Hamiltonian Formulation Research Articles

Downfolding from ab initio to interacting model Hamiltonians: comprehensive analysis and benchmarking of the DFT+cRPA approach

Model Hamiltonians are regularly derived from first principles to describe correlated matter. However, the standard methods for this contain a number of largely unexplored approximations. For a strongly correlated impurity model system, here we carefully compare a standard downfolding technique with the best possible ground-truth estimates for charge-neutral excited-state energies and wave functions using state-of-the-art first-principles many-body wave function approaches. To this end, we use the vanadocene molecule and analyze all downfolding aspects, including the Hamiltonian form, target basis, double-counting correction, and Coulomb interaction screening models. We find that the choice of target-space basis functions emerges as a key factor for the quality of the downfolded results, while orbital-dependent double-counting corrections diminish the quality. Background screening of the Coulomb interaction matrix elements primarily affects crystal-field excitations. Our benchmark uncovers the relative importance of each downfolding step and offers insights into the potential accuracy of minimal downfolded model Hamiltonians.

Entanglement Hamiltonian for inhomogeneous free fermions

We study the entanglement Hamiltonian for the ground state of one-dimensional free fermions in the presence of an inhomogeneous chemical potential. In particular, we consider a lattice with a linear, as well as a continuum system with a quadratic potential. It is shown that, for both models, conformal field theory predicts a Bisognano–Wichmann form for the entanglement Hamiltonian of a half-infinite system. Furthermore, despite being nonrelativistic, this result is inherited by our models in the form of operators that commute exactly with the entanglement Hamiltonian. After appropriate rescaling, they also yield an excellent approximation of the entanglement spectra, which becomes asymptotically exact in the bulk of the trapped Fermi gas. For the gradient chain, however, the conformal result is recovered only after taking a proper continuum limit.

Hamiltonian/Stroh formalism for reversible poroelasticity (and thermoelasticity)

Stroh’s sextic formalism represents the equilibrium equations of anisotropic elasticity in a particularly attractive form, that is most suitable for studying interface-dominated multilayered solids, composite materials and time-harmonic problems. Taking advantage of the fact that the Stroh formalism really amounts to the canonical form of the equations in the Hamiltonian sense, the case of Biot’s reversible (i.e. no fluid dissipation) poroelasticity is here addressed, in the absence of a fluid pressure gradient. This framework is the same as thermoelasticity of perfect conductors. Two Hamiltonian formulations are developed: the first describes both the solid and the fluid phases and it exhibits, besides energy conservation, momentum conservation, as a result of pressure uniformity (perfectly drained conditions). The second is restricted to the solid skeleton and perfectly parallels anisotropic elasticity, where the Stroh matrices refer to the effective stress tensor. The case of weak fluid–solid coupling is also considered and it produces a perturbation from anisotropic elasticity with the same structure as incompressibility, although in an “opposing” manner. This comparison suggests that the incompressibility limit introduced by Biot should be revised. The energy conservation integral and the edge impedance matrix are also illustrated.

Massive propagating modes of torsion

The dynamics of the torsion field is analyzed in the framework of the Covariant Canonical Gauge Theory of Gravity (CCGG), a De Donder–Weyl Hamiltonian formulation of gauge gravity. The action is quadratic in both, the torsion and the Riemann–Cartan tensor. Since the latter adds the derivative of torsion to the equations of motion, torsion is no longer identical to spin density, as in the Einstein–Cartan theory, but an additional propagating degree of freedom. As torsion turns out to be totally anti-symmetric, it can be parametrised via a single axial vector. It is shown in this paper that, in the weak torsion limit, the axial vector obeys a wave equation with an effective mass term which is partially dependent on the scalar curvature. The source of torsion is thereby given by the fermion axial current which is the net fermionic spin density of the system. Possible measurable effects and approaches to experimental analysis are addressed. For example, neutron star mergers could act as a dipoles or quadrupoles for torsional radiation, and an analysis of radiation of pulsars could lead to a detection of torsion wave background radiation.

Gravitational form factors of charmonia

We investigate the gravitational form factors of charmonium. Our method is based on a Hamiltonian formalism on the light front known as basis light-front quantization. The charmonium mass spectrum and light-front wave functions were obtained from diagonalizing an effective Hamiltonian that incorporates confinement from holographic QCD and one-gluon exchange interaction from light-front QCD. We proposed a quantum many-body approach to construct the hadronic matrix elements of the energy momentum tensor T++ and T+−, which are used to extract the gravitational form factors A(Q2) and D(Q2). The obtained form factors satisfy the known constraints, e.g., the von Laue condition. From these quantities, we also extract the energy, pressure and light-front energy distributions of the system. We find that hadrons are multilayer systems. Published by the American Physical Society 2024

Classical and quantum computing of shear viscosity for (2+1)D SU(2) gauge theory

We perform a nonperturbative calculation of the shear viscosity for (2+1)-dimensional SU(2) gauge theory by using the lattice Hamiltonian formulation. The retarded Green’s function of the stress-energy tensor is calculated from real time evolution via exact diagonalization of the lattice Hamiltonian with a local Hilbert space truncation, and the shear viscosity is obtained via the Kubo formula. When taking the continuum limit, we account for the renormalization group flow of the coupling but no additional operator renormalization. We find the ratio of the shear viscosity and the entropy density ηs is consistent with a well-known holographic result 14π at several temperatures on a 4×4 honeycomb lattice with the local electric representation truncated at jmax=12. We also find the ratio of the spectral function and frequency ρxy(ω)ω exhibits a peak structure when the frequency is small. Both the exact diagonalization method and simple matrix product state classical simulation method beyond jmax=12 on bigger lattices require exponentially growing resources. So we develop a quantum computing method to calculate the retarded Green’s function and analyze various systematics of the calculation including jmax truncation and finite size effects, Trotter errors and the thermal state preparation efficiency. Our thermal state preparation method still requires resources that grow exponentially with the lattice size, but with a very small prefactor at high temperature. We test our quantum circuit on both the Quantinuum emulator and the IBM simulator for a small lattice and obtain results consistent with the classical computing ones. Published by the American Physical Society 2024

Exergy and dynamics analyses in centrifugal turbomachinery pressurized long-distance natural gas pipelines based on Hamiltonian model

Natural gas pipelines represent intricate systems comprising diverse components, which is crucial to industry, yet demanding substantial energy consumption. However, the nuanced energy and exergy transfer dynamics within these pipelines, particularly during dynamic processes, remain relatively unexplored. Given the distributed nature and complicated component coupling among pipeline elements, coolers, and compressors, traditional energy analysis tools face challenges in their application. In this study, we introduce a novel approach utilizing Hamiltonian formalism to establish an energy and exergy analysis framework for natural gas pipelines. Specifically, we develop Hamiltonian energy functions for various components, including pipes, centrifugal compressors, and coolers, accounting for dynamic mechanisms. Through comprehensive simulations covering static and dynamic scenarios, we elucidate energy and exergy transfer processes, leveraging Hamiltonian energy function terms. Our findings validate the efficacy of the proposed analysis framework in accurately characterizing energy transfer and dissipation processes. Furthermore, validation with field data demonstrates the capability of our method to online exhibit energy transfer features effectively.

Z3 lattice gauge theory as a toy model for dense QCD

We propose the (3+1)-dimensional Z3 lattice gauge theory coupled with the 2-flavor Wilson-Dirac fermion as a toy model for studying quantum chromodynamics (QCD) at nonzero density. We study its phase diagram in the space of the lattice gauge couplings g2 and the quark chemical potentials μ and discuss the similarity and difference compared with anticipated behaviors of actual QCD. This model also provides a testing ground for various algorithms of the numerical Hamiltonian formalism as its Hilbert space is finite-dimensional in a finite box. Published by the American Physical Society 2024

The Herglotz variational principle for dissipative field theories

In the recent years, with the incorporation of contact geometry, there has been a renewed interest in the study of dissipative or non-conservative systems in physics and other areas of applied mathematics. The equations arising when studying contact Hamiltonian systems can also be obtained via the Herglotz variational principle. The contact Lagrangian and Hamiltonian formalisms for mechanical systems has also been generalized to field theories. The main goal of this paper is to develop a generalization of the Herglotz variational principle for first-order and higher-order field theories. In order to illustrate this, we study three examples: the damped vibrating string, the Korteweg–de Vries equation, and an academic example showing that the non-holonomic and the vakonomic variational principles are not fully equivalent.

Free scalar field theory on a Sobolev space over a bounded interval

This paper discusses several functional analytic issues relevant for field theories in the context of the Hamiltonian formulation for a free, massless, scalar field defined on a closed interval of the real line. The fields that we use belong to a Sobolev space with a scalar product. As we show this choice is useful because it leads to an explicit representation of the points in the fibers of the phase space (the cotangent bundle of the configuration space). The dynamical role of the boundary of the spatial manifold where the fields are defined is analyzed.

Quantum transitions, detailed balance, black holes, and nothingness

We consider vacuum transitions by bubble nucleation among 4D vacua with different values and signs of the cosmological constant Λ, including both up and down tunnelings. Following the Hamiltonian formalism, we explicitly compute the transition probabilities for all possible combinations of initial and final values of Λ and find that up tunneling is allowed starting not only from dS spacetime but also from AdS and Minkowski spacetimes. We trace the difference with the Euclidean approach, where these transitions are found to be forbidden, to the difference of treating the latter spacetimes as pure (vacuum) states rather than mixed states with correspondingly vanishing or infinite entropy. We point out that these transitions are best understood as limits of the corresponding transitions with black holes in the zero mass limit M→0. We find that detailed balance is satisfied provided we use the Hartle-Hawking sign of the wave function for nucleating spacetimes. In the formal limit Λ→−∞, the transition rates for anti–de Sitter (AdS) to dS agree with both the Hartle-Hawking and Vilenkin amplitudes for the creation of dS from nothing. This is consistent with a proposal of Brown and Dahlen to define “nothing” as AdS in this limit. For M≠0, the detailed balance is satisfied only in a range of mass values. We compute the bubble trajectory after nucleation and find that, contrary to the M=0 case, the trajectory does not correspond to the open universe slicing of dS. We briefly discuss the relevance of our results to the string landscape. Published by the American Physical Society 2024

Hamiltonian formalism for Bose excitations in a plasma with a non-Abelian interaction I: Plasmon – hard particle scattering

Hamiltonian theory for collective longitudinally polarized gluon excitations (plasmons) interacting with classical high-energy test color-charged particle propagating through a high-temperature gluon plasma is developed. A generalization of the Lie-Poisson bracket to the case of a continuous medium involving bosonic normal field variable ak⁎a and a non-Abelian color charge Qa is performed and the corresponding Hamilton equations are presented. The canonical transformations including simultaneously both bosonic degrees of freedom of the soft collective excitations and degree of freedom of hard test particle connecting with its color charge in the hot gluon plasma are written out. A complete system of the canonicity conditions for these transformations is derived. The notion of the plasmon number density Nkaa1′, which is a nontrivial matrix in the color space, is introduced. An explicit form of the effective fourth-order Hamiltonian describing the elastic scattering of a plasmon off a hard color particle is found and the self-consistent system of Boltzmann-type kinetic equations taking into account the time evolution of the mean value of the color charge of the hard particle is obtained. On the basis of these equations, a model problem of the interaction of two infinitely narrow wave packets is considered. A system of nonlinear first-order ordinary differential equations defining the dynamics of the interaction of the colorless Nkl and color Wkl components of the plasmon number density is derived. The problem of determining the third- and fourth-order coefficient functions entering into the canonical transformations of the original bosonic variable ak⁎a and color charge Qa is discussed. With the help of the coefficient functions obtained, a complete effective amplitude of the elastic scattering of plasmon off hard test particle is written out.

Heterotic integrable deformation of the principal chiral model

A novel classically integrable model is proposed. It is a deformation of the two-dimensional principal chiral model, embedded into a heterotic σ model by a particular heterotic gauge field. This is inspired by the bosonic part of the heterotic σ model and its recent Hamiltonian formulation in terms of O(d,d+n)-generalized geometry in Hatsuda [Gauged double field theory, current algebras and heterotic sigma models, ]. Classical integrability is shown by the construction of a Lax pair and a classical R matrix. The latter is almost of the canonical form with a twist function and solves the classical Yang-Baxter equation. Published by the American Physical Society 2024

Stochastic modified equations for symplectic methods applied to rough Hamiltonian systems

Abstract We investigate stochastic modified equations to explain the mathematical mechanism of symplectic methods applied to rough Hamiltonian systems. The contribution of this paper is threefold. First, we construct a new type of stochastic modified equation. For symplectic methods applied to rough Hamiltonian systems, the associated stochastic modified equations are proved to have Hamiltonian formulations. Secondly, the pathwise convergence order of the truncated modified equation to the numerical method is obtained by techniques in rough path theory. Thirdly, if increments of noises are simulated by truncated random variables, we show that the error can be made exponentially small with respect to the time step size.

Coarse graining with control points: a cubic-Bézier based approach to modeling athermal fibrous materials

Fibrous materials continue to be of central importance to modern life, including traditional materials, such as textiles and paper, modern materials, such as polymeric materials and glass fiber, and emerging materials, such as carbon nanotube yarns. Developing constitutive laws for the mechanical behavior of such materials can be challenging owing to the often non-periodic microstructure, resulting in fairly large and computationally intractable representative volume elements. Therefore, it is imperative to design novel coarse-grained models that can bridge the computational gap and capture accurate physics associated with the mechanical behavior of fibrous materials. In this work, cubic-Béziers are used to represent fiber-segments, allowing for up to C 2(curvature)-continuous representation of curved fibers. Equations of motion are derived in terms of the control points, which allow for minimization and time integration strategies with the Lagrangian and Hamiltonian formalism. This coarse-grained model promises faster and more accurate computational performance compared to discrete approaches.

Design of optimal controller for static compensator via Hamiltonian formalism for the multimachine system

Design of optimal controller for static compensator via Hamiltonian formalism for the multimachine system

A novel and efficient Hamiltonian dynamic analysis approach for constraint force determination in flexible multibody systems

In the dynamic analysis of flexible multibody systems, Hamiltonian formulations offer advantages in numerical stabilization and systematic handling of systems with varying mass. However, current approaches face challenges. Differential algebraic equations (DAEs) can directly express constraint forces but are computationally inefficient, while ordinary differential equations (ODEs) are more computationally efficient but cannot directly represent constraint forces due to the elimination of the Lagrange multiplier. This paper presents an innovative and efficient dynamic analysis approach based on the Hamiltonian formulation, incorporating velocity transformation and open-constraint coordinate methods. Compared to conventional DAE-based Hamiltonian formulations, our approach conducts analysis efficiently using only independent variables. Compared to conventional ODE-based Hamiltonian formulations, our approach effectively expresses constraint forces through Lagrange multiplier reformulation. Furthermore, when compared to the traditional ODE-based Lagrangian formulation, our approach exhibits superior computational efficiency. Numerical simulations assess our proposed formulation, showing agreement with conventional formulations, shorter calculation time, and alignment with analytical results, confirming the accuracy and usefulness of our approach.

An infinite dimensional KAM theorem with normal degeneracy

In this paper, we consider a classical Hamiltonian normal form with degeneracy in the normal direction. In previous results, one needs to assume that the perturbation satisfies certain non-degenerate conditions in order to remove the degeneracy in the normal form. Instead of that, we introduce a topological degree condition and a weak convexity condition, which are easy to verify, and we prove the persistence of lower dimensional tori without any restriction on perturbation but only smallness and analyticity.

Experiment-Driven Atomistic Materials Modeling: A Case Study Combining X-Ray Photoelectron Spectroscopy and Machine Learning Potentials to Infer the Structure of Oxygen-Rich Amorphous Carbon.

An important yet challenging aspect of atomistic materials modeling is reconciling experimental and computational results. Conventional approaches involve generating numerous configurations through molecular dynamics or Monte Carlo structure optimization and selecting the one with the closest match to experiment. However, this inefficient process is not guaranteed to succeed. We introduce a general method to combine atomistic machine learning (ML) with experimental observables that produces atomistic structures compatible with experiment by design. We use this approach in combination with grand-canonical Monte Carlo within a modified Hamiltonian formalism, to generate configurations that agree with experimental data and are chemically sound (low in energy). We apply our approach to understand the atomistic structure of oxygenated amorphous carbon (a-COx), an intriguing carbon-based material, to answer the question of how much oxygen can be added to carbon before it fully decomposes into CO and CO2. Utilizing an ML-based X-ray photoelectron spectroscopy (XPS) model trained from GW and density functional theory (DFT) data, in conjunction with an ML interatomic potential, we identify a-COx structures compliant with experimental XPS predictions that are also energetically favorable with respect to DFT. Employing a network analysis, we accurately deconvolve the XPS spectrum into motif contributions, both revealing the inaccuracies inherent to experimental XPS interpretation and granting us atomistic insight into the structure of a-COx. This method generalizes to multiple experimental observables and allows for the elucidation of the atomistic structure of materials directly from experimental data, thereby enabling experiment-driven materials modeling with a degree of realism previously out of reach.

Almost gauge-invariant states and the ground state of Yang-Mills theory

We consider the problem of the explicit description of the gauge-invariant subspace of pure lattice gauge theories in the Hamiltonian formulation, where the gauge group is either a compact Lie group or a finite group. The latter case is particularly interesting for quantum simulation. A basis of states where configurations are grouped according to their holonomies is shown to have several advantages over other descriptions. Using this basis, we compute some properties of interest for some non-Abelian finite groups on small lattices, and in particular we examine the question of whether a certain ansatz introduced long ago is a good approximation for the ground state. Published by the American Physical Society 2024