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Xiaodong Cao

Publications and source records attributed to Xiaodong Cao.

At least 37 records · Page 2Linked to original sources
arXiv2023

The Aubry-Andre Anderson model: Magnetic impurities coupled to a fractal spectrum

The Anderson model for a magnetic impurity in a one-dimensional quasicrystal is studied using the numerical renormalization group (NRG). The main focus is elucidating the physics at the critical point of the Aubry-Andre (AA) Hamiltonian, which exhibits a fractal spectrum with multifractal wave functions, leading to an AA Anderson (AAA) impurity model with an energy-dependent hybridization function defined through the multifractal local density of states at the impurity site. We first study a class of Anderson impurity models with uniform fractal hybridization functions that the NRG can solve to arbitrarily low temperatures. Below a Kondo scale $T_K$, these models approach a fractal strong-coupling fixed point where impurity thermodynamic properties oscillate with $\log_b T$ about negative average values determined by the fractal dimension of the spectrum. The fractal dimension also enters into a power-law dependence of $T_K$ on the Kondo exchange coupling $J_K$. To treat the AAA model, we combine the NRG with the kernel polynomial method (KPM) to form an efficient approach that can treat hosts without translational symmetry down to a temperature scale set by the KPM expansion order. The aforementioned fractal strong-coupling fixed point is reached by the critical AAA model in a simplified treatment that neglects the wave-function contribution to the hybridization. The temperature-averaged properties are those expected for the numerically determined fractal dimension of $0.5$. At the AA critical point, impurity thermodynamic properties become negative and oscillatory. Under sample-averaging, the mean and median Kondo temperatures exhibit power-law dependences on $J_K$ with exponents characteristic of different fractal dimensions. We attribute these signatures to the impurity probing a distribution of fractal strong-coupling fixed points with decreasing temperature.

cond-mat.str-el↗
arXiv2022

Rigidity of four-dimensional Kähler-Ricci solitons

In this article, we investigate four-dimensional gradient shrinking Ricci solitons close to a Kähler model. The first theorem could be considered as a rigidity result for the Kähler-Ricci soliton structure on $\mathbb{S}^2\times \mathbb{R}^2$ (in the sense of Remark 1). Moreover, we show that if the quotient of norm of the self-dual Weyl tensor and scalar curvature is close to that on a Kähler metric in a specific sense, then the gradient Ricci soliton must be either half-conformally flat or locally Kähler.

math.DG↗
arXiv2021

Curvature of the Second kind and a conjecture of Nishikawa

In this paper, we investigate manifolds for which the curvature of the second kind (following the terminology of Nishikawa) satisfies certain positivity conditions. Our main result settles Nishikawa's conjecture that manifolds for which the curvature (operator) of the second kind are positive are diffeomorphic to a sphere, by showing that such manifolds satisfy Brendle's PIC1 condition. In dimension four we show that curvature of the second kind has a canonical normal form, and use this to classify Einstein four-manifolds for which the curvature (operator) of the second kind is five-non-negative. We also calculate the normal form for some explicit examples in order to show that this assumption is sharp.

math.DG↗
arXiv2021

Continued fraction formulae involving ratios of three gamma functions

Via the MC-algorithm, in this paper we produce seven continued fraction formulae involving products and quotients of three gamma functions with three parameters, and another is an extension of Entry 34 in Chapter 12 of Ramanujan's second notebook. Five of them will be proved rigorously by the Bauer-Muir transformation. A crucial ingredient in the proofs of our five theorems is to employ the Bauer-Muir transformation twice with different nonlinear modifying factors.

math.NT↗
arXiv2021

Minimally Entangled Typical Thermal States Algorithms for Finite Temperature Matsubara Green Functions

We extend finite-temperature tensor network methods to compute Matsubara imaginary-time correlation functions, building on the minimally entangled typical thermal states (METTS) and purification algorithms. While imaginary-time correlation functions are straightforward to formulate with these methods, care is needed to avoid convergence issues that would result from naive estimators. As a benchmark, we study the single-band Anderson impurity model, even though the algorithm is quite general and applies to lattice models. The special structure of the impurity model benchmark system and our choice of basis enable techniques such as reuse of high-probability METTS for increasing algorithm efficiency. The results are competitive with state-of-the-art continuous time Monte Carlo. We discuss the behavior of computation time and error as a function of the number of purified sites in the Hamiltonian.

cond-mat.str-el↗
arXiv2020

Some mean value results related to Hardy's function

Let $ζ(s)$ and $Z(t)$ be the Riemann zeta function and Hardy's function respectively. We show asymptotic formulas for $\int_0^T Z(t)ζ(1/2+it)dt$ and $\int_0^T Z^2(t) ζ(1/2+it)dt$. Furthermore we derive an upper bound for $\int_0^T Z^3(t)χ^α(1/2+it)dt$ for $-1/2<α<1/2$, where $χ(s)$ is the function which appears in the functional equation of the Riemann zeta function: $ζ(s)=χ(s)ζ(1-s)$.

math.NT↗
arXiv2020

Average of Hardy's function at Gram points

Let $Z(t)=χ^{-1/2}(1/2+it)ζ(1/2+it)=e^{iθ(t)}ζ(1/2+it)$ be Hardy's function and $g(n)$ be the $n$-th Gram points defined by $θ(g(n))=πn$. Titchmarsh proved that $\sum_{n \leq N} Z(g(2n)) =2N+O(N^{3/4}\log^{3/4}N) $ and $\sum_{n \leq N} Z(g(2n+1)) =-2N+O(N^{3/4}\log^{3/4}N)$. We shall improve the error terms to $O(N^{1/4}\log^{3/4}N \log\log N)$.

math.NT↗
arXiv2019

Einstein four-manifolds of pinched sectional curvature

In this paper, we obtain classification of four-dimensional Einstein manifolds with positive Ricci curvature and pinched sectional curvature. In particular, the first result concerns with an upper bound of sectional curvature, improving a theorem of E. Costa. The second is a generalization of D. Yang's result assuming an upper bound on the difference between sectional curvatures.

math.DG↗
arXiv2019

Four-manifolds of Pinched Sectional Curvature

In this paper, we study closed four-dimensional manifolds. In particular, we show that under various new pinching curvature conditions (for example, the sectional curvature is no more than 5/6 of the smallest Ricci eigenvalue) then the manifold is definite. If restricting to a metric with harmonic Weyl tensor, then it must be self-dual or anti-self-dual under the same conditions. Similarly, if restricting to an Einstein metric, then it must be either the complex projective space with its Fubini-Study metric, the round sphere or their quotients. Furthermore, we also classify Einstein manifolds with positive intersection form and an upper bound on the sectional curvature.

math.DG↗
arXiv2017

A Differential Harnack Inequality for the Newell-Whitehead Equation

This paper will develop a Li-Yau-Hamilton type differential Harnack estimate for positive solutions to the Newell-Whitehead equation on $\mathbb{R}^n$. We then use our LYH-differential Harnack inequality to prove several properties about positive solutions to the equation, including deriving a classical Harnack inequality, and characterizing standing solutions and traveling wave solutions.

math.AP↗
arXiv2017

Chiral d-wave Superconductivity in a Triangular Surface Lattice Mediated by Long-range Interaction

Correlated ad-atom systems on the Si(111) surface have recently attracted an increased attention as strongly correlated systems with a rich phase diagram. We study these materials by a single band model on the triangular lattice including 1/r long-range interaction. Employing the recently proposed TRILEX method we find an unconventional superconducting phase of chiral d-wave symmetry in hole-doped systems. The superconductivity is driven simultaneously by both charge and spin fluctuations and is strongly enhanced by the long-range tail of the interaction. We provide an analysis of the relevant collective bosonic modes and explain how in triangular symmetry both charge and spin channels contribute to the Cooper-pairing.

cond-mat.supr-con↗
arXiv2017

A note on the almost one half holomorphic pinching

Motivated by a previous work of Zheng and the second named author, we study pinching constants of compact Kähler manifolds with positive holomorphic sectional curvature. In particular we prove a gap theorem following the work of Petersen and Tao on Riemannian manifolds with almost quarter-pinched sectional curvature.

math.DG↗
arXiv2017

Three-dimensional noncompact $κ$-solutions that are Type I forward and backward

As indicated by the third author in [19], there is a gap in the previous version of this paper by the first two authors [5]. We provide in this version an argument to fix the aforementioned gap. The main proposition, whose proof uses Perelman's techniques, is implied by Ding [9] and is covered by [19]. Our approach, however, is different from theirs. In addition, we prove a necessary and sufficient condition for a three-dimensional $κ$-solution to form a forward singularity. We hope that this condition is helpful in the classification of all three-dimensional $κ$-solutions. Up to now, the only main progress on such a classification, as conjectured by Perelman, is by Brendle [2].

math.DG↗
arXiv2016

Differential Harnack Estimates for Fisher's Equation

In this paper, we derive several differential Harnack estimates (also known as Li-Yau-Hamilton-type estimates) for positive solutions of Fisher's equation. We use the estimates to obtain lower bounds on the speed of traveling wave solutions and to construct classical Harnack inequalities.

math.AP↗
arXiv2015

Multiple-correction and summation of the rational series

The goal of this work is to formulate a systematical method for looking for the simple closed form or continued fraction representation of a class of rational series. As applications, we obtain the continued fraction representations for the alternating Mathieu series and some rational series. The main tools are multiple-correction and two of Ramanujan's continued fraction formulae involving the quotient of the gamma functions.

math.CA↗
arXiv2015

The fastest possible continued fraction approximations of a class of functions

The goal of this paper is to formulate a systematical method for constructing the fastest possible continued fraction approximations of a class of functions. The main tools are the multiple-correction method, the generalized Mortici's lemma and the Mortici-transformation. As applications, we will present some sharp inequalities, and the continued fraction expansions associated to the volume of the unit ball. In addition, we obtain a new continued fraction expansion of Ramanujan for a ratio of the gamma functions, which is showed to be the fastest possible. Finally, three conjectures are proposed.

math.CA↗
arXiv2015

Interfacial thermal conduction and negative temperature jump in one-dimensional lattices

We study the thermal boundary conduction in one-dimensional harmonic and $ϕ^{4}$ lattices, both of which consist of two segments coupled by a harmonic interaction. For the ballistic interfacial heat transport through the harmonic lattice, we use both theoretical calculation and molecular dynamics simulation to study the heat flux and temperature jump at the interface as to gain insights of the Kapitza resistance at the atomic scale. In the weak coupling regime, the heat current is proportional to the square of the coupling strength for the harmonic model as well as anharmonic models. Interestingly, there exists a negative temperature jump between the interfacial particles in particular parameter regimes. A nonlinear response of the boundary temperature jump to the externally applied temperature difference in the $ϕ^{4}$ lattice is observed. To understand the anomalous result, we then extend our studies to a model in which the interface is represented by a relatively small segment with gradually changing spring constants, and find that the negative temperature jump still exist. Finally, we show that the local velocity distribution at the interface is so close to the Gaussian distribution that the existence/absence of local equilibrium state seems unable to determine by numerics in this way.

cond-mat.stat-mech↗