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We provide generating functions, formulas, and asymptotic expressions for the number of Catalan words based on the number of runs of ascents (descents), runs of weak ascents (descents), $\ell$-valleys, valleys, symmetric valleys, $\ell$-peaks, peaks, and symmetric peaks. We also establish some bijections with restricted Dyck paths and ordered trees that transports some statistics.
Two-dimensional magnetism and valleys have recently emerged as two significant research areas, with intriguing properties and practical uses in advanced information technology. Considering the importance of these two areas and their couplings, controllable creations of both the magnetism and valley polarization are highly sought after. Based on first-principles calculations, we propose a new class of two-dimensional monolayers with a chemical formula of MAZ$_3$, which is viewed as a 2H-MZ$_2$ trilayer passivated by the A-Z bilayer on its one side. Taking VSiN$_3$ as an example, the MAZ$_3$ monolayers are found to exhibit tunable magnetism and valleys. For the intrinsic VSiN$_3$ monolayer, it is a non-magnetic semiconductor, with multiple degenerate valleys and trigonal warping near $K_\pm$ points in the band structure. Besides, the bands have spin splittings owing to the spin-orbit coupling. Under a moderate carrier doping, the monolayer becomes a Stoner ferromagnet, which enhances the spin splittings of the valence band and generates valley splittings. Moreover, the Berry curvature is valley contrasting, leading to distinct valley-spin related anomalous Hall currents as the doping
We consider a generalization of the parabolic Anderson model driven by space-time white noise, also called the stochastic heat equation, on the real line. High peaks of solutions have been extensively studied under the name of intermittency, but less is known about spatial regions between peaks, which may loosely refer to as valleys. We present two results about the valleys of the solution. Our first theorem provides information about the size of valleys and the supremum of the solution over a valley. More precisely, we show that the supremum of the solution over a valley vanishes as $t\to\infty$, and we establish an upper bound of $\exp\{-\text{const}\cdot t^{1/3}\}$ for the rate of decay. We demonstrate also that the length of a valley grows at least as $\exp\{+\text{const}\cdot t^{1/3}\}$ as $t\to\infty$. Our second theorem asserts that the length of the valleys are eventually infinite when the initial data has subgaussian tails.
A lattice path is called \emph{Delannoy} if its every step belongs to $\left\{N, E, D\right\}$, where $N=(0,1)$, $E=(1,0)$, and $D=(1,1)$ steps. \emph{Peak}, \emph{valley}, and \emph{deep valley} mean $NE$, $EN$, and $EENN$ on the lattice path, respectively. In this paper, we find a bijection between $\mathcal{P}_{n,m}(NE, EN)$ and a specific subset of ${\mathcal{P}_{n,m}}(D, EENN)$, where $\mathcal{P}_{n,m}(NE, EN)$ is the set of Delannoy paths from the origin to the points $(n,m)$ without peaks and valleys and ${\mathcal{P}_{n,m}}(D, EENN)$ is the set of Delannoy lattice paths from the origin to the points $(n,m)$ without diagonal steps and deep valleys. We also enumerate the number of Delannoy paths without peaks and valleys on the restricted region $\left\{ (x,y) \in \mathbb{Z}^2 : y \ge k x \right\}$ for a positive integer $k$.
The manipulation of the valley degree of freedom can boost the technological development of novel functional devices based on valleytronics. The current mainstream platform for valleytronics is to produce a monolayer with inversion asymmetry, in which the strain-band engineering through the substrates can serve to improve the performance of valley-based devices. However, pinpointing the effective role of strain is inevitable for the precise design of the desired valley structure. Here, we demonstrate the charge transport under continuously controllable external strain for bulk bismuth crystals with three equivalent electron valleys and one hole valley. The strain response of resistance, namely elastoresistance, exhibits the evolutions in both antisymmetric and symmetric channels with decreasing temperature. The elastoresistance behaviors mainly reflect the significant changes in valley density depending on the symmetry of induced strain, evidenced by our strain-dependent quantum oscillation measurements and first-principle band calculations under strain. These facts suggest the successful tune and evaluation of the valley populations through strain-dependent charge valley transport
The notion of symmetric and asymmetric peaks in Dyck paths was introduced by Flórez and Rodríguez, who counted the total number of such peaks over all Dyck paths of a given length. In this paper we generalize their results by giving multivariate generating functions that keep track of the number of symmetric peaks and the number of asymmetric peaks, as well as the widths of these peaks. We recover a formula of Denise and Simion as a special case of our results. We also consider the analogous but more intricate notion of symmetric valleys. We find a continued fraction expression for the generating function of Dyck paths with respect to the number of symmetric valleys and the sum of their widths, which provides an unexpected connection between symmetric valleys and statistics on ordered rooted trees. Finally, we enumerate Dyck paths whose peak or valley heights satisfy certain monotonicity and unimodality conditions, using a common framework to recover some known results, and relating our questions to the enumeration of certain classes of column-convex polyominoes.
In this paper, we focus on the enumeration of permutations by number of cyclic occurrence of peaks and valleys. We find several recurrence relations involving the number of permutations with a prescribed number of cyclic peaks, cyclic valleys, fixed points and cycles. Several associated permutation statistics and the corresponding generating functions are also studied. In particular, we establish a connection between cyclic valleys and Pell numbers as well as cyclic peaks and alternating runs.
Neural networks provide a rich class of high-dimensional, non-convex optimization problems. Despite their non-convexity, gradient-descent methods often successfully optimize these models. This has motivated a recent spur in research attempting to characterize properties of their loss surface that may explain such success. In this paper, we address this phenomenon by studying a key topological property of the loss: the presence or absence of spurious valleys, defined as connected components of sub-level sets that do not include a global minimum. Focusing on a class of two-layer neural networks defined by smooth (but generally non-linear) activation functions, we identify a notion of intrinsic dimension and show that it provides necessary and sufficient conditions for the absence of spurious valleys. More concretely, finite intrinsic dimension guarantees that for sufficiently overparametrised models no spurious valleys exist, independently of the data distribution. Conversely, infinite intrinsic dimension implies that spurious valleys do exist for certain data distributions, independently of model overparametrisation. Besides these positive and negative results, we show that, althoug
Conventionally, perturbative and non-perturbative calculations are performed independently. In this paper, valleys in the configuration space in quantum mechanics are investigated as a way to treat them in a unified manner. All the known results of the interplay of them are reproduced naturally. The prescription for separating the non-perturbative contribution from the perturbative is given in terms of the analytic continuation of the valley parameter. Our method is illustrated on a new series of examples with the asymmetric double-well potential. We obtain the non-perturbative part explicitly, which leads to the prediction of the large order behavior of the perturbative series. We calculate the first 200 perturbative coefficients for a wide range of parameters and confirm the agreement with the prediction of the valley method.
Momentum-resolved spin textures and potential valley-contrasting physical properties in the momentum space are two intriguing characteristics of noncentrosymmetric materials, and they have broad applications in spintronics and valleytronics. The realization of diverse spin textures within a single material, along with their further coupling to the valley degree of freedom, is highly desirable. Via first-principles calculations, we investigate electronic properties of Janus MP$_2$S$_3$Se$_3$ monolayers, which exhibits distinct spin textures at different valleys. While Ising-type spin textures are located at $K_\pm$ valleys, the symmetry breaking from the Janus structure brings about a coexistence of Weyl-type and Rashba-type spin textures at $Γ$ valley. In addition to valley-contrasting spin textures, valley dependence also occurs in Berry-curvature-driven anomalous Hall currents and optical selectivity. Besides, energy differences between $Γ$ and $K_\pm$, as well as band gaps, are highly tunable by applied strain. These findings present an intriguing coupling between diverse spin textures and multiple valleys, and pave the way for designing advanced electronic devices that leverage
Valleytronics harnesses the valley degree of freedom -- energy-degenerate extrema in the electronic band structure -- for information storage and processing. Valley Hall effect (VHE) is a cornerstone of valleytronics, enabling electric generation of pure valley currents. While extensively studied in systems with valleys located at time-reversal-breaking points, here, we shift the paradigm to valleytronic platforms with time-reversal-invariant valleys (TRIVs), revealing a novel phenomenon: eccentricity VHE. Unlike conventional VHE, the valley Hall angle for eccentricity VHE is an intrinsic geometric property, governed solely by the eccentricity of the valley Fermi surface, rendering it highly robust against variations in temperature or carrier density. Eccentricity VHE emerges universally across all 25 layer groups supporting TRIVs. We demonstrate these distinctive features in monolayer GeS$_{2}$ via first-principles calculations, predicting a significant valley Hall angle of 0.74. This effect can be detected through nonlocal transport measurements exhibiting characteristic scaling behavior, or, in certain cases, through valley-layer coupling. Our findings reveal a critical overlook
The selective control of specific momentum valleys lies at the core of valleytronics, a field that has thus far focused primarily on the $\mathbf{K}$ and $\mathbf{K'}$ valleys in transition metal dichalcogenides (TMDs). However, direct optical access to other low-lying yet conventionally inaccessible valleys such as the sixfold degenerate $\mathbf{Q}$ valleys has remained an outstanding challenge, fundamentally limiting the exploitation of the full valley degree of freedom for information processing. Here, we theoretically introduce an emergent light-wave valley selection rule that enables deterministic and high fidelity excitation of any single $\mathbf{Q}$ valley in monolayer TMDs. By coherently combining a circularly polarized pump pulse with a linearly polarized driver pulse, we engineer distinct quantum pathways that unambiguously excited electrons into a targeted $\mathbf{Q}$ valley, completely decoupled from the conventional $\mathbf{K}/\mathbf{K'}$ valleys. This all-optical scheme achieves near-unity ($\sim$100\%) valley polarization across an exceptionally broad ultrafast window, from the terahertz ($10^{12}$~Hz) to petahertz ($10^{15}$~Hz) regimes, enabling single $\mathb
Device stability is essential for quantum information technologies, where reliable control of electronic states is crucial. Diamond valleytronics offers a promising platform by exploiting the valley degree of freedom to store and manipulate information. In this work, we demonstrate a diamond-based valley transistor with a dual-gate, two-drain architecture that enables tunable valley-polarized transport via gate voltage modulation. By leveraging the significant effective-mass anisotropy of diamond's conduction band valleys, this architecture provides control over spatial distribution and transit times. We further demonstrate that valley-polarized transport in diamond is remarkably robust against thermal variations over macroscopic distances. These results demonstrate the resilience of valley states and highlight diamond's potential for energy-efficient valleytronic devices in next-generation quantum and high-power electronics.
Quantum electronic fluids with spin and valley degrees of freedom have a correlation driven tendency to flavor polarization (generalized ferromagnetism). To first order in the long-range Coulomb interactions -- i.e. in the Hartree-Fock approximation -- spin and valley polarization exhibit a spurious degeneracy. We show that to second order -- or more generally in the random-phase approximation -- this degeneracy is lifted in a way that depends only on the underlying symmetry relating the two valleys. In two spatial dimensions, if the valleys are related by an $n-$fold rotation ($n>2$) or by mirror reflection and each valley is invariant under $C_2$ or time reversal (as is the case in AlAs quantum wells) then valley polarization is preferred. If the valleys are related by time reversal or by $C_2$ rotation symmetry (as in multilayer graphene systems) then spin order is selected.
We reveal an instantaneous optical valley selection rule that illuminates the coupling between the instantaneous optical chirality of the driving laser field and the chirality of valley systems. Building on this principle, we propose and demonstrate that a single chirality-separated optical field, in which oppositely signed instantaneous optical chiralities are separated within an optical cycle, enables independent manipulation of currents from K and K' valleys. Based on this scheme, we highlight two key example applications: (1) complete separation of currents from different valleys, yielding 100%-purity valley-polarized currents, and (2) generation of pure valley current with zero net charge flow. Our work offers a robust and highly controllable all-optical strategy for ultrafast engineering valley currents in the optical cycle timescale, paving a new avenue for valleytronics and quantum information technologies.
Non-Hermitian physics characterized by complex band spectra has established a new paradigm in condensed matter systems and metamaterials. Recently, non-Hermitian gain and nonreciprocity are deliberately introduced to valley manipulation, leading to various phenomena beyond the Hermitian scenarios, such as the amplified topological whispering gallery modes as an acoustic laser. In contrast, pure loss is inevitable in practice and generally regarded as a detrimental factor. Here, we reveal that the coupling loss can manipulate valley degrees of freedom in a phononic metamaterial. Three distinct valley-related effects, including valley-resolved nonreciprocity that functions as a valley filter, valley-dependent skin effects where bulk states from different valleys localize at opposite boundaries, and valley-projected edge states with boundary-dependent lifetimes that leads to an anomalous beam splitting, are demonstrated through theoretical analysis and airborne sound experiments. Owing to the easy preparation of loss, our findings shed light on both non-Hermitian and valley physics and may facilitate innovative applications of valley-related devices.
The emerging field of valleytronics harnesses the valley degree of freedom of electrons, akin to how electronic and spintronic devices utilize the charge and spin degrees of freedom of electrons respectively. The engineering of valleytronic devices typically relies on the coupling between valley and other degrees of freedom such as spin, giving rise to valley-spintronics where an external magnetic field manipulates the information stored in valleys. Here, the valley gapless semiconductor is proposed as a potential electrically controlled valleytronic platform because the valley degree of freedom is coupled to the carrier type, i.e., electrons and holes. The valley degree of freedom can be electrically controlled by tuning the carrier type via the device gate voltage. We demonstrate the proposal for realizing a valley gapless semiconductor in the honeycomb lattice with the Haldane and modified Haldane models. The system's valley-carrier coupling is further studied for its transport properties in an all-electrically controlled valley filter device setting. Our work highlights the significance of the valley gapless semiconductor for valleytronic devices.
We present a Poisson/drift-diffusion model that includes valley scattering effects for simulating valley photovoltaic devices. The valley photovoltaic concept is a novel implementation of a hot-carrier solar cell and leverages the valley scattering effect under large electric field to potentially achieve high voltage and high efficiency. Fabricated devices have shown S-shaped current-voltage curves, low fill factor, and thus low efficiency. We hence develop the first device model for valley photovoltaics. Our model includes electric-field-dependent valley scattering rates extracted from previous ensemble Monte Carlo simulations. We show that the condition of nonequilibrium carrier populations in the satellite valleys is not enough for valley photovoltaics to achieve high efficiency. We also show that increasing the built-in electric field of the valley-scattering region does not improve efficiency, contrary to previous suggestion.
Topological valley photonics, which exploits valley degree of freedom to manipulate electromagnetic waves, offers a practical and effective pathway for various classical and quantum photonic applications across the entire spectrum. Current valley photonics, however, has been limited to two dimensions, which typically suffer from out-of-plane losses and can only manipulate the flow of light in planar geometries. Here, we have theoretically and experimentally developed a framework of three-dimensional (3D) topological valley photonics with a complete photonic bandgap and vectorial valley contrasting physics. Unlike the two-dimensional counterparts with a pair of valleys characterized by scalar valley Chern numbers, the 3D valley systems exhibit triple pairs of valleys characterized by valley Chern vectors, enabling the creation of vectorial bulk valley vortices and canalized chiral valley surface states. Notably, the valley Chern vectors and the circulating propagation direction of the valley surface states are intrinsically governed by the right-hand-thumb rule. Our findings reveal the vectorial nature of the 3D valley states and highlight their potential applications in 3D waveguid
We theoretically study valley-filtering in pristine graphene irradiated by bicircular counter-rotating laser drive. The dynamical symmetry of the graphene and laser drive disrupts graphene's inversion symmetry, which results distinct quasi-energy states and Floquet band occupations in the two valleys. Controlling the relative phase between the bicircular laser drive ultimately allows to blocks the contribution from one valley while allowing the opposite valley currents in the system. For practical realization of valley-based device, we propose configurational setup for valley filters and valley valve consisting of two graphene nanoribbons irradiated by two bicircular counter-rotating laser drives with a relative phase shift. It is observed that the relative phase between the two bicircular laser drives offer a control knob to generate valley-selective currents and transport responses with very high efficiency by an all-optical way. In addition, our findings about valley filter and valley valve are robust against moderate disorder and modest changes in driving laser parameters. Present work opens an avenue to realise light-based valleytronics devices in reality.