Hamiltonian Properties Research Articles

Dynamic of bifurcation, chaotic structure and multi soliton of fractional nonlinear Schrödinger equation arise in plasma physics

In this study, we examine the third-order fractional nonlinear Schrödinger equation (FNLSE) in \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(1+1)$$\\end{document}-dimensional, by employing the analytical methodology of the new extended direct algebraic method (NEDAM) alongside optical soliton solutions. In order to better understand high-order nonlinear wave behaviors in such systems, the researched model captures the physical and mathematical properties of nonlinear dispersive waves, with applications in plasma physics and optics. With the aid of above mentioned approach, we rigorously assess the novel optical soliton solutions in the form of dark, bright–dark, dark–bright, periodic, singular, rational, mixed trigonometric and hyperbolic forms. Additionally, stability assessments using conserved quantities, such as Hamiltonian property, and consistency checks were used to validate the solutions. The dynamic structure of the governing model is further examined using chaos, bifurcation, and sensitivity analysis. With the appropriate parameter values, 2D, 3D, and contour plots can all be utilized to graphically show the data. This work advances our knowledge of nonlinear wave propagation in Bose–Einstein condensates, ultrafast fibre optics, and plasma physics, among other areas with higher-order chromatic effects.

Largest and smallest eigenvalues of matrices and some Hamiltonian properties of graphs

Largest and smallest eigenvalues of matrices and some Hamiltonian properties of graphs

Largest and smallest eigenvalues of matrices and some Hamiltonian properties of graphs

Largest and smallest eigenvalues of matrices and some Hamiltonian properties of graphs

Revealing symmetries in quantum computing for many-body systems

Abstract We develop a method to deduce the symmetry properties of many-body Hamiltonians when they are prepared in Jordan-Wigner form in which they can act on multi-qubit states. 
Symmetries, such as point-group symmetries in molecules, are apparent in the standard second quantized form of the Hamiltonian.
They are, however,
masked when the Hamiltonian is translated into a Pauli matrix representation required for its operation on qubits.
To reveal these symmetries we prove a general theorem that provides a straightforward method to calculate the transformation of Pauli tensor strings under symmetry operations.
They are a subgroup of the Clifford group transformations and induce a corresponding group representation inside the symplectic matrices.
We finally give a simplified derivation of an affine qubit encoding scheme which allows for the removal of qubits due to Boolean symmetries and thus reduces effort in quantum computations
for many-body systems.

Hamiltonian Properties of Augmented k-Ary n-Cubes with Faulty Edges

The augmented [Formula: see text]-ary [Formula: see text]-cube [Formula: see text] is a recently proposed interconnection network that combines the structural characteristics of [Formula: see text]-ary [Formula: see text]-cube [Formula: see text] and augmented cube [Formula: see text], with better features. In this paper, we investigate the Hamiltonian properties of augmented [Formula: see text]-ary [Formula: see text]-cubes. We prove that the augmented [Formula: see text]-ary [Formula: see text]-cube with any [Formula: see text] (resp. [Formula: see text]) faulty edges admits a fault-free Hamilton cycle (resp. Hamilton path).

Phase diagram of the three-dimensional subsystem toric code

Subsystem quantum error-correcting codes typically involve measuring a sequence of noncommuting parity check operators. They can sometimes exhibit greater fault tolerance than conventional codes, which use commuting checks. However, unlike subspace codes, it is unclear if subsystem codes—in particular their advantages—can be understood in terms of ground-state properties of a physical Hamiltonian. In this paper, we address this question for the three-dimensional subsystem toric code (3D STC), as recently constructed by Kubica and Vasmer [], which exhibits single-shot error correction. Motivated by a conjectured relation between single-shot properties and thermal stability, we study the zero- and finite-temperature phases of an associated noncommuting Hamiltonian. By mapping the Hamiltonian model to a pair of 3D Z2 gauge theories coupled by a kinetic constraint, we find various phases at zero temperature, all separated by first-order transitions: There are 3D toric code-like phases with deconfined point-like excitations in the bulk, and there are phases with a confined bulk supporting a 2D toric code on the surface when appropriate boundary conditions are chosen. The latter is similar to the surface topological order present in 3D STC. However, the similarities between the single-shot correction in 3D STC and the confined phases are only partial: they share the same sets of degrees of freedom, but they are governed by different dynamical rules. Instead, we argue that the process of single-shot error correction can more suitably be associated with a path (rather than a point) in the zero-temperature phase diagram, a perspective, which inspires alternative measurement sequences enabling single-shot error correction. Moreover, since none of the above-mentioned phases survives at nonzero temperature, the single-shot error-correction property of the code does not imply thermal stability of the associated Hamiltonian phase. Published by the American Physical Society 2024

Sufficient conditions for hamiltonian properties of graphs based on the difference of Zagreb indices

Sufficient conditions for hamiltonian properties of graphs based on the difference of Zagreb indices

Kylin-V: An open-source package calculating the dynamic and spectroscopic properties of large systems.

Quantum dynamics simulation and computational spectroscopy serve as indispensable tools for the theoretical understanding of various fundamental physical and chemical processes, ranging from charge transfer to photochemical reactions. When simulating realistic systems, the primary challenge stems from the overwhelming number of degrees of freedom and the pronounced many-body correlations. Here, we present Kylin-V, an innovative quantum dynamics package designed for accurate and efficient simulations of dynamics and spectroscopic properties of vibronic Hamiltonians for molecular systems and their aggregates. Kylin-V supports various quantum dynamics and computational spectroscopy methods, such as time-dependent density matrix renormalization group and our recently proposed single-site and hierarchical mapping approaches, as well as vibrational heat-bath configuration interaction. In this paper, we introduce the methodologies implemented in Kylin-V and illustrate their performances through a diverse collection of numerical examples.

Exponentially improved efficient machine learning for quantum many-body states with provable guarantees

Solving the ground state and the ground-state properties of quantum many-body systems is generically a hard task for classical algorithms. For a family of Hamiltonians defined on an m-dimensional space of physical parameters, the ground state and its properties at an arbitrary parameter configuration can be predicted via a machine learning protocol up to a prescribed prediction error ɛ, provided that a sample set (of size N) of the states can be efficiently prepared and measured. In a recent work [Huang , ], a rigorous guarantee for such a generalization was proved. Unfortunately, an exponential scaling for the provable sample complexity, N=mO(1ɛ), was found to be universal for generic gapped Hamiltonians. This result applies to the situation where the dimension of the parameter space is large, while the scaling with the accuracy is not an urgent factor, not entering the realm of more precise learning and prediction. In this work, we consider an alternative relevant scenario, where the effective dimension m is a finite, not necessarily large constant, while the scaling with the prediction error becomes the central concern. By jointly preserving the fundamental properties of density matrices in the learning protocol and utilizing the continuity of quantum states in the parameter range of interest, we rigorously obtain a polynomial sample complexity for predicting quantum many-body states and their properties, with respect to the prediction error ɛ and the number of qubits, n, with N=poly(ɛ−1,n,ln1δ), where poly denotes a polynomial function, and (1−δ) is the probability of success. Moreover, if restricted to learning local quantum-state properties, the number of samples can be further reduced to N=poly(ɛ−1,lnnδ). Numerical demonstrations confirm our findings, and an alternative approach utilizing statistical learning theory with reproducing kernel Hilbert space achieves consistent results. The mere continuity assumption indicates that our results are not restricted to gapped Hamiltonian systems and properties within the same phase. Published by the American Physical Society 2024

Non-Hermitian dynamical topological winding in photonic mesh lattices.

Topological winding in non-Hermitian systems is generally associated to the Bloch band properties of lattice Hamiltonians. However, in certain non-Hermitian models, topological winding naturally arises from the dynamical evolution of the system and is related to a new form of geometric phase. Here we investigate dynamical topological winding in non-Hermitian photonic mesh lattices, where the mean survival time of an optical pulse circulating in coupled fiber loops is quantized and robust against Hamiltonian deformations. The suggested photonic model could provide an experimentally accessible platform for the observation of non-Hermitian dynamical topological windings.

On the mil graph of module over ring

Let $M$ be an unital left module over a ring $R$ with unity. We define an undirected (\textit{nil graphs}) for the module $M$ as a graph whose vertex set is $M^*=M-{0}$ and any two distinct vertices $x$ and $y$, in these graphs, are adjacent if and only if there exist $r \in R$ such that $r^{2} (x+y) = 0$ and $r(x+y)\neq 0$. In this paper, we study the graph's adjacency, diameter, radius, and eulerian and hamiltonian properties. We also defined another nil graph $\Gamma^{*}_{N} (M)$, in which we reduced the vertex set to $N(M^*)$, set of all non-zero nil elements of the module, and keep the adjacency relation same as that of $\Gamma_{N} (M)$. We investigate the adjacency, diameter, radius, eulerian and hamiltonian properties of the graph $\Gamma^{*}_{N} (\mathbb{Z}_{p^n})$ and compare these properties among both the graphs.

Higher-Order Topological In-Bulk Corner State in Pure Diffusion Systems.

Compared with conventional topological insulator that carries topological state at its boundaries, the higher-order topological insulator exhibits lower-dimensional gapless boundary states at its corners and hinges. Leveraging the form similarity between Schrödinger equation and diffusion equation, research on higher-order topological insulators has been extended from condensed matter physics to thermal diffusion. Unfortunately, all the corner states of thermal higher-order topological insulator reside within the band gap. Another kind of corner state, which is embedded in the bulk states, has not been realized in pure diffusion systems so far. Here, we construct higher-dimensional Su-Schrieffer-Heeger models based on sphere-rod structure to elucidate these corner states, which we term "in-bulk corner states." Because of the anti-Hermitian properties of diffusive Hamiltonian, we investigate the thermal behavior of these corner states through theoretical calculation, simulation, and experiment. Furthermore, we study the different thermal behaviors of in-bulk corner state and in-gap corner state. Our results would open a different gate for diffusive topological states and provide a distinct application for efficient heat dissipation.

Dynamical properties, chirped solutions, and chaotic behaviors of the extended nonlinear Schrödinger equation

In this paper, we focus on an extended nonlinear Schrödinger equation describing the pulse propagation in a nonlinear Schrödinger equation with self‐steepening and magneto‐optic effects. The existence of periodic and solitary solutions are proved based on the bifurcation method, and also, the Hamiltonian properties and the classification of its equilibrium points are obtained. The chirped solutions of the extended nonlinear Schrödinger equations are obtained by using the complete discrimination system for polynomial method, and under specific parameter conditions, three types of optical wave patterns are obtained to visualize the model. In particular, we consider the external perturbation terms to analyze the chaotic behavior of this equation.

Noval soliton solution, sensitivity and stability analysis to the fractional gKdV-ZK equation

This work examines the fractional generalized Korteweg-de-Vries-Zakharov-Kuznetsov equation (gKdV-ZKe) by utilizing three well-known analytical methods, the modified G′G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( \\frac{G^{'}}{G^2}\\right)$$\\end{document}-expansion method, 1G′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( \\frac{1}{G^{'}}\\right)$$\\end{document}-expansion method and the Kudryashov method. The gKdV-ZK equation is a nonlinear model describing the influence of magnetic field on weak ion-acoustic waves in plasma made up of cool and hot electrons. The kink, singular, anti-kink, periodic, and bright soliton solutions are observed. The effect of the fractional parameter on wave shapes have been analyzed by displaying various graphs for fractional-order values of β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document}. In addition, we utilize the Hamiltonian property to observe the stability of the attained solution and Galilean transformation for sensitivity analysis. The suggested methods can also be utilized to evaluate the nonlinear models that are being developed in a variety of scientific and technological fields, such as plasma physics. Findings show the effectiveness simplicity, and generalizability of the chosen computational approach, even when applied to complex models.

On the comaximal graph of a ring

In this paper, we prove that the study of the comaximal graph of a finite commutative ring with identity is the same as the study of the zero-divisor graph of the specially constructed lattice. Also, we introduce the blow-up of a Boolean lattice using finite chains. Furthermore, we prove that in the case of a reduced ring, the comaximal graph and the co-annihilating graph coincide. In the last section, we study the perfectness of the comaximal graphs and characterize the Hamiltonian property of the complement of a comaximal graph.

Asymptotic properties of non-coercive Hamiltonians with drift

Asymptotic properties of non-coercive Hamiltonians with drift

Large deviations for trajectory observables of diffusion processes in dimension d > 1 in the double limit of large time and small diffusion coefficient

For diffusion processes in dimension d > 1, the statistics of trajectory observables over the time-window [0,T] can be studied via the Feynman–Kac deformations of the Fokker–Planck generator, which can be interpreted as Euclidean non-Hermitian electromagnetic quantum Hamiltonians. It is interesting to compare the four regimes corresponding to the time T, either finite or large, and to the diffusion coefficient D, either finite or small. (1) For finite T and finite D, one needs to consider the full time-dependent quantum problem that involves the full spectrum of the Hamiltonian. (2) For large time T→+∞ and finite D, one only needs to consider the ground-state properties of the quantum Hamiltonian to obtain the generating function of rescaled cumulants and to construct the corresponding canonical conditioned processes. (3) For finite T and D → 0, one only needs to consider the dominant classical trajectory and its action satisfying the Hamilton–Jacobi equation, as in the semi-classical Wentzel–Kramers–Brillouin (WKB) approximation of quantum mechanics. (4) In the double limit T→+∞ and D → 0, the simplifications in the large deviations in TD of trajectory observables can be analyzed via the two orders of limits, i.e. either from the limit D → 0 of the ground-state properties of the quantum Hamiltonians of (2), or from the limit of long classical trajectories T→+∞ in the semi-classical WKB approximation of (3). This general framework is illustrated in dimension d = 2 with rotational invariance.

Eulerian and pancyclic zero-divisor graphs of ordered sets

In this paper, we determine when the zero-divisor graph of a special class of a finite pseudocomplemented poset is Eulerian. Also, we deal with Hamiltonian, vertex pancyclic, and edge pancyclic properties of the complement of a zero-divisor graph of finite 0-distributive posets. These results are applied to study the Eulerian, Hamiltonian and pancyclic properties of the zero-divisor graph, comaximal ideal graph, annihilating ideal graph, intersection ideal graph of a special class of commutative rings and intersection graph of a special class of groups.

Probing localization properties of many-body Hamiltonians via an imaginary vector potential

Probing localization properties of many-body Hamiltonians via an imaginary vector potential

Unusual isospectral factorizations of shape invariant Hamiltonians with Scarf II potential

In this paper, we search the factorizations of the shape invariant Hamiltonians with Scarf II potential. We find two classes: one of them is the standard real factorization which leads us to a real hierarchy of potentials and their energy levels; the other one is complex and it leads us naturally to a hierarchy of complex Hamiltonians. We will show some properties of these complex Hamiltonians: they are not parity-time (or PT) symmetric, but their spectrum is real and isospectral to the Scarf II real Hamiltonian hierarchy. The algebras for real and complex shift operators (also called potential algebras) are computed; they consist of su(1, 1) for each of them and the total potential algebra including both hierarchies is the direct sum su(1, 1) ⊕ su(1, 1).