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Wen Huang

Publications and source records attributed to Wen Huang.

At least 163 records · Page 9Linked to original sources
arXiv2018

Local stable and unstable sets for positive entropy $C^1$ dynamical systems

For any $C^1$ diffeomorphism on a smooth compact Riemannian manifold that admits an ergodic measure with positive entropy, a lower bound of the Hausdorff dimension for the local stable and unstable sets is given in terms of the measure-theoretic entropy and the maximal Lyapunov exponent. The mainline of our approach to this result is under the settings of topological dynamical systems, which is also applicable to infinite dimensional $C^1$ dynamical systems.

math.DS↗
arXiv2018

Spontaneous surface current in multi-component cubic superconductors with time-reversal symmetry breaking

In this work we present a comprehensive study of the spontaneous currents in time-reversal symmetry breaking (TRSB) multi-component superconductors with cubic crystalline symmetry. We argue, not limiting to cubic lattices, that spontaneous current on certain high-symmetry surfaces can exist only if the TRSB pairing simultaneously breaks a certain pair of mirror symmetries. This is shown to have exact correspondence with the Gingzburg-Landau (GL) theory and is verified by numerical Bogoliubov de-Gennes (BdG) calculations. In the course we extend the BdG to include effects of gap anisotropy and surface disorder, both of which could lead to a much suppressed current. The GL theory has been known to describe well the spontaneous current. However, we highlight a special case where it becomes less adequate, and show that a refined effective theory for low temperatures is needed. These results could shed light on the phenomenology of cubic superconductors such as U$_{1-x}$Th$_x$Be$_{13}$, the filled skutterudites PrOs$_4$Sb$_{12}$, PrPt$_4$Ge$_{12}$ and related compounds.

cond-mat.supr-con↗
arXiv2018

Blind Deconvolution by a Steepest Descent Algorithm on a Quotient Manifold

In this paper, we propose a Riemannian steepest descent method for solving a blind deconvolution problem. We prove that the proposed algorithm with an appropriate initialization will recover the exact solution with high probability when the number of measurements is, up to log-factors, the information-theoretical minimum scaling. The quotient structure in our formulation yields a simpler penalty term in the cost function compared to [LLSW16], which eases the convergence analysis and yields a natural implementation. Empirically, the proposed algorithm has better performance than the Wirtinger gradient descent algorithm and an alternating minimization algorithm in the sense that i) it needs fewer operations, such as DFTs and matrix-vector multiplications, to reach a similar accuracy, and ii) it has a higher probability of successful recovery in synthetic tests. An image deblurring problem is also used to demonstrate the efficiency and effectiveness of the proposed algorithm.

cs.IT↗
arXiv2018

Identifying the dominant pairing interaction in high-$T_c$ FeSe superconductors through Leggett modes

Heavily electron-doped and single-layer FeSe superconduct at much higher temperatures than bulk FeSe. There have been a number of proposals attempting to explain the origin of the enhanced transition temperature, including the proximity to magnetic, nematic and antiferro-orbital critical points, as well as possible strong interfacial phonon coupling in the case of single-layer FeSe. In this paper, we examine the effect of the various mechanisms in an effective two-band model. Within our model, the fluctuations associated with these instabilities contribute to different parts of the effective multiband interactions. We propose to use the collective phase fluctuation between the bands--the Leggett mode--as a tool to identify the dominant effective pairing interaction in these systems. The Leggett mode can be resolved by means of optical probes such as electronic Raman scattering. We point out that the Leggett mode in these systems, if present, shall manifest in the Raman $B_{1g}$ channel.

cond-mat.supr-con↗
arXiv2018

Superconducting pairing in Sr$_2$RuO$_4$ from weak to intermediate coupling

The unconventional superconductivity in Sr$_2$RuO$_4$ continues to attract considerable interest. While many measurements can be interpreted on the basis of chiral $p$-wave pairing with intriguing topological character, a number of exceptions hinder an unambiguous verification of such pairing. The pairing mechanism also remains under debate. In this paper, with effects of the sizable spin-orbit coupling accounted for, we reexamine the superconducting instabilities in Sr$_2$RuO$_4$ through systematic microscopic analysis within random phase approximation. Our calculations show that the odd-parity $p$-wave pairing is favored in the regime of extremely weak interactions, but that highly anisotropic even-parity pairings become most leading over a broad range of stronger interactions. These results could shed light on the nature of the enigmatic superconductivity in Sr$_2$RuO$_4$.

cond-mat.supr-con↗
arXiv2018

Entropy dimension of measure preserving systems

The notion of metric entropy dimension is introduced to measure the complexity of entropy zero dynamical systems. For measure preserving systems, we define entropy dimension via the dimension of entropy generating sequences. This combinatorial approach provides us with a new insight to analyze the entropy zero systems. We also define the dimension set of a system to investigate the structure of the randomness of the factors of a system. The notion of a uniform dimension in the class of entropy zero systems is introduced as a generalization of a K-system in the case of positive entropy. We investigate the joinings among entropy zero systems and prove the disjointness property among entropy zero systems using the dimension sets. Given a topological system, we compare topological entropy dimension with metric entropy dimension.

math.DS↗
arXiv2017

Interband interference effects at the edge of a multiband chiral p-wave superconductor

Chiral superconductors support chiral edge modes and potentially spontaneous edge currents at their boundaries. Motivated by the putative multiband chiral p-wave superconductor Sr$_2$RuO$_4$, we study the influence of the interference between different bands at the edges, which may appear in the presence of moderate edge disorder or in edge tunneling measurements. We show that interband interference can strongly modify the measurable quantities at the edges when the order parameter exhibits phase difference between the bands. This is illustrated by investigating the edge dispersion and the edge current distribution in the presence of interband mixing, as well as the conductance at a tunneling junction. The results are discussed in connection with the putative chiral p-wave superconductor Sr$_2$RuO$_4$. In passing, we also discuss similar interference effects in multiband models with other pairing symmetries.

cond-mat.supr-con↗
arXiv2017

Measure complexity and Möbius disjointness

In this paper, the notion of measure complexity is introduced for a topological dynamical system and it is shown that Sarnak's Möbius disjointness conjecture holds for any system for which every invariant Borel probability measure has sub-polynomial measure complexity. Moreover, it is proved that the following classes of topological dynamical systems $(X,T)$ meet this condition and hence satisfy Sarnak's conjecture: (1) Each invariant Borel probability measure of $T$ has discrete spectrum. (2) $T$ is a homotopically trivial $C^\infty$ skew product system on $\mathbb{T}^2$ over an irrational rotation of the circle. Combining this with the previous results it implies that the Möbius disjointness conjecture holds for any $C^\infty$ skew product system on $\mathbb{T}^2$. (3) $T$ is a continuous skew product map of the form $(ag,y+h(g))$ on $G\times \mathbb{T}^1$ over a minimal rotation of the compact metric abelian group $G$ and $T$ preserves a measurable section. (4) $T$ is a tame system.

math.DS↗
arXiv2017

Pointwise convergence of multiple ergodic averages and strictly ergodic models

By building some suitable strictly ergodic models, we prove that for an ergodic system $(X,\mathcal{X},μ, T)$, $d\in{\mathbb N}$, $f_1, \ldots, f_d \in L^{\infty}(μ)$, the averages $$\frac{1}{N^2} \sum_{(n,m)\in [0,N-1]^2} f_1(T^nx)f_2(T^{n+m}x)\ldots f_d(T^{n+(d-1)m}x) $$ converge $μ$ a.e. Deriving some results from the construction, for distal systems we answer positively the question if the multiple ergodic averages converge a.e. That is, we show that if $(X,\mathcal{X},μ, T)$ is an ergodic distal system, and $f_1, \ldots, f_d \in L^{\infty}(μ)$, then multiple ergodic averages $$\frac 1 N\sum_{n=0}^{N-1}f_1(T^nx)\ldots f_d(T^{dn}x) $$ converge $μ$ a.e.

math.DS↗
arXiv2017

Almost sure convergence of the multiple ergodic average for certain weakly mixing systems

The family of pairwise independently determined (PID) systems, i.e. those for which the independent joining is the only self joining with independent 2-marginals, is a class of systems for which the long standing open question by Rokhlin, of whether mixing implies mixing of all orders, has a positive answer. We show that in the class of weakly mixing PID one finds a positive answer for another long-standing open problem, whether the multiple ergodic averages \begin{equation*} \frac 1 N\sum_{n=0}^{N-1}f_1(T^nx)\cdots f_d(T^{dn}x), \quad N\to \infty, \end{equation*} almost surely converge.

math.DS↗
arXiv2016

ManifoldOptim: An R Interface to the ROPTLIB Library for Riemannian Manifold Optimization

Manifold optimization appears in a wide variety of computational problems in the applied sciences. In recent statistical methodologies such as sufficient dimension reduction and regression envelopes, estimation relies on the optimization of likelihood functions over spaces of matrices such as the Stiefel or Grassmann manifolds. Recently, Huang, Absil, Gallivan, and Hand (2016) have introduced the library ROPTLIB, which provides a framework and state of the art algorithms to optimize real-valued objective functions over commonly used matrix-valued Riemannian manifolds. This article presents ManifoldOptim, an R package that wraps the C++ library ROPTLIB. ManifoldOptim enables users to access functionality in ROPTLIB through R so that optimization problems can easily be constructed, solved, and integrated into larger R codes. Computationally intensive problems can be programmed with Rcpp and RcppArmadillo, and otherwise accessed through R. We illustrate the practical use of ManifoldOptim through several motivating examples involving dimension reduction and envelope methods in regression.

stat.CO↗
arXiv2016

Topological correspondence of multiple ergodic averages of nilpotent group actions

Let $(X,Γ)$ be a topological system, where $Γ$ is a nilpotent group generated by $T_1,\ldots, T_d$ such that for each $T\in Γ$, $T\neq e_Γ$, $(X,T)$ is weakly mixing and minimal. For $d,k\in \mathbb{N}$, let $p_{i,j}(n), 1\le i\le k, 1\le j\le d$ be polynomials with rational coefficients taking integer values on the integers and $p_{i,j}(0)=0$. We show that if the expressions $g_i(n)=T_1^{p_{i,1}(n)}\cdots T_d^{p_{i,d}(n)}$ depends nontrivially on $n$ for $i=1,2,\cdots,k$, and for all $i\neq j\in \{1,2,\ldots,k\}$ the expressions $g_i(n)g_j(n)^{-1}$ depend nontrivially on $n$, then there is a residual set $X_0$ of $X$ such that for all $x\in X_0$ \begin{equation*} \{(g_1(n)x, g_2(n)x,\ldots, g_k(n)x)\in X^k:n\in \mathbb{Z}\} \end{equation*} is dense in $X^k$.

math.DS↗
arXiv2016

Leggett modes and multi-band superconductivity in Sr2RuO4

Sr$_2$RuO$_4$ is a prototypical multi-band superconductor with three bands crossing the Fermi level. These bands exhibit distinct dimensional characteristics, with one quasi-2D $γ$-band and two quasi-1D $α$- and $β$-bands. This leads to the expectation that the superconductivity on the $γ$-band may be only weakly Josephson-coupled to that on the other two bands. Based on an explicit microscopic weak coupling calculation appropriate for Sr$_2$RuO$_4$, we study the collective Leggett modes associated with the relative phase oscillations between the bands and show that a relatively soft Leggett mode exists due to the comparatively weaker inter-band Josephson coupling. These calculations also provide insight into why the superconducting gap magnitudes may be comparable on all three bands, despite the noticeable differences between the $γ$ and $α, β$ bands. The analyses can be readily applied to other multi-band superconductors.

cond-mat.supr-con↗
arXiv2016

Concentration and limit behaviors of stationary measures

In this paper, we study limit behaviors of stationary measures of the Fokker-Planck equations associated with a system of ordinary differential equations perturbed by a class of multiplicative including additive white noises. As the noises are vanishing, various results on the invariance and concentration of the limit measures are obtained. In particular, we show that if the noise perturbed systems admit a uniform Lyapunov function, then the stationary measures form a relatively sequentially compact set whose weak$^*$-limits are invariant measures of the unperturbed system concentrated on its global attractor. In the case that the global attractor contains a strong local attractor, we further show that there exists a family of admissible multiplicative noises with respect to which all limit measures are actually concentrated on the local attractor; and on the contrary, in the presence of a strong local repeller in the global attractor, there exists a family of admissible multiplicative noises with respect to which no limit measure can be concentrated on the local repeller. Moreover, we show that if there is a strongly repelling equilibrium in the global attractor, then limit measures with respect to typical families of multiplicative noises are always concentrated away from the equilibrium. As applications of these results, an example of stochastic Hopf bifurcation is provided. Our study is closely related to the problem of noise stability of compact invariant sets and invariant measures of the unperturbed system.

math.DS↗
arXiv2016

Auslander-Yorke dichotomy theorem, multi-sensitivity and Lyapunov numbers

In this paper we study several stronger forms of sensitivity for continuous surjective selfmaps on compact metric spaces and relations between them. The main result of the paper states that a minimal system is either multi-sensitive or an almost one-to-one extension of its maximal equicontinuous factor, which is an analog of the Auslander-Yorke dichotomy theorem. For minimal dynamical systems, we also show that all notions of thick sensitivity, multi-sensitivity and thickly syndetical sensitivity are equivalent, and all of them are much stronger than sensitivity.

math.DS↗
arXiv2016

Dynamical compactness and sensitivity

To link the Auslander point dynamics property with topological transitivity, in this paper we introduce dynamically compact systems as a new concept of a chaotic dynamical system $(X,T)$ given by a compact metric space $X$ and a continuous surjective self-map $T:X \to X$. Observe that each weakly mixing system is transitive compact, and we show that any transitive compact M-system is weakly mixing. Then we discuss the relationships among it and other several stronger forms of sensitivity. We prove that any transitive compact system is Li-Yorke sensitive and furthermore multi-sensitive if it is not proximal, and that any multi-sensitive system has positive topological sequence entropy. Moreover, we show that multi-sensitivity is equivalent to both thick sensitivity and thickly syndetic sensitivity for M-systems. We also give a quantitative analysis for multi-sensitivity of a dynamical system.

math.DS↗
arXiv2016

Analogues of Auslander-Yorke theorems for multi-sensitivity

In this paper we study multi-sensitivity and thick sensitivity for continuous surjective selfmaps on compact metric spaces. We show that multi-sensitivity implies thick sensitivity, and the converse holds true for transitive systems. Our main result is an analog of the Auslander-Yorke dichotomy theorem: a minimal system is either multi-sensitive or an almost one-to-one extension of its maximal equicontinuous factor. Furthermore, we refine it by introducing the concept of syndetically equicontinuous points: a transitive system is either thickly sensitive or contains syndetically equicontinuous points.

math.DS↗