Non-diagonal Terms Research Articles

Method to discriminate between localized and chaotic quantum systems

We study whether a generic isolated quantum system initially set out of equilibrium can be considered as localized close to its initial state. Our approach considers the time evolution in the Krylov basis, which maps the system's dynamics onto that of a particle moving in a one-dimensional lattice where both the energy in the lattice sites and the tunneling from one lattice site to the next are inhomogeneous. By tying the dynamical propagation in the Krylov basis to that in the basis of microstates, we infer qualitative criteria that allow distinguishing systems that remain localized close to their initial state from systems that undergo quantum thermalization. These criteria are system-dependent and involve the expectation values and standard deviations of both the coupling strengths between Krylov states and their energy. We verify their validity by inspecting two cases: Anderson localization as a function of dimension and the out-of-equilibrium dynamics of a many-body dipolar spin system. We finally investigate the Wigner surmise and the eigenstate thermalization hypothesis that have both been proposed to characterize quantum chaotic systems. We show that when the average value of the nondiagonal terms in the Lanczos matrix is large compared to their fluctuations and to the fluctuations of the energy expectation values, which typically corresponds to delocalized quantum systems according to our criteria, there can be level repulsion of eigenenergies (also known as spectral rigidity), similar to that of the Wigner-Dyson statistics of energy levels; and we also demonstrate that in the same regime, the expectation values of typical local observables only weakly vary as a function of eigenstates, an important condition for the eigenstate thermalization hypothesis. Our demonstration assumes that, in the chaotic regime, the observable is sufficiently diagonal in the basis of Krylov states. Published by the American Physical Society 2024

Cosmological consequences of the Lorentz and Doppler transformations

The common opinion that the Lorentz transformation distorts the Minkowski spacetime arises from a mathematically incorrect neglect of its non-diagonal terms. When the non-diagonal Lorentz transformation is properly treated, the time dilation and the space distortion disappear. Consequently, the Lorentz metric produced by the Lorentz transformation is identical to the Minkowski metric. The spacetime distortion in moving inertial frames is properly described by the so-called Doppler metric produced by the Doppler transformation. This transformation assumes the existence of a preferred frame and depicts the Doppler shift of light for a moving light source and/or observer. The Doppler metric is Lorentz invariant and leaves the Maxwell’s equations unchanged from their form in the Minkowski spacetime. Moreover, the Doppler metric is experimentally confirmed because it yields the null result in the Michelson–Morley experiment and other interferometric experiments. The suitability of the Doppler metric for describing light properties in Special and General Relativity is supported by recent cosmological observations of the cosmic microwave background.

Ideals of submaximal minors of sparse symmetric matrices

We study the ideal of submaximal minors of a sparse generic symmetric matrix. This ideal is generated by all (n−1)-minors of a symmetric n×n matrix whose entries in the upper triangle are distinct variables or zeros, and the zeros are only allowed at off-diagonal places. The surviving off-diagonal entries are encoded in a simple graph G with n vertices. We prove that the minimal free resolution of this ideal is obtained from the case without any zeros via a simple pruning procedure, extending methods of Boocher. This allows us to compute all graded Betti numbers in terms of n and a single invariant of G. Moreover, it turns out that these ideals are always radical and have Cohen–Macaulay quotients if and only if G is either connected or has no edges at all. The key input are some new Gröbner basis results with respect to non-diagonal term orders associated to G.

A new generation of non-diagonal, renormalized self-energies for calculation of electron removal energies.

A new generation of diagonal self-energies for the calculation of electron removal energies of molecules and molecular ions that has superseded its predecessors with respect to accuracy, efficiency, and interpretability is extended to include non-diagonal self-energies that permit Dyson orbitals to be expressed as linear combinations of canonical Hartree-Fock orbitals. In addition, an improved algorithm for renormalized methods eliminates the convergence difficulties encountered in the first studies of the new, diagonal self-energies. A dataset of outer-valence, vertical ionization energies with almost full-configuration-interaction quality serves as a standard of comparison in numerical tests. The new non-diagonal, renormalized methods are slightly more accurate than their diagonal counterparts, with mean absolute errors between 0.10 and 0.06eV for outer-valence final states. This advantage is procured at the cost of an increase in the scaling of arithmetic bottlenecks that accompany the inclusion of non-diagonal self-energy terms. The new, non-diagonal, renormalized self-energies are also more accurate and efficient than their non-diagonal predecessors.

Multiband Superconductivity in High-Pressure Sulfur Hydrides

The temperature dependence of the two superconducting gaps in pressurised H3S at 155 GPa with a critical temperature of 203 K has been determined using a data analysis of the experimental curve of the upper critical magnetic field as a function of temperature in the framework of the two-band s-wave Eliashberg theory. Two different phonon-mediated intra-band Cooper pairing channels in a regime of moderate strong couplings have the key role of the pair-exchange interaction between the two gaps, giving the two non-diagonal terms of the coupling tensor, which are missing in the single-band s-wave Eliashberg theory. The results provide a prediction of the different temperature dependence of the small and large gaps as a function of temperature, which provides evidence of multigap superconductivity in H3S.

Algebraic Bethe Ansatz for the Open XXZ Spin Chain with Non-Diagonal Boundary Terms via $U_{\mathfrak{q}}\mathfrak{sl}_2$ Symmetry

We derive by the traditional algebraic Bethe ansatz method the Bethe equations for the general open XXZ spin chain with non-diagonal boundary terms under the Nepomechie constraint [J. Phys. A 37 (2004), 433-440, arXiv:hep-th/0304092]. The technical difficulties due to the breaking of $\mathsf{U}(1)$ symmetry and the absence of a reference state are overcome by an algebraic construction where the two-boundary Temperley-Lieb Hamiltonian is realised in a new $U_{\mathfrak{q}}\mathfrak{sl}_2$-invariant spin chain involving infinite-dimensional Verma modules on the edges [J. High Energy Phys. 2022 (2022), no. 11, 016, 64 pages, arXiv:2207.12772]. The equivalence of the two Hamiltonians is established by proving Schur-Weyl duality between $U_{\mathfrak{q}}\mathfrak{sl}_2$ and the two-boundary Temperley-Lieb algebra. In this framework, the Nepomechie condition turns out to have a simple algebraic interpretation in terms of quantum group fusion rules.

Experimental study of the turbulent flow in the wake of a horizontal axis tidal current turbine

A scaled-model of a horizontal axis tidal current turbine (HATCT) is tested in the CNR-INM Circulating Water Channel. The experiments are designed to establish in the first place the performances of the turbine at different working settings. The second goal is to investigate the hydrodynamics generated by the turbine in the near wake using the Particle Image Velocimetry (PIV) technique. For this purpose, velocity measurements are performed in a longitudinal plane and phase-locked to the rotor angle in order to resolve the wake structure at different working regimes. The analysis of the axial and radial velocity fields reveals the flow features of the slipstream, as well as its expansion and dependence on the turbine operating parameters. Analysis of the non-diagonal terms of the Reynolds stress tensor provides insight into the onset of tip vortex pairing and of vortex instabilities. Furthermore the separate contributions of transport, production and dissipation to the turbulent kinetic energy in the wake field are discussed in detail. The vortex unsteadiness is captured and correlated with the evolution of the kinetic energy transport and production terms. Understanding these phenomenologies is an important step to develop computational tools able to estimate the radiated noise or the potential impact of turbulence on performances of further rotors placed in the wake, as in an array of tidal turbines.

Particles of Negative and Zero Energy in Black Holes and Cosmological Models

Particles with negative energies are considered for three different cases: inside the horizon of a Schwarzschild black hole, Milne’s coordinates in flat Minkowski space–time (Milne’s universe using nonsynchronous coordinates) and in the cosmological Gödel model of the rotating universe. It is shown that, differently from the Gödel model with a nondiagonal term, where it occurs that negative energies are impossible, they are present in all other cases considered in the paper. Particles with zero energy are also possible in the first two cases.

Accurate characterization of the stochastic gravitational-wave background with pulsar timing arrays by likelihood reweighting

An isotropic stochastic background of nanohertz gravitational waves creates excess residual power in pulsar-timing-array datasets, with characteristic interpulsar correlations described by the Hellings-Downs function. These correlations appear as nondiagonal terms in the noise covariance matrix, which must be inverted to obtain the pulsar-timing-array likelihood. Searches for the stochastic background, which require many likelihood evaluations, are therefore quite computationally expensive. We propose a more efficient method: we first compute approximate posteriors by ignoring cross correlations and then reweight them to exact posteriors via importance sampling. We show that this technique results in accurate posteriors and marginal likelihood ratios, because the approximate and exact posteriors are similar, which makes reweighting especially accurate. The Bayes ratio between the marginal likelihoods of the exact and approximate models, commonly used as a detection statistic, is also estimated reliably by our method, up to ratios of at least ${10}^{6}$.

Modeling of spin decoherence in a Si hole qubit perturbed by a single charge fluctuator

Spin qubits in semiconductor quantum dots are one of the promizing devices to realize a quantum processor. A better knowledge of the noise sources affecting the coherence of such a qubit is therefore of prime importance. In this work, we study the effect of telegraphic noise induced by the fluctuation of a single electric charge. We simulate as realistically as possible a hole spin qubit in a quantum dot defined electrostatically by a set of gates along a silicon nanowire channel. Calculations combining Poisson and time-dependent Schr\"odinger equations allow to simulate the relaxation and the dephasing of the hole spin as a function of time for a classical random telegraph signal. We show that dephasing time $T_2$ is well given by a two-level model in a wide range of frequency. Remarkably, in the most realistic configuration of a low frequency fluctuator, the system has a non-Gaussian behavior in which the phase coherence is lost as soon as the fluctuator has changed state. The Gaussian description becomes valid only beyond a threshold frequency $\omega_{th}$, when the two-level system reacts to the statistical distribution of the fluctuator states. We show that the dephasing time $T_{2}(\omega_{th})$ at this threshold frequency can be considerably increased by playing on the orientation of the magnetic field and the gate potentials, by running the qubit along "sweet" lines. However, $T_{2}(\omega_{th})$ remains bounded due to dephasing induced by the non-diagonal terms of the stochastic perturbation Hamiltonian. Our simulations reveal that the spin relaxation cannot be described cleanly in the two-level model because the coupling to higher energy hole levels impacts very strongly the spin decoherence. This result suggests that multi-level simulations including the coupling to phonons should be necessary to describe the relaxation phenomenon in this type of qubit.

Numerical study of viscosity and heat flux role in heavy species dynamics in Hall thruster discharge

A two- and three-dimensional velocity space axisymmetric hybrid-PIC model of Hall thruster discharge called Hybrid2D has been developed. The particle-in-cell (PIC) method was used for neutrals and ions (heavy species), and fluid dynamics on a magnetic field-aligned (MFA) mesh was used for electrons. A time-saving method for heavy species moment interpolation on a MFA mesh was developed. The method comprises using regular rectangle and irregular triangle meshes, connected to each other on a pre-processing stage. The electron fluid model takes into account neither inertia terms nor viscous terms and includes an electron temperature equation with a heat flux term. The developed model was used to calculate all heavy species moments up to the third one in a stationary case. The analysis of the viscosity and the heat flux impact on the force and energy balance has shown that for the calculated geometry of the Hall thruster, the viscosity and the heat flux terms have the same magnitude as the other terms and could not be omitted. Also, it was shown that the heat flux is not proportional to the temperature gradient and, consequently, the highest moments should be calculated to close the neutral fluid equation system. At the same time, ions can only be modeled as a cold non-viscous fluid when the sole aim of modeling is the calculation of the operating parameters or distribution of the local parameters along the centerline of the discharge channel. This is because the magnitude of the viscosity and the temperature gradient terms are negligible at the centerline. However, when a simulation’s focus is either on the radial divergence of the plume or on magnetic pole erosion, three components of the ion temperature should be taken into consideration. The non-diagonal terms of ion pressure tensor have a lower impact than the diagonal terms. According to the study, a zero heat flux condition could be used to close the ion equation system in calculated geometry.

Nonlinear chirped interferometry for frequency-shift measurement and χ(3) spectroscopy

Four-wave mixing processes are ubiquitous in ultrafast optics and the determination of the coefficients of the χ(3) tensor is thus essential. We introduce a novel time-resolved ultrafast spectroscopic method to characterize the third-order nonlinearity on the femtosecond time-scale. This approach, coined as “nonlinear chirped interferometry,” makes use of the variation of the optical group delay of a transmitted probe under the effect of an intense pump pulse in the nonlinear medium of interest. The observable is the spectral interference between the probe and a reference pulse sampled upstream and the metric is the transient swing of the probe group delay. We show that the detected signal is enhanced when the pulses are weakly chirped, and that, although interferometric, the method is intrinsically less sensitive to environmental phase fluctuations and drifts. By chirping adequately the reference pulse, the transient frequency shift of the probe pulses is also detected in the time domain and the detected nonlinear signal is enhanced. Nonlinear phase shifts as low as 10 mrad, corresponding to a frequency shift of 30 GHz, i.e., 0.01% of the carrier frequency, are detected without heterodyne detection or active phase-stabilization. The diagonal and/or non-diagonal terms of reference glasses (SiO2) and crystals (Al2O3, BaF2, CaF2) are characterized. The method is finally applied to measure the soft vibration mode of KTiOAsO4 (KTA).

Universal dynamics of heavy operators in boundary CFT2

We derive a universal asymptotic formula for generic boundary conditions for the average value of the bulk-to-boundary and boundary Operator Product Expansion coefficients of any unitary, compact two-dimensional Boundary CFT (BCFT) with c > 1. The asymptotic limit consists of taking one or more boundary primary operators — which transform under a single copy of the Virasoro algebra — to have parametrically large conformal dimension for fixed central charge. In particular, we find a single universal expression that interpolates between distinct heavy regimes, exactly as in the case of bulk OPE asymptotics [1]. The expression depends universally on the boundary entropy and the central charge, and not on any other details of the theory. We derive these asymptotics by studying crossing symmetry of various correlation functions on higher genus Riemann surfaces with open boundaries. Essential in the derivation is the use of the irrational versions of the crossing kernels that relate holomorphic Virasoro blocks in different channels. Our results strongly suggest an extended version of the Eigenstate Thermalization Hypothesis for boundary OPE coefficients, where the hierarchy between the diagonal and non-diagonal term in the ansatz is further controlled by the boundary entropy. We finally comment on the applications of our results in the context of AdS3/BCFT2, as well as on the recent relation of BCFTs with lower dimensional models of evaporating black holes.

Novel computational mathematical algorithms for structural optimization using graph-theoretical methods

PurposeThe purpose is to reduce round-off errors in numerical simulations. In the numerical simulation, different kinds of errors may be created during analysis. Round-off error is one of the sources of errors. In numerical analysis, sometimes handling numerical errors is challenging. However, by applying appropriate algorithms, these errors are manageable and can be reduced. In this study, five novel topological algorithms were proposed in setting up a structural flexibility matrix, and five different examples were used in applying the proposed algorithms. In doing so round-off errors were reduced remarkably.Design/methodology/approachFive new algorithms were proposed in order to optimize the conditioning of structural matrices. Along with decreasing the size and duration of analyses, minimizing analytical errors is a critical factor in the optimal computer analysis of skeletal structures. Appropriate matrices with a greater number of zeros (sparse), a well structure and a well condition are advantageous for this objective. As a result, a problem of optimization with various goals will be addressed. This study seeks to minimize analytical errors such as rounding errors in skeletal structural flexibility matrixes via the use of more consistent and appropriate mathematical methods. These errors become more pronounced in particular designs with ill-suited flexibility matrixes; structures with varying stiffness are a frequent example of this. Due to the usage of weak elements, the flexibility matrix has a large number of non-diagonal terms, resulting in analytical errors. In numerical analysis, the ill-condition of a matrix may be resolved by moving or substituting rows; this study examined the definition and execution of these modifications prior to creating the flexibility matrix. Simple topological and algebraic features have been mostly utilized in this study to find fundamental cycle bases with particular characteristics. In conclusion, appropriately conditioned flexibility matrices are obtained, and analytical errors are reduced accordingly.Findings(1) Five new algorithms were proposed in order to optimize the conditioning of structural flexibility matrices. (2) A JAVA programming language was written for all five algorithms and a friendly GUI software tool is developed to visualize sub-optimal cycle bases. (3) Topological and algebraic features of the structures were utilized in this study.Research limitations/implicationsThis is a multi-objective optimization problem which means that sparsity and well conditioning of a matrix cannot be optimized simultaneously. In conclusion, well-conditioned flexibility matrices are obtained, and analytical errors are reduced accordingly.Practical implicationsEngineers always finding mathematical modeling of real-world problems and make them as simple as possible. In doing so, lots of errors will be created and these errors could cause the mathematical models useless. Applying decent algorithms could make the mathematical model as precise as possible.Social implicationsErrors in numerical simulations should reduce due to the fact that they are toxic for real-world applications and problems.Originality/valueThis is an original research. This paper proposes five novel topological mathematical algorithms in order to optimize the structural flexibility matrix.

A novel Bethe ansatz scheme for the one-dimensional Hubbard model

We propose a novel characterization of the exact solution of the one-dimensional Hubbard model with unparallel boundary magnetic fields. The non-diagonal boundary terms break the U(1) symmetry of the system and the number of electrons with fixed spin is not conserved. Thus it is difficult to study the physical quantities in the thermodynamic limit based on the previously obtained inhomogeneous T−Q relation and Bethe ansatz equations. In order to solve this problem, we propose the t−W scheme. The basic idea is that the eigenvalues of the transfer matrix can be expressed by its zero-roots instead of the usual Bethe roots. This method allows us to study thermodynamic properties of the system such as the ground state, surface energy, charge and spin excitations. The t−W scheme is universal and can be applied to the systems either with or without U(1) symmetry.

Exact solution of a quantum spin chain with competing bulk and boundary terms

We obtain the exact solution of an anisotropic quantum spin chain including nearest neighbor (NN), next nearest neighbor (NNN), chiral three-spin interactions and non-diagonal boundary terms. Because the NNN couplings are involved, we find that the off-diagonal boundary reflections can not only induce the anisotropic Dzyloshinsky–Moriya interactions at boundaries but also enhance the anisotropy of first and last bonds. These properties are absent if only the NN spin-exchanging interactions are considered. Due to that the U(1) symmetry is broken, we analytically solve the system by using the off-diagonal Bethe ansatz. The inhomogeneous T − Q relation, energy spectrum and associated Bethe ansatz equations are given explicitly. This paper provides an universal scheme to construct new integrable models with certain interesting interactions.

Semi-decoupled first-order correction for smoothed particle hydrodynamics

The Finite Particle Method (FPM) and the Decoupled Finite Particle Method (DFPM) are variants of the Smoothed Particle Hydrodynamics (SPH) in which the estimation of a field and its gradient for an arbitrary distribution of particles is performed by imposing first order consistency equations (C1). A modification of the kernel is introduced that involves the inversion of a correction matrix for each interpolation point in the system. In the FPM, the inversion is rigorously performed and therefore the method has first-order consistency (C1). However, in DPFM the vanishingly small non-diagonal terms in the correction matrix are neglected to obtain a more straightforward method, at expense of consistency. Following the idea of DFPM, the Semi-Decoupled Finite Particle Method (SDFPM) and the Corrected Semi Decoupled Finite Particle Method (CSDFPM) are introduced. In the SDFPM, the kernel is normalized by a Shepard factor and the first order consistency equations are solved by neglecting non-diagonal terms in the correction matrix as in the DFPM method. In the corrected version of this method (CSDFPM), the SDFPM estimation is used as the initial guess to get a second corrected estimation in the C1 equations. The precision of FPM, DFPM, SDFPM, and CSDFPM is tested by evaluating the gradient components of a field around a cylindrical obstacle and around a cylindrical hole by using (i) trial field functions with an ordered array of particles and (ii) the pressure field with a distribution of particles taken from a flow simulation. Drag coefficients for flow around a cylinder are obtained by the different methods and compared in a wide range of Reynolds numbers, including the laminar and turbulent regimes. The gradient components and drag coefficients calculated with the proposed methods show a precision comparable to the FPM at a lower computational cost.

Nondiagonal terms in the second moment of automorphic -functions

We investigate the exact formula for the second moment of central -values associated to primitive cusp forms of level one and weight , , which was proved by Kuznetsov in 1994. Nondiagonal terms in this formula are expressed in terms of several convolution sums of divisor functions. We investigate the main terms of the convolution sums and show that they cancel out. Bibliography: 8 titles.

Particles with Negative Energies in Nonrelativistic and Relativistic Cases

States of particles with negative energies are considered for the nonrelativistic and relativistic cases. In the nonrelativistic case it is shown that the decay close to the attracting center can lead to the situation similar to the Penrose effect for a rotating black hole when the energy of one of the fragments is larger than the energy of the initial body. This is known as the Oberth effect in the theory of the rocket movement. The realizations of the Penrose effect in the non-relativistic case in collisions near the attracting body and in the evaporation of stars from star clusters are indicated. In the relativistic case similar to the well known Penrose process in the ergosphere of the rotating black hole it is shown that the same situation as in ergosphere of the black hole occurs in rotating coordinate system in Minkowski space-time out of the static limit due to existence of negative energies. In relativistic cases differently from the nonrelativistic ones, the mass of the fragment can be larger than the mass of the decaying body. Negative energies for particles are possible in the relativistic case in cosmology of the expanding space when the coordinate system is used with a nondiagonal term in metrical tensor of the space-time. Friedmann metrics for three cases: open, close and quasieuclidian, are analyzed. The De Sitter space-time is shortly discussed.

Limits on non-minimal Lorentz violating parameters through FCNC and LFV processes

In this work we analyze a non-minimal Lorentz-violating extension of the electroweak theory in the fermionic sector. Firstly we analyze the relation between the CKM rotation in the quark sector and possible contributions of this new coupling to flavor changing neutral currents processes. In sequel we look for non-diagonal terms through possible lepton flavor violation decays. Strong bounds are presented to the Lorentz violating parameters of both the quark and the lepton sectors.