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We provide a novel methodological approach to the estimate of the change of the Quantum Vacuum electromagnetic energy density in a High critical Temperature superconducting metal bulk sample, when it undergoes the transition in temperature, from the superconducting to the normal phase. The various contributions to the Casimir energy in the two phases are highlighted and compared. While the TM polarization of the vacuum mode allows for a macroscopic description of the superconducting transition, the changes in the TE vacuum mode induced by the superconductive correlations are introduced within a microscopic model, which does not explicitly take into account the anisotropic structure of the material., 20 pages, 13 figures, to be published in Phys.Rev. B

The results of Cubitt et al. on the spectral gap problem add a new chapter to the issue of undecidability in physics, as they show that it is impossible to decide whether the Hamiltonian of a quantum many-body system is gapped or gapless. This implies, amongst other things, that a reductionist viewpoint would be untenable. In this paper, we examine their proof and a few philosophical implications, in particular ones regarding models and limitative results. In more detail, we examine the way these theorems model many-body quantum systems, and we question what, if anything, is the physical counterpart of the models used by Cubitt et al. We argue that these models are non-representational and that, even if they are so artificial that it is hard to imagine a physical system arising from them, they nonetheless offer an opportunity to learn about the world and the relation between mathematics and reality. On this basis, we draw the conclusion that their results do not undermine the reductionist viewpoint in a strong sense but leave the question open in a weak sense.

Anomalous metallic properties are often observed in the proximity of quantum critical points (QCPs), with violation of the Fermi Liquid paradigm. We propose a scenario where, due to the presence of a nearby QCP, dynamical fluctuations of the order parameter with finite correlation length mediate a nearly isotropic scattering among the quasiparticles over the entire Fermi surface. This scattering produces an anomalous metallic behavior, which is extended to the lowest temperatures by an increase of the damping of the fluctuations. We phenomenologically identify one single parameter ruling this increasing damping when the temperature decreases, accounting for both the linear-in-temperature resistivity and the seemingly divergent specific heat observed, e.g., in high-temperature superconducting cuprates and some heavy-fermion metals., Comment: 5 pages, 3 figures

In this paper we study the Casimir energy of a sample made by N cavities, with $$N\gg 1$$ N ≫ 1 , across the transition from the metallic to the superconducting phase of the constituting plates. After having characterised the energy for the configuration in which the layers constituting the cavities are made by dielectric and for the configuration in which the layers are made by plasma sheets, we concentrate our analysis on the latter. It represents the final step towards the macroscopical characterisation of a “multi-cavity” (with N large) necessary to fully understand the behaviour of the Casimir energy of a YBCO (or a GdBCO) sample across the transition. Our analysis is especially useful to the Archimedes experiment, aimed at measuring the interaction of the electromagnetic vacuum energy with a gravitational field. To this purpose, we aim at modulating the Casimir energy of a layered structure, the multi-cavity, by inducing a transition from the metallic to the superconducting phase. After having characterised the Casimir energy of such a structure for both the metallic and the superconducting phase, we give an estimate of the modulation of the energy across the transition.

Besides the mechanism responsible for high critical temperature superconductivity, the grand unresolved issue of the cuprates is the occurrence of a strange metallic state above the so-called pseudogap temperature $T^*$. Even though such state has been successfully described within a phenomenological scheme, the so-called Marginal Fermi-Liquid theory, a microscopic explanation is still missing. However, recent resonant X-ray scattering experiments identified a new class of charge density fluctuations characterized by low characteristic energies and short correlation lengths, which are related to the well-known charge density waves. These fluctuations are present over a wide region of the temperature-vs-doping phase diagram and extend well above $T^*$. Here we investigate the consequences of charge density fluctuations on the electron and transport properties and find that they can explain the strange metal phenomenology. Therefore, charge density fluctuations are likely the long-sought microscopic mechanism underlying the peculiarities of the metallic state of cuprates., 18 pages including Supplementary Informations, 10 figures

We report the tilt sensitivity reached by the ARCHIMEDES tiltmeter in the 2–20 Hz frequency region, where seismic noise is expected to give an important limitation to the sensitivity in the next future Gravitational Waves detection, particularly through Newtonian noise. The tilt noise level $$\tilde{\theta }(f)$$ θ ~ ( f ) is about $$10^{-12} \mathrm{rad}/\sqrt{\mathrm{Hz}}$$ 10 - 12 rad / Hz in most of the band, reaching the minimum of $$\tilde{\theta } = 7\cdot 10^{-13} \mathrm{rad}/\sqrt{\mathrm{Hz}}$$ θ ~ = 7 · 10 - 13 rad / Hz around 9 Hz. The tiltmeter is a beam balance with a 0.5 m suspended arm and interferometric optical readout, working in closed loop. The results have been obtained by a direct measurement of the ground tilt at the Sos Enattos site (Sardinia, Italy). This sensitivity is a requirement to use the tiltmeter as part of an effective Newtonian noise reduction system for present Gravitational Waves detectors, and also confirms that Sos Enattos is among the quietest sites in the world, suitable to host the third-generation Gravitational Waves detector Einstein Telescope.

Restoring strangeness Cuprate superconductors host a number of complex phases, and elucidating their relationship to each other may shed light on the nature of cuprate superconductivity. Two of these intriguing phases are the strange metal phase and the charge-density wave (CDW) order, which appear next to each other in the cuprate phase diagram. The hallmark of the strange metal phase, the linear dependence of electrical resistivity on temperature, disappears in the CDW phase. Wahlberg et al . strained thin films of the cuprate YBa 2 Cu 3 O 7–δ to suppress CDW order (see the Perspective by Le Tacon). Using resonant inelastic x-ray scattering and transport measurements, the researchers found that suppressing CDW restored the linear-in-temperature resistivity in the CDW portion of the phase diagram. —JS

Solid 4He may acquire superfluid characteristics due to the frustration of the solid phase at grain boundaries. Here, introducing a negative-U generalized Hubbard model and a coarse-grained semiclassical pseudospin model, we show that an analogous effect occurs in systems with competition among charge-density-waves (CDW) and superconductivity in the presence of disorder, as cuprate or dichalcogenide superconductors. The CDW breaks apart in domains with topologically protected filamentary superconductivity at the interfaces. Our transport measurements, carried out in underdoped La2−x Sr x CuO4, with the magnetic field acting as a control parameter, are shown to be in excellent agreement with our theoretical prediction. Assuming superconductivity and CDW phases have similar energies, at intermediate temperatures, the magnetic field drives the system from a fluctuating superconductor to a CDW as expected in the clean limit. Lowering the temperature, the expected clean quantum critical point is avoided and a filamentary phase appears, analogous to ‘glassy’ supersolid phenomena in 4He. The transition line ends at a second quantum critical point at high-fields. Within our scenario, the filamentary superconducting phase is parasitic with CDW and bulk superconducting phases playing the role of primary competing order parameters.

International audience; We present the status of the art of the Archimedes experiment, devoted to measuring the debated interaction of quantum vacuum fluctuations and gravity. The method is essentially the weighing of the transition energy of a layered superconductor where the contribution of vacuum energy to the transition energy is expected to be relevant. The transition is obtained by modulating the temperature of the superconducting sample at a frequency of about 10 mHz and the expected change of weight is measured with a suitably designed high sensitivity cryogenic beam balance. In this paper, we present an overview of the experiment, discussing the expected signal to be measured, and presenting in particular the result of a prototype balance operated in our present laboratory. In the frequency range of the measurement, the sensitivity is affected mainly by seismic, thermal, sensor, and control noise. We discuss these points showing in particular the design of the cryogenic apparatus, the final balance, and the quiet seismic site that will host the final measurement.

About 35 years ago Wheeler introduced the motto "law without law" to highlight the possibility that (at least a part of) Physics may be understood only following regularity principles and few relevant facts, rather than relying on a treatment in terms of fundamental theories. Such a proposal can be seen as part of a more general attempt (including the maximum entropy approach) summarized by the slogan "it from bit", which privileges the information as the basic ingredient. Apparently it seems that it is possible to obtain, without the use of physical laws, some important results in an easy way, for instance, the probability distribution of the canonical ensemble. In this paper we will present a general discussion on those ideas of Wheeler's that originated the motto "law without law". In particular we will show how the claimed simplicity is only apparent and it is rather easy to produce wrong results. We will show that it is possible to obtain some of the results treated by Wheeler in the realm of the statistical mechanics, using precise assumptions and nontrivial results of probability theory, mainly concerning ergodicity and limit theorems.

LaAlO3/SrTiO3 interfaces are a nice example of a two-dimensional electron gas, whose carrier density can be varied by top- and back-gating techniques. Due to the electron confinement near the interface, the two-dimensional band structure is split into sub-bands, and more than one sub-band can be filled when the carrier density increases. These interfaces also host superconductivity, and the interplay of two-dimensionality, multi-band character, with the possible occurrence of multi-gap superconductivity and disorder calls for a better understanding of finite-bandwidth effects on the superconducting critical temperature of heavily disordered multi-gap superconductors.

From systematic analysis of the high pulsed magnetic field resistance data of La$_{2-x}$Sr$_x$CuO$_{4}$ thin films, we extract an experimental phase diagram for several doping values ranging from the very underdoped to the very overdoped regimes. Our analysis highlights a competition between charge density waves and superconductivity which is ubiquitous between $x=0.08$ and $x=0.19$ and produces the previously observed double step transition. When suppressed by a strong magnetic field, superconductivity is resilient for two specific doping ranges centered around respectively $x\approx 0.09$ and $x\approx 0.19$ and the characteristic temperature for the onset of the competing charge density wave phase is found to vanish above $x = 0.19$. At $x=1/8$ the two phases are found to coexist exactly at zero magnetic field., 13 figures. Changes from previous version are in red. A few clarifications and a discussion about the different materials were added together with additional references

One of the hallmarks of the Berezinskii-Kosterlitz-Thouless (BKT) transition in two-dimensional superconductors is the universal jump of the superfluid density that can be indirectly probed via the nonlinear exponent of the current-voltage $I\text{\ensuremath{-}}V$ characteristics. Here, we compare the experimental measurements of $I\text{\ensuremath{-}}V$ characteristics in two cases, namely NbN thin films and ${\mathrm{SrTiO}}_{3}$-based interfaces. While the former display a paradigmatic example of BKT-like nonlinear effects, the latter do not seem to justify a BKT analysis. Rather, the observed $I\text{\ensuremath{-}}V$ characteristics can be well reproduced theoretically by modeling the effect of mesoscopic inhomogeneity of the superconducting state. Our results offer an alternative perspective on the spontaneous fragmentation of the superconducting background in confined two-dimensional systems.

Recent scanning tunnelling microscopy experiments in NbN thin disordered superconducting films found an emergent inhomogeneity at the scale of tens of nanometers. This inhomogeneity is mirrored by an apparent dimensional crossover in the paraconductivity measured in transport above the superconducting critical temperature $T_c$. This behavior was interpreted in terms of an anomalous diffusion of fluctuating Cooper pairs, that display a {\em quasi-confinement} (i.e., a slowing down of their diffusive dynamics) on length scales shorter than the inhomogeneity identified by tunnelling experiments. Here we assume this anomalous diffusive behavior of fluctuating Cooper pairs and calculate the effect of these fluctuations on the electron density of states above $T_c$. We find that the density of states is substantially suppressed up to temperatures well above $T_c$. This behavior, which is closely reminiscent of a pseudogap, only arises from the anomalous diffusion of fluctuating Cooper pairs in the absence of stable preformed pairs, setting the stage for an intermediate behavior between the two common paradigms in the superconducting-insulator transition, namely the localisation of Cooper pairs (the so-called bosonic scenario) and the breaking of Cooper pairs into unpaired electrons due to strong disorder (the so-called fermionic scenario)., 6 pages 7 figures

In multi-orbital materials, superconductivity can exhibit several coupled conden- sates. In this context, quantum con nement in two-dimensional superconducting oxide interfaces o ers new degrees of freedom to engineer the band structure and selectively control 3d-orbitals occupancy by electrostatic doping. Here, we use resonant microwave transport to extract the super uid sti ness of the (110)-oriented LaAlO3/SrTiO3 in- terface in the entire phase diagram. We evidence a transition from single-condensate to two-condensate superconductivity driven by continuous and reversible electrostatic doping, which we relate to the Lifshitz transition between 3d-bands based on numerical simulations of the quantum well. We nd that the superconducting gap is suppressed while the second band is populated, challenging the Bardeen-Cooper-Schrie er theory. We ascribe this behavior to the existence of superconducting order parameters with opposite signs in the two condensates, due to a repulsive coupling. Our ndings o er an innovative perspective on the possibility to tune and control multiple-orbitals physics in superconducting interfaces., The authors thank K. Behnia and J. Lorenzana for useful discussions. This work was supported by the Région Ile-de-France in the framework of CNano IdF, OXYMORE and Sesame programmes, by CNRS through a PICS programme (S2S) and ANR JCJC (Nano-SO2DEG). This work was supported by the Spanish MAT2017-85232-R, MAT2014-56063-C2-1-R and Severo Ochoa SEV-2015-0496 grants and the Generalitat de Catalunya (2017 SGR 1377). This work was supported by the Italian MAECI under the Italia–India collaborative project SUPERTOP-PGR04879. The authors acknowledge funding from the project Quantox of QuantERA ERA-NET Cofund in Quantum Technologies (grant agreement no. 731473) implemented within the European Union’s Horizon 2020 Programme. The authors also acknowledge the COST project Nanoscale Coherent Hybrid Devices for Superconducting Quantum Technologies–Action CA16218.

Charge density waves are a common occurrence in all families of high critical temperature superconducting cuprates. Although consistently observed in the underdoped region of the phase diagram and at relatively low temperatures, it is still unclear to what extent they influence the unusual properties of these systems. Using resonant x-ray scattering we carefully determined the temperature dependence of charge density modulations in (Y,Nd)Ba$_2$Cu$_3$O$_{7-{\delta}}$ for three doping levels. We discovered short-range dynamical charge density fluctuations besides the previously known quasi-critical charge density waves. They persist up to well above the pseudogap temperature T*, are characterized by energies of few meV and pervade a large area of the phase diagram, so that they can play a key role in shaping the peculiar normal-state properties of cuprates., Comment: 34 pages, 4 figures, 11 supplementary figures

We present a general scenario for high-temperature superconducting cuprates, based on the presence of dynamical charge density waves (CDWs) and to the occurrence of a CDW quantum critical point, which occurs, e.g., at doping p~0.16 in YBCO. In this framework, even the pseudogap temperature T* is interpreted in terms of a reduction of the density of states due to incipient CDW and, at lower temperature to the possible formation of incoherent superconducting pairs. The dynamically fluctuating character of CDW accounts for the different temperatures at which the pseudogap (T*), the CDW onset revealed by X-ray scattering (T_{ons}(p)), and the static three-dimensional CDW ordering appear. We also investigate the anisotropic character of the CDW-mediated scattering. We find that this is strongly anisotropic only close to the CDW quantum critical point (QCP) at low temperature and very low energy. It rapidly becomes nearly isotropic and marginal-Fermi-liquid-like away from the CDW QCP and at finite (even rather small) energies. This may reconcile the interpretation of Hall measurements in terms of anisotropic CDW scattering with recent photoemission experiments., 6 pages, 3 figures. arXiv admin note: text overlap with arXiv:1604.07852

It is shown that non-homogeneity in the spin-orbit coupling (SOC) arising from structure inversion asymmetry gives rise to a finite intrinsic spin Hall conductivity in contrast with the corresponding case of a homogeneous SOC in the disordered two-dimensional electron gas. In particular, we examine the inhomogeneity arising from a striped modulation of the Rashba type spin-orbit coupling. A corresponding structure could be realized at oxide interfaces with periodic top gating or in semiconductor heterostructures.

Inelastic Raman scattering is used to probe the critical spin fluctuations in an iron pnictide superconductor, providing insights into the origin of nematic order in this system. Nematic fluctuations and order play a prominent role in material classes such as the cuprates1, some ruthenates2 or the iron-based compounds3,4,5,6 and may be interrelated with superconductivity7,8,9,10,11. In iron-based compounds12 signatures of nematicity have been observed in a variety of experiments. However, the fundamental question as to the relevance of the related spin13, charge9,14 or orbital8,15,16 fluctuations remains open. Here, we use inelastic light (Raman) scattering and study Ba(Fe1−xCox)2As2 (0 ≤ x ≤ 0.085) for getting direct access to nematicity and the underlying critical fluctuations with finite characteristic wavelengths17,18,19,20,21. We show that the response from fluctuations appears only in B1g (x2 − y2) symmetry (1 Fe unit cell). The scattering amplitude increases towards the structural transition at Ts but vanishes only below the magnetic ordering transition at TSDW < Ts, suggesting a magnetic origin of the fluctuations. The theoretical analysis explains the selection rules and the temperature dependence of the fluctuation response. These results make magnetism the favourite candidate for driving the series of transitions.

We study one-dimensional structures that may be formed at the LaAlO 3 /SrTiO 3 oxide interface by suitable top gating. These structures are modeled via a single-band model with Rashba spin-orbit coupling, superconductivity and a magnetic field along the one-dimensional chain. We first discuss the conditions for the occurrence of a topological superconducting phase and the related formation of Majorana fermions at the chain endpoints, highlighting a close similarity between this model and the Kitaev model, which also reflects in a similar condition the formation of a topological phase. Solving the model in real space, we also study the spatial extension of the wave function of the Majorana fermions and how this increases with approaching the limit condition for the topological state. Using a scattering matrix formalism, we investigate the stability of the Majorana fermions in the presence of disorder and discuss the evolution of the topological phase with increasing disorder.

We evaluate the non-equilibrium spin polarization induced by an applied electric field for a tight-binding model of electron states at oxides interfaces in LAO/STO heterostructures. By a combination of analytic and numerical approaches we investigate how the spin texture of the electron eigenstates due to the interplay of spin-orbit coupling and inversion asymmetry determines the sign of the induced spin polarization as a function of the chemical potential or band filling, both in the absence and presence of local disorder. With the latter, we find that the induced spin polarization evolves from a non monotonous behavior at zero temperature to a monotonous one at higher temperature. Our results may provide a sound framework for the interpretation of recent experiments., Submitted to Proceedings of SPIE Nanoscience + Engineering 2018, Spintronics XI, 23 pages, 9 figures

International audience; Archimedes is an experiment devoted to measure the weight of zero point energy by weighing acascade of rigid Casimir cavitities costituted by a layered type II superconductor, when the reflec-tivity of the cavities is modulated and thus the energy inside. In this paper the working principleof the experiment is presented and the results of the existing prototype are presented, showingthe torque sensitivity of about 10−11 Nm/sqrtHz in the region of frequency from about 50 mHzup two 150 mHz. Moreover the major upgrades needed to reach the desired final sensitivity arediscussed.

The spin galvanic effect (SGE) describes the conversion of a non-equilibrium spin polarization into a charge current and has recently attracted renewed interest due to the large conversion efficiency observed in oxide interfaces. An important factor in the SGE theory is disorder which ensures the stationarity of the conversion. Through this paper, we propose a procedure for the evaluation of the SGE on disordered lattices which can also be readily implemented for multiband systems. We demonstrate the performance of the method for a single-band Rashba model and compare our results with those obtained within the self-consistent Born approximation for a continuum model.

In the last few years charge density waves (CDWs) have been ubiquitously observed in high-temperature superconducting cuprates and are now the most investigated among the competing orders in the still hot debate on these systems. A wealth of new experimental data raise several fundamental issues that challenge the various theoretical proposals. Here, we account for the complex experimental temperature vs. doping phase diagram and we provide a coherent scenario explaining why different CDW onset curves are observed by different experimental probes and seem to extrapolate at zero temperature into seemingly different quantum critical points (QCPs) in the intermediate and overdoped region. We also account for the pseudogap and its onset temperature T*(p) on the basis of dynamically fluctuating CDWs. The nearly singular anisotropic scattering mediated by these fluctuations also account for the rapid changes of the Hall number seen in experiments and provides the first necessary step for a possible Fermi surface reconstruction fully establishing at lower doping. Finally we show that phase fluctuations of the CDWs, which are enhanced in the presence of strong correlations near the Mott insulating phase, naturally account for the disappearance of the CDWs at low doping with yet another QCP., 13 pages, 7 figures

The spin-Hall effect is the generation of a transverse spin-current induced by a longitudinal electric field. Among the various scattering mechanisms which can induce a finite spin-Hall effect we focus on the intrinsic contribution arising from a Rashba-type spin–orbit (RSO) coupling which for couplings linear in momentum usually vanishes. Here we show that non-homogeneity in the spin–orbit coupling arising from structure inversion asymmetry gives rise to a finite spin-Hall effect which we exemplify for a system with striped Rashba spin–orbit coupling.

The spin galvanic effect (SGE) describes the conversion of a non-equilibrium spin polarization into a transverse charge current. Recent experiments have demonstrated a large conversion efficiency for the two-dimensional electron gas formed at the interface between two insulating oxides, LaAlO$_3$ and SrTiO$_3$. Here we analyze the SGE for oxide interfaces within a three-band model for the Ti t$_{2g}$ orbitals which displays an interesting variety of effective spin-orbit couplings in the individual bands that contribute differently to the spin-charge conversion. Our analytical approach is supplemented by a numerical treatment where we also investigate the influence of disorder and temperature, which turns out to be crucial to provide an appropriate description of the experimental data., Comment: 5 pages, 3 figures

Archimedes is a feasibility study of a future experiment to ascertain the interaction of vacuum fluctuations with gravity. The experiment should measure the force that the earth's gravitational field exerts on a Casimir cavity by using a small force detector. Here we analyse the main parameters of the experiment and we present its conceptual scheme, which overcomes in principle the most critical problems., Comment: 3 pages, MG14 Conference

The diverse phenomena associated with the two-dimensional electron gas (2DEG) that occurs at oxide interfaces include, among others, exceptional carrier mobilities, magnetism and superconductivity. Although these have mostly been the focus of interest for potential future applications, they also offer an opportunity for studying more fundamental quantum many-body effects. Here, we examine the magnetic-field-driven quantum phase transition that occurs in electrostatically gated superconducting LaTiO3/SrTiO3 interfaces. Through a finite-size scaling analysis, we show that it belongs to the (2+1)D XY model universality class. The system can be described as a disordered array of superconducting puddles coupled by a 2DEG and, depending on its conductance, the observed critical behaviour is single (corresponding to the long-range phase coherence in the whole array) or double (one related to local phase coherence, the other one to the array). A phase diagram illustrating the dependence of the critical field on the 2DEG conductance is constructed, and shown to agree with theoretical proposals. Moreover, by retrieving the coherence-length critical exponent ν, we show that the quantum critical behaviour can be clean or dirty according to the Harris criterion, depending on whether the phase-coherence length is smaller or larger than the size of the puddles.

Archimedes is a feasibility study to a future experiment to ascertain the interaction of vacuum fluctuations with gravity. The future experiment should measure the force that the earth's gravitational field exerts on a Casimir cavity by using a balance as the small force detector. The Archimedes experiment analizes the important parameters in view of the final measurement and experimentally explores solutions to the most critical problems.

Archimedes is an INFN-funded pathfinder experiment aimed at verifying the feasibility of measuring the interaction of vacuum fluctuations with gravity. The final experiment will measure the force exerted by the gravitational field on a Casimir cavity whose vacuum energy is modulated with a superconductive transition, by using a balance as a small force detector. Archimedes is a two-year project devoted to test the most critical experimental aspects, in particular the balance resonance frequency and quality factor, the thermal modulation efficiency and the superconductive sample realization., Comment: 4 pages, 2 figures

The microscopic nature of an insulating state in the vicinity of a superconducting state, in the presence of disorder, is a hotly debated question. While the simplest scenario proposes that Coulomb interactions destroy the Cooper pairs at the transition, leading to localization of single electrons, an alternate possibility supported by experimental observations suggests that Cooper pairs instead directly localize. The question of the homogeneity, granularity, or possibly glassiness of the material on the verge of this transition is intimately related to this fundamental issue. Here, by combining macroscopic and nano-scale studies of superconducting ultrathin NbN films, we reveal nanoscopic electronic inhomogeneities that emerge when the film thickness is reduced. In addition, while thicker films display a purely two-dimensional behaviour in the superconducting fluctuations, we demonstrate a zero-dimensional regime for the thinner samples precisely on the scale of the inhomogeneities. Such behavior is somehow intermediate between the Fermi and Bose insulator paradigms and calls for further investigation to understand the way Cooper pairs continuously evolve from a bound state of fermionic objects into localized bosonic entities., 29 pages 9 figures

Oxide heterostructures are of great interest both for fundamental and applicative reasons. In particular the two-dimensional electron gas at the LaAlO$_3$/SrTiO$_3$ or LaTiO$_3$/SrTiO$_3$ interfaces displays many different physical properties and functionalities. However there are clear indications that the interface electronic state is strongly inhomogeneous and therefore it is crucially relevant to investigate possible intrinsic electronic mechanisms underlying this inhomogeneity. Here the electrostatic potential confining the electron gas at the interface is calculated self-consistently, finding that the electron confinement at the interface may induce phase separation, to avoid a thermodynamically unstable state with a negative compressibility. This provides a generic robust and intrinsic mechanism for the experimentally observed inhomogeneous character of these interfaces., Comment: 8 pages and 4 figures

There is a persistent confusion about determinism and predictability. In spite of the opinions of some eminent philosophers (e.g., Popper), it is possible to understand that the two concepts are completely unrelated. In few words we can say that determinism is ontic and has to do with how Nature behaves, while predictability is epistemic and is related to what the human beings are able to compute. An analysis of the Lyapunov exponents and the Kolmogorov-Sinai entropy shows how deterministic chaos, although with an epistemic character, is non subjective at all. This should clarify the role and content of stochastic models in the description of the physical world.