Isoperiodic classical systems and their quantum counterparts

One-dimensional isoperiodic classical systems have been first analyzed by Abel. Abel’s characterization can be extended for singular potentials and potentials which are not defined on the whole real line. The standard shear equivalence of isoperiodic potentials can also be extended by using reflection and inversion transformations. We provide a full characterization of isoperiodic rational potentials showing that they are connected by translations, reflections or Joukowski transformations. Upon quantization many of these isoperiodic systems fail to exhibit identical quantum energy spectra. This anomaly occurs at order O(?²) because semiclassical corrections of energy levels of order O(?) are identical for all isoperiodic systems. We analyze families of systems where this quantum anomaly occurs and some special systems where the spectral identity is preserved by quantization. Conversely, we point out the existence of isospectral quantum systems which do not correspond to isoperiodic classical systems.     

Towards a dynamical temperature of finite Hamiltonian systems

With the aid of numerical simulations of the β Fermi-Pasta-Ulam (FPU) system, we compare the different definitions of dynamical temperature for Hamiltonian systems. We have shown that each definition gives different values of temperature for a system with a small number of degrees of freedom (DOF). Only for systems with a sufficiently large number of DOF, do all the definitions of dynamical temperature approach the same value.     

Invariants for E0-Semigroups on II1 Factors

We introduce four new cocycle conjugacy invariants for E ₀-semigroups on II ₁ factors: a coupling index, a dimension for the gauge group, a super product system and a C*-semiflow. Using noncommutative Itô integrals we show that the dimension of the gauge group can be computed from the structure of the additive cocycles. We do this for the Clifford flows and even Clifford flows on the hyperfinite II ₁ factor, and for the free flows on the free group factor ${L(F_\infty)}$ L ( F ∞ ) . In all cases the index is 0, which implies they have trivial gauge groups. We compute the super product systems for these families and, using this, we show they have trivial coupling index. Finally, using the C*-semiflow and the boundary representation of Powers and Alevras, we show that the families of Clifford flows and even Clifford flows contain infinitely many mutually non-cocycle-conjugate E ₀-semigroups.     

On the role of electron correlation and disorder on persistent currents in isolated one-dimensional rings

To understand the role of electron correlation and disorder on persistent currents in isolated 1D rings threaded by magnetic flux ?, we study the behavior of persistent currents in aperiodic and ordered binary alloy rings. These systems may be regarded as disordered systems with well-defined long-range order so that we do not have to perform any configuration averaging of the physical quantities. We see that in the absence of interaction, disorder suppresses persistent currents by orders of magnitude and also removes its discontinuity as a function of ?. As we introduce electron correlation, we get enhancement of the currents in certain disordered rings. Quite interestingly we observe that in some cases, electron correlation produces kink-like structures in the persistent current as a function of ?. This may be considered as anomalous Aharonov-Bohm oscillations of the persistent current and recent experimental observations support such oscillations. We find that the persistent current converges with the size of the rings.     

On the Distribution of the Wave Function for Systems in Thermal Equilibrium

For a quantum system, a density matrix ρ that is not pure can arise, via averaging, from a distribution μ of its wave function, a normalized vector belonging to its Hilbert space ?. While ρ itself does not determine a unique μ, additional facts, such as that the system has come to thermal equilibrium, might. It is thus not unreasonable to ask, which μ, if any, corresponds to a given thermodynamic ensemble? To answer this question we construct, for any given density matrix ρ, a natural measure on the unit sphere in ?, denoted GAP(ρ). We do this using a suitable projection of the Gaussian measure on ? with covariance ρ. We establish some nice properties of GAP(ρ) and show that this measure arises naturally when considering macroscopic systems. In particular, we argue that it is the most appropriate choice for systems in thermal equilibrium, described by the canonical ensemble density matrix ρ_β = (1/Z) exp (?β H). GAP(ρ) may also be relevant to quantum chaos and to the stochastic evolution of open quantum systems, where distributions on ? are often used.     

The N*(1710) as a resonance in the ππN system

We study the ππN system by solving the Faddeev equations, for which the input two-body t-matrices are obtained by solving the Bethe-Salpeter equation in the coupled-channel formalism. The potentials for the ππ, πN sub-systems and their coupled channels are obtained from chiral Lagrangians, which have been earlier used to study resonances in these systems successfully. In this work, we find a resonance in the ππN system with a mass of 1704 ? i375/2MeV and with quantum numbers I = 1/2, J ^π = 1/2⁺. We identify this state with the N ^*(1710). This peak is found where the energies of the ππ sub-system fall in the region of the σ-resonance. We do not find evidence for the Roper resonance in our study indicating a more complex structure for this resonance, nor for any state with total isospin I = 3/2 or 5/2.     

Frequency dependent conductivity of random one-dimensional chains

We examine the frequency dependent conductivity of an exactly soluble model of a random system, the one-dimensional quantum percolation model. Since this system does not fall into the category of weak scattering by the disorder, we do not find the asymptoticω ²ln²(ω/ω ₀) behavior suggested by Mott. For systems which are most disordered, we find that the conductivity spectrum consists of discrete levels, while for systems in which the disorder is much less dense, we may use a continuum approximation. The conductivity is then given by anω ^?3 cosech (2π/ωτ) law. We speculate that such behavior may be found in mixed valent planar compounds such as K₂Pt(CN₄)3(H₂O) with small amounts of Br or Cl added.     

On the derivation of power-law distributions within classical statistical mechanics far from the thermodynamic limit

We show that within classical statistical mechanics, without taking the thermodynamic limit, the most general Boltzmann factor for the canonical ensemble is a q-exponential function. The only assumption here is that microcanonical distributions have to be separated from the total system energy, which is the prerequisite for any sensible measurement. We derive that all separable distributions are parametrized by a mathematical separation constant Q, which can be related to the non-extensivity q-parameter in Tsallis distributions. We further demonstrate that nature fixes the separation constant Q to 1 for large dimensionality of Gibbs $\Gamma$-phase space. Our results will be relevant for systems with a low-dimensional $\Gamma$-space, for example nanosystems, comprised of a small number of particles, or for systems with a dimensionally collapsed phase space, which might be the case for a large class of complex systems.     

Relativistic Point Dynamics and Einstein Formula as a Property of Localized Solutions of a Nonlinear Klein-Gordon Equation

Einstein’s relation E = Mc ² between the energy E and the mass M is the cornerstone of the relativity theory. This relation is often derived in a context of the relativistic theory for closed systems which do not accelerate. By contrast, the Newtonian approach to the mass is based on an accelerated motion. We study here a particular neoclassical field model of a particle governed by a nonlinear Klein-Gordon (KG) field equation. We prove that if a solution to the nonlinear KG equation and its energy density concentrate at a trajectory, then this trajectory and the energy must satisfy the relativistic version of Newton’s law with the mass satisfying Einstein’s relation. Therefore the internal energy of a localized wave affects its acceleration in an external field as the inertial mass does in Newtonian mechanics. We demonstrate that the “concentration” assumptions hold for a wide class of rectilinear accelerating motions.     

Operator Systems from Discrete Groups

We express various sets of quantum correlations studied in the theoretical physics literature in terms of different tensor products of operator systems of discrete groups. We thus recover earlier results of Tsirelson and formulate a new approach for the study of quantum correlations. To do this we formulate a general framework for the study of operator systems arising from discrete groups. We study in detail the operator system of the free group ${\mathbb{F}_n}$ on n generators, as well as the operator systems of the free products of finitely many copies of the two-element group ${\mathbb{Z}_2}$ . We examine various tensor products of group operator systems, including the minimal, the maximal, and the commuting tensor products. We introduce a new tensor product in the category of operator systems and formulate necessary and sufficient conditions for its equality to the commuting tensor product in the case of group operator systems.     

Critique of generalised purity as an entanglement measure: Explicit failure in simple separable Bose–Hubbard models

The SU(2) and SU(3) Lie algebras lend themselves naturally to studies of two- and three-well Bose–Einstein condensates, with the group operators being expressed in terms of bosonic annihilation and creation operators at each site. The success of these representations has led to the purities associated with these algebras to be promoted as a measure of entanglement in these systems. In this work, we show that these purities do not provide an unambiguous measure of entanglement between wells, but instead give results which depend on the quantum statistical states of the atomic ensembles in each well. Using the example of totally uncoupled wells where the atoms in one have never interacted with the atoms in the other, we quantify these purities for different states and show that completely separable states can give values which have been claimed to indicate the presence of entanglement. We also consider claims that the generalised purities measure particle rather than mode entanglement, with emphasis on the case of indistinguishable bosons, as found in these systems.     

Entanglement as a Semantic Resource

The characteristic holistic features of the quantum theoretic formalism and the intriguing notion of entanglement can be applied to a field that is far from microphysics: logical semantics. Quantum computational logics are new forms of quantum logic that have been suggested by the theory of quantum logical gates in quantum computation. In the standard semantics of these logics, sentences denote quantum information quantities: systems of qubits (quregisters) or, more generally, mixtures of quregisters (qumixes), while logical connectives are interpreted as special quantum logical gates (which have a characteristic reversible and dynamic behavior). In this framework, states of knowledge may be entangled, in such a way that our information about the whole determines our information about the parts; and the procedure cannot be, generally, inverted. In spite of its appealing properties, the standard version of the quantum computational semantics is strongly “Hilbert-space dependent”. This certainly represents a shortcoming for all applications, where real and complex numbers do not generally play any significant role (as happens, for instance, in the case of natural and of artistic languages). We propose an abstract version of quantum computational semantics, where abstract qumixes, quregisters and registers are identified with some special objects (not necessarily living in a Hilbert space), while gates are reversible functions that transform qumixes into qumixes. In this framework, one can give an abstract definition of the notions of superposition and of entangled pieces of information, quite independently of any numerical values. We investigate three different forms of abstract holistic quantum computational logic.     

Long-range potentials and the electromagnetic polarizabilities

The long-range spin and velocity independent forces of electromagnetic origin which act between any two systems are studied for those cases in which no forces of this type exist to order e². It is shown that they are uniquely determined by the charge, magnetic moment, and polarizabilities of both systems, not only to the dominant order r^?n, but also to the next one r^?(n+1). These potentials provide the link between Compton scattering polarizabilities (response to real photons) and classically defined polarizabilities (response to static electromagnetic field). The two definitions are shown to be equivalent for neutral spinless systems; the problems arising for a neutral particle with magnetic moment are studied in detail. The r^?(n+1) terms have no classical counterpart, since they are due to the relativistic quantum propagation of the system which carries charge or magnetic moment. The results are of general validity with analyticity, crossing, unitarity, and gauge invariance as only inputs. The most general conclusion is that the polarizabilities represent electromagnetic properties of a system at order e², as the charge and magnetic moment do at order e. Thus they give the strength of the response to electric and magnetic fields, independently of the specific characteristics of the electromagnetic agent.     

Chimera states in a two–population network of coupled pendulum–like elements

More than a decade ago, a surprising coexistence of synchronous and asynchronous behavior called the chimera state was discovered in networks of nonlocally coupled identical phase oscillators. In later years, chimeras were found to occur in a variety of theoretical and experimental studies of chemical and optical systems, as well as models of neuron dynamics. In this work, we study two coupled populations of pendulum-like elements represented by phase oscillators with a second derivative term multiplied by a mass parameter m and treat the first order derivative terms as dissipation with parameter ? > 0. We first present numerical evidence showing that chimeras do exist in this system for small mass values 0 < m ? 1. We then proceed to explain these states by reducing the coherent population to a single damped pendulum equation driven parametrically by oscillating averaged quantities related to the incoherent population.     

Magnetocaloric effect: Overcoming the magnetic limit

We have studied anomalous peaks observed in magnetocaloric −ΔS(T) curves for systems that undergo first-order magnetostructural transitions. The origin of those peaks, which can exceed the conventional magnetic limit, R ln(2J+1), is discussed on thermodynamic bases by introducing an additional-exchange contribution (due to exchange constant variation arising from magnetostructural transition). We also applied a semiphenomenological model to include this additional-exchange contribution in Gd₅Si₂Ge₂- and MnAs-based systems, obtaining excellent results for the observed magnetocaloric effect.     

Ab initio studies of isolated hydrogen vacancies in graphane

We present a density functional study of various hydrogen vacancies located on a single hexagonal ring of graphane (fully hydrogenated graphene) considering the effects of charge states and the position of the Fermi level. We find that uncharged vacancies that lead to a carbon sublattice balance are energetically favorable and are wide band gap systems just like pristine graphane. Vacancies that do create a sublattice imbalance introduce spin polarized states into the band gap, and exhibit a half-metallic behavior with a magnetic moment of 1.00 μ_B per vacancy. The results show the possibility of using vacancies in graphane for novel spin-based applications. When charging such vacancy configurations, the deep donor (+1/0) and deep acceptor (0/−1) transition levels within the band gap are noted. We also note a half-metallic to metallic transition and a significant reduction of the induced magnetic moment due to both negative and positive charge doping.