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Yuan Zhou

Publications and source records attributed to Yuan Zhou.

At least 307 records · Page 17Linked to original sources
arXiv2017

Characterizations of topological superconductors: Chern numbers, edge states and Majorana zero modes

The topological properties in topological superconductors are usually characterized by the bulk Chern numbers, edge-state spectra, and Majorana zero modes. Whether they are equivalent or inequivalent is not well understood. Here, we investigate this issue with focus on a checkerboard-lattice model combining the Chern insulator and chiral $p$-wave superconductivity. Multiple topologically superconducting phases with Chern numbers up to $\mathcal{N}=4$ are produced. We explicitly demonstrate the mismatch between the Chern numbers, edge states and Majorana zero modes in this two-dimensional topological-superconductor model. The intrinsic reason is that some edge states in the superconducting phases inherited from the Chern-insulator phase are not protected by the particle-hole symmetry. We further check the mismatches in vortex states. Our results therefore clarify these different but complementary topological features and suggest that further considerations are required to characterize various topological superconductors.

cond-mat.mes-hall↗
arXiv2017

Quantum microwave-optical interface with nitrogen-vacancy centers in diamond

We propose an efficient scheme for a coherent quantum interface between microwave and optical photons using nitrogen-vacancy (NV) centers in diamond. In this setup, an NV center ensemble is simultaneously coupled to an optical and a microwave cavity. We show that, by using the collective spin excitation modes as an intermediary, quantum states can be transferred between the microwave cavity and the optical cavity through either a double-swap scheme or a dark-state protocol. This hybrid quantum interface may provide interesting applications in single microwave photon detections or quantum information processing.

quant-ph↗
arXiv2017

Width-tuned magnetic order oscillation on zigzag edges of honeycomb nanoribbons

Quantum confinement and interference often generate exotic properties in nanostructures. One recent highlight is the experimental indication of a magnetic phase transition in zigzag-edged graphene nanoribbons at the critical ribbon width of about 7 nm [G. Z. Magda et al., Nature \textbf{514}, 608 (2014)]. Here we show theoretically that with further increase in the ribbon width, the magnetic correlation of the two edges can exhibit an intriguing oscillatory behavior between antiferromagnetic and ferromagnetic, driven by acquiring the positive coherence between the two edges to lower the free energy. The oscillation effect is readily tunable in applied magnetic fields. These novel properties suggest new experimental manifestation of the edge magnetic orders in graphene nanoribbons, and enhance the hopes of graphene-like spintronic nanodevices functioning at room temperature.

cond-mat.mes-hall↗
arXiv2017

Phase Competition and Anomalous Thermal Evolution in High-Temperature Superconductors

The interplay of competing orders is relevant to high-temperature superconductivity known to emerge upon suppression of a parent antiferromagnetic order typically via charge doping. How such interplay evolves at low temperature---in particular at what doping level the zero-temperature quantum critical point (QCP) is located---is still elusive because it is masked by the superconducting state. The QCP had long been believed to follow a smooth extrapolation of the characteristic temperature $T^*$ for the strange normal state well above the superconducting transition temperature. However, recently the $T^*$ within the superconducting dome was reported to unexpectedly exhibit back-bending likely in the cuprate Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+δ}$. Here we show that the original and revised phase diagrams can be understood in terms of weak and moderate competitions, respectively, between superconductivity and a pseudogap state such as $d$-density-wave or spin-density-wave, based on both Ginzburg-Landau theory and the realistic $t$-$t^{\prime}$-$t^{\prime\prime}$-$J$-$V$ model for the cuprates. We further found that the calculated temperature and doping-level dependence of the quasiparticle spectral gap and Raman response qualitatively agrees with the experiments. In particular, the $T^*$ back-bending can provide a simple explanation of the observed anomalous two-step thermal evolution dominated by the superconducting gap and the pseudogap, respectively. Our results imply that the revised phase diagram is likely to take place in high-temperature superconductors.

cond-mat.str-el↗
arXiv2017

Adaptive Multiple-Arm Identification

We study the problem of selecting $K$ arms with the highest expected rewards in a stochastic $n$-armed bandit game. This problem has a wide range of applications, e.g., A/B testing, crowdsourcing, simulation optimization. Our goal is to develop a PAC algorithm, which, with probability at least $1-δ$, identifies a set of $K$ arms with the aggregate regret at most $ε$. The notion of aggregate regret for multiple-arm identification was first introduced in \cite{Zhou:14} , which is defined as the difference of the averaged expected rewards between the selected set of arms and the best $K$ arms. In contrast to \cite{Zhou:14} that only provides instance-independent sample complexity, we introduce a new hardness parameter for characterizing the difficulty of any given instance. We further develop two algorithms and establish the corresponding sample complexity in terms of this hardness parameter. The derived sample complexity can be significantly smaller than state-of-the-art results for a large class of instances and matches the instance-independent lower bound upto a $\log(ε^{-1})$ factor in the worst case. We also prove a lower bound result showing that the extra $\log(ε^{-1})$ is necessary for instance-dependent algorithms using the introduced hardness parameter.

cs.LG↗
arXiv2016

On the notions of facets, weak facets, and extreme functions of the Gomory-Johnson infinite group problem

We investigate three competing notions that generalize the notion of a facet of finite-dimensional polyhedra to the infinite-dimensional Gomory-Johnson model. These notions were known to coincide for continuous piecewise linear functions with rational breakpoints. We show that two of the notions, extreme functions and facets, coincide for the case of continuous piecewise linear functions, removing the hypothesis regarding rational breakpoints. We then separate the three notions using discontinuous examples.

math.OC↗
arXiv2016

Parameterized Algorithms for Constraint Satisfaction Problems Above Average with Global Cardinality Constraints

Given a constraint satisfaction problem (CSP) on $n$ variables, $x_1, x_2, \dots, x_n \in \{\pm 1\}$, and $m$ constraints, a global cardinality constraint has the form of $\sum_{i = 1}^{n} x_i = (1-2p)n$, where $p \in (Ω(1), 1 - Ω(1))$ and $pn$ is an integer. Let $AVG$ be the expected number of constraints satisfied by randomly choosing an assignment to $x_1, x_2, \dots, x_n$, complying with the global cardinality constraint. The CSP above average with the global cardinality constraint problem asks whether there is an assignment (complying with the cardinality constraint) that satisfies more than $(AVG+t)$ constraints, where $t$ is an input parameter. In this paper, we present an algorithm that finds a valid assignment satisfying more than $(AVG+t)$ constraints (if there exists one) in time $(2^{O(t^2)} + n^{O(d)})$. Therefore, the CSP above average with the global cardinality constraint problem is fixed-parameter tractable.

cs.DS↗
arXiv2016

A quasiconformal composition problem for the Q-spaces

Given a quasiconformal mapping $f:\mathbb R^n\to\mathbb R^n$ with $n\ge2$, we show that (un-)boundedness of the composition operator ${\bf C}_f$ on the spaces $Q_α(\mathbb R^n)$ depends on the index $α$ and the degeneracy set of the Jacobian $J_f$. We establish sharp results in terms of the index $α$ and the local/global self-similar Minkowski dimension of the degeneracy set of $J_f$. This gives a solution to [Problem 8.4, 3] and also reveals a completely new phenomenon, which is totally different from the known results for Sobolev, BMO, Triebel-Lizorkin and Besov spaces. Consequently, Tukia-Väisälä's quasiconformal extension $f:\mathbb R^n\to\mathbb R^n$ of an arbitrary quasisymmetric mapping $g:\mathbb R^{n-p}\to \mathbb R^{n-p}$ is shown to preserve $Q_α (\mathbb R^n)$ for any $(α,p)\in (0,1)\times[2,n)\cup(0,1/2)\times\{1\}$. Moreover, $Q_α(\mathbb R^n)$ is shown to be invariant under inversions for all $0<α<1$.

math.FA↗
arXiv2016

Lump solutions to nonlinear partial differential equations via Hirota bilinear forms

Lump solutions are analytical rational function solutions localized in all directions in space. We analyze a class of lump solutions, generated from quadratic functions, to nonlinear partial differential equations. The basis of success is the Hirota bilinear formulation and the primary object is the class of positive multivariate quadratic functions. A complete determination of quadratic functions positive in space and time is given, and positive quadratic functions are characterized as sums of squares of linear functions. Necessary and sufficient conditions for positive quadratic functions to solve Hirota bilinear equations are presented, and such polynomial solutions yield lump solutions to nonlinear partial differential equations under the dependent variable transformations u=2(ln f)_x and u=2(ln f)_{xx}, where x is one spatial variable. Applications are made for a few generalized KP and BKP equations.

nlin.SI↗
arXiv2016

Hypercontractive inequalities via SOS, and the Frankl--Rödl graph

Our main result is a formulation and proof of the reverse hypercontractive inequality in the sum-of-squares (SOS) proof system. As a consequence we show that for any constant $0 < γ\leq 1/4$, the SOS/Lasserre SDP hierarchy at degree $4\lceil \frac{1}{4γ}\rceil$ certifies the statement "the maximum independent set in the Frankl--Rödl graph $\mathrm{FR}^{n}_γ$ has fractional size~$o(1)$". Here $\mathrm{FR}^{n}_γ = (V,E)$ is the graph with $V = \{0,1\}^n$ and $(x,y) \in E$ whenever $Δ(x,y) = (1-γ)n$ (an even integer). In particular, we show the degree-$4$ SOS algorithm certifies the chromatic number lower bound "$χ(\mathrm{FR}^{n}_{1/4}) = ω(1)$", even though $\mathrm{FR}^{n}_{1/4}$ is the canonical integrality gap instance for which standard SDP relaxations cannot even certify "$χ(\mathrm{FR}^{n}_{1/4}) > 3$". Finally, we also give an SOS proof of (a generalization of) the sharp $(2,q)$-hypercontractive inequality for any even integer $q$.

cs.CC↗
arXiv2015

Satisfiability of Ordering CSPs Above Average

We study the satisfiability of ordering constraint satisfaction problems (CSPs) above average. We prove the conjecture of Gutin, van Iersel, Mnich, and Yeo that the satisfiability above average of ordering CSPs of arity $k$ is fixed-parameter tractable for every $k$. Previously, this was only known for $k=2$ and $k=3$. We also generalize this result to more general classes of CSPs, including CSPs with predicates defined by linear inequalities. To obtain our results, we prove a new Bonami-type inequality for the Efron-Stein decomposition. The inequality applies to functions defined on arbitrary product probability spaces. In contrast to other variants of the Bonami Inequality, it does not depend on the mass of the smallest atom in the probability space. We believe that this inequality is of independent interest.

cs.DS↗
arXiv2015

A spatial compositional model (SCM) for linear unmixing and endmember uncertainty estimation

The normal compositional model (NCM) has been extensively used in hyperspectral unmixing. However, most of the previous research has focused on estimation of endmembers and/or their variability. Also, little work has employed spatial information in NCM. In this paper, we show that NCM can be used for calculating the uncertainty of the estimated endmembers with spatial priors incorporated for better unmixing. This results in a spatial compositional model (SCM) which features (i) spatial priors that force neighboring abundances to be similar based on their pixel similarity and (ii) a posterior that is obtained from a likelihood model which does not assume pixel independence. The resulting algorithm turns out to be easy to implement and efficient to run. We compared SCM with current state-of-the-art algorithms on synthetic and real images. The results show that SCM can in the main provide more accurate endmembers and abundances. Moreover, the estimated uncertainty can serve as a prediction of endmember error under certain conditions.

cs.CV↗
arXiv2015

Uniqueness of absolute minimizers for $L^\fz$-functionals involving Hamiltonians $H(x,p)$

For a bounded domain $U\subset\rn$, consider the $L^\fz$-functional involving a nonnegative Hamilton function $H:\overline U\times\rn\to [0,\fz)$. In this paper, we will establish the uniqueness of absolute minimizers $u\in W^{1,\fz}_\loc(U)\cap C(\overline U)$ for $H$, under the Dirichlet boundary value $g\in C(\partial U)$, provided \noindent (A1) $H$ is lower semicontinuous in $\overline U\times\rn$, and $H(x,\cdot)$ is convex for any $x\in\overline U$. \noindent (A2) $\displaystyle H(x,0)=\min_{p\in \rn}H(x,p)=0$ for any $ x\in \overline U$, and $\displaystyle\bigcup_{x\in \overline U}\big\{p: H(x,p)=0\big\}$ is contained in a hyperplane of $\rn$. \noindent (A3) For any $\lz>0$, there exist $\displaystyle 0 0\ \mbox{and}\ x\in \overline U.$$ This generalizes the uniqueness theorem by \cite{j93, jwy, acjs} and \cite{ksz} to a large class of Hamiltonian functions $H(x,p)$ with $x$-dependence. As a corollary, we confirm an open question on the uniqueness of absolute minimizers posed by {\cite{jwy}}. The proofs rely on geometric structure of the action function $\mathcal L_t(x,y)$ induced by $H$, and the identification of the absolute subminimality of $u$ with convexity of the Hamilton-Jacobi flow $t\mapsto T^tu(x)$

math.AP↗
arXiv2014

Hypercontractivity, Sum-of-Squares Proofs, and their Applications

We study the computational complexity of approximating the 2->q norm of linear operators (defined as ||A||_{2->q} = sup_v ||Av||_q/||v||_2), as well as connections between this question and issues arising in quantum information theory and the study of Khot's Unique Games Conjecture (UGC). We show the following: 1. For any constant even integer q>=4, a graph $G$ is a "small-set expander" if and only if the projector into the span of the top eigenvectors of G's adjacency matrix has bounded 2->q norm. As a corollary, a good approximation to the 2->q norm will refute the Small-Set Expansion Conjecture--a close variant of the UGC. We also show that such a good approximation can be obtained in exp(n^(2/q)) time, thus obtaining a different proof of the known subexponential algorithm for Small Set Expansion. 2. Constant rounds of the "Sum of Squares" semidefinite programing hierarchy certify an upper bound on the 2->4 norm of the projector to low-degree polynomials over the Boolean cube, as well certify the unsatisfiability of the "noisy cube" and "short code" based instances of Unique Games considered by prior works. This improves on the previous upper bound of exp(poly log n) rounds (for the "short code"), as well as separates the "Sum of Squares"/"Lasserre" hierarchy from weaker hierarchies that were known to require omega(1) rounds. 3. We show reductions between computing the 2->4 norm and computing the injective tensor norm of a tensor, a problem with connections to quantum information theory. Three corollaries are: (i) the 2->4 norm is NP-hard to approximate to precision inverse-polynomial in the dimension, (ii) the 2->4 norm does not have a good approximation (in the sense above) unless 3-SAT can be solved in time exp(sqrt(n) polylog(n)), and (iii) known algorithms for the quantum separability problem imply a non-trivial additive approximation for the 2->4 norm.

cs.CC↗