One striking anomaly of water ice has been largely neglected and never explained. Replacing hydrogen ( 1 H) by deuterium ( 2 H) causes ice to expand, whereas the "normal" isotope effect is volume contraction with increased mass. Furthermore, the anomaly increases with temperature T , even though a normal isotope shift should decrease with T and vanish when T is high enough to use classical nuclear motions. In this study, we show that these effects are very well described by ab initio density functional theory. Our theoretical modeling explains these anomalies, and allows us to predict and to experimentally confirm a counter effect, namely that replacement of 16 =0 [15]. This "normal" isotope effect corresponds to a ∼12% zero-point expansion of 20 Ne relative to a hypothetical "classical" or "frozen" lattice [16,17]. Since H 2 O and Ne have similar molecular masses, one might expect similar effects. However, the volume of H 2 O at T = 0 is ∼0.1% smaller than that of D 2 O [12,13]. It has rarely been mentioned in the literature that this is opposite to the usual behavior, and no explanation has been offered.In this paper, we explain this effect as an interesting coupling between quantum nuclear motion and hydrogen bonding, that may be relevant also to the structure of liquid water. Our analysis shows that, despite the anomalous isotope effect, quantum ice actually has a volume 1% larger than it would have with classical nuclei. The effects are smaller than in Ne mostly because of delicate cancellations. We exploit these cancellations to make critical comparisons of: (i) quasiharmonic theory versus fully anharmonic path-integral molecular dynamics (PIMD); (ii) ab initio forces versus flexible and polarizable empirical force fields (EFF); and (iii) various flavors of ab initio density-functional theory (DFT) exchange and correlation (XC) density functionals (DF) with and without inclusion of van der Waals (vdW) interactions. We find: (i) quasiharmonic theory is satisfactory for this problem; (ii) present state of the art EFFs are not good enough to describe nuclear quantum effects in water; and (iii) all the DFs considered describe qualitatively the anomalous effects, although some versions perform better than others.Within the volume-dependent quasiharmonic approximation (QHA), the equilibrium volume V (T ) is obtained by minimizing at each T the Helmholtz free energy F (V, T ) [18,19]:where E 0 (V ) is the energy for classical (T = 0 or frozen) nuclei, at the relaxed atomic coordinates for each volume. ω k are the phonon frequencies, with k combining the branch index and the phonon wave vector within the Brillouin zone. Their volume dependence is linearized as:whereis the Grüneisen parameter of the mode, and V 0 is the equilibrium volume of E 0 (V ). ω k (V 0 ) and γ k (V 0 ) are obtained by diagonalizing the dynamical matrix, computed by finite differences from the atomic forces in a (3 × 3 × 3) supercell, at two volumes slightly below and above V 0 . As shown in the supplementary information [20] (SI), this li...

Path-integral molecular dynamics (PIMD) simulations have been carried out to study the influence of quantum dynamics of carbon atoms on the properties of a single graphene layer. Finitetemperature properties were analyzed in the range from 12 to 2000 K, by using the LCBOPII effective potential. To assess the magnitude of quantum effects in structural and thermodynamic properties of graphene, classical molecular dynamics simulations have been also performed. Particular emphasis has been laid on the atomic vibrations along the out-of-plane direction. Even though quantum effects are present in these vibrational modes, we show that at any finite temperature classical-like motion dominates over quantum delocalization, provided that the system size is large enough. Vibrational modes display an appreciable anharmonicity, as derived from a comparison between kinetic and potential energy of the carbon atoms. Nuclear quantum effects are found to be appreciable in the interatomic distance and layer area at finite temperatures. The thermal expansion coefficient resulting from PIMD simulations vanishes in the zero-temperature limit, in agreement with the third law of thermodynamics.

The Ising model in small-world networks generated from two-and three-dimensional regular lattices has been studied. Monte Carlo simulations were carried out to characterize the ferromagnetic transition appearing in these systems. In the thermodynamic limit, the phase transition has a mean-field character for any finite value of the rewiring probability p, which measures the disorder strength of a given network. For small values of p, both the transition temperature and critical energy change with p as a power law. In the limit p → 0, the heat capacity at the transition temperature diverges logarithmically in two-dimensional (2D) networks and as a power law in 3D.

Ice Ih has been studied by path-integral molecular dynamics simulations, using the effective q-TIP4P/F potential model for flexible water. This has allowed us to analyze finite-temperature quantum effects in this solid phase from 25 to 300 K at ambient pressure. Among these effects we find a negative thermal expansion of ice at low temperatures, which does not appear in classical molecular dynamics simulations. The compressibility derived from volume fluctuations gives results in line with experimental data. We have analyzed isotope effects in ice Ih by considering normal, heavy, and tritiated water. In particular, we studied the effect of changing the isotopic mass of hydrogen on the kinetic energy and atomic delocalization in the crystal as well as on structural properties such as interatomic distances and molar volume. For D(2)O ice Ih at 100 K we obtained a decrease in molar volume and intramolecular O-H distance of 0.6% and 0.4%, respectively, as compared to H(2)O ice.

The out-of-plane fluctuations of carbon atoms in a graphene sheet have been studied by means of classical molecular dynamic simulations with an empirical force-field as a function of temperature. The Fourier analysis of the out-of-plane fluctuations often applied to characterize the acoustic bending mode of graphene is extended to the optical branch, whose polarization vector is perpendicular to the graphene layer. This observable is inaccessible in a continuous elastic model of graphene but it is readily obtained by the atomistic treatment. Our results suggest that the long-wavelength limit of the acoustic out-of-plane fluctuations of a free layer without stress is qualitatively similar to that predicted by a harmonic model under a tensile stress. This conclusion is a consequence of the anharmonicity of both in-plane and out-of-plane vibrational modes of the lattice. The most striking anharmonic effect is the presence of a linear term, ωA = vAk, in the dispersion relation of the acoustic bending band of graphene at long wavelengths (k → 0). This term implies a strong reduction of the amplitude of out-of-plane oscillations in comparison to a flexural mode with a k 2 -dependence in the long-wavelength limit. Our simulations show an increase of the sound velocity associated to the bending mode, as well as an increase of its bending constant, κ, as the temperature increases. Moreover, the frequency of the optical bending mode, ωO(Γ), also increases with the temperature. Our results are in agreement with recent analytical studies of the bending modes of graphene using either perturbation theory or an adiabatic approximation in the framework of continuous layer models.

We study strong electron tunneling in the single-electron box, a small metallic island coupled to an electrode by a tunnel junction, by means of quantum Monte Carlo simulations. We obtain results, at arbitrary tunneling strength, for the free energy of this system and the average charge on the island as a function of an external bias voltage. In much of the parameter range an extrapolation to the ground state is possible. Our results for the effective charging energy for strong tunneling are compared to earlier -in part controversial -theoretical predictions and Monte Carlo simulations.

The Ising model in uncorrelated scale-free networks has been studied by means of Monte Carlo simulations. These networks are characterized by a degree (or connectivity) distribution P (k) ∼ k −γ . The ferromagnetic-paramagnetic transition temperature has been studied as a function of the parameter γ. For γ > 3 our results agree with earlier analytical calculations, which found a phase transition at a temperature Tc(γ) in the thermodynamic limit. For γ ≤ 3, a ferromagnetic-paramagnetic crossover occurs at a size-dependent temperature Tco, and the system is in the ordered ferromagnetic state at any temperature for a system size N → ∞. For γ = 3 and large enough N , the crossover temperature is found to be Tco ≈ A ln N , with a prefactor A proportional to the mean degree. For 2 < γ < 3, we obtain Tco ∼ k N z , with an exponent z that decreases as γ increases. This exponent is found to be lower than predicted by earlier calculations.PACS numbers: 64.60. Cn, 05.50.+q, 89.75.Hc,84.35.+i Complex networks describe several kinds of natural and artificial systems (social, biological, technological, economic), and are currently employed as models to study various processes taking place in real-life systems [1][2][3]. In last years, new models of complex networks have been introduced, motivated by empirical data in different fields. Thus, the so-called small-world [4] and scale-free (SF) networks [5] incorporate various aspects of real systems. These complex networks provide us with the underlying topological structure to study processes such as spread of infections [6], signal propagation [1,7], and cooperative phenomena [8][9][10][11].In a SF network the degree distribution, P (k), where k is the number of links connected to a node, has a powerlaw decay P (k) ∼ k −γ . This kind of networks have been found in particular in social systems [12], in protein interaction networks [13], in the internet [14], and in the world-wide web [15]. In both natural and artificial networks, the exponent γ controlling the degree distribution is usually in the range 2 < γ < 3 [3,16].Cooperative phenomena in complex networks are expected to display unusual characteristics, associated to the peculiar topology of these systems. In this context, the Ising model on SF networks has been studied with several theoretical techniques [9,17,18], and its critical behavior was found to be dependent on the exponent γ. In particular, when k 2 is finite, there appears a ferromagnetic (FM) to paramagnetic (PM) transition at a finite temperature T c . On the contrary, when k 2 diverges (as happens for γ ≤ 3), the system remains in its ordered FM phase at any temperature, and no phase transition occurs in the thermodynamic limit.Here we investigate the FM-PM transition for the Ising model in scale-free networks with various values of the exponent γ. We employ Monte Carlo (MC) simulations to obtain the transition temperature, and compare it with that predicted in earlier calculations. Our results confirm those of analytical calculations for γ > 3, and are u...

Combining nonperturbative techniques with Monte Carlo simulations we demonstrate that quantum coherence effects for a particle on a ring are suppressed beyond a finite length Lϕ even at zero temperature if the particle is coupled to a diffusive electron gas by means of long range Coulomb interaction. This length is consistent with Lϕ derived from weak-localization-type of analysis.