Mobile Bipolarons in the Adiabatic Holstein-Hubbard Model in 1 and 2 dimensions

The bound states of two electrons in the adiabatic Holstein-Hubbard model are studied numerically in one and two dimensions from the anticontinuous limit. This model involves a competition between a local electron-phonon coupling (with a classical lattice) which tends to form pairs of electrons and the repulsive Hubbard interaction $U \geq 0$ which tends to break them. In 1D, the ground-state always consists in a pair of localized polarons in a singlet state. They are located at the same site for U=0. Increasing U, there is a first order transition at which the bipolaron becomes a spin singlet pair of two polarons bounded by a magnetic interaction. The pinning mode of the bipolaron soften in the vicinity of this transition leading to a higher mobility of the bipolaron which is tested numerically. In 2D, and for any $U$, the electron-phonon coupling needs to be large enough in order to form small polarons or bipolarons instead of extended electrons. We calculate the phase diagram of the bipolaron involving first order transitions lines with a triple point. A pair of polarons can form three types of bipolarons: a) on a single site at small $U$, b) a spin singlet state on two nearest neighbor sites for larger $U$ as in 1D and c) a new intermediate state obtained as the resonant combination of four 2-sites singlet states sharing a central site, called quadrisinglet. The breathing and pinning internal modes of bipolarons in 2D generally only weakly soften and thus, they are practically not mobile. On the opposite, in the vicinity of the triple point involving the quadrisinglet, both modes exhibit a significant softening. However, it was not sufficient for allowing the existence of a classical mobile bipolaron (at least in that model)

Bounds on the volume entropy and simplicial volume in Ricci curvature LpL^p bounded from below

Let $(M,g)$ be a compact manifold with Ricci curvature almost bounded from below and $\pi:\bar{M}\to M$ be a normal, Riemannian cover. We show that, for any nonnegative function $f$ on $M$, the means of f\o\pi on the geodesic balls of $\bar{M}$ are comparable to the mean of $f$ on $M$. Combined with logarithmic volume estimates, this implies bounds on several topological invariants (volume entropy, simplicial volume, first Betti number and presentations of the fundamental group) in Ricci curvature $L^p$-bounded from below

``Critical'' phonons of the supercritical Frenkel-Kontorova model: renormalization bifurcation diagrams

The phonon modes of the Frenkel-Kontorova model are studied both at the pinning transition as well as in the pinned (cantorus) phase. We focus on the minimal frequency of the phonon spectrum and the corresponding generalized eigenfunction. Using an exact decimation scheme, the eigenfunctions are shown to have nontrivial scaling properties not only at the pinning transition point but also in the cantorus regime. Therefore the phonons defy localization and remain critical even where the associated area-preserving map has a positive Lyapunov exponent. In this region, the critical scaling properties vary continuously and are described by a line of renormalization limit cycles. Interesting renormalization bifurcation diagrams are obtained by monitoring the cycles as the parameters of the system are varied from an integrable case to the anti-integrable limit. Both of these limits are described by a trivial decimation fixed point. Very surprisingly we find additional special parameter values in the cantorus regime where the renormalization limit cycle degenerates into the above trivial fixed point. At these ``degeneracy points'' the phonon hull is represented by an infinite series of step functions. This novel behavior persists in the extended version of the model containing two harmonics. Additional richnesses of this extended model are the one to two-hole transition line, characterized by a divergence in the renormalization cycles, nonexponentially localized phonons, and the preservation of critical behavior all the way upto the anti-integrable limit.Comment: 10 pages, RevTeX, 9 Postscript figure

Standing wave instabilities in a chain of nonlinear coupled oscillators

We consider existence and stability properties of nonlinear spatially periodic or quasiperiodic standing waves (SWs) in one-dimensional lattices of coupled anharmonic oscillators. Specifically, we consider Klein-Gordon (KG) chains with either soft (e.g., Morse) or hard (e.g., quartic) on-site potentials, as well as discrete nonlinear Schroedinger (DNLS) chains approximating the small-amplitude dynamics of KG chains with weak inter-site coupling. The SWs are constructed as exact time-periodic multibreather solutions from the anticontinuous limit of uncoupled oscillators. In the validity regime of the DNLS approximation these solutions can be continued into the linear phonon band, where they merge into standard harmonic SWs. For SWs with incommensurate wave vectors, this continuation is associated with an inverse transition by breaking of analyticity. When the DNLS approximation is not valid, the continuation may be interrupted by bifurcations associated with resonances with higher harmonics of the SW. Concerning the stability, we identify one class of SWs which are always linearly stable close to the anticontinuous limit. However, approaching the linear limit all SWs with nontrivial wave vectors become unstable through oscillatory instabilities, persisting for arbitrarily small amplitudes in infinite lattices. Investigating the dynamics resulting from these instabilities, we find two qualitatively different regimes for wave vectors smaller than or larger than pi/2, respectively. In one regime persisting breathers are found, while in the other regime the system rapidly thermalizes.Comment: 57 pages, 21 figures, to be published in Physica D. Revised version: Figs. 5 and 12 (f) replaced, some new results added to Sec. 5, Sec.7 (Conclusions) extended, 3 references adde

Conformal harmonic forms, Branson-Gover operators and Dirichlet problem at infinity

For odd dimensional Poincar\'e-Einstein manifolds $(X^{n+1},g)$, we study the set of harmonic $k$-forms (for k<\ndemi) which are $C^m$ (with m\in\nn) on the conformal compactification $\bar{X}$ of $X$. This is infinite dimensional for small $m$ but it becomes finite dimensional if $m$ is large enough, and in one-to-one correspondence with the direct sum of the relative cohomology H^k(\bar{X},\pl\bar{X}) and the kernel of the Branson-Gover \cite{BG} differential operators $(L_k,G_k)$ on the conformal infinity (\pl\bar{X},[h_0]). In a second time we relate the set of $C^{n-2k+1}(\Lambda^k(\bar{X}))$ forms in the kernel of $d+\delta_g$ to the conformal harmonics on the boundary in the sense of \cite{BG}, providing some sort of long exact sequence adapted to this setting. This study also provides another construction of Branson-Gover differential operators, including a parallel construction of the generalization of $Q$ curvature for forms.Comment: 35 page

On the maximum number of rational points on singular curves over finite fields

We give a construction of singular curves with many rational points over finite fields. This construction enables us to prove some results on the maximum number of rational points on an absolutely irreducible projective algebraic curve defined over Fq of geometric genus g and arithmetic genus $\pi$

Small Bipolarons in the 2-dimensional Holstein-Hubbard Model. II Quantum Bipolarons

We study the effective mass of the bipolarons and essentially the possibility to get both light and strongly bound bipolarons in the Holstein-Hubbard model and some variations in the vicinity of the adiabatic limit. Several approaches to investigate the quantum mobility of polarons and bipolarons are proposed for this model. It is found that the bipolaron mass generally remains very large except in the vicinity of the triple point of the phase diagram, where the bipolarons have several degenerate configurations at the adiabatic limit (single site (S0), two sites (S1) and quadrisinglet (QS)), while the polarons are much lighter. This degeneracy reduces the bipolaron mass significantly. The triple point of the phase diagram is washed out by the lattice quantum fluctuations which thus suppress the light bipolarons. We show that some model variations, for example a phonon dispersion may increase the stability of the (QS) bipolaron against the quantum lattice fluctuations. The triple point of the phase diagram may be stable to quantum lattice fluctuations and a very sharp mass reduction may occur, leading to bipolaron masses of the order of 100 bare electronic mass for realistic parameters. Thus such very light bipolarons could condense as a superconducting state at relatively high temperature when their interactions are not too large, that is, their density is small enough. This effect might be relevant for understanding the origin of the high Tc superconductivity of doped cuprates far enough from half filling.Comment: accepted Eur. Phys. J. B (january 2000) Ref. B960

Differentially 4-uniform functions

We give a geometric characterization of vectorial boolean functions with differential uniformity less or equal to 4

Small Bipolarons in the 2-dimensional Holstein-Hubbard Model. I The Adiabatic Limit

The spatially localized bound states of two electrons in the adiabatic two-dimensional Holstein-Hubbard model on a square lattice are investigated both numerically and analytically. The interplay between the electron-phonon coupling g, which tends to form bipolarons and the repulsive Hubbard interaction $\upsilon \geq 0$, which tends to break them, generates many different ground-states. There are four domains in the $g,\upsilon$ phase diagram delimited by first order transition lines. Except for the domain at weak electron-phonon coupling (small g) where the electrons remain free, the electrons form bipolarons which can 1) be mostly located on a single site (small $\upsilon$, large g); 2) be an anisotropic pair of polarons lying on two neighboring sites in the magnetic singlet state (large $\upsilon$, large g); or 3) be a "quadrisinglet state" which is the superposition of 4 electronic singlets with a common central site. This quadrisinglet bipolaron is the most stable in a small central domain in between the three other phases. The pinning modes and the Peierls-Nabarro barrier of each of these bipolarons are calculated and the barrier is found to be strongly depressed in the region of stability of the quadrisinglet bipolaron