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We introduce a minimal model to describe the charge degree of freedom in cuprates with the on-site Hilbert space reduced to only the three valence states CuO$_4^{7-,6-,5-}$ (nominally Cu$^{1+,2+,3+}$) and make use of the S=1 pseudospin formalism. The formalism constitutes a powerful method to study complex phenomena in interacting quantum systems characterized by the coexistence and competition of various ordered states. Overall, such a framework provides a simple and systematic methodology to predict and discover new kinds of orders. In particular, the pseudospin formalism provides the most effective way to describe different topological structures, in particular, due to a possibility of a geometrical two-vector description of the on-site states. We introduce and analyze effective pseudospin Hamiltonian with on-site and inter-site charge correlations, two types of a correlated one-particle transfer and two-particle, or the composite boson transfer. The 2D S=1 pseudospin system is prone to a creation of different topological structures, which form topologically protected inhomogeneous distributions of the eight local S=1 pseudospin order parameters. We present a short overview of localized topological structures, typical for S=1 (pseudo)spin systems, focusing on unexpected antiphase domain walls in parent cuprates and so-called quadrupole skyrmion, which are believed to be candidates for a topological charge excitation in parent or underdoped cuprates. Puzzlingly, these unconventional structures can be characterized by an uniform distribution of the mean on-site charge, that makes these invisible for X-rays. Quasiclassical approximation and computer simulation are applied to analyze localized topological defects and evolution of the domain structures in "negative-$U$" model under charge order-superfluid phase transition., 25 pages, 7 figures. arXiv admin note: text overlap with arXiv:1409.5064

In this chapter, we look at the energy of electrons in an atom and determine how the quantum state of the atom changes over time. This leads us to introduce the energy operator, or Hamiltonian, and the Schrodinger equation that governs the behaviour of quantum systems. We show how light can interact with an atom, and as a demonstration of the quantum theory developed so far we explain how atomic clocks work.

We investigate the long-range, two-body interactions between rubidium and potassium atoms in highly excited ($n=70$) Rydberg states. After establishing properly symmetrized asymptotic basis states, we diagonalize an interaction Hamiltonian consisting of the standard Coulombic potential expansion and atomic fine structure to calculate electronic potential energy curves. We find that when both atoms are excited to either the $70s$ state or the $70p$ state, both the $\Omega=0+$ symmetry interactions and the $\Omega=0-$ symmetry interactions demonstrate a deep potential well capable of supporting many bound levels; the size of the corresponding dimer states are on the order of 2.25 $\mu$m. We estabish $n$-scaling relations for the equilibrium separation $R_e$ and the dissociation energy $D_e$ and find these relations to be consistent with similar calculations involving the homonuclear interactions between rubidium and cesium. We discuss the specific effects of $\ell$-mixing and the exact composition of the calculated potential well \textit{via} the expansion coefficients of the asymptotic basis states. Finally, we apply a Landau-Zener treatment to show that the dimer states are stable with respect to predissociation., Comment: 8 pages, 4 figures, 4 tables

In the current research, the structural parameters of MgxCd1−xX (X = S, Se, Te) ternary semiconductor compounds are determined. Structures of binary compounds are also defined, using the computational approach in zinc blende (B3) structure. For determination of structural parameters, the first principles calculations for energy optimization have been done. As the concentration of Mg dopant goes up, the parameter lattice constant and bulk modulus decreases. The Linear Combination of Atomic Orbital method was used within DFT using CRYSTAL code. The Kohn-Sham Hamiltonian was being constructed using HYBRID B3LYP scheme. The structural parameters are studied in terms of bulk modulus and lattice constant. All the observations, graphs and obtained values are in good coordination with the previously done work.

In this paper we study the effect of solvent potential interaction on the dynamics of Peyrard Bishop model of DNA. The potential energy in the Hamiltonian arises from the morse potential beacause of hydrogen bonding and solvent potential due to environmental interaction. We invoke the semi discrete approximation and the dynamics is found to be governed by the Nonlinear Schrodinger equation(NLS). The propagation of envelope soliton in DNA is analyzed for various contributing factors.

We study the nanoscale magnetization evolution of a one dimensional ferromagnetic nanowire with the inclusion of next-nearest-neighbor interaction and Dzyaloshinkii-Moriya (DM) interaction. The system is modelled using Heisenberg Hamiltonian in the quasi classical approximation of spin operators. The dynamics of the system is found to be governed by generalized discrete nonlinear Schrodinger equation. Further we investigate analytically the modulational instability in ferromagnetic nanowire system, where the effect of next-nearest neighbour (NNN) interaction and DM interaction are taken into an account to show graphically the effect of interactions on Modulational Instability (MI) gain spectra.

The inhomogeneous magneto-electro-elastic (MEE) coupling element-free Galerkin method (IMC-EFGM) is proposed for solving static behaviors of MEE structures. Plates made of homogeneous or inhomogeneous MEE materials are analyzed by IMC-EFGM, and the mechanical behaviors in different temperature fields are simulated. Displacement, magnetic potential and electric potential are studied using moving least squares (MLS) approximation. For inhomogeneous MEE materials, the constitutive equation in the MEE-thermal environment is first obtained by using Gibbs energy, and then used together with the Hamiltonian principle to determine the system governing equation. Since the MLS approximation does not satisfy the principle of Kronecker delta, the penalty function method is used to impose the natural boundary. The value of material properties at the point of integration is taken into account when calculating inhomogeneous materials. Compared with the analytical solution and finite element method (FEM), IMC-EFGM is straightforward and has greater precision than FEM. Some numerical examples are shown for the static FG-MEE plate with different hole shapes. Homogeneous MEE materials and inhomogeneous MEE materials are mainly used in intelligent structures and have good application perspectives. This research will promote the future use of inhomogeneous MEE materials.

The results of the calculation on magnetization (M) for diluted magnetic semiconductor Pb1-xEuxSe are reported. Our calculation includes two distinct contributions namely contribution from the localized magnetic impurity (Mi) and contribution from the band effect (Mb). Mi has been calculated using a three-spin cluster model which comprises the clusters consisting of one-spin (M1), two-spins (M2) and three-spins (M3). Moreover, M3 includes both open triplets (Mot) and closed triplets (Mct). The statistical distribution of the different clusters has been considered using random distribution approximation in the dilute limit. The contributions from these clusters to the magnetization have been evaluated using a modified Heisenberg's exchange Hamiltonian in the nearest approximation. The magnetization has been calculated as a function of magnetic field and impurity concentration. The band effect includes the contribution from lattice diamagnetism( χ l a t t ) . This mechanism has been considered in the effective mass formalism using a two-band k → . π → model. Our calculated values of M compare very well with experimental data where available.

This paper focuses on the study of ground states and nontrivial solutions for the following Hamiltonian elliptic system: { − Δ u + V ( x ) u = f 1 ( x , v ) , x ∈ R 2 , − Δ v + V ( x ) v = f 2 ( x , u ) , x ∈ R 2 , where V ∈ C ( R 2 , ( 0 , ∞ ) ) and f 1 , f 2 : R 2 × R → R have critical exponential growth. The strongly indefinite features together with the critical exponent bring some new difficulties in our analysis. In this paper, we develop a direct approach and use an approaching argument to seek Cerami sequences for the energy functional and estimate the minimax levels of such sequences. In particular, under some general assumptions imposed on the nonlinearity f i , we obtain the existence of ground states and nontrivial solutions for the above system as well as the following system in bounded domain, { − Δ u = f 1 ( v ) , x ∈ Ω , − Δ v = f 2 ( u ) , x ∈ Ω , u = 0 , v = 0 , x ∈ ∂ Ω . Our results improve and extend the related results of de Figueiredo-do O-Ruf (2004) [20] ; (2011) [21] , of Lam-Lu (2014) [30] , and of de Figueiredo-do O-Zhang (2020) [22] .

The combination of free energy simulations in the alchemical and configurational spaces provides a feasible route to access the thermodynamic profiles under a computationally demanding target Hamiltonian. Normally, due to the significant differences between the computational cost of ab initio quantum mechanics (QM) calculations and those of semi-empirical quantum mechanics (SQM) and molecular mechanics (MM), this indirect method could be used to obtain the QM thermodynamics by combining the SQM or MM results and the SQM-to-QM or MM-to-QM corrections. In our previous works, a multi-dimensional nonequilibrium pulling framework for Hamiltonian variations has been introduced based on bidirectional pulling and bidirectional reweighting. The method performs nonequilibrium free energy simulations in the configurational space to obtain the thermodynamic profile along the conformational change pathway under a selected computationally efficient Hamiltonian, and uses the nonequilibrium alchemical method to correct or perturb the thermodynamic profile to that under the target Hamiltonian. The BAR-based method is designed to achieve the best generality and transferability and thus leads to modest (~20 folds) speedup. In this work, we explore the possibility of further accelerating the nonequilibrium free energy simulation by employing unidirectional pulling and using the selection criterion to obtain the initial configurations used to initiate nonequilibrium trajectories following the idea of adaptive steered molecular dynamics (ASMD). A single initial condition is used to seed the whole multi-dimensional nonequilibrium free energy simulation and the sampling is performed fully in the nonequilibrium ensemble. Introducing very short ps-length equilibrium sampling to grab more initial seeds could also be helpful. The ASMD scheme estimates the free energy difference with the unidirectional exponential average (EXP), but it does not follow exactly the requirements of the EXP estimator. Another deficiency of the seeding simulation is the inherently sequential or serial pulling due to the inter-segment dependency, which triggers some problems in the parallelizability of the simulation. Numerical tests are performed to grasp some insights and guidelines for using this selection-criterion-based ASMD scheme. The presented selection-criterion-based multi-dimensional ASMD scheme follows the same perturbation network of the BAR-based method, and thus could be used in various Hamiltonian-variation cases.

By a bi-variational system we mean any system of equations generated by two different Hamiltonian actions. A connection between their variational symmetries is established. The effective use of the nonclassical Hamiltonian actions for the construction of approximate solutions with the high accuracy for the given dissipative problem is demonstrated. We also investigate the potentiality of the given operator equation with the second-order time derivative, construct the corresponding functional and find necessary and sufficient conditions for the operator S to be a generator of symmetry of the constructed functional. Theoretical results are illustrated by some examples.

International audience; Many-particle time domain methods are rising alternatives to particle-in-cell (PIC) or frequency methods to simulate the wave–beam interactions in traveling wave tubes (TWTs). We focus on two of those: our Hamiltonian discrete model DIMOHA is compared analytically against the pseudospectral method RUBEUS. Although based on two completely different approaches—the Gel’fand transform for DIMOHA and the telegraphist circuit for RUBEUS—we surprisingly find out that they share perfectly parallel sets of equations and variables. However, we conclude that DIMOHA is more flexible than RUBEUS in terms of pitch tapering and absorbing boundary conditions. It also shows excellent stability for steady-state simulation, allowing us to explain some discrepancies of RUBEUS with experimental results. These come from a standing wave pattern which is detectable in the vicinity of the sever. Finally, DIMOHA is tested for the first time with ultra-short pulses and exhibits excellent agreement with RUBEUS.

Given a linear operator T on a Hilbert space, the pseudo-point spectrum, the pseudo-residual spectrum, and the pseudo-continuous spectrum of T are defined, and some of their properties are studied....

The emergence of long-range order at low temperatures in atomistic systems with continuous symmetry is a fundamental, yet poorly understood phenomenon in Physics. To address this challenge we study a discrete microscopic model for an elastic crystal with dislocations in three dimensions, previously introduced by Ariza and Ortiz. The model is rich enough to support some realistic features of three-dimensional dislocation theory, most notably grains and the Read-Shockley law for grain boundaries, which we rigorously derive in a simple, explicit, geometry. We analyze the model at positive temperatures, in terms of a Gibbs distribution with energy function given by the Ariza-Ortiz Hamiltonian plus a contribution from the dislocation cores. Our main result is that the model exhibits long range positional order at low temperatures. The proof is based on the tools of discrete exterior calculus, together with cluster expansion techniques., 54 pages, 8 figures. Final version accepted for publication on JEMS. Compared to version v2: minor corrections and new latex format

In quantum field theory it is believed that the spontaneous decay of excited atomic or molecular level is due to the interaction with continuum of field modes. Besides, the atom makes a transition from upper level to lower one so that the probability to find the atom in the excited state tends to zero. In this paper it will be shown that the mathematical model in single-photon approximation may predict another behavior of this probability generally. Namely, the probability to find the atom in the excited state may tend to a nonzero constant so that the atom is not in the pure state finally. This effect is due to that the spectrum of the complete Hamiltonian is not purely absolutely continuous and has a discrete level outside the continuous part. Namely, we state that in the corresponding invariant subspace, determining the time evolution, the spectrum of the complete Hamiltonian when the field is considered in three dimensions may be not purely absolutely continuous and may have an eigenvalue. The appearance of eigenvalue has a threshold character. If the field is considered in two dimensions the spectrum always has an eigenvalue and the decay is absent.

We calculate the conductance through strongly correlated T-shaped molecular or quantum dot systems under the influence of phonons. The system is modelled by the extended Anderson-Holstein Hamiltonian. The finite-U mean-field slave boson approach is used to study many-body effects. Phonons influence both interference and correlations. Depending on the dot unperturbed energy and the strength of electron-phonon interaction the system is occupied by a different number of electrons which effectively interact with each other repulsively or attractively. This leads together with the interference effects to different spin or charge Fano-Kondo effects, Comment: 16 pages, 11 figures

This article is devoted to the study of a two-dimensional $(2D)$ quasi-periodically forced beam equation $ u_{tt}+\Delta^2 u+ \varepsilon\phi(t)(u+{u}^3) = 0, \quad x\in\mathbb{T}^2, \quad t\in\mathbb{R} $ under periodic boundary conditions, where $\varepsilon$ is a small positive parameter, $\phi(t)$ is a real analytic quasi-periodic function in $t$ with frequency vector $\omega = (\omega_1, \omega_2 \ldots, \omega_m)$. We prove that the equation possesses a Whitney smooth family of small-amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional Hamiltonian system. The proof is based on an infinite dimensional KAM theorem and Birkhoff normal form. By solving the measure estimation of infinitely many small divisors, we construct a symplectic coordinate transformation which can reduce the linear part of Hamiltonian system to constant coefficients. And we construct some conversion of coordinates which can change the Hamiltonian of the equation into some Birkhoff normal form depending sparse angle-dependent terms, which can be achieved by choosing the appropriate tangential sites. Lastly, we prove that there are many quasi-periodic solutions for the above equation via an abstract KAM theorem.

In a dissipative quantum system, we report the dynamic mode decomposition (DMD) analysis of damped oscillation signals. We used a reflection-type pump-probe method to observe time-domain signals, including the coupled modes of long-lived longitudinal optical phonons and quickly damped plasmons (LOPC) at various pump powers. The Fourier transformed spectra of the observed damped oscillation signals show broad and asymmetric modes, making it difficult to evaluate their frequencies and damping rates. We then used DMD to analyze the damped oscillation signals by precisely determining their frequencies and damping rates. We successfully identified the LOPC modes. The obtained frequencies and damping rates were shown to depend on the pump power, which implies photoexcited carrier density. We compared the pump-power dependence of the frequencies and damping rates of the LOPC modes with the carrier density dependence of the complex eigen-energies of the coupled modes by using the non-Hermitian phenomenological effective Hamiltonian. Good agreement was obtained between the observed and calculated dependences, demonstrating that DMD is an effective alternative to Fourier analysis which often fails to estimate effective damping rates.

Exchange anisotropy effects in equal-spin s(1) = s(2) dimeric single-molecule magnets exhibiting C2 point- group symmetry have been studied. The Hamiltonian, which is written in two-spin 4 × 4 Pauli matrixes representation in the appropriate noncollinear local coordinates systems, has been transformed to common coordinates and used to derive energy levels and spin eigenstates. An anomalous variation of the magnetic system average spin depending on the misalignment of the local anisotropy axes has been found. Particularly, a fully nonmagnetic state of an s = 1/2 spin dimer and “silent” EPR mode has been predicted. The angular boundaries of the EPR existence have been determined for any possible mutual deflections of the local ion axes for each of the 6 possible transitions between the levels of the spin system, both in zero and in a nonzero magnetic field.

We compute the next-to-leading order impact factor for inclusive dijet production in deeply inelastic electron-nucleus scattering at small $x_{\rm Bj}$. Our computation, performed in the framework of the Color Glass Condensate effective field theory, includes all real and virtual contributions in the gluon shock wave background of all-twist lightlike Wilson line correlators. We demonstrate explicitly that the rapidity evolution of these correlators, to leading logarithmic accuracy, is described by the JIMWLK Hamiltonian. When combined with the next-to-leading order JIMWLK Hamiltonian, our results for the impact factor improve the accuracy of the inclusive dijet cross section to $\mathcal{O}(\alpha_s^2 \ln(x_f/x_{\rm Bj}))$, where $x_f$ is a rapidity factorization scale. These results are an essential ingredient in assessing the discovery potential of inclusive dijets to uncover the physics of gluon saturation at the Electron-Ion Collider., Comment: v3: typos corrected post-publication, a list of these will be published separately as an Erratum. v2: matches the published version. v1: 112 pages, 11 figures, 4 tables

Structural DNA nanotechnology is a promising approach to create chromophore networks with modular structures and Hamiltonians to control the material's functions. The functional behaviors of these systems depend on the interactions of the chromophores' vibronic states, as well as interactions with their environment. To optimize their functions, it is necessary to characterize the chromophore network's structural and energetic properties, including the electronic delocalization in some cases. In this study, parameters of interest are deduced in DNA-scaffolded Cyanine 3 and Cyanine 5 dimers. The methods include steady-state optical measurements, physical modeling, and a genetic algorithm approach. The parameters include the chromophore network's vibronic Hamiltonian, molecular positions, transition dipole orientations, and environmentally induced energy broadening. Additionally, the study uses temperature-dependent optical measurements to characterize the spectral broadening further. These combined results reveal the quantum mechanical delocalization, which is important for functions like coherent energy transport and quantum information applications.

In this paper, we show a q-deformed spin $-\frac {1}{2}$ Haldane-Shastry (HS) chain with the symmetry of quantum group algebra, and find that the Hamiltonian of the system can be constructed from the Temperley-Lieb (TL) algebra generator. Based on this, it is shown that the topological basis states are the two eigenstaes of the four-qubit q-deformed spin $-\frac {1}{2}$ HS chain. Specifically, the spin singlet states of the system all fall on the topological basis states. And the number of the spin singlet states of the system is equal to the number of the topological basis states. Meanwhile, in the case of antiferromagnetism, the energy ground state of the system falls on the topological subspace.

The accurate computational determination of chemical, materials, biological, and atmospheric properties has a critical impact on a wide range of health and environmental problems, but is deeply limited by the computational scaling of quantum mechanical methods. The complexity of quantum chemical studies arises from the steep algebraic scaling of electron correlation methods and the exponential scaling in studying nuclear dynamics and molecular flexibility. To date, efforts to apply quantum hardware to such quantum chemistry problems have focused primarily on electron correlation. Here, we provide a framework that allows for the solution of quantum chemical nuclear dynamics by mapping these to quantum spin-lattice simulators. Using the example case of a short-strong hydrogen-bonded system, we construct the Hamiltonian for the nuclear degrees of freedom on a single Born-Oppenheimer surface and show how it can be transformed to a generalized Ising model Hamiltonian. We then demonstrate a method to determine the local fields and spin-spin couplings needed to identically match the molecular and spin-lattice Hamiltonians. We describe a protocol to determine the on-site and intersite coupling parameters of this Ising Hamiltonian from the Born-Oppenheimer potential and nuclear kinetic energy operator. Our approach represents a paradigm shift in the methods used to study quantum nuclear dynamics, opening the possibility to solve both electronic structure and nuclear dynamics problems using quantum computing systems.

Let $(Z,\omega)$ be a connected Kahler manifold with an holomorphic action of the complex reductive Lie group $U^{\mathbb C}$, where $U$ is a compact connected Lie group acting in a hamiltonian fashion. Let $G$ be a closed compatible Lie group of $U^{\mathbb C}$ and let $M$ be a $G$-invariant connected submanifold of $Z$. Let $x\in M$. If $G$ is a real form of $U^{\mathbb C}$, we investigate conditions such that $G\cdot x$ compact implies $U^{\mathbb C} \cdot x$ is compact as well. The vice-versa is also investigated. We also characterize $G$-invariant real submanifolds such that the norm square of the gradient map is constant. As an application, we prove a splitting result for real connected submanifolds of $(Z,\omega)$ generalizing a result proved in \cite{pg}, see also \cite{bg,bs}., Comment: 9 opages

Finite basis vibrational configuration interaction theory (VCI) is a highly accurate method for the variational calculation of state energies and related properties, but suffers from fast growing computational costs in dependence of the size of the correlation space. In this series of papers, concepts and techniques will be presented, which diminish the computational demands and thus broaden the applicability of this method to larger molecules or more complex situations. This first part focuses on a highly efficient implementation of the vibrational angular momentum (VAM) terms as occurring in the Watson Hamiltonian and the prediagonalization of initial subspaces within an iterative configuration selective VCI implementation. Working equations and benchmark calculations are provided, the latter demonstrating the increased performance of the new algorithm.

Nonadiabatic (NA) molecular dynamics (MD) goes beyond the adiabatic Born-Oppenheimer approximation to account for transitions between electronic states. Such processes are common in molecules and materials used in solar energy, optoelectronics, sensing, and many other fields. NA-MD simulations are much more expensive compared to adiabatic MD due to the need to compute excited state properties and NA couplings (NACs). Similarly, application of machine learning (ML) to NA-MD is more challenging compared with adiabatic MD. We develop an NA-MD simulation strategy in which an adiabatic MD trajectory, which can be generated with a ML force field, is used to sample excitation energies and NACs for a small fraction of geometries, while the properties for the remaining geometries are interpolated with kernel ridge regression (KRR). This ML strategy allows for one to perform NA-MD under the classical path approximation, increasing the computational efficiency by over an order of magnitude. Compared to neural networks, KRR requires little parameter tuning, saving efforts on model building. The developed strategy is demonstrated with two metal halide perovskites that exhibit complicated MD and are actively studied for various applications.

Using the example of the harmonic oscillator, we illustrate the use of hybrid dynamical brackets in analyzing quantum-classical interaction. We only assume that a hybrid dynamical bracket exists, is bilinear, and reduces to the pure quantum or classical bracket when acting on pure quantum or classical variables. Any hybrid bracket obeying these natural requirements will produce the same dynamics for pure classical or quantum variables, given a hybrid Hamiltonian. Backreaction is manifested in the evolution of a nonvanishing commutator between classical variables. The more massive the classical system is, the less it is affected by backreaction. Interestingly, we show that while pure variables evolve to violate the pure canonical relations, they always obey the hybrid canonical relations. The dynamics of hybrid variables, on the other hand, is shown to require a fully specified and consistent hybrid bracket, otherwise evolution cannot be defined.

The present work investigates the vibrational properties of a DNA-like structure by means of a harmonic Hamiltonian and the Green’s function formalism. The DNA sequence is considered as a quasi one-dimensional system in which the mass-spring pairs are randomly distributed inside each crystalline unit. The sizes of the units inside the system are increased, in a step-by-step approach, so that the actual condition of the DNA could be modeled more accurately. The linear-elastic forces mimicking the bonds between the pairs are initially considered constant along the entire length of the system. In the next step, these forces are randomly shuffled so as to take into account the inherent randomness of the DNA. The results reveal that increasing the number of mass-spring pairs in the crystalline structure decreases the influence of randomness on the mechanical behavior of the structure. This also holds true for systems with larger crystalline units. The obtained results can be used to investigate the mechanical behavior of similar macro-systems.

New spin-dependent components of the cross sections for nonresonant and resonant X-ray scattering by atoms of matter have been found using quasi-relativistic expansion of the Dirac Hamiltonian in the Foldy–Wouthuysen representation. Spin-dependent components of resonant scattering, which are maximum at the K-absorption edge in magnetic materials, contain information about the spin density of p states in the continuous spectrum of material atoms.

A novel approach to solve the problem of distributed state estimation of linear time-invariant systems is proposed in this article. It relies on the application of parameter-estimation-based observers, where the state observation task is reformulated as a parameter estimation problem. In contrast with existing results, our solution achieves convergence in finite time, without injection of high gain, and imposes very weak assumptions on the communication graph—namely, the existence of an open Hamiltonian walk. It is shown that this assumption is strictly weaker than the usual strong connectivity requirement. The scheme is shown to be robust vis-a-vis external disturbances and communication delays.

Let G be a graph, ν ( G ) the order of G , κ ( G ) the connectivity of G and k a positive integer such that k ≤ ( ν ( G ) − 2 ) ∕ 2 . Then G is said to be k -extendable if it has a matching of size k and every matching of size k extends to a perfect matching of G . A graph G is Hamiltonian if it contains a Hamiltonian cycle. In this paper, we define a parameter c ( G ) and show some of its properties in bipartite graphs. Let G be a bipartite graph with bipartition ( X , Y ) such that | X | = | Y | and c ( G ) > 0 , and C a longest cycle in G . We show that 0 c ( G ) ≤ κ ( G ) − 1 , G is Hamiltonian if ν ( G ) ≤ 2 κ ( G ) + 4 c ( G ) − 2 , and | V ( C ) | ≥ 6 c ( G ) if ν ( G ) > 6 c ( G ) . We show some of its corollaries in k -extendable, bipartite graphs and two conjectures in k -extendable graphs.

The n-dimensional hypercube Q n is one of the most attractive interconnection networks for multiprocessor systems and it is a bipartite graph. Let F v be a set of the end-nodes of k independent edges in Q n and F e be a set of f edges in Q n − F v . Given a linear forest L of Q n − F v − F e , in this paper, we prove that (i) Q n − F v − F e admits a hamiltonian cycle passing through L if | E ( L ) | + k + f ≤ n − 2 ; and ( i i ) for any two nodes x and y of the opposite partite sets in Q n − F v − F e such that none of the paths in L has x or y as internal node or both of them as end-nodes, Q n − F v − F e admits a hamiltonian path between x and y passing through L if | E ( L ) | + k + f ≤ n − 3 ; and ( i i i ) for any two distinct nodes u and v of the partite set not containing w in Q n − F v − F e − w such that none of the paths in L has u or v as internal node or both of them as end-nodes, Q n − F v − F e − w admits a hamiltonian path between u and v passing through L if | E ( L ) | + k + f ≤ n − 3 , where w is an arbitrary node in Q n − F v − F e .

Within the framework of an idealized theoretical model, we study the effect of external static homogeneous exchange and magnetic field on the spin part of the singlet wave function of two electrons. We begin by revising the traditional (text- book) approach to the spin singlet. Basing our own approach solely on the property of invariance under rotations of the coordinate system and using the theory of spinor in- variants, we derive a generalized representation of the spin singlet whose main feature is that the spins are in mutually time-reversed states. We show that exactly this fea- ture predetermines the actual form of the Hamiltonian of interaction with the external field and stipulates time evolution of the singlet. Some applications of these results to the theory of superconductivity and spin chemistry are presented. In particular, it is shown that the case of ferromagnetic superconductors constitutes a good illustration of the validity of our quantum-mechanical consideration., 14 pages

The notion of topological phases extended to dynamical systems stimulates extensive studies, of which the characterization of nonequilibrium topological invariants is a central issue and usually necessitates the information of quantum dynamics in both the time and momentum dimensions. Here, we propose the topological holographic quench dynamics in synthetic dimension, and also show it provides a highly efficient scheme to characterize photonic topological phases. A pseudospin model is constructed with ring resonators in a synthetic lattice formed by frequencies of light, and the quench dynamics is induced by initializing a trivial state, which evolves under a topological Hamiltonian. Our key prediction is that the complete topological information of the Hamiltonian is encoded in quench dynamics solely in the time dimension, and is further mapped to lower-dimensional space, manifesting the holographic features of the dynamics. In particular, two fundamental time scales emerge in the dynamical evolution, with one mimicking the topological band on the momentum dimension and the other characterizing the residue time evolution of the state after the quench. For this, a universal duality between the quench dynamics and the equilibrium topological phase of the spin model is obtained in the time dimension by extracting information from the field evolution dynamics in modulated ring systems in simulations. This work also shows that the photonic synthetic frequency dimension provides an efficient and powerful way to explore the topological nonequilibrium dynamics., Topological holographic quench dynamics in a synthetic frequency dimension studied in modulated ring system shows the universal bulk-surface duality with complete topological information being encoded in the single time variable.

The pyrochlore lattice is associated with several potential and actual spin liquid phases as a result of its strong geometric frustration. At finite temperature, these can exhibit an unusually broad cross-over regime to a conventional paramagnet. Here, we study this regime analytically by showing how a single-tetrahedron Hamiltonian can extrapolate beyond the first term of a high-temperature expansion and yield insights into the build-up of correlations. We discuss how this unusual behaviour is brought about by the structure of the eigenspaces of the coupling matrix. Further interesting behaviour can appear for parameter values located near phase transitions: we find coexistence of $(111)$ rods and $(220)$ peaks in the structure factor, as observed in neutron scattering experiments on Yb$_2$Ti$_2$O$_7$., Comment: 7 pages, 4 figures

Polymer self-consistent field theory techniques are used to derive quantum density functional theory without the use of the theorems of density functional theory. Instead, a free energy is obtained from a partition function that is constructed directly from a Hamiltonian, so that the results are, in principle, valid at finite temperatures. The main governing equations are found to be a set of modified diffusion equations, and the set of self-consistent equations are essentially identical to those of a ring polymer system. The equations are shown to be equivalent to Kohn-Sham density functional theory, and to reduce to classical density functional theory, each under appropriate conditions. The obtained non-interacting kinetic energy functional is, in principle, exact, but suffers from the usual orbital-free approximation of the Pauli exclusion principle in additional to the exchange-correlation approximation. The equations are solved using the spectral method of polymer self-consistent field theory, which allows the set of modified diffusion equations to be evaluated for the same computational cost as solving a single diffusion equation. A simple exchange-correlation functional is chosen, together with a shell-structure-based Pauli potential, in order to compare the ensemble average electron densities of several isolated atom systems to known literature results. The agreement is excellent, justifying the alternative formalism and numerical method. Some speculation is provided on considering the time-like parameter in the diffusion equations, which is related to temperature, as having dimensional significance, and thus picturing point-like quantum particles instead as non-local, polymer-like, threads in a higher dimensional thermal-space. A consideration of the double-slit experiment from this point of view is speculated to provide results equivalent to the Copenhagen interpretation., 32 pages, 5 figures

All results concern characteristic 2. Two procedures that to every simple Lie algebra assign simple Lie superalgebras, most of the latter new, are offered. We prove that every simple finite-dimensional Lie superalgebra is obtained as the result of one of these procedures, so we classified all simple finite-dimensional Lie superalgebras modulo non-existing at the moment classification of simple finite-dimensional Lie algebras. This result concerns Lie superalgebras considered naively, as vector spaces. To obtain classification of simple Lie superalgebras in the category of supervarieties, one should list the non-isomorphic deforms (results of deformations) with odd parameter. This problem is open bar several examples described in arXiv~0807.3054. For Lie algebras, in addition to the known ---"classical" --- restrictedness, we introduce a (2,4)-structure on the two non-alternating series: orthogonal and of Hamiltonian vector fields. For Lie superalgebras, the classical restrictedness of Lie algebras has two analogs: $(2|4)$- and $(2|2)$-structures, one more analog --- a $(2,4)|4$-structure on Lie superalgebras is the analog of (2,4)-structure on Lie algebras.

Spin-orbit coupling (SOC) in conjunction with broken inversion symmetry acts as a key ingredient for several intriguing quantum phenomena viz. persistent spin textures, topological surface states and Rashba-Dresselhaus (RD) effects. The coexistence of spontaneous polarization and the RD effect in ferroelectric materials enables the electrical control of spin degree of freedom. In light of this, we explore here the ferroelectric lead halide perovskite viz. CsPbF$_3$ as a potential candidate in the field of spintronics by employing state-of-the-art first-principles based methodologies viz. density functional theory (DFT) with semi-local and hybrid functional (HSE06) combined with spin-orbit coupling (SOC) and many-body perturbation theory (G$_0$W$_0$). For a deeper understanding of the observed spin-splitting, we have analyzed the spin textures within the combined framework of DFT and $\textbf{k.p}$ model Hamiltonian. The latter confirms that there is no out of plane spin component indicating that the Rashba splitting dominates over Dresselhaus splitting in this system. Owing to the presence of Pb-6$p$ orbital in conduction band, the large value of Rashba coefficient ($\alpha_R$) at conduction band minimum (CBm) is noticed in comparison to that of at the valence band maximum (VBM). Notably, we also observe that strength of Rashba spin-splitting can be substancially tuned on application of uniaxial strain ($\pm5\%$). This finding will further pave the path for perovskite-based spintronics devices., Comment: 34 pages, 11 figures

Let $$\xi $$ be a Gaussian white noise on $${\mathbb {R}}^d$$ ( $$d=1,2,3$$ ). Let $$(\xi _\varepsilon )_{\varepsilon >0}$$ be continuous Gaussian processes such that $$\xi _\varepsilon \rightarrow \xi $$ as $$\varepsilon \rightarrow 0$$ , defined by convolving $$\xi $$ against a mollifier. We consider the asymptotics of the parabolic Anderson model (PAM) with noise $$\xi _{\varepsilon (t)}$$ for large time $$t\gg 1$$ , and the Dirichlet eigenvalues of the Anderson Hamiltonian (AH) with potential $$\xi _{\varepsilon (t)}$$ on large boxes $$(-t,t)^d$$ , where the parameter $$\varepsilon (t)$$ vanishes as $$t\rightarrow \infty $$ . We prove that the asymptotics in question exhibit a phase transition in the rate at which $$\varepsilon (t)$$ vanishes, which distinguishes between the behavior observed in the AH/PAM with continuous Gaussian noise and white noise. By comparing our main theorems with previous results on the AH/PAM with white noise, our results show that some asymptotics of the latter can be accessed with solely elementary methods, and we obtain quantitative estimates on the difference between the AH/PAM with white noise and its continuous-noise approximations as $$t\rightarrow \infty $$ .

Nonlinear propagation of stationary large-amplitude electromagnetic (EM) solitary waves in a magnetized electron–positron (EP) plasma is studied using a fully relativistic two-fluid hydrodynamic model. In the first step, we wrote the one-body particle and N-body particle fluid equations for the electron–positron fluid, taking into account the spin effects and regardless of the thermal effects. These equations are the first to be set for electron and positron fluids. The particle-antiparticle nature of the relativistic particles is explicit in this fluid theory. The nonrelativistic limit is shown to be in agreement with previous attempts to develop a spin plasma theory derived from the Pauli Hamiltonian. In the second step, we investigated the thermal effects without considering the spin effects and according to the enthalpy of the system. In the last step, we studied the electromagnetic mode propagation for this type of plasma and determined the range of wave transition.

Let a reductive group $G$ act on a smooth affine complex algebraic variety $X.$ Let $\mathfrak{g}$ be the Lie algebra of $G$ and $\mu:T^*(X)\to \mathfrak{g}$ be the moment map. If the moment map is flat, and for a generic character $\chi:\mathfrak{g}\to\mathbb{C}$, the action of $G$ on $\mu^{-1}(\chi)$ is free, then we show that for very generic characters $\chi$ the corresponding quantum Hamiltonian reduction of the ring of differential operators $D(X)$ is simple., Comment: 5 pages, to appear in Comptes Rendus Math

A system of four point vortices on a plane is considered. Its motion is described by the Kirchhoff equations. Three vortices have unit intensity and one vortex has arbitrary intensity $$\varkappa$$ . We study the stability problem for the stationary rotation of a vortex quadrupole consisting of three identical vortices located uniformly on a circle around a fourth vortex. It is known that for $$\varkappa>1$$ the regime under study is unstable, and in the case of $$\varkappa

In this note, we show a sharp lower bound of $$\min \left\{{\sum\nolimits_{i = 1}^k {{d_G}({u_i}):{u_1}{u_2} \ldots {u_k}}} \right.$$ min { ∑ i = 1 k d G ( u i ) : u 1 u 2 … u k is a path of (2-)connected G on its order such that (k-1)-iterated line graphs Lk−1(G) are hamiltonian.

The non-relativistic equations of motion for a dipole in a stationary non-homogeneous electromagnetic field are derived and analysed. It is shown that they are Hamiltonian with respect to a certain degenerated Poisson structure. Described by them dynamics is complex because the motion of the centre of mass of the dipole is coupled with its rotational motion. The problem of the existence of linear in momenta first integrals which can be useful for the separation of rotational motion is discussed. The presence of such first integral appears to be related with a linear symmetry of electric and magnetic fields. Also results of search of quadratic in momenta first integrals for uniform and stationary electromagnetic fields are reported. Deriving equations of motion of a dipole in arbitrary stationary electromagnetic fields and analysis of described by them dynamics is important for the construction of electromagnetic traps for polar particles.

Exciton dynamics, lifetimes, and scattering are directly related to the exciton dispersion or band structure. Here, we present a general theory for exciton band structure within both ab initio and model Hamiltonian approaches. We show that contrary to common assumption, the exciton band structure contains nonanalytical discontinuities-a feature which is impossible to obtain from the electronic band structure alone. These discontinuities are purely quantum phenomena, arising from the exchange scattering of electron-hole pairs. We show that the degree of these discontinuities depends on materials' symmetry and dimensionality, with jump discontinuities occurring in 3D and different orders of removable discontinuities in 2D and 1D, whose details depend on the exciton degeneracy and material thickness. We connect these features to the early stages of exciton dynamics, radiative lifetimes, and diffusion constants, in good correspondence with recent experimental observations, revealing that the discontinuities in the band structure lead to ultrafast ballistic transport and suggesting that measured exciton diffusion and dynamics are influenced by the underlying exciton dispersion.

We study the semiclassical limit of the sine-Gordon (sG) equation with below threshold pure impulse initial data of Klaus-Shaw type. The Whitham averaged approximation of this system exhibits a gradient catastrophe in finite time. In accordance with a conjecture of Dubrovin, Grava and Klein, we found that in a $\mathcal{O}(\epsilon^{4/5})$ neighborhood near the gradient catastrophe point, the asymptotics of the sG solution are universally described by the Painleve I tritronquee solution. A linear map can be explicitly made from the tritronquee solution to this neighborhood. Under this map: away from the tritronquee poles, the first correction of sG is universally given by the real part of the Hamiltonian of the tritronquee solution; localized defects appear at locations mapped from the poles of tritronquee solution; the defects are proved universally to be a two parameter family of special localized solutions on a periodic background for the sG equation. We are able to characterize the solution in detail. Our approach is the rigorous steepest descent method for matrix Riemann--Hilbert problems, substantially generalizing Bertola and Tovbis's results on the nonlinear Schrodinger equation to establish universality beyond the context of solutions of a single equation.

We demonstrate that quantum Hamiltonian operator for a free transverse field within the framework of the second quantization reveals an alternative set of states satisfying the eigenstate functional equations. The construction is based upon extensions of the quadratic form of the transverse Laplace operator which are used as a source of spherical basis functions with singularity at the origin. This basis then naturally takes place of the one of plane or spherical waves in the process of Fourrier or spherical variable separation., Comment: 22 pages

The generalized quantum master equation (GQME) provides a powerful framework for simulating electronic energy, charge, and coherence transfer dynamics in molecular systems. Within this framework, the effect of the nuclear degrees of freedom on the time evolution of the electronic reduced density matrix is fully captured by a memory kernel superoperator. However, the actual memory kernel depends on the choice of projection operator and is therefore not unique. Furthermore, calculating the memory kernel can be done in multiple ways that use different forms of projection-free inputs. Although the electronic dynamics is invariant to those choices when quantum-mechanically exact projection-free inputs are used, this is not the case when they are obtained via more feasible semiclassical or mixed quantum-classical approximate methods. Furthermore, the accuracy and numerical stability of the resulting electronic dynamics has been observed to be sensitive to the above-mentioned choices when approximate methods are used to calculate the projection-free inputs. In this article, we provide a systematic road map to 30 possible pathways for calculating the memory kernel and highlight how they are related as well as the ways in which they differ. We also compare the performance of different pathways in the context of the spin-boson benchmark model, with the projection-free inputs obtained via a mapping Hamiltonian linearized semiclassical method. In this case, we find that expressing the memory kernel with an exponential operator where the projection operator precedes the Liouvillian yields the most accurate and most numerically stable results.

In this paper, we present a general method to solve the non-hermetic potentials with PT symmetry using the definition of two $$\eta $$ -pseudo-hermetic and first-order operators. This generator applies to the Dirac equation which consists of two spinor wave functions and non-hermetic potential. Mass is considered a constant, and the Hamiltonian hierarchy method and the shape invariance property are used to perform calculations. Furthermore, we show the correlations between the potential parameters with transmission probabilities where $$\eta $$ -pseudo-hermetic utilizing the change of focal points on Hamiltonian can be formalized based on Schrodinger-like equation. We employ this method for some solvable potentials such as deformed Woods–Saxon potential and show that these real potentials can be decomposed into complex potentials consisting of eigenvalues of a class of $$\eta $$ -pseudo-hermetic generator.

An approach for evaluating spectra from ground state probability amplitudes (GSPA) obtained from diffusion Monte Carlo (DMC) simulations is extended to improve the description of excited state energies and allow for coupling among vibrational excited states. This approach is applied to studies of the protonated water trimer and tetramer, and their deuterated analogs. These ions provide models for solvated hydronium, and analysis of these spectra provides insights into spectral signatures of proton transfer in aqueous environments. In this approach, we obtain a separable set of internal coordinates from the DMC ground state probability amplitude. A basis is then developed from products of the DMC ground state wave function and low-order polynomials in these internal coordinates. This approach provides a compact basis in which the Hamiltonian and dipole moment matrix are evaluated and used to obtain the spectrum. The resulting spectra are in good agreement with experiment and in many cases provide comparable agreement to the results obtained using much larger basis sets. In addition, the compact basis allows for interpretation of the spectral features and how they evolve with cluster size and deuteration.