Simple Ansatz Research Articles

Anomalous dimension and quasinormal modes of flavor branes

We study scalar quasinormal modes in a D3/D7 system holographically dual to a quantum field theory with chiral symmetry breaking at finite temperature. From the bottom-up approach, we consider a nontrivial dilaton profile which is responsible for the anomalous dimension of the quark condensate. It depends on a new parameter q in the model. By varying this parameter, we study the behavior of the massive and massless scalar quasinormal modes. The numerical method that we use is the spectral method, and we find that there is no pure imaginary mode for the massless case but it appears by increasing the parameter q. It is known that this mode becomes tachyonic for massive cases. Then we turn on a pseudoscalar field and using a simple ansatz study its effect on the quasinormal modes of the scalar field. By varying the parameter of the nontrivial dilaton profile in the model, we qualitatively study quasinormal modes in walking theories.

Reconstructing generalized parton distributions from the lattice off-forward Compton amplitude

We present a determination of the structure functions of the off-forward Compton amplitude H1 and E1 from the Feynman-Hellmann method in lattice QCD. At leading twist, these structure functions give access to the generalized parton distributions (GPDs) H and E, respectively. This calculation is performed for an unphysical pion mass of mπ=412 MeV and four values of the soft momentum transfer, t≈0,−0.3,−0.6,−1.1 GeV2, all at a hard momentum scale of Q¯2≈5 GeV2. Using these results, we test various methods to determine properties of the real-time scattering amplitudes and GPDs: (1) we fit their Mellin moments, and (2) we use a simple GPD ansatz to reconstruct the entire distribution. Our final results show promising agreement with phenomenology and other lattice results and highlight specific systematics in need of control. Published by the American Physical Society 2024

Elasticity of Diametrically Compressed Microfabricated Woodpile Lattices

Modulus–porosity relationships are invaluable to rational material design of porous and structured solids. When struts in a lattice are compressed diametrically, the mechanics is rather complex. Herein, the problem of modulus–porosity in the spirit of scaling arguments and analyses based on simple ansatz followed by variational minimization of the elastic potential energy is addressed. Using scaling arguments, a simple power law where the apparent modulus of elasticity scales quadratically with the volume fraction for diametrically compressed elastic lattices is obtained. The modulus–porosity relationship is found to be consistent with computations and laboratory experiments on additively manufactured woodpile lattices with various cross‐sectional shapes and lattice spacing. It is also shown that the persistence length of diametrically pinched elastic rods is small, so that the effect of compressive strain from neighboring sites can be ignored. The decay behavior is surprisingly accurately captured by the variational approach and is consistent with computations. Finally, the range of validity of the quadratic power law presented here, up to relative density ~80%, is identified. On the apparent modulus–porosity plane, the experimental data aligns well with the power law for modulus–porosity predicted from simple analyses and finite element calculations.

Dibaryons and where to find them

In recent years, there has been tremendous progress in the investigation of bound systems of quarks with multiplicities beyond the more usual two- and three-quark systems. Experimental and theoretical progress has been made in the four-, five- and even six-quark sectors. In this paper, we review the possible lightest six-quark states using a simple ansatz based on SU(3) symmetry and evaluate the most promising decay branches. The work will be useful to help focus future experimental searches in this six-quark sector.

Basis Set Calculations of Heavy Atoms

Most modern calculations of many-electron atoms use basis sets of atomic orbitals. An accurate account for electronic correlations in heavy atoms is a very difficult computational problem, and an optimization of the basis sets can reduce computational costs and increase final accuracy. Here, we propose a simple differential ansatz to form virtual orbitals from the Dirac–Fock orbitals of the core and valence electrons. We use basis sets with such orbitals to calculate different properties in Cs including hyperfine structure constants and QED corrections to the valence energies and to the E1 transition amplitudes.

Amplitude amplification-inspired QAOA: improving the success probability for solving 3SAT

The Boolean satisfiability problem (SAT), in particular 3SAT with its bounded clause size, is a well-studied problem since a wide range of decision problems can be reduced to it. The Quantum Approximate Optimization Algorithm (QAOA) is a promising candidate for solving 3SAT for Noisy Intermediate-Scale Quantum devices in the near future due to its simple quantum ansatz. However, although QAOA generally exhibits a high approximation ratio, there are 3SAT problem instances where the algorithm’s success probability when obtaining a satisfying variable assignment from the approximated solution drops sharply compared to the approximation ratio. To address this problem, in this paper, we present variants of the algorithm that are inspired by the amplitude amplification algorithm to improve the success probability for 3SAT. For this, (i) three amplitude amplification-inspired QAOA variants are introduced and implemented, (ii) the variants are experimentally compared with a standard QAOA implementation, and (iii) the impact on the success probability and ansatz complexity is analyzed. The experiment results show that an improvement in the success probability can be achieved with only a moderate increase in circuit complexity.

Study of the positional and orientational contributions to the Helmholtz free energy of a finite hard-disk system. A molecular dynamics analysis of its hexatic transition

Based on an equilibrium thermodynamic scheme, this work studies the separation of the Helmholtz free energy of a hard-disk system into its positional and orientational contributions. For the positional part, an accurate two dimensional version of the Carnahan-Starling equation of state is proposed, while a simple ansatz of the one particle partition function for the orientational part is proposed. The latter expression, which contains information about the probability of the norm of the six-fold orientational order parameter, is obtained through molecular dynamics simulations. Perhaps due to finite size effects, the ordering of the fluid is not a sudden phenomenon: the fluid-hexatic phase transition is preceded by a linear increase in the orientational contribution to the pressure of the fluid. A similar result is obtained for the hexatic-solid phase transition. Thus, the hexatic first-order phase transition is located between two linear ordering processes, consistent with the KTHNY theory. The proposed separation also predicts the existence of first order fluid-hexatic phase transition by using two procedures: the first, based on a purely bimodal probability density function, produces a pressure with a van del Waals-like first-order phase transition, while the second, considering density fluctuations, results in a flattened pressure phase transition.

Control of neural transport for normalising flows

Inspired by normalising flows, we analyse the bilinear control of neural transport equations by means of time-dependent velocity fields restricted to fulfil, at any time instance, a simple neural network ansatz. The L1 approximate controllability property is proved, showing that any probability density can be driven arbitrarily close to any other one in any time horizon. The control vector fields are built explicitly and inductively and this provides quantitative estimates on their complexity and amplitude. This also leads to statistical error bounds when only random samples of the target probability density are available.

Fake Z

Recently introduced connections between quantum codes and Narain CFTs provide a simple ansatz to express a modular-invariant function Zleft(tau, overline{tau}right) in terms of a multivariate polynomial satisfying certain additional properties. These properties include algebraic identities, which ensure modular invariance of Zleft(tau, overline{tau}right) , and positivity and integrality of coefficients, which imply positivity and integrality of the \U0001d532(1)n × \U0001d532(1)n character expansion of Zleft(tau, overline{tau}right) . Such polynomials come naturally from codes, in the sense that each code of a certain type gives rise to the so-called enumerator polynomial, which automatically satisfies all necessary properties, while the resulting Zleft(tau, overline{tau}right) is the partition function of the code CFT — the Narain theory unambiguously constructed from the code. Yet there are also “fake” polynomials satisfying all necessary properties, that are not associated with any code. They lead to Zleft(tau, overline{tau}right) satisfying all modular bootstrap constraints (modular invariance and positivity and integrality of character expansion), but whether they are partition functions of any actual CFT is unclear. We consider the group of the six simplest fake polynomials and denounce the corresponding Z’s as fake: we show that none of them is the torus partition function of any Narain theory. Moreover, four of them are not partition functions of any unitary 2d CFT; our analysis for other two is inconclusive. Our findings point to an obvious limitation of the modular bootstrap approach: not every solution of the full set of torus modular bootstrap constraints is due to an actual CFT. In the paper we consider six simple examples, keeping in mind that thousands more can be constructed.

Narain CFTs and quantum codes at higher genus

Code CFTs are 2d conformal field theories defined by error-correcting codes. Recently, Dymarsky and Shapere generalized the construction of code CFTs to include quantum error-correcting codes. In this paper, we explore this connection at higher genus. We prove that the higher-genus partition functions take the form of polynomials of higher-weight theta functions, and that the higher-genus modular group acts as simple linear transformations on these polynomials. We explain how to solve the modular constraints explicitly, which we do for genus 2. The result is that modular invariance at genus 1 and genus 2 is much more constraining than genus 1 alone. This allows us to drastically reduce the space of possible code CFTs. We also consider a number of examples of “isospectral theories” — CFTs with the same genus 1 partition function — and we find that they have different genus 2 partition functions. Finally, we make connection to some 2d CFTs known from the modular bootstrap. The n = 4 theory conjectured to have the largest possible gap in Virasoro characters, the SO(8) WZW model, is a code CFT, allowing us to give an expression for its genus 2 partition function. We also find some other known CFTs which are not code theories but whose partition functions satisfy the same simple polynomial ansatz as the code theories. This leads us to speculate about the usefulness of the code polynomial form beyond the study of code CFTs.

Long-term degradation of high molar mass poly(ethylene oxide) in a turbulent pilot-scale pipe flow

The long-term drag reduction capability of poly(ethylene oxide) with a nominal molar weight of Mw=4×106 g/mol dissolved in water was investigated in a pilot-scale pipe flow device (inner diameter of test section 26 mm) at a Reynolds number of 105. A total loss of the initially high (75%) drag reduction capability was observed over a flow distance of several ∼10 km while the molar weight of the polymer was still Mw∼5×105 g/mol. Mechanical degradation in the turbulent flow as well as ageing of the polymer dissolved in water caused this loss in drag reduction capability. A simple ansatz of two independent, statistical polymer chain scission mechanisms was used to describe the polymer degradation empirically using a modified Brostow model. This empirical description was applied successfully and suggested that the polymer exhibited at least 15 cleavage points for mechanical degradation.

Novel solutions to the coupled KdV equations and the coupled system of variant Boussinesq equations

This article examines the coupled KdV equations and the coupled system of variant Boussinesq equations with beta time derivative and explores their travelling wave solutions. This work explains the evolution of waves with fractional parameter. The simple ansatz approach produces a variety of novel solutions in terms of hyperbolic and periodic functions. Graphical representation of solutions are also presented.

Density-matrix functional theory of the attractive Hubbard model: Statistical analogy of pairing correlations

The ground-state properties of the Hubbard model with attractive local pairing interactions are investigated in the framework of lattice density-functional theory. A remarkable correlation is revealed between the interaction-energy functional $W[\mathbit{\ensuremath{\eta}}]$ corresponding to the Bloch-state occupation-number distribution ${\ensuremath{\eta}}_{\ensuremath{\kappa}\ensuremath{\sigma}}$ and the entropy $S[\mathbit{\ensuremath{\eta}}]$ of a system of noninteracting fermions having the same ${\ensuremath{\eta}}_{\mathbit{k}\ensuremath{\sigma}}$. The relation between $W[\mathbit{\ensuremath{\eta}}]$ and $S[\mathbit{\ensuremath{\eta}}]$ is shown to be approximately linear for a wide range of ground-state representable occupation-number distributions ${\ensuremath{\eta}}_{\mathbit{k}\ensuremath{\sigma}}$. Taking advantage of this statistical analogy, a simple explicit ansatz for $W[\mathbit{\ensuremath{\eta}}]$ of the attractive Hubbard model is proposed, which can be applied to arbitrary periodic systems. The accuracy of this approximation is demonstrated by calculating the main ground-state properties of the model on several 1D and 2D bipartite and nonbipartite lattices and by comparing the results with exact diagonalizations.

Early time behavior of spatial and momentum anisotropies in kinetic theory across different Knudsen numbers

We investigate the early time development of the anisotropic transverse flow and spatial eccentricities of a fireball with various particle-based transport approaches using a fixed initial condition. In numerical simulations ranging from the quasi-collisionless case to the hydrodynamic regime, we find that the onset of v_n and of related measures of anisotropic flow can be described with a simple power-law ansatz, with an exponent that depends on the amount of rescatterings in the system. In the few-rescatterings regime we perform semi-analytical calculations, based on a systematic expansion in powers of time and the cross section, which can reproduce the numerical findings.

Modeling Spin-Dependent Nonadiabatic Dynamics with Electronic Degeneracy: A Phase-Space Surface-Hopping Method.

Nuclear Berry curvature effects emerge from electronic spin degeneracy and can lead to nontrivial spin-dependent (nonadiabatic) nuclear dynamics. However, such effects are not captured fully by any current mixed quantum-classical method such as fewest-switches surface hopping. In this work, we present a phase-space surface-hopping (PSSH) approach to simulate singlet-triplet intersystem crossing dynamics. We show that with a simple pseudodiabatic ansatz, a PSSH algorithm can capture the relevant Berry curvature effects and make predictions in agreement with exact quantum dynamics for a simple singlet-triplet model Hamiltonian. Thus, this approach represents an important step toward simulating photochemical and spin processes concomitantly, as relevant to intersystem crossing and spin-lattice relaxation dynamics.

Theory-agnostic framework for inspiral tests of general relativity with higher-harmonic gravitational waves

Recent gravitational wave observations show evidence for the presence of higher harmonics, thus possibly indicating that these waves were generated in the inspiral of compact objects with asymmetric mass ratios. Signals with higher harmonics contain a trove of information that can lead to a better estimation of system parameters and possibly to more stringent tests of general relativity. Gravitational wave model that include higher harmonics, however, have only been developed within general relativity, while models to test theory-agnostic deviations from general relativity have been purely based on the signal's dominant mode. We here extend the parameterized post-Einsteinian framework to include the $\ell=2, 3$ and $4$ higher harmonics to first post-Newtonian order, therefore providing a ready-to-use Fourier-domain waveform model for tests of general relativity with higher harmonics. We find that the deformations to the higher harmonics of the Fourier phase can be easily mapped to the deformation of the dominant harmonic, while the deformations to the higher-harmonics of the Fourier amplitude in general cannot in a theory-agnostic way. Nonetheless, we develop a simple ansatz for the deformations of the waveform amplitude (through a re-scaling deformation of the time-domain amplitude) that both minimizes the number of independent amplitude deformations parameters and captures the predictions of all known modified theories to date.

Phenomenological extraction of a universal TMD fragmentation function from single hadron production in $e^+ e^-$ annihilations

Factorizing the cross section for single hadron production in e^+e^-e+e− annihilations, differential in z_hzh, P_TPT and thrust, is a highly non trivial task. We have devised a factorization scheme that allows us to recast the e^+e^- \to hXe+e−→hX cross section in the convolution of a calculable hard coefficient and a universal Transverse Momentum Dependent (TMD) Fragmentation Function (FF). The predictions obtained from our NLO-LL perturbative computation, together with a simple ansatz to model the non-perturbative part of the TMD, are applied to the experimental measurements of the BELLE Collaboration for the phenomenological extraction of this process independent TMD FF.

Blast wave kinematics: theory, experiments, and applications

Measurements of the time of arrival of shock waves from explosions can serve as powerful markers of the evolution of the shock front for determining crucial parameters driving the blast. Using standard theoretical tools and a simple ansatz for solving the hydrodynamics equations, a general expression for the Mach number of the shock front is derived. Dimensionless coordinates are introduced allowing a straightforward visualization and direct comparison of blast waves produced by a variety of explosions, including chemical, nuclear, and laser-induced plasmas. The results are validated by determining the yield of a wide range of explosions, using data from gram-size charges to thermonuclear tests.

Schur indices of class S and quasimodular forms

We investigate the unflavoured Schur indices of class $\mathcal S$ theories of modest rank, and in the case of $\mathcal{N}=4$ super Yang-Mills theory with special unitary gauge group of somewhat more general rank, with an eye towards their modular properties. We find closed form expressions for many of these theories in terms of quasimodular forms of level one or two, with the curious feature that in general they are sums of quasimodular forms of different weights. For type $\mathfrak{a}_1$ theories, the index can be fixed by taking a simple Ansatz for the family of quasimodular forms appearing in the expansion of this type and demanding that the result be sufficiently regular as $q\to0$. For higher rank cases, an equally simple construction is lacking, but we nevertheless find that these indices can be expressed in terms of mixed-weight quasimodular forms.

Efficient Measure for the Expressivity of Variational Quantum Algorithms.

The superiority of variational quantum algorithms (VQAs) such as quantum neural networks (QNNs) and variational quantum eigensolvers (VQEs) heavily depends on the expressivity of the employed Ansätze. Namely, a simple Ansatz is insufficient to capture the optimal solution, while an intricate Ansatz leads to the hardness of trainability. Despite its fundamental importance, an effective strategy of measuring the expressivity of VQAs remains largely unknown. Here, we exploit an advanced tool in statistical learning theory, i.e., covering number, to study the expressivity of VQAs. Particularly, we first exhibit how the expressivity of VQAs with an arbitrary Ansätze is upper bounded by the number of quantum gates and the measurement observable. We next explore the expressivity of VQAs on near-term quantum chips, where the system noise is considered. We observe an exponential decay of the expressivity with increasing circuit depth. We also utilize the achieved expressivity to analyze the generalization of QNNs and the accuracy of VQE. We numerically verify our theory employing VQAs with different levels of expressivity. Our Letter opens the avenue for quantitative understanding of the expressivity of VQAs.