Diagrammatic Scheme Research Articles

Geometric phase predicts locomotion performance in undulating living systems across scales

Self-propelling organisms locomote via generation of patterns of self-deformation. Despite the diversity of body plans, internal actuation schemes and environments in limbless vertebrates and invertebrates, such organisms often use similar traveling waves of axial body bending for movement. Delineating how self-deformation parameters lead to locomotor performance (e.g. speed, energy, turning capabilities) remains challenging. We show that a geometric framework, replacing laborious calculation with a diagrammatic scheme, is well-suited to discovery and comparison of effective patterns of wave dynamics in diverse living systems. We focus on a regime of undulatory locomotion, that of highly damped environments, which is applicable not only to small organisms in viscous fluids, but also larger animals in frictional fluids (sand) and on frictional ground. We find that the traveling wave dynamics used by mm-scale nematode worms and cm-scale desert dwelling snakes and lizards can be described by time series of weights associated with two principal modes. The approximately circular closed path trajectories of mode weights in a self-deformation space enclose near-maximal surface integral (geometric phase) for organisms spanning two decades in body length. We hypothesize that such trajectories are targets of control (which we refer to as "serpenoid templates"). Further, the geometric approach reveals how seemingly complex behaviors such as turning in worms and sidewinding snakes can be described as modulations of templates. Thus, the use of differential geometry in the locomotion of living systems generates a common description of locomotion across taxa and provides hypotheses for neuromechanical control schemes at lower levels of organization.

Diagrammatics for the inverse problem in spin systems and simple liquids

Modeling complex systems, like neural networks, simple liquids or flocks of birds, often works in reverse to textbook approaches: given data for which averages and correlations are known, we try to find the parameters of a given model consistent with it. In general, no exact calculation directly from the model is available and we are left with expensive numerical approaches. A particular situation is that of a perturbed Gaussian model with polynomial corrections for continuous degrees of freedom. Indeed perturbation expansions for this case have been implemented in the last 60 years. However, there are models for which the exactly solvable part is non-Gaussian, such as independent Ising spins in a field, or an ideal gas of particles. We implement a diagrammatic perturbative scheme in weak correlations around a non-Gaussian yet solvable probability weight. This applies in particular to spin models (Ising, Potts, Heisenberg) with weak couplings, or to a simple liquid with a weak interaction potential. Our method casts systems with discrete degrees of freedom and those with continuous ones within the same theoretical framework. When the core theory is Gaussian it reduces to the well-known Feynman diagrammatics.

Erratum: Pseudogauge field driven acoustoelectric current in two-dimensional hexagonal Dirac materials [Phys. Rev. B 105 , 125407 (2022)

Using a diagrammatic scheme, we study the acoustoelectric effects in two-dimensional (2D) hexagonal Dirac materials due to the sound-induced pseudo-gauge field. We analyze both uniform and {\em spatially dispersive} currents in response to copropagating and counterpropagating sound waves, respectively. In addition to the longitudinal acoustoelectric current, we obtain an exotic {\em transverse} charge current flowing perpendicular to the sound propagation direction owing to the interplay of transverse and longitudinal gauge field components $j_T\propto A_L A^\ast_T$. In contrast to the almost isotropic directional profile of the longitudinal uniform current, a highly anisotropic transverse component $j_T\sim\sin(6\theta)$ is achieved that stems from the inherited three-fold symmetry of the hexagonal lattice. However, both longitudinal and transverse parts of the dispersive current are predicted to be strongly anisotropic $\sim\sin^2(3\theta)$ or $\cos^2(3\theta)$. We quantitatively estimate the pseudogauge field contribution to the acoustoelectric current that can be probed in future experiments in graphene and other 2D hexagonal Dirac materials.

Pseudogauge field driven acoustoelectric current in two-dimensional hexagonal Dirac materials

Using a diagrammatic scheme, we study the acoustoelectric effects in two-dimensional (2D) hexagonal Dirac materials due to the sound-induced pseudogauge field. We analyze both uniform and spatially dispersive currents in response to copropagating and counterpropagating sound waves, respectively. In addition to the longitudinal acoustoelectric current, we obtain an exotic transverse charge current flowing perpendicular to the sound propagation direction owing to the interplay of transverse and longitudinal gauge field components ${j}_{T}\ensuremath{\propto}{A}_{L}{A}_{T}^{*}$. In contrast to the almost isotropic directional profile of the longitudinal uniform current, a highly anisotropic transverse component ${j}_{T}\ensuremath{\sim}sin(6\ensuremath{\theta})$ is achieved that stems from the inherited threefold symmetry of the hexagonal lattice. However, both longitudinal and transverse parts of the dispersive current are predicted to be strongly anisotropic $\ensuremath{\sim}{sin}^{2}(3\ensuremath{\theta})$ or ${cos}^{2}(3\ensuremath{\theta})$. We quantitatively estimate the pseudogauge field contribution to the acoustoelectric current that can be probed in future experiments in graphene and other 2D hexagonal Dirac materials.

A diagrammatic approach to variational quantum ansatz construction

Variational quantum eigensolvers (VQEs) are a promising class of quantum algorithms for preparing approximate ground states in near-term quantum devices. Minimizing the error in such an approximation requires designing ansatzes using physical considerations that target the studied system. One such consideration is size-extensivity, meaning that the ground state quantum correlations are to be compactly represented in the ansatz. On digital quantum computers, however, the size-extensive ansatzes usually require expansion via Trotter-Suzuki methods. These introduce additional costs and errors to the approximation. In this work, we present a diagrammatic scheme for the digital VQE ansatzes, which is size-extensive but does not rely on Trotterization. We start by designing a family of digital ansatzes that explore the entire Hilbert space with the minimum number of free parameters. We then demonstrate how one may compress an arbitrary digital ansatz, by enforcing symmetry constraints of the target system, or by using them as parent ansatzes for a hierarchy of increasingly long but increasingly accurate sub-ansatzes. We apply a perturbative analysis and develop a diagrammatic formalism that ensures the size-extensivity of generated hierarchies. We test our methods on a short spin chain, finding good convergence to the ground state in the paramagnetic and the ferromagnetic phase of the transverse-field Ising model.

Triangular Scheme Revisited in the Light of n-permutable Categories

The first diagrammatic scheme was developed by H.P. Gumm under the name Shifting Lemma in case to characterize congruence modularity. A diagrammatic scheme is developed for the generalized semi distributive law in Mal'tsev categories. In this paper we study this diagrammatic scheme in the context of $n$-permutable, and of Mal'tsev categories in particular. Several remarks concerning the Triangular scheme case are included.

The end of primary moult as an indicator of global warming effects in the Red-legged Partridge Alectoris rufa, a medium sized, sedentary species

Global warming affects ecosystem services, communities and populations, influencing the physiology, behaviour or environment of living beings, and hence impacts its survival or breeding. Identifying species susceptibility to warming is relevant in assessing risks to animal populations and ecological processes. The progressive increase in ambient temperature as a result of global warming might have an effect on the timing of primary moult. This could affect a bird’s annual cycle, influencing reproductive success and population dynamics.We describe a method to examine the potential effects of global warming on the primary moult process in a sedentary population of Red-legged Partridges (Alectoris rufa). We organised the factors that might influence the timing of moult end into a network and distinguished between environmental and intrinsic factors. We sorted the factors according to their contribution to quantitative moult models and constructed a diagrammatic scheme showing their interactions and effect on the end of primary moult over the annual cycle.In Red-legged Partridges, the timing of the end of moult varies according to age-sex class. We found no timing differences by age, but found significant timing differences by adult sex. More females overlap their moult with juveniles than males because female parental effort is higher, more females incubate and brood chicks. The timing of the end of moult varies by year due to conspecific interactions that change according to influences of the weather, habitat, and social and flock conditions. Parent birds synchronize their primary moult with the chick's growth, degree of cover and food resources. From the time of hatching to the following year, the date (day-length), social factors (conspecific interactions), and weather (resources) affect the timing of moult and the birds’ annual cycle.Global warming affects the timing of the end of moult and that of the annual cycle. If the extent of the breeding period is shortened, there could be a negative effect on population outcomes. Middle size prey species are key in trophic chains. Our results suggest that the timing of the end of moult could be used as a proxy measure of warming impacts on wildlife and ecosystems and also as a tool for the management of game birds.

Decision Theory, Relative Plausibility and the Criminal Standard of Proof

The evolution of the understanding of evidence-based proof and decision processes in the law, especially criminal law, and standards of proof in this area, has a long-standing and controversial history. Competing accounts cause the legal scholarship to engage in critical and thoughtful exchanges. Some of the divergent views reflect different methodological perspectives similarly recognized in other fields, such as applied psychology and economy, and the broader interdisciplinary research fields of judgment and decision-making, system analysis and decision science. One such methodological perspective asserts that accounts of juridical proof should provide a description and explanation of how the legal system actually works as a whole. Other—more mathematical and analytical accounts—concentrate on how, ideally, legal decision-making under uncertainty ought to be made in order be considered sensible. This paper focuses on the relative plausibility (RP) account advocated by Professors Allen and Pardo as an example of the former perspective. Its logical structure and argumentative implications are analysed using elements of decision theory, which is the prime representative of the latter, more mathematical approach to legal proof. Using formal diagrammatic schemes to depict the structural relationships between the core elements of the two accounts, it is demonstrated in what sense they can be considered logically related and congruent. The demonstration shows that the principal disagreements among the proponents of the two examined theories derive from differences in (1) the criteria used to judge the adequacy of competing accounts of legal decision-making, and (2) the level of formalization of the bases of decisions in each candidate account. This structural analysis supports the view that adherence to one or the other of the examined perspectives does not imply a contradiction, but reflects the coverage of different aspects of the same overall decision architecture. Using decision-theoretic notions, our analyses also provide a way to explain RP decisions through an explicit criterion, thus providing a reply to the recurrent critique that RP theory lacks specific means to justify its decisional framework.

Spectra of heavy polarons and molecules coupled to a Fermi sea

We study the spectrum of an impurity coupled to a Fermi sea (e.g., minority atom in an ultracold gas, exciton in a solid) by attraction strong enough to form a molecule/trion. We introduce a diagrammatic scheme which allows treating a finite mass impurity while reproducing the Fermi edge singularity in the immobile limit. For large binding energies the spectrum is characterized by a semi-coherent repulsive polaron and an incoherent molecule-hole continuum, which is the lowest-energy feature in the single-particle spectrum. The previously predicted attractive polaron seems not to exist for strong binding.

Landau Effective Interaction between Quasiparticles in a Bose-Einstein Condensate

Landau's description of the excitations in a macroscopic system in terms of quasiparticles stands out as one of the highlights in quantum physics. It provides an accurate description of otherwise prohibitively complex many-body systems, and has led to the development of several key technologies. In this paper, we investigate theoretically the Landau effective interaction between quasiparticles, so-called Bose polarons, formed by impurity particles immersed in a Bose-Einstein condensate (BEC). In the limit of weak interactions between the impurities and the BEC, we derive rigorous results for the effective interaction. They show that it can be strong even for weak impurity-boson interaction, if the transferred momentum/energy between the quasiparticles is resonant with a sound mode in the BEC. We then develop a diagrammatic scheme to calculate the effective interaction for arbitrary coupling strengths, which recovers the correct weak coupling results. Using this, we show that the Landau effective interaction in general is significantly stronger than that between quasiparticles in a Fermi gas, mainly because a BEC is more compressible than a Fermi gas. The interaction is particularly large near the unitarity limit of the impurity-boson scattering, or when the quasiparticle momentum is close to the threshold for momentum relaxation in the BEC. Finally, we show how the Landau effective interaction leads to a sizeable shift of the quasiparticle energy with increasing impurity concentration, which should be detectable with present day experimental techniques.

Topological Constraints and Their Conformational Entropic Penalties on RNA Folds

Functional RNAs can fold into intricate structures using a number of different secondary and tertiary structural motifs. Many factors contribute to the overall free energy of the target fold. This study aims at quantifying the entropic costs coming from the loss of conformational freedom when the sugar-phosphate backbone is subjected to constraints imposed by secondary and tertiary contacts. Motivated by insights from topology theory, we design a diagrammatic scheme to represent different types of RNA structures so that constraints associated with a folded structure may be segregated into mutually independent subsets, enabling the total conformational entropy loss to be easily calculated as a sum of independent terms. We used high-throughput Monte Carlo simulations to simulate large ensembles of single-stranded RNA sequences in solution to validate the assumptions behind our diagrammatic scheme, examining the entropic costs for hairpin initiation and formation of many multiway junctions. Our diagrammatic scheme aids in the factorization of secondary/tertiary constraints into distinct topological classes and facilitates the discovery of interrelationships among multiple constraints on RNA folds. This perspective, which to our knowledge is novel, leads to useful insights into the inner workings of some functional RNA sequences, demonstrating how they might operate by transforming their structures among different topological classes.

Bose Polarons at Finite Temperature and Strong Coupling.

A mobile impurity coupled to a weakly interacting Bose gas, a Bose polaron, displays several interesting effects. While a single attractive quasiparticle is known to exist at zero temperature, we show here that the spectrum splits into two quasiparticles at finite temperatures for sufficiently strong impurity-boson interaction. The ground state quasiparticle has minimum energy at T_{c}, the critical temperature for Bose-Einstein condensation, and it becomes overdamped when T≫T_{c}. The quasiparticle with higher energy instead exists only below T_{c}, since it is a strong mixture of the impurity with thermally excited collective Bogoliubov modes. This phenomenology is not restricted to ultracold gases, but should occur whenever a mobile impurity is coupled to a medium featuring a gapless bosonic mode with a large population for finite temperature.

Kinetic field theory: effects of momentum correlations on the cosmic density-fluctuation power spectrum

In earlier work, we have developed a kinetic field theory (KFT) for cosmological structure formation and showed that the nonlinear density-fluctuation power spectrum known from numerical simulations can be reproduced quite well even if particle interactions are taken into account to first order only. Besides approximating gravitational interactions, we had to truncate the initial correlation hierarchy of particle momenta at the second order. Here, we substantially simplify KFT. We show that its central object, the free generating functional, can be factorised, taking the full hierarchy of momentum correlations into account. The factors appearing in the generating functional, which we identify as nonlinearly evolved density-fluctuation power spectra, have a universal form and can thus be tabulated for fast access in perturbation schemes. In this paper, we focus on a complete evaluation of the free generating functional of KFT, not including particle interactions yet. This implies that the nonlinearly evolved power spectra contain a damping term which reflects that structures are being wiped out at late times by free streaming. Once particle interactions will be taken into account, they will compensate this damping. If we suppress this damping in a way suggested by the fluctuation-dissipation relations of KFT, our results show that the complete hierarchy of initial momentum correlations is responsible for a large part of the characteristic nonlinear deformation and the mode transport in the density-fluctuation power spectrum. Without any adjustable parameters, KFT accurately reproduces the scale at which nonlinear evolution sets in. Finally, we further develop perturbation theory based on the factorisation of the generating functional and propose a diagrammatic scheme for the perturbation terms.

Steady-state skewness and kurtosis from renormalized cumulants in (2 + 1)-dimensional stochastic surface growth

The phenomenon of stochastic growth of a surface on a two-dimensional substrate occurs in Nature in a variety of circumstances and its statistical characterization requires the study of higher order cumulants. Here, we consider the statistical cumulants of height fluctuations governed by the (2 + 1)-dimensional KPZ equation for flat geometry. We follow a diagrammatic scheme to derive the expressions for renormalized cumulants up to fourth order in the stationary state. Assuming a value for the roughness exponent from reliable numerical predictions, we calculate the second, third and fourth cumulants, yielding skewness S = 0.2879 and kurtosis Q = 0.1995. These values agree well with the available numerical estimations.

Shifted-action expansion and applicability of dressed diagrammatic schemes

While bare diagrammatic series are merely Taylor expansions in powers of interaction strength, dressed diagrammatic series, built on fully or partially dressed lines and vertices, are usually constructed by reordering the bare diagrams, which is an a priori unjustified manipulation, and can even lead to convergence to an unphysical result [Kozik, Ferrero and Georges, PRL 114, 156402 (2015)]. Here we show that for a broad class of partially dressed diagrammatic schemes, there exists an action $S^{(\xi)}$ depending analytically on an auxiliary complex parameter $\xi$, such that the Taylor expansion in $\xi$ of correlation functions reproduces the original diagrammatic series. The resulting applicability conditions are similar to the bare case. For fully dressed skeleton diagrammatics, analyticity of $S^{(\xi)}$ is not granted, and we formulate a sufficient condition for converging to the correct result.

On the dangers of partial diagrammatic summations: Benchmarks for the two-dimensional Hubbard model in the weak-coupling regime

We study the two-dimensional Hubbard model in the weak-coupling regime and compare the self-energy obtained from various approximate diagrammatic schemes to the result of diagrammatic Monte Carlo simulations, which sum up all weak-coupling diagrams up to a given order. While dynamical mean-field theory provides a good approximation for the local part of the self-energy, including its frequency dependence, the partial summation of bubble and/or ladder diagrams typically yields worse results than second order perturbation theory. Even widely used self-consistent schemes such as GW or the fluctuation-exchange approximation (FLEX) are found to be unreliable. Combining the dynamical mean-field self-energy with the nonlocal component of GW in GW+DMFT yields improved results for the local self-energy and nonlocal self-energies of the correct order of magnitude, but here, too, a more reliable scheme is obtained by restricting the nonlocal contribution to the second order diagram. FLEX+DMFT is found to give accurate results in the low-density regime, but even worse results than FLEX near half-filling.

Kurtosis of height fluctuations in (1 + 1) dimensional KPZ dynamics

We study the fourth order normalized cumulant of height fluctuations governed by 1 + 1 dimensional Kardar–Parisi–Zhang (KPZ) equation for a growing surface. Following a diagrammatic renormalization scheme, we evaluate the kurtosis Q from the connected diagrams leading to the value Q = 0.1523 in the large-scale long-time limit.

A visual formalism for weights satisfying reverse inequalities

In this expository article we introduce a diagrammatic scheme to represent reverse classes of weights and some of their properties.

Magnon self-energy in the correlated ferromagnetic Kondo lattice model: Spin-charge coupling effects on magnon excitations in manganites

Magnon self-energy due to spin-charge coupling is calculated for the correlated ferromagnetic Kondo lattice model (FKLM) including the intraorbital Coulomb repulsion term $U$ for the band electrons using a diagrammatic expansion scheme. By systematically incorporating correlation effects in the form of self-energy and vertex corrections, the expansion scheme explicitly preserves the continuous spin-rotation symmetry and hence the Goldstone mode. Due to a near cancellation of the correlation-induced quantum correction terms at intermediate coupling and optimal band filling relevant for ferromagnetic manganites, the renormalized magnon energies for the correlated FKLM are nearly independent of correlation term. Even at higher-band fillings, despite exhibiting overall non-Heisenberg behavior, magnon dispersion in the $\ensuremath{\Gamma}$-$X$ direction retains nearly Heisenberg form. Therefore, the experimentally observed doping-dependent zone-boundary magnon softening must be ascribed to spin-orbital coupling effects.

Finite-frequency noise properties of the nonequilibrium Anderson impurity model

We analyze the spectrum of the electric-current autocorrelation function (noise power) in the Anderson impurity model biased by a finite transport voltage. Special emphasis is placed on the interplay of non-equilibrium effects and electron-electron interactions. Analytic results are presented for a perturbation expansion in the interaction strength $U$. Compared to the non-interacting setup we find a suppression of noise for finite frequencies in equilibrium and an amplification in non-equilibrium. Furthermore, we use a diagrammatic resummation scheme to obtain non-perturbative results in the regime of intermediate $U$. At finite voltage, the noise spectrum shows sharp peaks at positions related to the Kondo temperature instead of the voltage.