On the spectrum of the twisted Dolbeault Laplacian over K\"ahler manifolds

We use Dirac operator techniques to a establish sharp lower bound for the first eigenvalue of the Dolbeault Laplacian twisted by Hermitian-Einstein connections on vector bundles of negative degree over compact K\"ahler manifolds.Comment: 14 pages. Completely revised: estimates corrected and shown to be shar

Scaling properties of a ferromagnetic thin film model at the depinning transition

In this paper, we perform a detailed study of the scaling properties of a ferromagnetic thin film model. Recently, interest has increased in the scaling properties of the magnetic domain wall (MDW) motion in disordered media when an external driving field is present. We consider a (1+1)-dimensional model, based on evolution rules, able to describe the MDW avalanches. The global interface width of this model shows Family-Vicsek scaling with roughness exponent $\zeta\simeq 1.585$ and growth exponent $\beta\simeq 0.975$. In contrast, this model shows scaling anomalies in the interface local properties characteristic of other systems with depinning transition of the MDW, e.g. quenched Edwards-Wilkinson (QEW) equation and random-field Ising model (RFIM) with driving. We show that, at the depinning transition, the saturated average velocity $v_\mathrm{sat}\sim f^\theta$ vanished very slowly (with $\theta\simeq 0.037$) when the reduced force $f=p/p_\mathrm{c}-1\to 0^{+}$. The simulation results show that this model verifies all accepted scaling relations which relate the global exponents and the correlation length (or time) exponents, valid in systems with depinning transition. Using the interface tilting method, we show that the model, close to the depinning transition, exhibits a nonlinearity similar to the one included in the Kardar-Parisi-Zhang (KPZ) equation. The nonlinear coefficient $\lambda\sim f^{-\phi}$ with $\phi\simeq -1.118$, which implies that $\lambda\to 0$ as the depinning transition is approached, a similar qualitatively behaviour to the driven RFIM. We conclude this work by discussing the main features of the model and the prospects opened by it.Comment: 10 pages, 5 figures, 1 tabl

Weak and strong typicality in quantum systems

We study the properties of mixed states obtained from eigenstates of many-body lattice Hamiltonians after tracing out part of the lattice. Two scenarios emerge for generic systems: (i) the diagonal entropy becomes equivalent to the thermodynamic entropy when a few sites are traced out (weak typicality); and (ii) the von Neumann (entanglement) entropy becomes equivalent to the thermodynamic entropy when a large fraction of the lattice is traced out (strong typicality). Remarkably, the results for few-body observables obtained with the reduced, diagonal, and canonical density matrices are very similar to each other, no matter which fraction of the lattice is traced out. Hence, for all physical quantities studied here, the results in the diagonal ensemble match the thermal predictions.Comment: 6 pages, 7 figures, as publishe

Cold uniform spherical collapse revisited

We report results of a study of the Newtonian dynamics of N self-gravitating particles which start in a quasi-uniform spherical configuration, without initial velocities. These initial conditions would lead to a density singularity at the origin at a finite time when N \rightarrow \infty, but this singularity is regulated at any finite N (by the associated density fluctuations). While previous studies have focussed on the behaviour as a function of N of the minimal size reached during the contracting phase, we examine in particular the size and energy of the virialized halo which results. We find the unexpected result that the structure decreases in size as N increases, scaling in proportion to N^{-1/3}, a behaviour which is associated with an ejection of kinetic energy during violent relaxation which grows in proportion to N^{1/3}. This latter scaling may be qualitatively understood, and if it represents the asymptotic behaviour in N implies that this ejected energy is unbounded above. We discuss also tests we have performed which indicate that this ejection is a mean-field phenomenon (i.e. a result of collisionless dynamics).Comment: 10 pages, 9 figures; proceedings of "Invisible Universe" conference, Paris, July 200