Non-vanishing Theorems for Quadratic Twists of Elliptic Curves

In this paper, we show that, by applying some results on modular symbols, for a family of certain elliptic curves defined over $\mathbb Q$, there is a large class of explicit quadratic twists whose complex $L$-series does not vanish at $s=1$, and for which the $2$-part of Birch-Swinnerton-Dyer conjecture holds.Comment: Published versio

Kibble-Zurek scaling in one-dimensional localization transitions

In this work, we explore the driven dynamics of the one-dimensional ($1$D) localization transitions. By linearly changing the strength of disorder potential, we calculate the evolution of the localization length $\xi$ and the inverse participation ratio (IPR) in a disordered Aubry-Andr\'{e} (AA) model, and investigate the dependence of these quantities on the driving rate. At first, we focus on the limit in the absence of the quasiperiodic potential. We find that the driven dynamics from both ground state and excited state can be described by the Kibble-Zurek scaling (KZS). Then, the driven dynamics near the critical point of the AA model is studied. Here, since both the disorder and the quasiperiodic potential are relevant directions, the KZS should include both scaling variables. Our present work not only extends our understanding of the localization transitions but also generalize the application of the KZS.Comment: 7 pages, 6 figure

Nonequilibrium dynamics of the localization-delocalization transition in the non-Hermitian Aubry-Andr\'{e} model

In this paper, we investigate the driven dynamics of the localization transition in the non-Hermitian Aubry-Andr\'{e} model with the periodic boundary condition. Depending on the strength of the quasi-periodic potential $\lambda$, this model undergoes a localization-delocalization phase transition. We find that the localization length $\xi$ satisfies $\xi\sim \varepsilon^{-\nu}$ with $\varepsilon$ being the distance from the critical point and $\nu=1$ being a universal critical exponent independent of the non-Hermitian parameter. In addition, from the finite-size scaling of the energy gap between the ground state and the first excited state, we determine the dynamic exponent $z$ as $z=2$. The critical exponent of the inverse participation ratio (IPR) for the $n$th eigenstate is also determined as $s=0.1197$. By changing $\varepsilon$ linearly to cross the critical point, we find that the driven dynamics can be described by the Kibble-Zurek scaling (KZS). Moreover, we show that the KZS with the same set of the exponents can be generalized to the localization phase transitions in the excited states