Hofstadter's Cocoon

Hofstadter showed that the energy levels of electrons on a lattice plotted as a function of magnetic field form an beautiful structure now referred to as "Hofstadter's butterfly". We study a non-Hermitian continuation of Hofstadter's model; as the non-Hermiticity parameter $g$ increases past a sequence of critical values the eigenvalues successively go complex in a sequence of "double-pitchfork bifurcations" wherein pairs of real eigenvalues degenerate and then become complex conjugate pairs. The associated wavefunctions undergo a spontaneous symmetry breaking transition that we elucidate. Beyond the transition a plot of the real parts of the eigenvalues against magnetic field resembles the Hofstadter butterfly; a plot of the imaginary parts plotted against magnetic fields forms an intricate structure that we call the Hofstadter cocoon. The symmetries of the cocoon are described. Hatano and Nelson have studied a non-Hermitian continuation of the Anderson model of localization that has close parallels to the model studied here. The relationship of our work to that of Hatano and Nelson and to PT transitions studied in PT quantum mechanics is discussed

Revisiting Pollock's Drip Paintings

We investigate the contentions that Jackson Pollock's drip paintings are fractals produced by the artist's Levy distributed motion and that fractal analysis may be used to authenticate works of uncertain provenance[1-5]. We find that the paintings exhibit fractal characteristics over too small a range to be usefully considered as fractal; their limited fractal characteristics are easily generated without Levy motion, both by freehand drawing and gaussian random motion. Several problems must therefore be addressed before fractal analysis can be used to authenticate paintings.Comment: Appeared in Nature Nov.2006. Previously unavailable on arxiv. Figures here are low resolution versio

Relativistic Non-Hermitian Quantum Mechanics

We develop relativistic wave equations in the framework of the new non-hermitian ${\cal PT}$ quantum mechanics. The familiar Hermitian Dirac equation emerges as an exact result of imposing the Dirac algebra, the criteria of ${\cal PT}$-symmetric quantum mechanics, and relativistic invariance. However, relaxing the constraint that in particular the mass matrix be Hermitian also allows for models that have no counterpart in conventional quantum mechanics. For example it is well-known that a quartet of Weyl spinors coupled by a Hermitian mass matrix reduces to two independent Dirac fermions; here we show that the same quartet of Weyl spinors, when coupled by a non-Hermitian but $\cal{PT}$ symmetric mass matrix, describes a single relativistic particle that can have massless dispersion relation even though the mass matrix is non-zero.The ${\cal PT}$-generalized Dirac equation is also Lorentz invariant, unitary in time, and CPT respecting, even though as a non-interacting theory it violates ${\cal P}$ and ${\cal T}$ individually. The relativistic wave equations are reformulated as canonical fermionic field theories to facilitate the study of interactions, and are shown to maintain many of the canonical structures from Hermitian field theory, but with new and interesting new possibilities permitted by the non-hermiticity parameter $m_2$

The Effect of Forcing on Vacuum Radiation

Vacuum radiation has been the subject of theoretical study in both cosmology and condensed matter physics for many decades. Recently there has been impressive progress in experimental realizations as well. Here we study vacuum radiation when a field mode is driven both parametrically and by a classical source. We find that in the Heisenberg picture the field operators of the mode undergo a Bogolyubov transformation combined with a displacement, in the Schr\"odinger picture the oscillator evolves from the vacuum to a squeezed coherent state. Whereas the Bogolyubov transformation is the same as would be obtained if only the parametric drive were applied the displacement is determined by both the parametric drive and the force. If the force is applied well after the parametric drive then the displacement is the same as would be obtained by the action of the force alone and it is essentially independent of $t_f$, the time lag between the application of the force and the parametric drive. If the force is applied well before the parametric drive the displacement is found to oscillate as a function of $t_f$. This behavior can be understood in terms of quantum interference. A rich variety of behavior is observed for intermediate values of $t_f$. The oscillations can turn off smoothly or grow dramatically and decrease depending on strength of the parametric drive and force and the durations for which they are applied. The displacement depends only on the Fourier component of the force at a single resonant frequency when the forcing and the parametric drive are well separated in time. However for a weak parametric drive that is applied at the same time as the force we show that the displacement responds to a broad range of frequencies of the force spectrum. Implications of our findings for experiments are briefly discussed

Reply to Comment on "Drip Paintings and Fractal Analysis" by Micolich et al (arXiv:0712.165v1 [cond-mat.stat-mech])

We reply to the comment of Micolich et al and demonstrate that their criticisms are unfounded. In particular we provide a detailed discussion of our box-counting algorithm and of the interpretation of multi-layered paintings. We point out that in their entire body of work, Taylor et al have not provided the scientific community with sufficient empirical support of their claims, nor have they adequately addressed any of the problems we have identified with the application of fractal analysis to drip paintings.Comment: 4 pages, 1 figur

Implications of Broken Symmetry for Superhorizon Conservation Theorems in Cosmology

Inflation produces super-horizon sized perturbations that ultimately return within the horizon and are thought to form the seeds of all observed large scale structure in the Universe. But inflationary predictions can only be compared with present day observations if, as conventional wisdom dictates, they remain unpolluted by subsequent sub-horizon causal physical processes and therefore remain immune from the vicissitudes of unknown universal dynamics in the intervening period. Here we demonstrate that conventional wisdom need not be correct, and as a result cosmological signatures arising from intervening unknown non-inflationary processes may confuse the interpretation of observational data today.Comment: changed titl

Chameleon Effects on Small Scale Structure and the Baryonic Jeans Mass

In the framework of Newtonian cosmology or general relativity it is simple to derive a mass scale below which collapsed structures are relatively devoid of baryons. We examine how the inclusion of a chameleon scalar field affects this baryonic Jeans mass, bearing in mind both the canonical case of a gravitational-strength coupling between the scalar field and matter, as well as the strong coupling regime wherein the coupling is very large. We find that baryon effects persist down to smaller scales in a chameleon theory than they do in ordinary general relativity, especially in the case of strong coupling. Several potentially observable consequences of this are identified

A 'Dysonization' Scheme for Identifying Particles and Quasi-Particles Using Non-Hermitian Quantum Mechanics

In 1956 Dyson analyzed the low-energy excitations of a ferromagnet using a Hamiltonian that was non-Hermitian with respect to the standard inner product. This allowed for a facile rendering of these excitations (known as spin waves) as weakly interacting bosonic quasi-particles. More than 50 years later, we have the full denouement of non-Hermitian quantum mechanics formalism at our disposal when considering Dyson's work, both technically and contextually. Here we recast Dyson's work on ferromagnets explicitly in terms of two inner products, with respect to which the Hamiltonian is always self-adjoint, if not manifestly "Hermitian". Then we extend his scheme to doped antiferromagnets described by the t-J model, in hopes of shedding light on the physics of high-temperature superconductivity

An Electrostatic Analogy for Symmetron Gravity

The symmetron model is a scalar-tensor theory of gravity with a screening mechanism that suppresses the effect of the symmetron field at high densities characteristic of the solar system and laboratory scales but allows it to act with gravitational strength at low density on the cosmological scale. We elucidate the screening mechanism by showing that in the quasi-static Newtonian limit there are precise analogies between symmetron gravity and electrostatics for both strong and weak screening. For strong screening we find that large dense bodies behave in a manner analogous to perfect conductors in electrostatics. Based on this analogy we find that the symmetron field exhibits a lightning rod effect wherein the field gradients are enhanced near the ends of pointed or elongated objects. An ellipsoid placed in a uniform symmetron gradient is shown to experience a torque. By symmetry there is no gravitational torque in this case. Hence this effect unmasks the symmetron and might serve as the basis for future laboratory experiments. The symmetron force between a point mass and a large dense body includes a component corresponding to the interaction of the point mass with its image in the larger body. None of these effects have counterparts in the Newtonian limit of Einstein gravity. We discuss the similarities between symmetron gravity and the chameleon model as well as the differences between the two.Comment: Matches journal version; notably this version has a clarified discussion of the possibility of a net repulsive force in Section II

Contact interactions and Kronig-Penney Models in Hermitian and PT-symmetric Quantum Mechanics

The delta function potential is a simple model of zero-range contact interaction in one dimension. The Kronig-Penney model is a one-dimensional periodic array of delta functions that models the energy bands in a crystal. Here we investigate contact interactions that generalize the delta function potential and corresponding generalizations of the Kronig-Penney model within conventional and parity-time symmetric quantum mechanics (PTQM). In conventional quantum mechanics we determine the most general contact interaction compatible with self-adjointness; in PTQM we consider interactions that are symmetric under the combined transformation PT. In both cases we find that the most general interaction has four independent real parameters; depending on the parameter values the interaction can support more bound states than the conventional delta function. In the PT case the two bound state energies can be both real or a complex conjugate pair, with the transition corresponding to the breaking of PT-symmetry. The scattering states for the PT case are also found to exhibit spontaneous breaking of PT-symmetry. We investigate the energy bands when the generalized contact interactions are repeated periodically in space in one dimension. In the Hermitian case we find that the two bound states result in two narrow bands generically separated by a gap. These bands intersect at a single point in the Brillouin zone as the interaction parameters are varied. Near the intersection the bands form a massless Dirac cone. In the PT-symmetric case, as the parameters of the contact interaction are varied the two bound state bands undergo a PT-symmetry breaking transition wherein the two band energies go from being real to being a complex conjugate pair. The PT-symmetric Kronig-Penney model provides a simple soluble example of the transition which has the same form as in other models of PT-symmetric crystals