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We uncovered for the first time the spin dynamics involved in the birth and growth of giant spin polarons in a magnetic semiconductor. For this purpose, we developed a new measurement technique, which for the first time provides direct access to the spin dynamics, irrespective of phonons and carriers involved in the process. Moreover, we solved for the first time the Landau-Liftshitz equation in the specific scenario of spin polarons, which fits our data excellently, and demonstrates for the first time that the spin polaron growth slows down dramatically when the sample is cooled in the paramagnetic phase. Finally, temperature dependent Monte Carlo simulations were performed, which are in excellent agreement with the observed slowdown, which demonstrates for the first time that fluctuations in the Weiss field play a decisive role in spin coherence generation induced by light in magnetic materials. These results offer a new tool and new insight for spin dynamics investigations.

In intrinsic magnetic semiconductors, the absorption of a single photon can generate a spin polaron, whose magnetic moment reaches many thousands of Bohr magnetons [1.2]. Here we investigate photoinduced spin polarons, using Monte Carlo simulations. In antiferromagnetic semiconductors, photoinduced spin polarons are most efficiently generated in the whole temperature interval up to the phase transition, whereas in ferromagnetic semiconductors much larger spin polarons can be photoinduced, but only around the phase transition temperature. Because Monte Carlo simulations are computationally expensive, we developed an analytical model, based on the Weiss field theory. Although the Weiss model does not provide as much information as a Monte Carlo simulation, such as spin texture and fluctuations, it yields formulas that can be used to estimate instantly the expected photoinduced spin polaron size in many intrinsic magnetic semiconductors., Comment: Accepted for publication in Phys. Rev. B

We find that in the ferromagnetic semiconductor EuS, near its Curie temperature, a single band-edge photon generates a spin polaron (SP), whose magnetic moment approaches 20,000 Bohr magnetons. This is much larger than the supergiant photoinduced SPs in antiferromagnetic europium chalcogenides, reported previously. The larger SP in ferromagnetic EuS, and still larger expected for EuO, is explained by a larger Bohr radius of the photoexcited electron state, which encircles and polarizes a greater number of lattice spins. However, because the wave function of the photoexcited electron spreads over a greater volume, the photoexcited electron exchange interaction with individual lattice spins weakens, which makes the SP more easily quenched thermally., Comment: Accepted for publication in Applied Physics Letters

We present a simple semiclassical model to sustain that in europium chalcogenides (EuX), Faraday rotation (FR) in the transparency gap is proportional to the magnetization of the sample, irrespective of the material's magnetic phase, temperature, or applied magnetic field. The model is validated by FR and magnetization measurements in EuSe in the temperature interval 1.7-300K, covering all EuSe magnetic phases (paramagnetic, antiferromagnetic type I or type II, ferrimagnetic and ferromagnetic). Furthermore, by combining the semiclassical model with the explicit electronic energy structure of EuX, the proportionality coefficient between magnetization and FR is shown to be dependent only on the wavelength and the band gap. Due to its simplicity, the model has didactic value, moreover, it provides a working tool for converting FR into magnetization in EuX. Possible extension of the model to other intrinsic magnetic semiconductors is discussed.

We demonstrate that light resonant with the bandgap forces the antiferromagnetic semiconducor EuSe to enter ferromagnetic alignment in the picosecond time scale. A photon generates an electron-hole pair, whose electron forms a supergiant spin polaron of magnetic moment of nearly 6,000 Bohr magnetons. By increasing the light intensity, the whole of the sample can be fully magnetized. The key to the novel large photoinduced magnetization mechanism is the huge enhancement of the magnetic susceptibility when both antiferromagnetic and ferromagnetic interactions are present in the material, and are of nearly equal magnitude, as is the case in EuSe.

A time-resolved optical pump-probe technique has been applied for studying the ultrafast dynamics in the magnetic semiconductor EuTe near the absorption band gap. We show that application of external magnetic field up to 6 T results in crossover from the inverse Faraday effect taking place on the femtosecond time scale to the optical orientation phenomenon with an evolution in the picosecond time domain. We propose a model which includes both these processes possessing different spectral and temporal properties. The circularly polarized optical pumping induces the optical electronic transition $4f^75d^0 \rightarrow 4f^65d^1$ forming the absorption band gap in EuTe. The observed crossover is related to a strong magnetic-field shift of the band gap in EuTe at low temperatures. It was found that manipulation of spin states on intrinsic defect levels takes place on a time scale of 19 ps in the applied magnetic field of 6 T., Comment: 11 pages, 4 figures

A probabilistic propositional logic, endowed with an epistemic component for asserting (non-)compatibility of diagonizable and bounded observables, is presented and illustrated for reasoning about the random results of projective measurements made on a given quantum state. Simultaneous measurements are assumed to imply that the underlying observables are compatible. A sound and weakly complete axiomatization is provided relying on the decidable first-order theory of real closed ordered fields. The proposed logic is proved to be a conservative extension of classical propositional logic.

A polytope $P$ is a {\em model} for a combinatorial problem on finite graphs $G$ whose variables are indexed by the edge set $E$ of $G$ if the points of $P$ with (0,1)-coordinates are precisely the characteristic vectors of the subset of edges inducing the feasible configurations for the problem. In the case of the (simple) MaxCut Problem, which is the one that concern us here, the feasible subsets of edges are the ones inducing the bipartite subgraphs of $G$. In this paper we introduce a new polytope $\mathbb{P}_{12} \subset \mathbb{R}^{|E|}$ given by at most $11|E|$ inequalities, which is a model for the MaxCut Problem on $G$. Moreover, the left side of each inequality is the sum of at most 4 edge variables with coefficients $\pm1$ and right side 0,1, or 2. We restrict our analysis to the case of $G=K_{z}$, the complete graph in $z$ vertices, where $z$ is an even positive integer $z\ge 4$. This case is sufficient to study because the simple MaxCut problem for general graphs $G$ can be reduced to the complete graph $K_z$ by considering the obective function of the associated integer programming as the characteristic vector of the edges in $G \subseteq K_z$. This is a polynomial algorithmic transformation., Comment: 14 pages, 5 figures

In a pioneer work, John Nagle has shown that an Ising chain with competing short and long-range interactions displays second and first-order phase transitions separated by a tricritical point. More recently, it has been claimed that Nagle's model provides an example of the inequivalence between canonical and microcanonical calculations. We then revisit Nagle's original solution, as well as the usual formulation of the problem in a canonical ensemble, which lead to the same results. Also, in contrast to recent claims, we show that an alternative formulation in the microcanonical ensemble, with the adequate choice of the fixed thermodynamic extensive variables, leads to equivalent thermodynamic results., Comment: 10 pages, no figures

Doubts are raised concerning the usual interpretation of the alleged failure, by quantum mechanics, of the distributive law of classical logic. The difficulty raised by incompatible sets of observables is overcome within an epistemic enrichment of classical logic that provides the means for distinguishing between the value of a variable and its observation while retaining the classical connectives.

The equivalence of thermodynamic results in the canonical and the microcanonical ensembles has been questioned in some calculations for spin models with long-range interactions. We show that these claims of inequivalence are related to an inadequate definition of the independent (density) variables in the microcanonical ensemble. We illustrate this point with the example of a simple spin-1 ideal paramagnet, and then revisit the original calculations of Barr\'e, Mukamel, and Ruffo, for a mean-field spin-1 Blume-Capel model. If the microcanonical ensemble is defined in terms of adequate density variables, we show that there is no disagreement with the calculations in the usual canonical ensemble (with fixed thermodynamic field variables)., Comment: 10 pages and no figures

This paper gives an explicit formula for the multiplier ideals, and consequently for the log canonical thresholds, of any GL(V)xGL(W)-invariant ideal in the symmetric algebra S of the tensor product of V with the dual of W, where V and W are vector spaces over a field of characteristic 0. This characterization is done in terms of a polytope constructed from the set of Young diagrams corresponding to the Schur modules generating the ideal. Our approach consists in computing the test ideals of some invariant ideals of S in positive characteristic: Namely, we will compute the test ideals (and so the F-pure thresholds) of any sum of products of determinantal ideals. Even in characteristic 0, not all the invariant ideals are as the latter, but they are up to integral closure, and this is enough to reach our goals. The results concerning the test ideals are obtained as a consequence of general results holding true in a special situation. Within such framework fall determinantal objects of a generic matrix, as well as of a symmetric matrix and of a skew-symmetric one. Similar results are thus deduced for the GL(V)-invariant ideals in the symmetric algebra of the second symmetric (or alternating) power of V. (Also monomial ideals fall in this framework, thus we recover Howald's formula for their multiplier ideals and, more generally, we get the formula for their test ideals). During the proof, we introduce the notion of "floating test ideals", a property that in a sense is satisfied by ideals defining schemes with singularities as nice as possible. As we will see, products of determinantal ideals, and by passing to characteristic 0 ideals generated by a single Schur module, have this property.

Lipid monolayers and bilayers have been used as experimental models for the investigation of membrane thermal transitions. The main transition takes place near ambient temperatures for several lipids and reflects the order-disorder transition of lipid hydrocarbonic chains, which is accompanied by a small density gap. Equivalence between the transitions in the two systems has been argued by several authors. The two-state statistical model adopted by numerous authors for different properties of the membrane, such as permeability, diffusion, mixture or insertion of cholesterol or protein, is inadequate for the description of charged membranes, since it lacks a proper description of surface density. We propose a lattice solution model which adds interactions with water molecules to lipid-lipid interactions and obtain its thermal properties under a mean-field approach. Density variations, although concomitant with chain order variations, are independent of the latter. The model presents both chain order and gas-liquid transitions, and extends the range of applicability of previous models, yielding Langmuir isotherms in the full range of pressures and areas., Comment: 26 pages, 11 figures

We explore the consequences of loop quantum cosmology (inverse-volume corrections) in the spectrum of the gravitational waves using the method of the Bogoliubov coefficients. These corrections are taken into account at the background level of the theory as well as at the first order in the perturbations theory framework. We show that these corrections lead to an intense graviton production during the loop super-inflationary phase prior to the standard slow-roll era, which leave their imprints through new features on the energy spectrum of the gravitational waves as would be measured today, including a new maximum on the low frequency end of the spectrum., Comment: 8 pages, 4 figures. RevTex4-1 Expanded version. Version accepted in PRD

A {\em blink} is a plane graph with an arbitrary bipartition of its edges. As a consequence of a recent result of Martelli, I show that the homeomorphisms classes of closed oriented 3-manifolds are in 1-1 correspondence with specific classes of blinks. In these classes, two blinks are equivalent if they are linked by a finite sequence of local moves, where each one appears in a concrete list of 64 moves: they organize in 8 types, each being essentially the same move on 8 simply related configurations. The size of the list can be substantially decreased at the cost of loosing symmetry, just by keeping a very simple move type, the {\em ribbon moves} denoted $\pm \mu_{11}^{\pm}$ (which are in principle redundant). The inclusion of $\pm \mu_{11}^{\pm}$ implies that all the moves corresponding to plane duality (the starred moves), except for $\mu_{20}^\star$ and $\mu_{02}^\star$, are redundant and the coin calculus is reduced to 36 moves on 36 coins. A {\em residual fraction link} or a {\em flink}, is a new object which generalizes {\em blackboard-framed link}. It plays an important role in this work. I try to make the topological exposition as complete as possible: about half of the exposition deals with the topological preliminaries. The objective is to make it easier for the combinatorially oriented readers to understand the paper. It is in the aegis of this work to find new important connections between 3-manifolds and plane graphs., Comment: 20 pages, 15 figures, 28 references. Version 9 fixes the figure explaining how to get a 3-manifold from a flink

We present an inflationary model preceded by a bounce in a metric $f(R)$ theory. In this model, modified gravity affects only the early stages of the universe. We analyse the predicted spectrum of the gravitational waves in this scenario using the method of the Bogoliubov coefficients. We show that there are distinctive (oscillatory) signals on the spectrum for very low frequencies; i.e., corresponding to modes that are currently entering the horizon., Comment: 4 pages, 2 figures. Contribution to the Spanish Relativity Meeting in Portugal 2012 (ERE2012), Guimaraes, Portugal

We present an inflationary model preceded by a bounce in a metric theory a l\'{a} $f(R)$ where $R$ is the scalar curvature of the space-time. The model is asymptotically de Sitter such that the gravitational action tends asymptotically to a Hilbert-Einstein action, therefore modified gravity affects only the early stages of the universe. We then analyse the spectrum of the gravitational waves through the method of the Bogoliubov coefficients by two means: taking into account the gravitational perturbations due to the modified gravitational action in the $f(R)$ setup and by simply considering those perturbations inherent to the standard Hilbert-Einstein action. We show that there are distinctive (oscillatory) signals on the spectrum for very low frequencies; i.e. corresponding to modes that are currently entering the horizon., Comment: 13 pages, 7 figures. Extensively expanded version. References added. Version accepted in PRD

We address the investigation of the solvation properties of the minimal orientational model for water, originally proposed by Bell and Lavis. The model presents two liquid phases separated by a critical line. The difference between the two phases is the presence of structure in the liquid of lower density, described through orientational order of particles. We have considered the effect of small inert solute on the solvent thermodynamic phases. Solute stabilizes the structure of solvent, by the organization of solvent particles around solute particles, at low temperatures. Thus, even at very high densities, the solution presents clusters of structured water particles surrounding solute inert particles, in a region in which pure solvent would be free of structure. Solute intercalates with solvent, a feature which has been suggested by experimental and atomistic simulation data. Examination of solute solubility has yielded a minimum in that property, which may be associated with the minimum found for noble gases. We have obtained a line of minimum solubility (TmS) across the phase diagram, accompanying the line of maximum in density (TMD). This coincidence is easily explained for non-interacting solute and it is in agreement with earlier results in the literature. We give a simple argument which suggests that interacting solute would dislocate TmS to higher temperatures.

In this paper we investigate the solubility of a hard - sphere gas in a solvent modeled as an associating lattice gas (ALG). The solution phase diagram for solute at 5% is compared with the phase diagram of the original solute free model. Model properties are investigated thr ough Monte Carlo simulations and a cluster approximation. The model solubility is computed via simulations and shown to exhibit a minimum as a function of temperature. The line of minimum solubility (TmS) coincides with the line of maximum density (TMD) for different solvent chemical potentials., Comment: pages, 13 figures

We calculate the full spectrum, as observed today, of the cosmological gravitational waves generated within a model based on loop quantum cosmology. It is assumed that the universe, after the transition to the classical regime, undergoes a period of inflation driven by a scalar field with a chaotic-type potential. Our analysis shows that, for certain conditions, loop quantum effects leave a clear signature on the spectrum, namely, an over-production of low-frequency gravitational waves. One of the aims of our work is to show that loop quantum cosmology models can be tested and that, more generally, pre-inflationary physical processes, contrary to what is usually assumed, leave their imprint in those spectra and can also be tested., Comment: 7 pages, 8 figures, revtex4

Extending a notion defined for surjective maps by Blanco, Majadas, and Rodicio, we introduce and study a class of homomorphisms of commutative noetherian rings, which strictly contains the class of locally complete intersection homomorphisms, while sharing many of its remarkable properties., Comment: Final version, to appear in the special issue of Pure and Applied Mathematics Quarterly dedicated to Andrey Todorov. The material in the first four sections has been reorganized and slightly expanded

In this paper we investigate the dynamic properties of the minimal Bell-Lavis (BL) water model and their relation to the thermodynamic anomalies. The Bell-Lavis model is defined on a triangular lattice in which water molecules are represented by particles with three symmetric bonding arms interacting through van der Waals and hydrogen bonds. We have studied the model diffusivity in different regions of the phase diagram through Monte Carlo simulations. Our results show that the model displays a region of anomalous diffusion which lies inside the region of anomalous density, englobed by the line of temperatures of maximum density (TMD). Further, we have found that the diffusivity undergoes a dynamic transition which may be classified as fragile-to-strong transition at the critical line only at low pressures. At higher densities, no dynamic transition is seen on crossing the critical line. Thus evidence from this study is that relation of dynamic transitions to criticality may be discarded.

We investigate the generation of gravitational waves in the hybrid quintessential inflationary model. The full gravitational-wave energy spectrum is calculated using the method of continuous Bogoliubov coefficients. The post-inflationary kination period, characteristic of quintessential inflationary models, leaves a clear signature on the spectrum, namely, a peak at high frequencies. The maximum of the peak is firmly located at the MHz-GHz region of the spectrum and corresponds to $\Omega_{GW} \simeq 10^{-12}$. This peak is substantially smaller than the one appearing in the gravitational-wave energy spectrum of the original quintessential inflationary model, therefore avoiding any conflict with the nucleosynthesis constraint on $\Omega_\Omega_{GW}$., Comment: 10 pages, 11 figures, one reference added

A new scenario for the early era of the Universe is proposed. It corresponds to a smooth transition between a de Sitter-like phase and a radiation dominated era. We calculate the production of gravitons in this model., Comment: 3 pages, 2 figures. To appear in the proceedings of 12th Marcel Grossmann Meeting, Paris (France)

We propose a new scenario for the early universe where there is a smooth transition between an early de Sitter-like phase and a radiation dominated era. In this model, the matter content is modelled by a new type of generalised Chaplygin gas for the early universe, with an underlying scalar field description. We study the gravitational waves generated by the quantum fluctuations. In particular, we calculate the gravitational wave power spectrum, as it would be measured today, following the method of the Bogoliubov coefficients. We show that the high frequencies region of the spectrum depends strongly on one of the parameters of the model. On the other hand, we use the number of e-folds, along with the power spectra and spectral index of the scalar perturbations, to constrain the model observationally., Comment: 8 pages, 6 figures, RevTeX 4. Published in PRD

The Bell-Lavis model for liquid water is investigated through numerical simulations. The lattice-gas model on a triangular lattice presents orientational states and is known to present a highly bonded low density phase and a loosely bonded high density phase. We show that the model liquid-liquid transition is continuous, in contradiction with mean-field results on the Husimi cactus and from the cluster variational method. We define an order parameter which allows interpretation of the transition as an order-disorder transition of the bond-network. Our results indicate that the order-disorder transition is in the Ising universality class. Previous proposal of an Ehrenfest second order transition is discarded. A detailed investigation of anomalous properties has also been undertaken. The line of density maxima in the HDL phase is stabilized by fluctuations, absent in the mean-field solution.

Spectroscopy of the centrosymmetric magnetic semiconductors EuTe and EuSe reveals spin-induced optical second harmonic generation (SHG) in the band gap vicinity at 2.1-2.4eV. The magnetic field and temperature dependence demonstrates that the SHG arises from the bulk of the materials due to a novel type of nonlinear optical susceptibility caused by the magnetic dipole contribution combined with spontaneous or induced magnetization. This spin-induced susceptibility opens access to a wide class of centrosymmetric systems by harmonics generation spectroscopy., Comment: 5 pages, 3 figures, submitted to PRL

Let R be a local ring with maximal ideal m admitting a non-zero element a\in\fm for which the ideal (0:a) is isomorphic to R/aR. We study minimal free resolutions of finitely generated R-modules M, with particular attention to the case when m^4=0. Let e denote the minimal number of generators of m. If R is Gorenstein with m^4=0 and e\ge 3, we show that \Poi MRt is rational with denominator \HH R{-t} =1-et+et^2-t^3, for each finitely generated R-module M. In particular, this conclusion applies to generic Gorenstein algebras of socle degree 3., Comment: 19 pages; new title, minor changes and extended bibliography

We investigate a cosmological model, based on the Salam-Sezgin six-dimensional supergravity theory and on previous work by Anchordoqui, Goldberg, Nawata, and Nu\~nez. Assuming a period of warm inflation, we show that it is possible to extend the evolution of the model back in time, to include the inflationary period, thus unifying inflation, dark matter, and dark energy within a single framework. Like the previous authors, we were not able to obtain the full dark matter content of the Universe from the Salam-Sezgin scalar fields. However, even if only partially successful, this work shows that present-day theories, based on superstrings and supergravity, may eventually lead to a comprehensive modelling of the evolution of the Universe. We find that the gravitational-wave spectrum of the model has a non-constant negative slope in the frequency range 10^(-15)-10^6 rad/s, and that, unlike standard (cold) inflation models, it shows no structure in the MHz/GHz range of frequencies., Comment: 10 pages, 10 figures

We investigate the generation of gravitational waves in power-law inflationary models. The energy spectrum of the gravitational waves is calculated using the method of continuous Bogoliubov coefficients. We show that, by looking at the interval of frequencies between 10^(-5) and 10^5 Hz and also at the GHz range, important information can be obtained, both about the inflationary period itself and about the thermalization regime between the end of inflation and the beginning of the radiation-dominated era. We thus deem the development of gravitational wave detectors, covering the MHz/GHz range of frequencies, to be an important task for the future., Comment: 8 pages, 6 figures, added 3 references, text improved

We investigate the production of gravitational waves due to quantum fluctuations of the vacuum during the transition from the inflationary to the radiation-dominated eras of the universe, assuming this transition to be dominated by the phenomenon of parametric resonance. The energy spectrum of the gravitational waves is calculated using the method of continuous Bogoliubov coefficients, which avoids the problem of overproduction of gravitons at large frequencies. We found, on the sole basis of the mechanism of quantum fluctuations, that the resonance field leaves no explicit and distinctive imprint on the gravitational-wave energy spectrum, apart an overall upward or downward translation. Therefore, the main features in the spectrum are due to the inflaton field, which leaves a characteristic imprint at frequencies of the order of MHz/GHz., Comment: 8 pages, 7 figures

We report an experimental study of miniband magnetoconduction in semiconducting InAs/GaSb superlattices. For samples with miniband widths below the longitudinal optical phonon energy we identify a new superlattice magnetophonon resonance (SLMPR) caused by resonant scattering of electrons across the mini-Brillouin zone. This new resonant feature arises directly from the drift velocity characteristics of the superlattice dispersion and total magnetic quantisation of the superlattice Landau level minibands., Comment: 9 pages, 8 figures, submitted to Phys. Rev. B

We investigate the phase diagram of a three-dimensional associating gas $(ALG)$ model. This model combines orientational ice-like interactions and ``van der Waals'' that might be repulsive, representing, in this case, a penalty for distortion of hydrogen bonds. These interactions can be interpreted as two competing distances making the connection between this model and continuous isotropic soft-core potentials. We present Monte Carlo studies of the $ALG$ model showing the presence of two liquid phase, two critical points and A density anomaly.

We have investigated the effects of either distorting hydrogen bonds or removing proton degeneracy on the thermodynamic properties of a minimal model for associating liquids. The presence of two liquid phases and a density anomaly is unaffected in both cases. Increasing the degeneracy of bonded structures leads to lower temperature critical points and a steeper liquid-liquid coexistence line, implying a low density liquid of larger entropy. Analysis of the hydrogen bond net accross the phase diagram indicates that the density anomaly is accompanied by a steep reduction of hydrogen bond density, which introduces a restriction on a correlation which has been preconized long ago. This feature is present independent of bond distortion or of the presence of proton entropy.

We present some results concerning the large volume limit of loop quantum cosmology in the flat homogeneous and isotropic case. We derive the Wheeler-De Witt equation in this limit. Looking for the action from which this equation can also be obtained, we then address the problem of the modifications to be brought to the Friedman's equation of motion and to the equation of motion of the scalar field, in the classical limit., Comment: 17 pages; 5 figures

We analyze the possible phase diagrams of a simple model for an associating liquid proposed previously. Our two-dimensional lattice model combines oreintati onal ice-like interactions and \"{}Van der Waals\"{} interactions which may be repulsive, and in this case represent a penalty for distortion of hydrogen bonds in the presence of extra molecules. These interactions can be interpreted in terms of two competing distances, but not necessarily soft-core. We present mean -field calculations and an exhaustive simulation study for different parameters which represent relative strength of the bonding interaction to the energy penalty for its distortion. As this ratio decreases, a smooth disappearance of the doubl e criticality occurs. Possible connections to liquid-liquid transitions of molecul ar liquids are suggested.

We present a simple model for an associating liquid in which polymorphism and density anomaly are connected. Our model combines a two dimensional lattice gas with particles interacting through a soft core potential and orientational degrees of freedom represented through thermal \char`\"{}ice variables\char`\"{} . The competition between the directional attractive forces and the soft core potential leads to a phase diagram in which two liquid phases and a density anomaly are present. The coexistence line between the low density liquid and the high density liquid has a positive slope contradicting the surmise that the presence of a density anomaly implies that the high density liquid is more entropic than the low density liquid.

We have studied a 50/50\AA superlattice of GaAs/Al$_{0.21}$Ga$_{0.79}$As composition, modulation-doped with Si, to produce $n=1.4\times 10^{12}$ cm$^{-2}$ electrons per superlattice period. The modulation-doping was tailored to avoid the formation of Tamm states, and photoluminescence due to interband transitions from extended superlattice states was detected. By studying the effects of a quantizing magnetic field on the superlattice photoluminescence, the miniband energy width, the reduced effective mass of the electron-hole pair, and the band gap renormalization could be deduced., Comment: minor typing errors (minus sign in eq. (5))