Table of contents

The Heisenberg and Fisher-information-based uncertainty relations are improved for stationary states of single-particle systems in a D-dimensional central potential. The improvement increases with the squared orbital hyperangular quantum number. The new uncertainty relations saturate for the isotropic harmonic oscillator wavefunction.

Inspired by a design originally proposed by Bullard (1955 Proc. Camb. Phil. Soc.51 744), we report a simple experimental arrangement which generates a turbulent fluid dynamo in the laboratory. It is based on a von Kármán gallium flow and it effectively models an 'alpha–omega' dynamo process, in which the alpha component is realized by external wirings, while the omega contribution fully incorporates the flow turbulence. In the dynamo state, we observe intermittent reversals of the dipole field, as well as excursions. The bifurcation develops through an 'on–off' regime.

We uncover a new type of unitary operation for quantum mechanics on the half-line which yields a transformation to 'hyperbolic phase space' (η, p_η). We show that this new unitary change of basis from the position x on the half line to the hyperbolic momentum p_η, transforms the wavefunction via a Mellin transform on to the critical line s = 1/2−ip_η. We utilize this new transform to find quantum wavefunctions whose hyperbolic momentum representation approximate a class of higher transcendental functions, and in particular, approximate the Riemann–Zeta function. We finally give possible physical realizations to perform an indirect measurement of the hyperbolic momentum of a quantum system on the half-line.

We show that some of the Josephson couplings of junctions arranged to form an inhomogeneous network undergo a non-perturbative renormalization provided that the network's connectivity is pertinently chosen. As a result, the zero-voltage Josephson critical currents I_c turn out to be enhanced along directions selected by the network's topology. This renormalization effect is possible only on graphs whose adjacency matrix admits a hidden spectrum (i.e. a set of localized states disappearing in the thermodynamic limit). We provide a theoretical and experimental study of this effect by comparing the superconducting behaviour of a comb-shaped Josephson junction network and a linear chain made with the same junctions: we show that the Josephson critical currents of the junctions located on the comb's backbone are bigger than those of the junctions located on the chain. Our theoretical analysis, based on a discrete version of the Bogoliubov–de Gennes equation, leads to results which are in good quantitative agreement with experimental results.

The 1998 discovery that the universe is accelerating set off an enormous amount of activity, both theoretical and observational. The original result, from two groups observing the Hubble diagram of Type Ia supernovae, has since been verified by a variety of independent types of observations. It seems clear that our universe really is accelerating; what remains a mystery is why.

The most straightforward explanation for the universe's acceleration is the presence of a dark energy component comprising 70% of the universe. In order to fit the data, dark energy must have two features: it should be smoothly distributed, so as not to show up in dynamical studies of galaxies and clusters, and its density should be nearly constant as the universe expands, to provide the persistent impulse necessary to make the universe accelerate.

The simplest candidate for dark energy is vacuum energy, equivalent to Einstein's cosmological constant. Simple estimates from quantum field theory indicate that vacuum energy should exist – indeed, in an amount larger than what we observe by a factor of 10¹²⁰. This discrepancy, the 'cosmological constant problem', led to a widespread assumption that some mysterious mechanism worked to set the vacuum energy precisely to zero. If the dark energy really is a cosmological constant, we must find a mechanism to suppress its natural value without driving it all the way to vanishing.

Alternatively, the dark energy could be a dynamical field, albeit one that changes very gradually with time. Such a case is observationally distinguishable, at least in principle, from that of a truly constant vacuum energy. Constraints on the evolution of the dark-energy density come from a variety of measurements, and improving the precision of these techniques is a major goal of the next decade in cosmology.

Most dramatically, there might not be any dark energy at all, even if the universe is accelerating – a possibility that is well-explored in this focus issue of New Journal of Physics. Two possibilities present themselves. On the one hand, general relativity could break down on cosmological scales, forcing us to a new theory of gravity. In that case, we may use other observed phenomena to put limits on the way in which gravity could deviate from Einstein's theory. On the other hand, general relativity could be correct, but differ in its true predictions from the simple approximations we are used to applying. Such a possibility is both dramatically different from more conventional approaches, and yet radically conservative, attempting to explain all of the accumulated observations with nothing more than matter particles and ordinary general relativity. Only a great deal of additional theoretical investigation and observational progress will be able to distinguish which of these possibilities explains the behaviour of our universe on large scales.

The articles below represent the first contributions and further additions will appear.

Focus on Dark Energy Contents

Cosmic clocks, cosmic variance and cosmic averagesDavid L Wiltshire

Dark energy, a cosmological constant, and type Ia supernovaeLawrence M Krauss, Katherine Jones-Smith and Dragan Huterer

Cosmological dark energy: prospects for a dynamical theoryIgnatios Antoniadis, Pawel O Mazur and Emil Mottola

Predictive power of strong coupling in theories with large distance modified gravity G Dvali

Constraints on dynamical dark energy: an update Alessandro Melchiorri, Barbara Paciello, Paolo Serra and Anze Slosar

Scaling solutions to 6D gauged chiral supergravity Andrew Tolley, C P Burgess, Claudia de Rham and D Hoover

Modified-source gravity and cosmological structure formation Sean Carroll, I Sawicki, A Silvestri and M Trodden

On cosmic acceleration without dark energy E W Kolb, Sabino Matarrese and A Riotto

Sean Carroll, California Institute of Technology, Pasadena, USA

We consider theories that modify gravity at cosmological distances, and show that any such theory must exhibit a strong coupling phenomenon, or else it is either inconsistent or is already ruled out by solar system observations. We show that all the ghost-free theories that modify dynamics of spin-2 graviton on asymptotically flat backgrounds, automatically have this property. Due to the strong coupling effect, modification of the gravitational force is source-dependent, and for lighter sources sets in at shorter distances. This universal feature makes modified gravity theories predictive and potentially testable not only by cosmological observations, but also by precision gravitational measurements at scales much shorter than the current cosmological horizon.

By combining data from cosmic microwave background experiments (most notably from the latest WMAP 3 years' results) with large scale structure data, luminosity measurements of type Ia supernovae and Lyman-α observations, we place new constraints on the dark energy equation of state parameter w_X and its evolution. Using the dark energy parameterization introduced by Hannestad and Mortsell (2004 J. Cosmol. Astropart. Phys. JCAP09(2004)001), we constrain the equation of state parameters to −1.23 < w₀ < −0.88 and −1.29 < w₁ < −0.78 at 95% C.L. Although our limits on w₀ and w₁ are improved with respect to previous analyses, cosmological data do not strongly rule out the possibility of a varying-with-redshift equation of state.

We construct explicitly time-dependent exact solutions to the field equations of six-dimensional (6D) gauged chiral supergravity, compactified to 4D in the presence of up to two three-branes situated within the extra dimensions. The solutions we find are scaling solutions, and are plausibly attractors which represent the late-time evolution of a broad class of initial conditions. By matching their near-brane boundary conditions to physical brane properties, we argue that these solutions (together with the known maximally symmetric solutions and a new class of non-Lorentz-invariant static solutions, which we also present here) describe the bulk geometry between a pair of three-branes with non-trivial on-brane equations of state.

One way to account for the acceleration of the universe is to modify general relativity, rather than introducing dark energy. Typically, such modifications introduce new degrees of freedom. It is interesting to consider models with no new degrees of freedom, but with a modified dependence on the conventional energy-momentum tensor; the Palatini formulation of f(R) theories is one example. Such theories offer an interesting testing ground for investigations of cosmological modified gravity. In this paper we study the evolution of structure in these 'modified-source gravity' (MSG) theories. In the linear regime, density perturbations exhibit scale dependent runaway growth at late times and, in particular, a mode of a given wavenumber goes nonlinear at a higher redshift than in the standard lambda-cold dark matter (ΛCDM) model. We discuss the implications of this behaviour and why there are reasons to expect that the growth will be cut off in the nonlinear regime. Assuming that this holds in a full nonlinear analysis, we briefly describe how upcoming measurements may probe the differences between the modified theory and the standard ΛCDM model.

We elaborate on the proposal that the observed acceleration of the Universe is the result of the backreaction of cosmological perturbations, rather than the effect of a negative-pressure dark-energy fluid or a modification of general relativity. Through the effective Friedmann equations describing an inhomogeneous Universe after smoothing, we demonstrate that acceleration in our local Hubble patch is possible even if fluid elements do not individually undergo accelerated expansion. This invalidates the no-go theorem that there can be no acceleration in our local Hubble patch if the Universe only contains irrotational dust. We then study perturbatively the time behaviour of general-relativistic cosmological perturbations, applying, where possible, the renormalization group to regularize the dynamics. We show that an instability occurs in the perturbative expansion involving sub-Hubble modes. Whether this is an indication that acceleration in our Hubble patch originates from the backreaction of cosmological perturbations on observable scales requires a fully non-perturbative approach.

In the framework of the renormalization-group (RG) theory of critical phenomena, a quantitative description of many continuous phase transitions can be obtained by considering an effective Φ⁴ theories, having an N-component fundamental field Φ_i and containing up to fourth-order powers of the field components. Their RG flow is usually characterized by several fixed points (FPs). We give here strong arguments in favour of the following conjecture: the stable FP corresponds to the fastest decay of correlations, that is, is the one with the largest values of the critical exponent η describing the power-law decay of the two-point function at criticality. We prove this conjecture in the framework of the ε-expansion. Then, we discuss its validity beyond the ε-expansion. We present several lower-dimensional cases, mostly three-dimensional, which support the conjecture. We have been unable to find a counterexample.

First-order phase transitions in many-fermion systems are not detected in the susceptibility analysis of common renormalization-group (RG) approaches. Here, we introduce a counterterm technique within the functional renormalization-group (fRG) formalism which allows access to all stable and metastable configurations. It becomes possible to study symmetry-broken states which occur through first-order transitions as well as hysteresis phenomena. For continuous transitions, the standard results are reproduced. As an example, we study discrete-symmetry breaking in a mean-field model for a commensurate charge-density wave. An additional benefit of the approach is that away from the critical temperature for the breaking of discrete symmetries large interactions can be avoided at all RG scales.

Spontaneous emission of positronium negative ions from polycrystalline tungsten surfaces has been observed for the first time. It is a new channel for the fate of thermalized positrons near the surface. The obtained formation probability is (7 ± 2) × 10⁻⁵. This method provides a new source of monoenergetic positronium negative ions for future applications.

The polarization of graphene is calculated exactly within the random phase approximation for arbitrary frequency, wavevector and doping. At finite doping, the static susceptibility saturates to a constant value for low momenta. At q = 2k_F it has a discontinuity only in the second derivative. In the presence of a charged impurity this results in Friedel oscillations which decay with the same power law as the Thomas–Fermi contribution, the latter being always dominant. The spin density oscillations in the presence of a magnetic impurity are also calculated. The dynamical polarization for low q and arbitrary ω is employed to calculate the dispersion relation and the decay rate of plasmons and acoustic phonons as a function of doping. The low screening of graphene, combined with the absence of a gap, leads to a significant stiffening of the longitudinal acoustic lattice vibrations.

We study interfaces between highly ionic crystals with different crystal structure by means of first-principles total-energy calculations in the repeated-slab approximation, and compare the results with experimental data extracted from high-resolution transmission electron micrographs. Despite the same Bravais lattices and the electrostatic neutrality of each atomic plane, the (110) interface between rocksalt and zinc-blende crystals gives rise to the most remarkable effect, a lateral spatial offset between the two crystals due to rebonding across the interface. A strong variation is observed for the separation of the two polar (001) interfaces depending on their cation- or anion-termination. In general, the long-range electrostatic forces lead to opposite atomic displacements along the interface normal independently of the interface orientation.

Based on phase separation, quasiperiodic PbZr_0.4 Ti_0.6O₃ (PZT) multilayers with alternating PZT and porous-PZT layers were fabricated by using a single precursor, and the ferroelectric and dielectric behaviours of the PZT multilayers were investigated. The PZT multilayers have an averaged remanent polarization of 42.3 μC cm⁻² and an average coercive field of 43 kV cm⁻¹. Two distinct dielectric relaxation phenomena were observed in the frequency range from 100 Hz to 1 MHz: the one at lower frequency is attributed to space charge polarization, while the one at higher frequency, with an activation energy of 0.49 eV, is expected to be associated with dipolar defect complexes related to oxygen vacancies.

We have measured the magnetization of strongly coupled bilayer two-dimensional electron systems. When sweeping the magnetic field steps appear in the magnetization whenever a transition between discrete energy levels takes place. At magnetic fields for which the Landau-level splitting dominates, the steps occurring at total filling factors ν_T = 4j are related to transitions between adjacent Landau levels j − 1 and j; such steps can also be observed in single layers. Additional magnetization steps showing up in bilayers at half Landau-level filling ν_T = 4j + 2 are associated with transitions from a symmetric to an antisymmetric state inside the same Landau level j. The observed size of the Landau-level steps in bilayers is considerably lower than expected theoretically. These findings are explained using a model with a large background density of states.

Dissociative electron attachment to the unstable CS molecule, carbon monosulfide, produced in a microwave discharge of carbon disulfide and helium has been observed in the electron energy range 0–10 eV. The S⁻ ion is formed in two electron attachment bands with maxima at 5.43 ± 0.15 eV and ∼6.70 ± 0.15 eV and C⁻ is formed in a band with a maximum of ∼6.40 ± 0.15 eV. Absolute cross sections for these processes are estimated to be 0.025, 0.003 and 0.002 Ų respectively. These interactions of CS with low energy electrons will play a role in technological plasmas containing carbon and sulfur and extraterrestrial environments where electrons are present with 5–7 eV energy. New results are also presented for the molecules S₂O, C₃S₂ and S₂F, which were also formed in the microwave discharge. For S₂O the previously reported S⁻ band at 1.8 ± 0.3 eV is confirmed and a new O⁻ attachment band at ∼5.4 eV is tentatively assigned to S₂O. Ionization thresholds of C₃S₂ (9.4 ± 0.3 eV) and S₂F (10.3 ± 0.4 eV) have been measured, apparently for the first time.

We study the depinning of driven drops on a heterogeneous substrate using a long-wave evolution equation for the film thickness profile. The heterogeneity is incorporated into an additional pressure term describing the interaction of the liquid with the substrate or with an external field. A drop may be pinned by a hydrophilic defect at the back or a hydrophobic defect at the front. Two types of depinning occur depending on the strength of the driving. The first occurs via a saddle-node (sniper) bifurcation, while the second involves a Hopf bifurcation. The parameter dependence of the depinning process is studied using linear stability theory, and direct numerical simulations are used to explore the dynamical properties of the stick-slip motion of the drop after depinning.

We cast a suitable model to describe the 3D dynamics of the coherent field in a monolithic multi-quantum-well (MQW) microresonator within and beyond the mean field limit (MFL) and provide a stability analysis and discriminating criteria to predict 3D pattern formation. While for fast media spontaneous self-confinement leads to the formation of 3D dissipative addressable spatial solitons, we show that for carrier dynamics compatible with GaAs/GaAlAs MQW devices longitudinal self-confinement is hindered by carrier 'sleuth'. We discuss turnaround strategies thereof.

We study the dynamics of Bose–Einstein condensed atoms in a one-dimensional (1D) optical lattice potential in a regime where the collective (Josephson) tunnelling energy is comparable with the on-site interaction energy, and the number of particles per lattice site is mesoscopically large. By directly imaging the motion of atoms in the lattice, we observe an abrupt suppression of atom transport through the array for a critical ratio of these energies, consistent with quantum fluctuation induced localization. Directly below the onset of localization, the frequency of the observed superfluid transport can be explained by a phonon excitation but deviates substantially from that predicted by the hydrodynamic/Gross–Pitaevskii equations.

A complete degradability analysis of one-mode bosonic Gaussian channels is presented. We show that apart from the class of channels which are unitarily equivalent to the channels with additive classical noise, these maps can be characterized in terms of weak- and/or anti-degradability. Furthermore a new set of channels which have null quantum capacity is identified. This is done by exploiting the composition rules of one-mode Gaussian maps and the fact that anti-degradable channels cannot be used to transfer quantum information.

We propose the novel combination of a laser guide and magnetic lens to transport a cold atomic cloud. We have modelled the loading and guiding of a launched cloud of cold atoms with the optical dipole force. We discuss the optimum strategy for loading typically 30% of the atoms from a magneto-optical trap (MOT) and guiding them vertically through 22 cm. However, although the atoms are tightly confined transversely, thermal expansion in the propagation direction still results in a density loss of two orders of magnitude. By combining the laser guide with a single impulse from a magnetic lens we show one can actually increase the density of the guided atoms by a factor of 10.

We consider an extension of the voter model in which a set of interacting elements (agents) can be in either of two equivalent states (A or B) or in a third additional mixed (AB) state. The model is motivated by studies of language competition dynamics, where the AB state is associated with bilingualism. We study the ordering process and associated interface and coarsening dynamics in regular lattices and small world networks. Agents in the AB state define the interfaces, changing the interfacial noise driven coarsening of the voter model to curvature driven coarsening. This change in the coarsening mechanism is also shown to originate for a class of perturbations of the voter model dynamics. When interaction is through a small world network the AB agents restore coarsening, eliminating the metastable states of the voter model. The characteristic time to reach the absorbing state scales with system size as τ ∼ lnN to be compared with the result τ ∼ N for the voter model in a small world network.

We study the level spacing distribution p(s) in the spectrum of random networks. According to our numerical results, the shape of p(s) in the Erds–Rényi (E–R) random graph is determined by the average degree ⟨k⟩ and p(s) undergoes a dramatic change when ⟨k⟩ is varied around the critical point of the percolation transition, ⟨k⟩ = 1. When ⟨k⟩ ≫ 1, the p(s) is described by the statistics of the Gaussian orthogonal ensemble (GOE), one of the major statistical ensembles in Random Matrix Theory, whereas at ⟨k⟩ = 1 it follows the Poisson level spacing distribution. Closely above the critical point, p(s) can be described in terms of an intermediate distribution between Poisson and the GOE, the Brody-distribution. Furthermore, below the critical point p(s) can be given with the help of the regularized Gamma-function. Motivated by these results, we analyse the behaviour of p(s) in real networks such as the internet, a word association network and a protein–protein interaction network as well. When the giant component of these networks is destroyed in a node deletion process simulating the networks subjected to intentional attack, their level spacing distribution undergoes a similar transition to that of the E–R graph.

We describe experiments aimed at characterizing complex plasma crystals and their interaction with the space charge surrounding an RF discharge. By analysing the complex plasma radiofrequency impedance, we were able to measure the density of the free electrons of the crystal. The density of the ions was derived by Langmuir probe measurements in the bulk plasma combined with a model of the RF plasma sheath. Surprisingly the full characterization of the crystal reveals that the 3D ensemble is quasi-neutral, i.e. a three component plasma and the particles levitate in their self-built electric field, much lower than the sheath field. This result is inherent to the 3D geometry.

In this work, we develop several new simulation algorithms for one-dimensional (1D) many-body quantum mechanical systems combining the Matrix Product State variational ansatz (vMPS) with Taylor, Padé and Arnoldi approximations to the evolution operator. By comparing with previous techniques based on vMPS and Density Matrix Renormalization Group (DMRG), we demonstrate that the Arnoldi method is the best one, reaching extremely good accuracy with moderate resources. Finally, we apply this algorithm to studying how correlations are transferred from the atomic to the molecular cloud when crossing a Feschbach resonance with two-species hard-core bosons in a 1D optical lattice.

A superfluid atomic Fermi system may support a giant vortex if the trapping potential is anharmonic. In such a potential, the single-particle spectrum has a positive curvature as a function of angular momentum. A tractable model is presented in which the lowest and next lowest Landau levels are occupied. Different parameter regimes are identified and characterized. Due to the dependence of the interaction on angular momentum quantum number, the Cooper pairing is at its strongest not only close to the Fermi level, but also close to the energy minimum. It is shown that the gas is superfluid in the interior of the toroidal density distribution and normal in the outer regions. Furthermore, the pairing may give rise to a localized density depression in configuration space.

The OPERA neutrino detector at the underground Gran Sasso Laboratory (LNGS) was designed to perform the first detection of neutrino oscillations in appearance mode, through the study of ν_μ → ν_τ oscillations. The apparatus consists of a lead/emulsion-film target complemented by electronic detectors. It is placed in the high-energy, long-baseline CERN to LNGS beam (CNGS) 730 km away from the neutrino source. In August 2006, a first run with CNGS neutrinos was successfully conducted. A first sample of neutrino events was collected, statistically consistent with the integrated beam intensity. After a brief description of the beam and of the various sub-detectors, we report on the achievement of this milestone, presenting the first data and some analysis results.

Bell theorems show how to experimentally falsify local realism. Conclusive falsification is highly desirable as it would provide support for the most profoundly counterintuitive feature of quantum theory—nonlocality. Despite the preponderance of evidence for quantum mechanics, practical limits on detector efficiency and the difficulty of coordinating space-like separated measurements have provided loopholes for a classical worldview; these loopholes have never been simultaneously closed. A number of new experiments have recently been proposed to close both loopholes at once. We show these novel designs fail in the most basic way, by not ruling out local hidden variable models, and we provide an explicit classical model to demonstrate this. These experiments share a common flaw, which reveals a basic misunderstanding of how nonlocality proofs work. Given the time and resources now being devoted to such experiments, theoretical clarity is essential. Our explanation is presented in terms of simple logic and should serve to correct misconceptions and avoid future mistakes.

A new diagnostic method for the determination of negative hydrogen ion densities H⁻ (D⁻), namely the H_α/H_β line ratio method, is presented and applied to high power (P≈100 kW) negative ion sources operating at low pressure (p < 1 Pa). The basis of the method is the mutual neutralization process which enhances Balmer line radiation selectively. Detailed parameter studies carried out with a collisional radiative model show that the Balmer line ratio H_α/H_β is very well suited to obtain negative ion densities in low temperature hydrogen plasmas with typical plasma parameters of T_e ≈ 1–5 eV and n_e ≈ 10¹⁶–5 × 10¹⁸ m⁻³. The H_α/H_β line ratio method has the great advantage that it can be accomplished easily with the non-invasive and in situ plasma diagnostic method of optical emission spectroscopy. The method is applied to the RF ion sources of IPP Garching, currently developed for the neutral beam injection system of ITER. Line-of-sight averaged negative hydrogen densities in the range of 10¹⁶–10¹⁷ m⁻³ are measured close to the extraction system in hydrogen and deuterium discharge. Absolute values depend on pressure, power and caesium conditioning of the source. Negative ion densities measured with this novel technique in front of the grid correlate very well with the extracted negative ion current densities.

We have investigated a possible application of single-electron transistor (SET) devices for use as a differential voltage amplifier. The device consists of a box-SET and probe-SET coupled with each other through a tunnel junction, with the gate electrodes of the two SETs acting as differential signal inputs. The voltage across the probe-SET at a fixed bias current provides information about the charge states of both the probe-SET and the box-SET, which was confirmed by simulations based on the orthodox theory of single-electron tunnelling. When operated as a differential amplifier, the output probe-SET voltage signal was measured as a function of the two gate input signals. While the output signal was found to be proportional to the difference in the two input signals, it remained unchanged for input signals of the same amplitude (referred to as the common mode signal), and the common-mode rejection ratio was found to be 27.5 dB.

Structural properties of the spherically averaged total charge density ρ_λ(r) of a degenerate free electron gas distorted by the action of an attractive, screened potential λV(r) are studied. The total charge density is described as the sum over the occupied bound and scattering states of independent electrons moving in an effective central field. Each of the individual (bound, scattering) terms in the density presents discontinuities at threshold values of λ for which new bound states appear. Nevertheless, it is mathematically proven that these discontinuities cancel each other and that the density ρ_λ(0) at the origin, as well as its two radial derivatives ρ_λ'(0) and ρ_λ''(0), are differentiable functions of λ for any λ ⩾ 0. The smooth behaviour implies that the curvature ρ_λ''(0) at the origin does not provide a criterion for the bound-to-free transition in metallic environments.

Experimental results for the sequential transport through a lateral quantum dot in a perpendicular magnetic field are compared with theory. Regular patterns of negative differential conductances are observed in the nonlinear regime. We attempt to reproduce theoretically these patterns in a simplified model which captures the essential features of the experimental system. Orbital and spin effects are treated in terms of the Fock–Darwin model. The transport properties are described by employing a master equation with tunable tunnelling and relaxation rates. We show that the essential physics underlying the experiment can be described within our approach if the timescales of the different transport channels are well separated.

The Lipkin model of arbitrary particle number N is studied in terms of exact differential-operator representation of spin-operators from which we obtain the low-lying energy spectrum with the instanton method of quantum tunnelling. Our new observation is that the well-known quantum phase transition (QPT) can also occur in the finite-N model only if N is an odd number. We furthermore demonstrate a new type of QPT characterized by level-crossing which is induced by the geometric phase interference and is marvellously periodic with respect to the coupling parameter. Finally, the conventional QPT is understood intuitively from the tunnelling formulation in the thermodynamic limit.