Nodal domain distributions for quantum maps

The statistics of the nodal lines and nodal domains of the eigenfunctions of quantum billiards have recently been observed to be fingerprints of the chaoticity of the underlying classical motion by Blum et al. (Phys. Rev. Lett., Vol. 88 (2002), 114101) and by Bogomolny and Schmit (Phys. Rev. Lett., Vol. 88 (2002), 114102). These statistics were shown to be computable from the random wave model of the eigenfunctions. We here study the analogous problem for chaotic maps whose phase space is the two-torus. We show that the distributions of the numbers of nodal points and nodal domains of the eigenvectors of the corresponding quantum maps can be computed straightforwardly and exactly using random matrix theory. We compare the predictions with the results of numerical computations involving quantum perturbed cat maps.Comment: 7 pages, 2 figures. Second version: minor correction

A new correlator in quantum spin chains

We propose a new correlator in one-dimensional quantum spin chains, the $s-$Emptiness Formation Probability ($s-$EFP). This is a natural generalization of the Emptiness Formation Probability (EFP), which is the probability that the first $n$ spins of the chain are all aligned downwards. In the $s-$EFP we let the spins in question be separated by $s$ sites. The usual EFP corresponds to the special case when $s=1$, and taking $s>1$ allows us to quantify non-local correlations. We express the $s-$EFP for the anisotropic XY model in a transverse magnetic field, a system with both critical and non-critical regimes, in terms of a Toeplitz determinant. For the isotropic XY model we find that the magnetic field induces an interesting length scale.Comment: 6 pages, 1 figur

On the Nodal Count Statistics for Separable Systems in any Dimension

We consider the statistics of the number of nodal domains aka nodal counts for eigenfunctions of separable wave equations in arbitrary dimension. We give an explicit expression for the limiting distribution of normalised nodal counts and analyse some of its universal properties. Our results are illustrated by detailed discussion of simple examples and numerical nodal count distributions.Comment: 21 pages, 4 figure

Coarse-graining protein energetics in sequence variables

We show that cluster expansions (CE), previously used to model solid-state materials with binary or ternary configurational disorder, can be extended to the protein design problem. We present a generalized CE framework, in which properties such as energy can be unambiguously expanded in the amino-acid sequence space. The CE coarse grains over nonsequence degrees of freedom (e.g., side-chain conformations) and thereby simplifies the problem of designing proteins, or predicting the compatibility of a sequence with a given structure, by many orders of magnitude. The CE is physically transparent, and can be evaluated through linear regression on the energies of training sequences. We show, as example, that good prediction accuracy is obtained with up to pairwise interactions for a coiled-coil backbone, and that triplet interactions are important in the energetics of a more globular zinc-finger backbone.Comment: 10 pages, 3 figure

Probing neutrino masses with CMB lensing extraction

We evaluate the ability of future cosmic microwave background (CMB) experiments to measure the power spectrum of large scale structure using quadratic estimators of the weak lensing deflection field. We calculate the sensitivity of upcoming CMB experiments such as BICEP, QUaD, BRAIN, ClOVER and PLANCK to the non-zero total neutrino mass M_nu indicated by current neutrino oscillation data. We find that these experiments greatly benefit from lensing extraction techniques, improving their one-sigma sensitivity to M_nu by a factor of order four. The combination of data from PLANCK and the SAMPAN mini-satellite project would lead to sigma(M_nu) = 0.1 eV, while a value as small as sigma(M_nu) = 0.035 eV is within the reach of a space mission based on bolometers with a passively cooled 3-4 m aperture telescope, representative of the most ambitious projects currently under investigation. We show that our results are robust not only considering possible difficulties in subtracting astrophysical foregrounds from the primary CMB signal but also when the minimal cosmological model (Lambda Mixed Dark Matter) is generalized in order to include a possible scalar tilt running, a constant equation of state parameter for the dark energy and/or extra relativistic degrees of freedom.Comment: 13 pages, 4 figures. One new figure and references added. Version accepted for publicatio

On the multiplicativity of quantum cat maps

The quantum mechanical propagators of the linear automorphisms of the two-torus (cat maps) determine a projective unitary representation of the theta group, known as Weil's representation. We prove that there exists an appropriate choice of phases in the propagators that defines a proper representation of the theta group. We also give explicit formulae for the propagators in this representation.Comment: Revised version: proof of the main theorem simplified. 21 page

Quantization of multidimensional cat maps

In this work we study cat maps with many degrees of freedom. Classical cat maps are classified using the Cayley parametrization of symplectic matrices and the closely associated center and chord generating functions. Particular attention is dedicated to loxodromic behavior, which is a new feature of two-dimensional maps. The maps are then quantized using a recently developed Weyl representation on the torus and the general condition on the Floquet angles is derived for a particular map to be quantizable. The semiclassical approximation is exact, regardless of the dimensionality or of the nature of the fixed points.Comment: 33 pages, latex, 6 figures, Submitted to Nonlinearit

Signatures of homoclinic motion in quantum chaos

Homoclinic motion plays a key role in the organization of classical chaos in Hamiltonian systems. In this Letter, we show that it also imprints a clear signature in the corresponding quantum spectra. By numerically studying the fluctuations of the widths of wavefunctions localized along periodic orbits we reveal the existence of an oscillatory behavior, that is explained solely in terms of the primary homoclinic motion. Furthermore, our results indicate that it survives the semiclassical limit.Comment: 5 pages, 4 figure

On the resonance eigenstates of an open quantum baker map

We study the resonance eigenstates of a particular quantization of the open baker map. For any admissible value of Planck's constant, the corresponding quantum map is a subunitary matrix, and the nonzero component of its spectrum is contained inside an annulus in the complex plane, $|z_{min}|\leq |z|\leq |z_{max}|$. We consider semiclassical sequences of eigenstates, such that the moduli of their eigenvalues converge to a fixed radius $r$. We prove that, if the moduli converge to $r=|z_{max}|$, then the sequence of eigenstates converges to a fixed phase space measure $\rho_{max}$. The same holds for sequences with eigenvalue moduli converging to $|z_{min}|$, with a different limit measure $\rho_{min}$. Both these limiting measures are supported on fractal sets, which are trapped sets of the classical dynamics. For a general radius $|z_{min}|< r < |z_{max}|$, we identify families of eigenstates with precise self-similar properties.Comment: 32 pages, 2 figure

Competitiveness and sustainability: can ‘smart city regionalism’ square the circle?

Increasingly, the widely established, globalisation-driven agenda of economic competitiveness meets a growing concern with sustainability. Yet, the practical and conceptual co-existence—or fusion—of these two agendas is not always easy. This includes finding and operationalising the ‘right’ scale of governance, an important question for the pursuit of the distinctly transscalar nature of these two policy fields. ‘New regionalism’ has increasingly been discussed as a pragmatic way of tackling the variable spatialities associated with these policy fields and their changing articulation. This paper introduces ‘smart (new) city-regionalism’, derived from the principles of smart growth and new regionalism, as a policy-shaping mechanism and analytical framework. It brings together the rationales, agreed principles and legitimacies of publicly negotiated polity with collaborative, network-based and policy-driven spatiality. The notion of ‘smartness’, as suggested here as central feature, goes beyond the implicit meaning of ‘smart’ as in ‘smart growth’. When introduced in the later 1990s the term embraced a focus on planning and transport. Since then, the adjective ‘smart’ has become used ever more widely, advocating innovativeness, participation, collaboration and co-ordination. The resulting ‘smart city regionalism’ is circumscribed by the interface between the sectorality and territoriality of policy-making processes. Using the examples of Vancouver and Seattle, the paper looks at the effects of the resulting specific local conditions on adopting ‘smartness’ in the scalar positioning of policy-making