This content will become publicly available on August 22, 2024
 Award ID(s):
 2011767
 NSFPAR ID:
 10476185
 Publisher / Repository:
 American Physical Society
 Date Published:
 Journal Name:
 Physical Review A
 Volume:
 108
 Issue:
 2
 ISSN:
 24699926
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Abstract We characterize initial value problems for the defocusing Manakov system (coupled twocomponent nonlinear Schrödinger equation) with nonzero background and welldefined spatial parity symmetry (i.e., when each of the components of the solution is either even or odd), corresponding to boundary value problems on the half line with Dirichlet or Neumann boundary conditions at the origin. We identify the symmetries of the eigenfunctions arising from the spatial parity of the solution, and we determine the corresponding symmetries of the scattering data (reflection coefficients, discrete spectrum and norming constants). All parity induced symmetries are found to be more complicated than in the scalar (i.e., onecomponent) case. In particular, we show that the discrete eigenvalues giving rise to dark solitons arise in symmetric quartets, and those giving rise to dark–bright solitons in symmetric octets. We also characterize the differences between the purely even or purely odd case (in which both components are either even or odd functions of x ) and the ‘mixed parity’ cases (in which one component is even while the other is odd). Finally, we show how, in each case, the spatial symmetry yields a constraint on the possible existence of selfsymmetric eigenvalues, corresponding to stationary solitons, and we study the resulting behavior of solutions.more » « less

Abstract We study the behavior of solutions to the incompressible 2
d Euler equations near two canonical shear flows with critical points, the Kolmogorov and Poiseuille flows, with consequences for the associated Navier–Stokes problems. We exhibit a large family of new, nontrivial stationary states that are arbitrarily close to the Kolmogorov flow on the square torus in analytic regularity. This situation contrasts strongly with the setting of some monotone shear flows, such as the Couette flow: there the linearized problem exhibits an “inviscid damping” mechanism that leads to relaxation of perturbations of the base flows back to nearby shear flows. Our results show that such a simple description of the longtime behavior is not possible for solutions near the Kolmogorov flow on$$\mathbb {T}^2$$ ${T}^{2}$ . Our construction of the new stationary states builds on a degeneracy in the global structure of the Kolmogorov flow on$$\mathbb {T}^2$$ ${T}^{2}$ , and we also show a lack of correspondence between the linearized description of the set of steady states and its true nonlinear structure. Both the Kolmogorov flow on a rectangular torus and the Poiseuille flow in a channel are very different. We show that the only stationary states near them must indeed be shears, even in relatively low regularity. In addition, we show that this behavior is mirrored closely in the related Navier–Stokes settings: the linearized problems near the Poiseuille and Kolmogorov flows both exhibit an enhanced rate of dissipation. Previous work by us and others shows that this effect survives in the full, nonlinear problem near the Poiseuille flow and near the Kolmogorov flow on rectangular tori, provided that the perturbations lie below a certain threshold. However, we show here that the corresponding result cannot hold near the Kolmogorov flow on$$\mathbb {T}^2$$ ${T}^{2}$ .$${\mathbb T}^2$$ ${T}^{2}$ 
Abstract The semiclassical (small dispersion) limit of the focusing nonlinear Schrödinger equation with periodic initial conditions (ICs) is studied analytically and numerically. First, through a comprehensive set of numerical simulations, it is demonstrated that solutions arising from a certain class of ICs, referred to as “periodic single‐lobe” potentials, share the same qualitative features, which also coincide with those of solutions arising from localized ICs. The spectrum of the associated scattering problem in each of these cases is then numerically computed, and it is shown that such spectrum is confined to the real and imaginary axes of the spectral variable in the semiclassical limit. This implies that all nonlinear excitations emerging from the input have zero velocity, and form a coherent nonlinear condensate. Finally, by employing a formal Wentzel‐Kramers‐Brillouin expansion for the scattering eigenfunctions, asymptotic expressions for the number and location of the bands and gaps in the spectrum are obtained, as well as corresponding expressions for the relative band widths and the number of “effective solitons.” These results are shown to be in excellent agreement with those from direct numerical computation of the eigenfunctions. In particular, a scaling law is obtained showing that the number of effective solitons is inversely proportional to the small dispersion parameter.

BACKGROUND Landau’s Fermi liquid theory provides the bedrock on which our understanding of metals has developed over the past 65 years. Its basic premise is that the electrons transporting a current can be treated as “quasiparticles”—electronlike particles whose effective mass has been modified, typically through interactions with the atomic lattice and/or other electrons. For a long time, it seemed as though Landau’s theory could account for all the manybody interactions that exist inside a metal, even in the socalled heavy fermion systems whose quasiparticle mass can be up to three orders of magnitude heavier than the electron’s mass. Fermi liquid theory also lay the foundation for the first successful microscopic theory of superconductivity. In the past few decades, a number of new metallic systems have been discovered that violate this paradigm. The violation is most evident in the way that the electrical resistivity changes with temperature or magnetic field. In normal metals in which electrons are the charge carriers, the resistivity increases with increasing temperature but saturates, both at low temperatures (because the quantized lattice vibrations are frozen out) and at high temperatures (because the electron mean free path dips below the smallest scattering pathway defined by the lattice spacing). In “strange metals,” by contrast, no saturation occurs, implying that the quasiparticle description breaks down and electrons are no longer the primary charge carriers. When the particle picture breaks down, no local entity carries the current. ADVANCES A new classification of metallicity is not a purely academic exercise, however, as strange metals tend to be the hightemperature phase of some of the best superconductors available. Understanding hightemperature superconductivity stands as a grand challenge because its resolution is fundamentally rooted in the physics of strong interactions, a regime where electrons no longer move independently. Precisely what new emergent phenomena one obtains from the interactions that drive the electron dynamics above the temperature where they superconduct is one of the most urgent problems in physics, attracting the attention of condensed matter physicists as well as string theorists. One thing is clear in this regime: The particle picture breaks down. As particles and locality are typically related, the strange metal raises the distinct possibility that its resolution must abandon the basic building blocks of quantum theory. We review the experimental and theoretical studies that have shaped our current understanding of the emergent strongly interacting physics realized in a host of strange metals, with a special focus on their posterchild: the copper oxide hightemperature superconductors. Experiments are highlighted that attempt to link the phenomenon of nonsaturating resistivity to parameterfree universal physics. A key experimental observation in such materials is that removing a single electron affects the spectrum at all energy scales, not just the lowenergy sector as in a Fermi liquid. It is observations of this sort that reinforce the breakdown of the singleparticle concept. On the theoretical side, the modern accounts that borrow from the conjecture that strongly interacting physics is really about gravity are discussed extensively, as they have been the most successful thus far in describing the range of physics displayed by strange metals. The foray into gravity models is not just a pipe dream because in such constructions, no particle interpretation is given to the charge density. As the breakdown of the independentparticle picture is central to the strange metal, the gravity constructions are a natural tool to make progress on this problem. Possible experimental tests of this conjecture are also outlined. OUTLOOK As more strange metals emerge and their physical properties come under the scrutiny of the vast array of experimental probes now at our disposal, their mysteries will be revealed and their commonalities and differences cataloged. In so doing, we should be able to understand the universality of strange metal physics. At the same time, the anomalous nature of their superconducting state will become apparent, offering us hope that a new paradigm of pairing of nonquasiparticles will also be formalized. The correlation between the strength of the linearintemperature resistivity in cuprate strange metals and their corresponding superfluid density, as revealed here, certainly hints at a fundamental link between the nature of strange metallicity and superconductivity in the cuprates. And as the gravityinspired theories mature and overcome the challenge of projecting their powerful mathematical machinery onto the appropriate crystallographic lattice, so too will we hope to build with confidence a complete theory of strange metals as they emerge from the horizon of a black hole. Curved spacetime with a black hole in its interior and the strange metal arising on the boundary. This picture is based on the string theory gaugegravity duality conjecture by J. Maldacena, which states that some strongly interacting quantum mechanical systems can be studied by replacing them with classical gravity in a spacetime in one higher dimension. The conjecture was made possible by thinking about some of the fundamental components of string theory, namely Dbranes (the horseshoeshaped object terminating on a flat surface in the interior of the spacetime). A key surprise of this conjecture is that aspects of condensed matter systems in which the electrons interact strongly—such as strange metals—can be studied using gravity.more » « less

Great progress has been made in recent years towards understanding the properties of disordered electronic systems. In part, this is made possible by recent advances in quantum effective medium methods which enable the study of disorder and electronelectronic interactions on equal footing. They include dynamical meanfield theory and the Coherent Potential Approximation, and their cluster extension, the dynamical cluster approximation. Despite their successes, these methods do not enable the firstprinciples study of the strongly disordered regime, including the effects of electronic localization. The main focus of this review is the recently developed typical medium dynamical cluster approximation for disordered electronic systems. This method has been constructed to capture disorderinduced localization and is based on a mapping of a lattice onto a quantum cluster embedded in an effective typical medium, which is determined selfconsistently. Unlike the average effective mediumbased methods mentioned above, typical mediumbased methods properly capture the states localized by disorder. The typical medium dynamical cluster approximation not only provides the proper order parameter for Anderson localized states, but it can also incorporate the full complexity of DensityFunctional Theory (DFT)derived potentials into the analysis, including the effect of multiple bands, nonlocal disorder, and electronelectron interactions. After a brief historical review of other numerical methods for disordered systems, we discuss coarsegraining as a unifying principle for the development of translationally invariant quantum cluster methods. Together, the Coherent Potential Approximation, the Dynamical MeanField Theory and the Dynamical Cluster Approximation may be viewed as a single class of approximations with a muchneeded small parameter of the inverse cluster size which may be used to control the approximation. We then present an overview of various recent applications of the typical medium dynamical cluster approximation to a variety of models and systems, including single and multiband Anderson model, and models with local and offdiagonal disorder. We then present the application of the method to realistic systems in the framework of the DFT and demonstrate that the resulting method can provide a systematic firstprinciples method validated by experiment and capable of making experimentally relevant predictions. We also discuss the application of the typical medium dynamical cluster approximation to systems with disorder and electronelectron interactions. Most significantly, we show that in the limits of strong disorder and weak interactions treated perturbatively, that the phenomena of 3D localization, including a mobility edge, remains intact. However, the metalinsulator transition is pushed to larger disorder values by the local interactions. We also study the limits of strong disorder and strong interactions capable of producing moment formation and screening, with a nonperturbative local approximation. Here, we find that the Anderson localization quantum phase transition is accompanied by a quantumcritical fan in the energydisorder phase diagram.more » « less