123516619
Nonanalytic twist maps and Frenkel-Kontorova models
We give a review of our recent works on a class of twist maps and Frenkel-Kontorova (FK) models in which the external potential function is nonanalytic (C1) and endowed with a variable degree of inflection z. When z > 3, recurrence of invariant curves has been observed. An “inverse residue criterion” for the reappearance of an invariant curve — complementary to the “residue criterion” for the disappearance of an invariant curve — is introduced to make a precise determination of the reappearance point. We have also studied the local and global scaling behaviors of these curves. The critical exponents, the singularity spectrum, and the generalized dimensions are found to vary with z when 2 ⩽ z < 3, but remain the same for z ⩾ 3. The degree of inflection thus plays a role reminiscent of that of dimensionality in phase transitions with z = 2 and 3 corresponding respectively to the lower and upper critical dimensions. When the same nonanalytic function is substituted for the sinusoidal external potential in the standard FK model, the same conclusions are reached: there exists a sequence of pinning-depinning transitions, and the critical exponents of the phonon gap, the coherence length, and the Peierls-Nabarro barrier all show the same dependence on z.
117912244
Galilean structure of the Poincaré group on the light-cone
SummaryIt is observed that the isomorphism between the Poincaré subgroup in the L.C. (light-cone) co-ordinates and the two-dimensional Galilean group can be trivially extended to higher dimensions. The case of five dimension is explicitly shown, and the generalization to any finite dimensionn (> 2) is immediate.RiassuntoSi osserva che l’isomorfismo fra il sottogruppo di Poincaré nelle coordinate del cono di luce (L.C.) ed il gruppo galileano bidimensionale può essere esteso in modo banale a dimensioni superiori. Si presenta esplicitamente il caso delle cinque dimensioni, e la generalizzazione a qualunque dimensione finitan (>2) è immediata.РеэюмеОтмечается, что иэоморфиэм между подгруппой Пуанкаре в координата светового конуса и двумерной группой Галилея может быть тривиально распространен на случай больщего числа иэмерений. Явным обраэом исследуется пятимерный случай. Обобшение на случай любого числа иэмеренийп (>2) получается непосредственно.
120428117
Crossover phenomena in the multifractal behavior of invariant circles
Abstract The singularity spectrum f(α) and the generalized dimension Dq of the critical invariant circle in an area-preserving twist map with a variable degree of inflection z is studied. The scaling behavior is found to depend on z, and there is a crossover from the simple fixed point to the standard fixed point when z varies from 2 to 3. Thus z plays a role quite similar to that of dimensionality in phase transition, with z = 2 and 3 corresponding to the lower and upper critical dimensions respectively.
209934211
Anomalous dimensions of tensor operators in 4-epsilon dimensions
122878082
Induced Topological Terms, Hidden Local Symmetry, and Anyon Superconductivity
We show how the topological Chern-Simons term in three dimensions and the θ-term, in four dimensions can be dynamically generated due to quantum fluctuations in Nambu-Jona-Lasinio models. These results give a vivid demonstration of hidden local symmetry at work. The fluctuation-induced Chern-Simons term provides a natural way to incorporate fractional statistics and thus a more direct connection with anyon superconductivity. Possible phenomenological implications of this approach in particle physics are also discussed.
14771278
Nonadiabatic geometric phase for the cyclic evolution of a time-dependent Hamiltonian system
The geometric phases of the cyclic states of a generalized harmonic oscillator with nonadiabatic timeperiodic parameters are discussed in the framework of squeezed state. It is shown that the cyclic and quasicyclic squeezed states correspond to the periodic and quasiperiodic solutions of an effective Hamiltonian defined on an extended phase space, respectively. The geometric phase of the cyclic squeezed state is found to be a phase-space area swept out by a periodic orbit. Furthermore, a class of cyclic states is expressed as a superposition of an infinite number of squeezed states. Their geometric phases are found to be independent of \, and equal to 2(n11/2) times the classical nonadiabatic Hannay angle. @S1050-2947~98!03311-3# The Berry phase is one of the most important findings on the fundamental problem of quantum mechanics in recent years @1#. It reveals the gauge structure associated with a phase shift in adiabatic processes in quantum mechanics. This quantal phase is connected with a classical angle, namely, the Hannay angle, by a simple and elegant expression in the semiclassical limit @2#. The next important advance was the relaxation of the adiabatic approximation @3,4#. Aharanov and Anandan studied the phase associated with a cyclic evolution in quantum mechanics, which occurs when a state returns to its initial condition. They showed that the phase is a geometrical property of the curve in the projective Hilbert space which is naturally associated with the motion @3#. For the special case of an adiabatic evolution, this phase factor is a gauge-invariant generalization of the Berry phase. The significance of Aharanov and Anandan’s generalization lies in the following: ~i! the cyclic evolution of a physical system is of the most interest in physics both experimentally and theoretically; ~ii! the universal existence of the cyclic evolution is guaranteed for any quantum system. The second point can be easily recognized by considering the eigenvectors of the unitary evolution operator for a quantum system. An explicit example is a time-periodic Hamiltonian system where the Floquet theorem applies. The eigenfunctions of the Floquet operator, which are the socalled Bloch wave functions in condensed-matter physics, are obviously cyclic solutions and are of great interest in physics. However, unlike the adiabatic case, in the nonadiabatic case calculating the eigenvectors and extracting the nonadiabatic geometric phase from the quasienergy term for a time-dependent Hamiltonian is far from a trivial task, except in the special case of the spin particle in a magnetic field. Recent works of Ge and Child@5# made a further step in this direction. They found a special cyclic state of the Gaussian wave packet’s form for a generalized harmonic oscillator. The nonadiabatic geometric phase is explicitly calculated and found to be one-half of the classical nonadiabatic Hannay angle. In this paper, we would like to suggest an alternative approach—the squeezed-state approach to study the general cyclic evolutions of the generalized harmonic oscillator. In particular, we shall construct a class of quantum states based on the superposition of an infinite number of squeezed states. We find that the condition for them to be cyclic evolutions is nothing but the quantization rule without Maslov-Morse correction. The nonadiabatic geometric phases are obtained analytically, and found to be related to the classical Hannay angle by an explicit expression.
16694899
Near thermal radiation in detectors, mirrors and black holes: A Stochastic approach
In analyzing the nature of thermal radiance experienced by an accelerated observer (Unruh effect), an eternal black hole (Hawking effect), and in certain types of cosmological expansion, one of us proposed a unifying viewpoint that these can be understood as arising from the vacuum fluctuations of the quantum field being subjected to an exponential scale transformation in these systems. This viewpoint, together with our recently developed stochastic theory of particle-field interaction understood as quantum open systems described by the influence functional formalism, can be used effectively to address situations where the spacetime possesses an event horizon only asymptotically, or none at all. Examples studied here include detectors moving at uniform acceleration only asymptotically or for a finite time, a moving mirror, and a two-dimensional collapsing mass. We show that in such systems radiance indeed is observed, albeit not in a precise Planckian spectrum. The deviation therefrom is determined by a parameter which measures the departure from uniform acceleration or from exact exponential expansion. The methodology illustrated here is expected to be useful for the investigation of nonequilibrium black hole thermodynamics and the linear response regime of back reaction problems in semiclassical gravity.
15446534
Critical Dynamics in the Early Universe ∗
Methods and concepts for the study of phase transitions mediated by a time-dependent order-parameter field in curved spacetimes are discussed. A practical example is the derivation of an effective (quasi)potential for the description of ‘slow-roll’ inflation in the early universe. We first summarize our early results on viewing the symmetry behavior of constant background fields in curved but static spacetimes as finite size effect, and the use of derivative expansions for constructing effective actions for slowly-varying background fields. We then introduce the notion of dynamical finite size effect to explain how an exponential expansion of the scale factor imparts a finite size to the system and how the symmetry behavior in de Sitter space can be understood qualitatively in this light. We reason why the exponential inflation can be described equivalently by a scale transformation, thus rendering this special class of dynamics as effectively static. Finally we show how, in this view, one can treat the class of ‘slow-roll’ inflation as a dynamic perturbation off the effectively static class of exponential inflation and understand it as a dynamical critical phenomenon
119533257
Hydrodynamic transport functions from quantum kinetic field theory
Starting from the quantum kinetic field theory [E. Calzetta and B. L. Hu, Phys. Rev. D 37, 2878 (1988)] constructed from the closed-time-path (CTP), two-particle-irreducible (2PI) effective action we show how to compute from first principles the shear and bulk viscosity functions in the hydrodynamic-thermodynamic regime. For a real scalar field with $\ensuremath{\lambda}{\ensuremath{\Phi}}^{4}$ self-interaction we need to include four-loop graphs in the equation of motion. This work provides a microscopic field-theoretical basis to the ``effective kinetic theory'' proposed by Jeon and Yaffe [S. Jeon and L. G. Yaffe, Phys. Rev. D 53, 5799 (1996)], while our result for the bulk viscosity reproduces their expression derived from linear-response theory and the imaginary-time formalism of thermal field theory. Though unavoidably involved in calculations of this sort, we feel that the approach using fundamental quantum kinetic field theory is conceptually clearer and methodically simpler than the effective kinetic theory approach, as the success of the latter requires a clever rendition of diagrammatic resummations which is neither straightforward nor fail-safe. Moreover, the method based on the CTP-2PI effective action illustrated here for a scalar field can be formulated entirely in terms of functional integral quantization, which makes it an appealing method for a first-principles calculation of transport functions of a thermal non-Abelian gauge theory, e.g., QCD quark-gluon plasma produced from heavy ion collisions.
119199048
Effective field theory of cosmic acceleration : An implementation in CAMB
We implement the effective field theory (EFT) approach to dark energy and modified gravity in the public Einstein-Boltzmann solver CAMB. The resulting code, which we dub EFTCAMB, is a powerful and versatile tool that can be used for several objectives. It can be employed to evolve the full dynamics of linear scalar perturbations in any given single field dark energy or modified gravity model, once the latter is mapped into the EFT formalism. It offers a numerical implementation of EFT as a model-independent framework to test gravity on cosmological scales. EFTCAMB has a built-in check for the fulfillment of general stability conditions such as the absence of ghost and superluminal propagation of perturbations. It handles phantom-divide crossing models and does not contain any quasi-static approximation, but rather evolves the full dynamics of perturbations on linear scales. As we will show, the latter is an important feature in view of the accuracy and scale range of upcoming surveys. We show the reliability and applicability of our code by evolving the dynamics of linear perturbations and extracting predictions for power spectra in several models. In particular we perform a thorough analysis of f(R) theories, comparing our outputs with those of an existing code for LCDM backgrounds, and finding an agreement that can reach 0.1% for models with a Compton wavelength consistent with current cosmological data. We then showcase the flexibility of our code studying two different scenarios. First we produce new results for designer f(R) models with a time-varying dark energy equation of state. Second, we extract predictions for linear observables in some parametrized EFT models with a phantom-divide crossing equation of state for dark energy.
15718551
Early Universe Quantum Processes in BEC Collapse Experiments
We show that in the collapse of a Bose–Einstein condensate (BEC)4certain processes involved and mechanisms at work share a common origin with corresponding quantum field processes in the early universe such as particle creation, structure formation, and spinodal instability. Phenomena associated with the controlled BEC collapse observed in the experiment of Donley et al. (Donley, E., et al. (2001), Nature412, 295; Claussen, N. (2003), PhD Thesis, University of Colorado; Claussen, N., et al. (2003), Physical Review A67, 060701(R))(they call it “Bose–Nova,” see also Chin, J., Vogels, J., and Ketterle, W. (2003), Physical Review Letters90, 160405) such as the appearance of bursts and jets can be explained as a consequence of the squeezing and amplification of quantum fluctuations above the condensate by the dynamics of the condensate. Using the physical insight gained in depicting these cosmological processes, our analysis of the changing amplitude and particle contents of quantum excitations in these BEC dynamics provides excellent quantitative fits with the experimental data on the scaling behavior of the collapse time and the amount of particles emitted in the jets. Because of the coherence properties of BEC and the high degree of control and measurement precision in atomic and optical systems, we see great potential in the design of tabletop experiments for testing out general ideas and specific (quantum field) processes in the early universe, thus opening up the possibility for implementing “laboratory cosmology.”