Structure of Helicity and Global Solutions of Incompressible Navier-Stokes Equation

In this paper we derive a new energy identity for the three-dimensional incompressible Navier-Stokes equations by a special structure of helicity. The new energy functional is critical with respect to the natural scalings of the Navier-Stokes equations. Moreover, it is conditionally coercive. As an application we construct a family of finite energy smooth solutions to the Navier-Stokes equations whose critical norms can be arbitrarily large.Comment: To appear in ARM

Superconductivity in a molecular graphene

We propose that constructing a molecule super-lattice on a superconducting ultrathin film is a promising way to manipulate superconductivity in experiment. We theoretically study superconductivity in a molecule graphene system, which is built by fabricating a hexagonal molecule super-lattice on 2-dimensional electron gas. The super-lattice potential dramatically changes the electron density of states, which oscillates as function of the energy. We show that such a molecular graphene may increase superconducting gap by a few times, which may open a new route to realize high temperature superconductivity.Comment: 4 pages, 4 figure

Unified Spin Order Theory via Gauge Landau-Lifshitz Equation

The continuum limit of the tilted SU(2) spin model is shown to give rise to the gauge Landau-Lifshitz equation which provides a unified description for various spin orders. For a definite gauge, we find a double periodic solution, where the conical spiral, in-plane spiral, helical, and ferromagnetic spin orders become special cases, respectively. For another gauge, we obtain the skyrmion-crystal solution. By simulating the influence of magnetic field and temperature for our covariant model, we find a spontaneous formation of skyrmion-fragment lattice and obtain a wider range of skyrmion-crystal phase in comparison to the conventional Dzyaloshinsky-Moriya model.Comment: RevTeX4, 4 pages, 3 figure

Scalable Stochastic Alternating Direction Method of Multipliers

Stochastic alternating direction method of multipliers (ADMM), which visits only one sample or a mini-batch of samples each time, has recently been proved to achieve better performance than batch ADMM. However, most stochastic methods can only achieve a convergence rate $O(1/\sqrt T)$ on general convex problems,where T is the number of iterations. Hence, these methods are not scalable with respect to convergence rate (computation cost). There exists only one stochastic method, called SA-ADMM, which can achieve convergence rate $O(1/T)$ on general convex problems. However, an extra memory is needed for SA-ADMM to store the historic gradients on all samples, and thus it is not scalable with respect to storage cost. In this paper, we propose a novel method, called scalable stochastic ADMM(SCAS-ADMM), for large-scale optimization and learning problems. Without the need to store the historic gradients, SCAS-ADMM can achieve the same convergence rate $O(1/T)$ as the best stochastic method SA-ADMM and batch ADMM on general convex problems. Experiments on graph-guided fused lasso show that SCAS-ADMM can achieve state-of-the-art performance in real application

Scaling Submodular Maximization via Pruned Submodularity Graphs

We propose a new random pruning method (called "submodular sparsification (SS)") to reduce the cost of submodular maximization. The pruning is applied via a "submodularity graph" over the $n$ ground elements, where each directed edge is associated with a pairwise dependency defined by the submodular function. In each step, SS prunes a $1-1/\sqrt{c}$ (for $c>1$) fraction of the nodes using weights on edges computed based on only a small number ($O(\log n)$) of randomly sampled nodes. The algorithm requires $\log_{\sqrt{c}}n$ steps with a small and highly parallelizable per-step computation. An accuracy-speed tradeoff parameter $c$, set as $c = 8$, leads to a fast shrink rate $\sqrt{2}/4$ and small iteration complexity $\log_{2\sqrt{2}}n$. Analysis shows that w.h.p., the greedy algorithm on the pruned set of size $O(\log^2 n)$ can achieve a guarantee similar to that of processing the original dataset. In news and video summarization tasks, SS is able to substantially reduce both computational costs and memory usage, while maintaining (or even slightly exceeding) the quality of the original (and much more costly) greedy algorithm

Superconductivity, pair density wave, and Neel order in cuprates

We investigate in underdoped cuprates possible coexistence of the superconducting (SC) order at zero momentum and pair density wave (PDW) at momentum ${\bf Q}=(\pi, \pi)$ in the presence of a Neel order. By symmetry, the $d$-wave uniform singlet pairing $dS_0$ can coexist with the $d$-wave triplet PDW $dT_{\bf Q}$, and the $p$-wave singlet PDW $pS_{\bf Q}$ can coexist with the $p$-wave uniform triplet $pT_0$. At half filling, we find the novel $pS_{\bf Q}+pT_0$ state is energetically more favorable than the $dS_0+dT_{\bf Q}$ state. At finite doping, however, the $dS_0+dT_{\bf Q}$ state is more favorable. In both types of states, the variational triplet parameters, $dT_{\bf Q}$ and $pT_0$, are of secondary significance. Our results point to a fully symmetric $\mathrm{Z_2}$ quantum spin liquid with spinon Fermi surface in proximity to the Neel order at zero doping, and to intertwined $d$-wave triplet PDW fluctuations and spin moment fluctuations along with the dominant $d$-wave singlet SC at finite doping. The results are obtained by variational quantum Monte Carlo simulations.Comment: 9 pages, 4 figures, minor typos remove

Testing the dx2−y2∗d^{*}_{x^{2}-y^{2}}-wave pairing symmetry by quasiparticle interference in BiS2_{2}-based superconductors

The quasiparticle interference (QPI) patterns in BiS$_{2}$-based superconductors are theoretically investigated by taking into account the spin-orbital coupling and assuming the recently proposed $d^{*}_{x^{2}-y^{2}}$-wave pairing symmetry. We found two distinct scattering wave vectors whose evolution can be explained based on the evolution of the constant-energy contours. The QPI spectra presented in this paper can thus be compared with future scanning tunneling microscopy experiments to test whether the pairing symmetry is $d^{*}_{x^{2}-y^{2}}$-wave in BiS$_{2}$-based superconductors.Comment: Due to the size limit, the original high-resolution figures can be asked for by emailling [email protected]

Hidden sign-changing ss-wave superconductivity in monolayer FeSe

Combining the recent scanning tunneling microscopy (STM) and angle-resolved photoemission spectroscopy (ARPES) measurements, we construct a tight-binding model suitable for describing the band structure of monolayer FeSe grown on SrTiO$_{3}$. Then we propose a possible pairing function, which can well describe the gap anisotropy observed by ARPES and has a hidden sign-changing characteristic. At last, as a test of this pairing function we further study the nonmagnetic impurity-induced bound states, to be verified by future STM experiments

Possible pairing symmetry in the FeSe-based superconductors determined by quasiparticle interference

We study the momentum-integrated quasiparticle interference (QPI) in the FeSe-based superconductors. This method was recently proposed theoretically and has been applied to determine the pairing symmetry in these materials experimentally. Our findings suggest that, if the incipient bands and the superconducting (SC) pairing on them are taken into consideration, then the experimentally measured bound states and momentum-integrated QPI can be well fitted, even if the SC order parameter does not change sign on the Fermi surfaces. Therefore, we offer an alternative explanation to the experimental data, calling for more careful identification of the pairing symmetry that is important for the pairing mechanism

A magnetic Impurity in a Weyl semimetal

We utilize the variational method to study the Kondo screening of a spin-$1/2$ magnetic impurity in a three-dimensional (3D) Weyl semimetal with two Weyl nodes along the $k_z$-axis. The model reduces to a 3D Dirac semimetal when the separation of the two Weyl nodes vanishes. When the chemical potential lies at the nodal point, $\mu=0$, the impurity spin is screened only if the coupling between the impurity and the conduction electron exceeds a critical value. For finite but small $\mu$, the impurity spin is weakly bound due to the low density of state, which is proportional to $\mu^2$, contrary to that in a 2D Dirac metal such as graphene and 2D helical metal where the density of states is proportional to $|\mu|$. The spin-spin correlation function $J_{uv}(\mathbf{r})$ between the spin $v$-component of the magnetic impurity at the origin and the spin $u$-component of a conduction electron at spatial point $\mathbf{r}$, is found to be strongly anisotropic due to the spin-orbit coupling, and it decays in the power-law. The main difference of the Kondo screening in 3D Weyl semimetals and in Dirac semimetals is in the spin $x$- ($y$-) component of the correlation function in the spatial direction of the $z$-axis.Comment: 8 pages, 5 figure