Zeta-function Approach Research Articles

Thermodynamic properties of substrate-induced graphene superlattice in homogeneous magnetic field

In this article we study the thermodynamic properties of a substrate-induced graphene superlattice in the presence of uniform magnetic field B. We derive an expression for the partition function of the system with uniform magnetic field B using the zeta function approach and find all related thermodynamic functions such as Helmholtz free energy, total energy, specific heat capacity and entropy for three cases related to the sublattice potential. We also show that the Dulong-Petit law is verified for all cases in the high-temperature limit. These results may shed light on the study of graphene superlattices in the developement of thermo-electric devices.

Critical behaviour of an effective relativistic mean field model in the presence of magnetic background and boundaries

In the present work we investigate the combined influence of magnetic background and boundaries on the thermodynamic properties of effective relativistic mean field models, like the so-called Walecka model. This is done by making use of generalized zeta-function approach and mean-field approximation at effective chemical equilibrium, focusing on the dependence with the size of compactified spatial dimension, the temperature and the magnetic field strength. The findings suggest a rich phase structure in the parameter space. The maintenance of long-range correlations is strongly affected under the change of these parameters, with the symmetric phase being favoured due to both inverse magnetic catalysis effect and the reduction of size of compactified dimension.

Analogues of the Prime Number Theorem and Mertens’ Theorem for Closed Orbits of the Motzkin Shift

In this paper, the prime orbit theorem and Mertens’ orbit theorem are proved for a shift dynamical system of infinite type called the Motzkin shift. Instead of using the zeta function approach to find the asymptotic formulas, different and more direct methods are used in the proof without any complicated theoretical discussion.

THE CASIMIR-POLDER INTERACTION OF AN ATOM WITH SPHERICAL SHELL

The Casimir-Polder and van der Waals interaction energy of an atom with infinitely thin sphere with finite conductivity is investigated in the framework of the hydrodynamic approach. We put the sphere into spherical cavity inside the infinite dielectric media, then calculate the energy of vacuum fluctuations in the context of the zeta-function approach. The energy for a single atom is obtained by rarefying media. The Casimir-Polder expression for an atom and plate is recovered in the limit of the infinite radius of the sphere. Assuming a finite radius of the sphere, the interaction energy of an atom falls down monotonic as third power of distance between atom and sphere for short distance and as seventh power for large distance from the sphere.

Casimir force in brane worlds: Coinciding results from Green’s and zeta function approaches

Casimir force encodes the structure of the field modes as vacuum fluctuations and so it is sensitive to the extra dimensions of brane worlds. Now, in flat spacetimes of arbitrary dimension the two standard approaches to the Casimir force, Green's function and zeta function, yield the same result, but for brane world models this was only assumed. In this work we show both approaches yield the same Casimir force in the case of Universal Extra Dimensions and Randall-Sundrum scenarios with one and two branes added by p compact dimensions. Essentially, the details of the mode eigenfunctions that enter the Casimir force in the Green's function approach get removed due to their orthogonality relations with a measure involving the right hyper-volume of the plates and this leaves just the contribution coming from the Zeta function approach. The present analysis corrects previous results showing a difference between the two approaches for the single brane Randall-Sundrum; this was due to an erroneous hyper-volume of the plates introduced by the authors when using the Green's function. For all the models we discuss here, the resulting Casimir force can be neatly expressed in terms of two four dimensional Casimir force contributions: one for the massless mode and the other for a tower of massive modes associated with the extra dimensions.

Quantum instability for charged scalar particles on charged Nariai and ultracold black hole manifolds

We analyze in detail the quantum instability which characterizes charged scalar field on three special de Sitter charged black hole backgrounds. In particular, we compute exactly the imaginary part of the effective action for scalar charged fields on the ultracold I, ultracold II and Nariai charged black hole backgrounds. Both the transmission coefficient approach and the zeta-function approach are exploited. Thermal effects on this quantum instability are also taken into account in the presence of a non-zero black hole temperature (ultracold I and Nariai).

Pair-production of charged Dirac particles on charged Nariai and ultracold black hole manifolds

Spontaneous loss of charge by charged black holes by means of pair-creation of charged Dirac particles is considered. We provide three examples of exact calculations for the spontaneous discharge process for 4D charged black holes by considering the process on three special non-rotating de Sitter black hole backgrounds, which allow to bring back the problem to a Kaluza-Klein reduction. Both the zeta-function approach and the transmission coefficient approach are taken into account. A comparison between the two methods is also provided, as well as a comparison with WKB results. In the case of non-zero temperature of the geometric background, we also discuss thermal effects on the discharge process.

Simplified vacuum energy expressions for radial backgrounds and domain walls

We extend our previous results of simplified expressions for functional determinants for radial Schrödinger operators to the computation of vacuum energy, or mass corrections, for static but spatially radial backgrounds, and for domain wall configurations. Our method is based on the zeta function approach to the Gel'fand–Yaglom theorem, suitably extended to higher-dimensional systems on separable manifolds. We find new expressions that are easy to implement numerically, for both zero and non-zero temperatures.

Zeta-function approach to Casimir energy with singular potentials

In the framework of zeta-function approach the Casimir energy for three simple model system: single delta potential, step function potential and three delta potentials are analyzed. It is shown that the energy contains contributions which are peculiar to the potentials. It is suggested to renormalize the energy using the condition that the energy of infinitely separated potentials is zero which corresponds to subtraction all terms of asymptotic expansion of zeta-function. The energy obtained in this way obeys all physically reasonable conditions. It is finite in the Dirichlet limit, and it may be attractive or repulsive depending on the strength of potential. The effective action is calculated, and it is shown that the surface contribution appears. The renormalization of the effective action is discussed.

Zero Point Energy of a Massless Scalar Field in the Cosmic String Space-Time

We consider the ground state energy of a massive scalar field in the space-time of a thick cosmic string in the 2+1 dimensional case for arbitrary angle deficit by using the zeta-function approach. Final numerical calculations were made in the massless case, only. We show that the zero point energy is negative, and for small angle deficit it is proportional to the fourth degree of the deficit.

Semiclassical wormholes

Smooth-throat wormholes are treated on as possessing quantum fluctuation energy with scalar massive field as its source. Heat kernel coefficients of the Laplace operator are calculated in background of the arbitrary-profile throat wormhole with the help of the zeta-function approach. Two specific profile are considered. Some arguments are given that the wormholes may exist. It serves as a solution of semiclassical Einstein equations in the range of specific values of length and certain radius of wormhole's throat and constant of non-minimal connection.

Thermal energy of a scalar field in a one-dimensional compact space

We discuss some controversial aspects of the evaluation of the thermal energy of a scalar field in a one-dimensional compact space. The calculations are carried out using a generalized zeta function approach.

Complete Zeta-Function Approach to the Electromagnetic Casimir Effect for Spheres and Circles

A method for evaluating the electromagnetic Casimir energy in situations involving spherical or circular boundaries is presented. Zeta function regularization is unambiguously used from the start and the properties of Bessel and related zeta functions are central. Nontrivial results concerning these functions are given. While part of their application agrees with previous knowledge, new results like the zeta-regularized electromagnetic Casimir energy for a circular wire are included.

A zeta function approach to the relation between the numbers of symmetry planes and axes of a polytope

A derivation of the Cesàro–Fedorov relation from the Selberg trace formula on an orbifolded 2-sphere is elaborated and extended to higher dimensions using the known heat-kernel coefficients for manifolds with piecewise-smooth boundaries. Several results are obtained that relate the coefficients, bi, in the Shephard–Todd polynomial to the geometry of the fundamental domain. For the 3-sphere, it is shown that b4 is given by the ratio of the volume of the fundamental tetrahedron to its Schläfli reciprocal.

A zeta function approach to the semiclassical quantization of maps

The quantum analogue of an area-preserving map on a compact phase-space is a unitary (evolution) operator which can be represented by a matrix of dimension L α ℏ −1. The semiclassical theory for the spectrum of the evolution operator will be reviewed with special emphasis on developing a dynamical zeta function approach, similar to the one introduced recently for the semiclassical quantization of hamiltonian flows.

An effective action for a variable electromagnetic field

This work consists essentially of two parts. The first part is an analysis of the one-loop effective action using the zeta-function approach. This gives a simple expression for the effective action in terms of the background field propagator. The next-of-kin to the zeta-function, the heat kernel, is given in terms of B. DeWitt's proper time expansion (also known as P. B. Gilkey's theorem). It is calculated in the second part for fermions interacting with an external electromagnetic field to first nonvanishing order in the variations of the gauge field.