Spectral Analysis of Universal Conductance Fluctuations

Journal of Experimental and Theoretical Physics - Universal conductance fluctuations are usually observed in the form of aperiodic oscillations in the magnetoresistance of thin wires under...     

Evidence for Geometry-Dependent Universal Fluctuations of the Kardar-Parisi-Zhang Interfaces in Liquid-Crystal Turbulence

We provide a comprehensive report on scale-invariant fluctuations of growing interfaces in liquid-crystal turbulence, for which we recently found evidence that they belong to the Kardar-Parisi-Zhang (KPZ) universality class for 1+1 dimensions [Takeuchi and Sano in Phys. Rev. Lett. 104:230601, 2010; Takeuchi et al. in Sci. Rep. 1:34, 2011]. Here we investigate both circular and flat interfaces and report their statistics in detail. First we demonstrate that their fluctuations show not only the KPZ scaling exponents but beyond: they asymptotically share even the precise forms of the distribution function and the spatial correlation function in common with solvable models of the KPZ class, demonstrating also an intimate relation to random matrix theory. We then determine other statistical properties for which no exact theoretical predictions were made, in particular the temporal correlation function and the persistence probabilities. Experimental results on finite-time effects and extreme-value statistics are also presented. Throughout the paper, emphasis is put on how the universal statistical properties depend on the global geometry of the interfaces, i.e., whether the interfaces are circular or flat. We thereby corroborate the powerful yet geometry-dependent universality of the KPZ class, which governs growing interfaces driven out of equilibrium.     

Equivariant PT-symmetric real Chern insulators

It was understood that Chern insulators cannot be realized in the presence of PT symmetry.In this paper,we reveal a new class of PT-symmetric Chern insulators,which has internal degrees of freedom forming real representations of a symmetry group with a complex endomorphism field.As a generalization to the conventional 2n-dimensional Chern insulators with integer n≥1,these PT-symmetric Chern insulators have the n-th complex Chern number as their topological invariant,and have a Z classification given by the equivariant orthogonal K theory.Thus,in a fairly different sense,there exist ubiquitously Chern insulators with PT.symmetry.By generalizing the Thouless charge pump argument,we find that,for a PT-symmetric Chern insulator with Chern number v.there are equally many v flavors of coexisting left-and right-handed chiral modes.Chiral modes with opposite chirality are complex conjugates to each other as complex representations of the internal symmetry group,but are not isomorphic.For the physical dimensionality d=2,the PT-symmetric Chern insulators may be realized in artificial systems including photonic crystals and periodic mechanical systems.     

The Chern classes of Sobolev connections

AssumeF is the curvature (field) of a connection (potential) onR ⁴ with finiteL ² norm . We show the chern number (topological quantum number) is an integer. This generalizes previous results which showed that the integrality holds forF satisfying the Yang-Mills equations. We actually prove the natural general result in all even dimensions larger than 2.     

Second-Order Five-Dimensional Chern–Simons Gravity

Continuing our previous discussion of the canonical covariant formalism (Zandron, O. S. (in press). International Journal of Theoretical Physics), the second-order canonical fünfbein formalism of the topological five-dimensional Chern–Simons gravity is constructed. Since this gravity model naturally contains a Gauss–Bonnet term quadratic in curvature, the second-order formalism requires the implementation of the Ostrogradski transformation in order to introduce canonical momenta. This is due to the presence of second time-derivatives of the fünfbein field. By performing the space–time decomposition of the manifold M ⁵, the set of first-class constraints that determines all the Hamiltonian gauge symmetries can be found. The total Hamiltonian as generator of time evolution is constructed, and the apparent gauge degrees of freedom are unambiguously removed, leaving only the physical ones.     

Heat Kernel Expansions on the Integers

In the case of the heat equation u _t =u _xx +Vu on the real line, there are some remarkable potentials V for which the asymptotic expansion of the fundamental solution becomes a finite sum and gives an exact formula.We show that a similar phenomenon holds when one replaces the real line by the integers. In this case the second derivative is replaced by the second difference operator L ₀. We show if L denotes the result of applying a finite number of Darboux transformations to L ₀ then the fundamental solution of u _t =Lu is given by a finite sum of terms involving the Bessel function I of imaginary argument.     

Revealing Chern number from quantum metric

Chern number is usually characterized by Berry curvature. Here, by investigating the Dirac model of even-dimensional Chern insulator, we give the general relation between Berry curvature and quantum metric, which indicates that the Chern number can be encoded in quantum metric as well as the surface area of the Brillouin zone on the hypersphere embedded in Euclidean parameter space. We find that there is a corresponding relationship between the quantum metric and the metric on such a hypersphere. We give the geometrical property of quantum metric. Besides, we give a protocol to measure the quantum metric in the degenerate system.     

二维三角格点系统中的拓扑陈数

利用紧束缚模型对二维三角周期格点中各能带的陈数分布进行研究.通过严格对角化方法得到体系能量本征值和对应的本征态,再利用Kubo公式计算出量子化的霍尔电导、态密度及各扩展态对应的陈数.在傅里叶变换下将哈密顿量转换到k空间从而得到体系的能谱分布.研究表明：次近邻格点之间的跳跃积分t'的不同取值影响体系各能带对应的陈数分布,计算得到当t'=1/2时体系三个能带从低到高对应的陈数分布为{-4,5,-1},t'=-1/2时其对应陈数分布变化为{2,-4,2},而t'=±1/4时对应的陈数分布都为{2,-1,-1}.同时发现：能谱帯隙的宽度和对应霍尔平台的宽度一致,并且k空间的能带越平坦,其对应的在霍尔电导跳跃处的态密度峰就越高越尖锐,而该处霍尔电导跳跃就越陡峭.     

Transgression and the Chern character of finite-dimensional K-cycles

It is shown that the [JLO] entire cocycle of a finitely summable unbounded Fredholm module can be retracted to a periodic cocycle. Moreover, the retracted cocycle admits a zero-temperature limit, which provides the extension of the transgressed cocycle of [CM1] from the invertible case to the general case.Dedicated to Huzihiro ArakiResearch supported in part by NSF Grant DMS-9101557     

Energy bands: Chern numbers and symmetry

Energy bands formed by rotation–vibrational states of molecules in the presence of symmetry and their qualitative modifications under variation of some control parameters are studied within the semi-quantum model. Rotational variables are treated as classical whereas a finite set of vibrational states is considered as quantum. In the two-state approximation the system is described in terms of a fiber bundle with the base space being a two-dimensional sphere, the classical phase space for rotational variables. Generically this rank 2 complex vector bundle can be decomposed into two complex line bundles characterized by a topological invariant, the first Chern class. A general method of explicit calculation of Chern classes and of their possible modifications under variation of control parameters in the presence of symmetry is suggested. The construction of iso-Chern diagrams which split the space of control parameters into connected domains with fixed Chern numbers is suggested. A detailed analysis of the rovibrational model Hamiltonian for a D₃ invariant molecule possessing two vibrational states transforming according to the two-dimensional irreducible representation is done to illustrate non-trivial restrictions imposed by symmetry on possible values of Chern classes.     

Multiple Instantons Representing Higher-Order Chern–Pontryagin Classes

It has been shown in the work of Chakrabarti, Sherry and Tchrakian that the chiral SO _±(4 p) Yang–Mills theory in the Euclidean 4 p (p≥ 2) dimensions allows an axially symmetric self-dual system of equations similar to Witten's instanton equations in the classical 4-dimensional SU(2)∼SO _±(4) theory and the solutions represent a new class of instantons. However the rigorous existence of these higher-dimensional instanton solutions has remained open except for the solution of unit charge representing a single instanton. In this paper we establish an existence and uniqueness theorem for multi-instantons of arbitrary charges in the case p≥ 2. These solutions are the first known instantons, with the Chern–Pontryagin index greater than one, of the Yang–Mills model in higher dimensions. Our approach is a study of a nonlinear variational equation defined on the Poincaré half plane. Received: 20 May 1996 / Accepted: 30 April 1997     

Fast adiabatic method for measuring topological Chern number

正Topological phases play an increasingly central role in condensed matter physics[1,2]and fault-tolerant quantum computation[3].The global nature can be characterized by certain topological invariants,many among them can be defined as the integrals of some geometric quantities.A well-known example is the Chern number[4].It is the integral of Berry curvature over a surface without boundary and is thus closely related to Berry phase[5].The integer Chern number can be     

Chern numbers for spin models of transition metal nanomagnets

We argue that ferromagnetic transition metal nanoparticles with fewer than approximately 100 atoms can be described by an effective Hamiltonian with a single giant spin degree of freedom. The total spin S of the effective Hamiltonian is specified by a Berry curvature Chern number that characterizes the topologically nontrivial dependence of a nanoparticle's many-electron wave function on magnetization orientation. The Berry curvatures and associated Chern numbers have a complex dependence on spin-orbit coupling in the nanoparticle and influence the semiclassical Landau-Liftshitz equations that describe magnetization-orientation dynamics.     

Curvature on determinant bundles and first Chern forms

The Quillen–Bismut–Freed construction associates a determinant line bundle with connection to an infinite dimensional super vector bundle with a family of Dirac-type operators. We define the regularized first Chern form of the infinite dimensional bundle, and relate it to the curvature of the Bismut–Freed connection on the determinant bundle. In finite dimensions, these forms agree (up to sign), but in infinite dimensions there is a correction term, which we express in terms of Wodzicki residues.

We illustrate these results with a string theory computation. There is a natural super vector bundle over the manifold of smooth almost complex structures on a Riemannian surface. The Bismut–Freed superconnection is identified with classical Teichmüller theory connections, and its curvature and regularized first Chern form are computed.     

Chern number spins of Mn acceptor magnets in GaAs

We determine the effective total spin J of local moments formed from acceptor states bound to Mn ions in GaAs by evaluating their magnetic Chern numbers. When individual Mn atoms are close to the sample surface, the total spin changes from J=1 to J=2, due to quenching of the acceptor orbital moment. For Mn pairs in bulk, the total J depends on pair orientation in the GaAs lattice and on the separation between the Mn atoms. We point out that Berry curvature variation as a function of local moment orientation can profoundly influence the quantum-spin dynamics of these magnetic entities.