investigated in the tight-binding approximation using the method of the level statistics. This is a self-similar structure consisting of three elementary structural units. The “central” and “nodal” decoration of this structure are examined. It is shown that the electronic energy spectrum of a two-dimensional Fibonacci lattice contains a singular part, but in contrast to a one-dimensional Fibonacci lattice the spectrum does not contain a hierarchical gap structure. The measure of allowed states (Lebesgue measure) of the spectrum is different from zero, and for “central” decoration it is close to 1. The character of the localization of the wave functions is investigated, and it is found that the wave functions are “critical.” Zh. éksp. Teor. Fiz. 116, 1834–1842 (November 1999)     

Off-centering of Pb and Sn impurities in GeTe induced by strong local stress

Noncentrality of large impurity atoms—Pb and Sn atoms—substituting for Ge atoms in a GeTe lattice has been discovered by means of EXAFS investigations in Ge_0.9Pb_0.1Te and Ge_0.85Sn_0.15Te samples. The transition of impurity atoms into a noncentral position under conditions of a strong local stress is explained by the participation of an unshared electron pair from the impurity atoms in the formation of the chemical bond. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 8, 600–603 (25 April 1996)     

Evolution of large-scale correlations in a strongly nonequilibrium Bose condensation process

The evolution of off-diagonal correlation functions (for the example of a single-particle density matrix) in the process of Bose condensation of an initially nonequilibrium interacting gas is discussed. Special attention is given to the character of the decay of the density matrix at distances much greater than the size of the quasicondensate region. Specifically, it is shown that the exponential decay of the density matrix necessarily presupposes the presence of a chaotic vortex structure — a tangle of vortex lines — in the system. When topological order is established but there is no off-diagonal long-range order, the density matrix decays with distance according to a power law. Pis’ma Zh. éksp. Teor. Fiz. 67, No. 7, 495–501 (10 April 1998)     

Wavevector-Dependent Susceptibility in <Emphasis Type="Italic">Z</Emphasis>-Invariant Pentagrid Ising Model

We study the q-dependent susceptibility χ(q) of a Z-invariant ferromagnetic Ising model on a Penrose tiling, as first introduced by Korepin using de Bruijn's pentagrid for the rapidity lines. The pair-correlation function for this model can be calculated exactly using the quadratic difference equations from our previous papers. Its Fourier transform χ(q) is studied using a novel way to calculate the joint probability for the pentagrid neighborhoods of the two spins, reducing this calculation to linear programming. Since the lattice is quasiperiodic, we find that χ(q) is aperiodic and has everywhere dense peaks, which are not all visible at very low or high temperatures. More and more peaks become visible as the correlation length increases—that is, as the temperature approaches the critical temperature. Supported in part by NSF Grant No. PHY 01-00041.     

A Lorentz-Poincaré Type Interpretation of the Weak Equivalence Principle

The validity of the Weak Equivalence Principle relative to a local inertial frame is detailed in a scalar-vector gravitation model with Lorentz-Poincaré type interpretation. Given the previously established first Post-Newtonian concordance of dynamics with General Relativity, the principle is to this order compatible with GRT. The gravitationally modified Lorentz transformations, on which the observations in physical coordinates depend, are shown to provide a physical interpretation of parallel transport. A development of ‘geodesic’ deviation in terms of the present model is given as well. PACS subject classifications. 04.20.-q, 04.50.+h     

An approximate expression for the galvanomagnetic tensor

Using the iterative solution to the Boltzmann equation for electrons in d.c. electric and magnetic fields, an expression for the resistivity tensor can be obtained in the form of an infinite series. This series can be approximated by retaining only the first two terms. In the cases where relaxation times exist — in the sense that the collision term in the Boltzmann equation can be written asg(k)/τ(k), whereτ(k) is the relaxation time, andf (k) = f _E(ɛ k) + [∂f _E(εk)/∂ε]·g(k) the distribution function for electrons with wavevectork — this approximation is exact. For polyvalent metals in the one-OPW approximation, the complete galvanomagnetic tensor can be obtained using this approximation and the result differs from that obtained by using a time of relaxation given by an expression suggested byZiman. A calculation for a simple model Fermi surface, with screened Coulomb scattering, is carried out and the results compared with those of the relaxation time approximation.     

Interface design in nanosystems of Advanced Superionic Conductors

First attempt of practical realization of new interface engineering approach “from advanced materials to advanced devices” for nanosystems of Advanced Superionic Conductors (ASICs), based on AgI (CuI) compounds is presented. Crystallochemical method of symmetry perfect ASIC//electrode interface searching is developed. Some new theoretical results of ASIC//indifferent electrode conjugated commensurate heteropairs with coherent interfaces and preliminary experimental results of the creation of thin-film supercapacitor — prototype based on the lattice matched heterojunction — are given. Future perspectives of the ASIC//electrode interface design suited for micro(nano)electronics and microsystem technology (MST) are discussed. Paper presented at the Patras Conference on Solid State Ionics — Transport Properties, Patras, Greece, Sept. 14–18, 2004.     

Symmetry and Topology in Quantum Logic

A test space is a collection of non-empty sets, usually construed as the catalogue of (discrete) outcome sets associated with a family of experiments. Subject to a simple combinatorial condition called algebraicity, a test space gives rise to a “quantum logic”—that is, an orthoalgebra. Conversely, all orthoalgebras arise naturally from algebraic test spaces. In non-relativistic quantum mechanics, the relevant test space is the set ℱ F(H) of frames (unordered orthonormal bases) of a Hilbert space H. The corresponding logic is the usual one, i.e., the projection lattice L(H) of H. The test space ℱ F(H) has a strong symmetry property with respect to the unitary group of H, namely, that any bijection between two frames lifts to a unitary operator. In this paper, we consider test spaces enjoying the same symmetry property relative to an action by a compact topological group. We show that such a test space, if algebraic, gives rise to a compact, atomistic topological orthoalgebra. We also present a construction that generates such a test space from purely group-theoretic data, and obtain a simple criterion for this test space to be algebraic. PACS: 02.10.Ab; 02.20.Bb; 03.65.Ta.     

Antipolarization effect in quantum wells in a strong external alternating field

It is shown that when a strong ac electric field acts on an electron in a double quantum well, the dipole moment is an almost periodic function of the dc voltage applied to the structure. An antipolarization effect — the structure is polarized in a direction opposite to the external field — appears during one half of the period. Pis’ma Zh. éksp. Teor. Fiz. 70, No. 6, 386–391 (25 September 1999)     

On the interrelation of the characteristic scales of depolarization and decorrelation of optical fields under multiple-scattering conditions

The interrelation of depolarization and decorrelation of optical fields in multiply scattering Brownian media is studied on the basis of the notion of the probability density of optical path lengths of the partial components of the scattered field under multiple-scattering conditions. To describe such media a universal parameter that is independent of the density (concentration) of scattering particles is introduced — the characteristic correlation time. Experimental results obtained with aqueous suspensions of polystyrene beads as model media are presented which demonstrate the constancy of this parameter at different concentrations of scattering particles. Pis’ma Zh. éksp. Teor. Fiz. 67, No. 7, 455–460 (10 April 1998)     

Poisson Approximations for the Ising Model

A d-dimensional Ising model on a lattice torus is considered. As the size n of the lattice tends to infinity, a Poisson approximation is given for the distribution of the number of copies in the lattice of any given local configuration, provided the magnetic field a = a(n) tends to −∞ and the pair potential b remains fixed. Using the Stein-Chen method, a bound is given for the total variation error in the ferromagnetic case. AMS SUBJECT CLASSIFICATION: 60F05, 82B20.     

Non-physical consequences of the muffin-tin-type intra-molecular potential

We demonstrate, using a simple model, that, in the frame of muffin-tin-like potential, non-physical peculiarities appear in molecular photoionization cross-sections that are a consequence of “jumps” in the potential and its first derivative at some radius. The magnitude of non-physical effects is of the same order as the physical oscillations in the cross-section of a diatomicmolecule. The role of the size of these “jumps” is illustrated by choosing three values for it. The results obtained are connected to the previously studied effect of non-analytic behavior as a function of r, the potential V(r) acting upon a particle on its photoionization cross-section. In reality, such potential has to be analytic in magnitude and have a first derivative function in r. The introduction of non-analytic features in model V(r) leads to non-physical features — oscillations, additional maxima, and so forth — in the corresponding cross-section.