Diamagnetic susceptibility obtained from the six-vertex model and its implications for the high-temperature diamagnetic state of cuprate superconductors

We study the diamagnetism of the 6-vertex model with the arrows as directed bond currents. To our knowledge, this is the first study of the diamagnetism of this model. A special version of this model, called F model, describes the thermal disordering transition of an orbital antiferromagnet, known as d-density wave (DDW), a proposed state for the pseudogap phase of the high-Tc cuprates. We find that the F model is strongly diamagnetic and the susceptibility may diverge in the high temperature critical phase with power law arrow correlations. These results may explain the surprising recent observation of a diverging low-field diamagnetic susceptibility seen in some optimally doped cuprates within the DDW model of the pseudogap phase.Comment: 4.5 pages, 2 figures, revised version accepted in Phys. Rev. Let

Self-Similar Blowup Solutions to the 2-Component Camassa-Holm Equations

In this article, we study the self-similar solutions of the 2-component Camassa-Holm equations% \begin{equation} \left\{ \begin{array} [c]{c}% \rho_{t}+u\rho_{x}+\rho u_{x}=0 m_{t}+2u_{x}m+um_{x}+\sigma\rho\rho_{x}=0 \end{array} \right. \end{equation} with \begin{equation} m=u-\alpha^{2}u_{xx}. \end{equation} By the separation method, we can obtain a class of blowup or global solutions for $\sigma=1$ or $-1$. In particular, for the integrable system with $\sigma=1$, we have the global solutions:% \begin{equation} \left\{ \begin{array} [c]{c}% \rho(t,x)=\left\{ \begin{array} [c]{c}% \frac{f\left( \eta\right) }{a(3t)^{1/3}},\text{ for }\eta^{2}<\frac {\alpha^{2}}{\xi} 0,\text{ for }\eta^{2}\geq\frac{\alpha^{2}}{\xi}% \end{array} \right. ,u(t,x)=\frac{\overset{\cdot}{a}(3t)}{a(3t)}x \overset{\cdot\cdot}{a}(s)-\frac{\xi}{3a(s)^{1/3}}=0,\text{ }a(0)=a_{0}% >0,\text{ }\overset{\cdot}{a}(0)=a_{1} f(\eta)=\xi\sqrt{-\frac{1}{\xi}\eta^{2}+\left( \frac{\alpha}{\xi}\right) ^{2}}% \end{array} \right. \end{equation} where $\eta=\frac{x}{a(s)^{1/3}}$ with $s=3t;$ $\xi>0$ and $\alpha\geq0$ are arbitrary constants.\newline Our analytical solutions could provide concrete examples for testing the validation and stabilities of numerical methods for the systems.Comment: 5 more figures can be found in the corresponding journal paper (J. Math. Phys. 51, 093524 (2010) ). Key Words: 2-Component Camassa-Holm Equations, Shallow Water System, Analytical Solutions, Blowup, Global, Self-Similar, Separation Method, Construction of Solutions, Moving Boundar

The various manifestations of collisionless dissipation in wave propagation

The propagation of an electrostatic wave packet inside a collisionless and initially Maxwellian plasma is always dissipative because of the irreversible acceleration of the electrons by the wave. Then, in the linear regime, the wave packet is Landau damped, so that in the reference frame moving at the group velocity, the wave amplitude decays exponentially with time. In the nonlinear regime, once phase mixing has occurred and when the electron motion is nearly adiabatic, the damping rate is strongly reduced compared to the Landau one, so that the wave amplitude remains nearly constant along the characteristics. Yet, we show here that the electrons are still globally accelerated by the wave packet, and, in one dimension, this leads to a non local amplitude dependence of the group velocity. As a result, a freely propagating wave packet would shrink, and, therefore, so would its total energy. In more than one dimension, not only does the magnitude of the group velocity nonlinearly vary, but also its direction. In the weakly nonlinear regime, when the collisionless damping rate is still significant compared to its linear value, this leads to an effective defocussing effect which we quantify, and which we compare to the self-focussing induced by wave front bowing.Comment: 23 pages, 6 figure

The Modulation of Multiple Phases Leading to the Modified KdV Equation

This paper seeks to derive the modified KdV (mKdV) equation using a novel approach from systems generated from abstract Lagrangians that possess a two-parameter symmetry group. The method to do uses a modified modulation approach, which results in the mKdV emerging with coefficients related to the conservation laws possessed by the original Lagrangian system. Alongside this, an adaptation of the method of Kuramoto is developed, providing a simpler mechanism to determine the coefficients of the nonlinear term. The theory is illustrated using two examples of physical interest, one in stratified hydrodynamics and another using a coupled Nonlinear Schr\"odinger model, to illustrate how the criterion for the mKdV equation to emerge may be assessed and its coefficients generated.Comment: 35 pages, 5 figure

Shock waves in strongly interacting Fermi gas from time-dependent density functional calculations

Motivated by a recent experiment [Phys. Rev. Lett. 106, 150401 (2011)] we simulate the collision between two clouds of cold Fermi gas at unitarity conditions by using an extended Thomas-Fermi density functional. At variance with the current interpretation of the experiments, where the role of viscosity is emphasized, we find that a quantitative agreement with the experimental observation of the dynamics of the cloud collisions is obtained within our superfluid effective hydrodynamics approach, where density variations during the collision are controlled by a purely dispersive quantum gradient term. We also suggest different initial conditions where dispersive density ripples can be detected with the available experimental spatial resolution.Comment: 5 pages, 4 figures, to be published in Phys. Rev.

Renormalized waves and thermalization of the Klein-Gordon equation: What sound does a nonlinear string make?

We study the thermalization of the classical Klein-Gordon equation under a u^4 interaction. We numerically show that even in the presence of strong nonlinearities, the local thermodynamic equilibrium state exhibits a weakly nonlinear behavior in a renormalized wave basis. The renormalized basis is defined locally in time by a linear transformation and the requirement of vanishing wave-wave correlations. We show that the renormalized waves oscillate around one frequency, and that the frequency dispersion relation undergoes a nonlinear shift proportional to the mean square field. In addition, the renormalized waves exhibit a Planck like spectrum. Namely, there is equipartition of energy in the low frequency modes described by a Boltzmann distribution, followed by a linear exponential decay in the high frequency modes.Comment: 13 pages, 13 figure

Orthogonality catastrophe and shock waves in a non-equilibrium Fermi gas

A semiclassical wave-packet propagating in a dissipationless Fermi gas inevitably enters a "gradient catastrophe" regime, where an initially smooth front develops large gradients and undergoes a dramatic shock wave phenomenon. The non-linear effects in electronic transport are due to the curvature of the electronic spectrum at the Fermi surface. They can be probed by a sudden switching of a local potential. In equilibrium, this process produces a large number of particle-hole pairs, a phenomenon closely related to the Orthogonality Catastrophe. We study a generalization of this phenomenon to the non-equilibrium regime and show how the Orthogonality Catastrophe cures the Gradient Catastrophe, providing a dispersive regularization mechanism. We show that a wave packet overturns and collapses into modulated oscillations with the wave vector determined by the height of the initial wave. The oscillations occupy a growing region extending forward with velocity proportional to the initial height of the packet. We derive a fundamental equation for the transition rates (MKP-equation) and solve it by means of the Whitham modulation theory.Comment: 5 pages, 1 figure, revtex4, pr

Three-layer flows in the shallow water limit

We formulate and discuss the shallow water limit dynamics of the layered flow with three layers of immiscible fluids of different densities bounded above and below by horizontal walls. We obtain a resulting system of four equations, which may be nonlocal in the non‐Boussinesq case. We provide a systematic way to pass to the Boussinesq limit, and then study those equations, which are first‐order PDEs of mixed type, more carefully. We show that in a symmetric case the solutions remain on an invariant surface and using simple waves we illustrate that this is not the case for nonsymmetric cases. Reduced models consisting of systems of two equations are also proposed and compared to the full system

Thermodynamic phase transitions and shock singularities

We show that under rather general assumptions on the form of the entropy function, the energy balance equation for a system in thermodynamic equilibrium is equivalent to a set of nonlinear equations of hydrodynamic type. This set of equations is integrable via the method of the characteristics and it provides the equation of state for the gas. The shock wave catastrophe set identifies the phase transition. A family of explicitly solvable models of non-hydrodynamic type such as the classical plasma and the ideal Bose gas are also discussed.Comment: revised version, 18 pages, 6 figure

Anomalous Scaling and Solitary Waves in Systems with Non-Linear Diffusion

We study a non-linear convective-diffusive equation, local in space and time, which has its background in the dynamics of the thickness of a wetting film. The presence of a non-linear diffusion predicts the existence of fronts as well as shock fronts. Despite the absence of memory effects, solutions in the case of pure non-linear diffusion exhibit an anomalous sub-diffusive scaling. Due to a balance between non-linear diffusion and convection we, in particular, show that solitary waves appear. For large times they merge into a single solitary wave exhibiting a topological stability. Even though our results concern a specific equation, numerical simulations supports the view that anomalous diffusion and the solitary waves disclosed will be general features in such non-linear convective-diffusive dynamics.Comment: Corrected typos, added 3 references and 2 figure