Landau Energy Research Articles

Gradient flows and randomised thresholding: sparse inversion and classification* *The author thanks the anonymous reviewers for their helpful remarks.

Sparse inversion and classification problems are ubiquitous in modern data science and imaging. They are often formulated as non-smooth minimisation problems. In sparse inversion, we minimise, e.g., the sum of a data fidelity term and an L1/LASSO regulariser. In classification, we consider, e.g., the sum of a data fidelity term and a non-smooth Ginzburg–Landau energy. Standard (sub)gradient descent methods have shown to be inefficient when approaching such problems. Splitting techniques are much more useful: here, the target function is partitioned into a sum of two subtarget functions—each of which can be efficiently optimised. Splitting proceeds by performing optimisation steps alternately with respect to each of the two subtarget functions. In this work, we study splitting from a stochastic continuous-time perspective. Indeed, we define a differential inclusion that follows one of the two subtarget function’s negative subdifferential at each point in time. The choice of the subtarget function is controlled by a binary continuous-time Markov process. The resulting dynamical system is a stochastic approximation of the underlying subgradient flow. We investigate this stochastic approximation for an L1-regularised sparse inversion flow and for a discrete Allen–Cahn equation minimising a Ginzburg–Landau energy. In both cases, we study the longtime behaviour of the stochastic dynamical system and its ability to approximate the underlying subgradient flow at any accuracy. We illustrate our theoretical findings in a simple sparse estimation problem and also in low- and high-dimensional classification problems.

The Use of Dirac Oscillator as Medium Substrate for Quantum Heat Engine

The research on Dirac oscillators has been increasing recently. In this paper, a quantum heat engine by means of a Dirac oscillator that interacts with the external magnetic field is proposed. Relativistic Landau energy levels are used to perform the iso-energetic cycle. The large magnetic field is set to obtain a perfect thermodynamic cycle. Some stable performances at a certain magnitude of magnetic field and expansion parameter range are obtained. When the value of efficiency is compared with the non-relativistic case, an opposite result occurs. Therefore, a quantum heat engine using a Dirac oscillator doesn’t govern like a classical oscillator.

Open bottom mesons and upsilon states in hot magnetized strange hadronic matter

The masses of open bottom mesons ([Formula: see text]([Formula: see text],[Formula: see text]), [Formula: see text]([Formula: see text],[Formula: see text]), [Formula: see text]([Formula: see text], [Formula: see text])) and upsilon states ([Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]) are investigated in the isospin asymmetric strange hadronic medium at finite temperature in the presence of strong magnetic fields using a chiral effective Lagrangian approach. Here the chiral SU(3) Lagrangian is generalized to include the bottom sector to incorporate the interactions of the open bottom mesons with the magnetized medium. At finite temperature, the number density and scalar density of baryons are expressed in terms of thermal distribution functions. For charged baryons, the magnetic field introduces contribution from Landau energy levels. The masses of the open bottom mesons get modified through their interactions with the baryons and the scalar mesons, which undergo modifications in a magnetized medium. The charged [Formula: see text], [Formula: see text] mesons have additional positive mass shifts due to Landau quantization in the presence of the magnetic field. The medium mass shift of the upsilon states originates from the modification of the gluon condensates simulated by the variation of dilaton field ([Formula: see text]) and a quark mass term in the magnetized medium. The open bottom mesons and upsilon states experience a mass drop in the magnetized medium. The masses of these mesons initially increase with a rise in temperature, and beyond a high value of temperature, their masses are observed to drop. When the temperature is below 90[Formula: see text]MeV, the in-medium masses of the mesons increase with an increase in the magnetic field. However, at high temperatures (T > 90 MeV), the masses are observed to drop with an increase in the magnetic field. These in-medium modifications can have observable effects in asymmetric heavy-ion collisions experiments.

On the stability of the Ginzburg–Landau vortex

We introduce a functional framework taylored to investigate the minimality and stability properties of the Ginzburg–Landau vortex of degree one on the whole plane. We prove that a renormalized Ginzburg–Landau energy is well defined in that framework and that the vortex is its unique global minimizer up to the invariances by translation and phase shift. Our main result is a nonlinear coercivity estimate for the renormalized energy around the vortex, from which we can deduce its orbital stability as a solution to the Gross–Pitaevskii equation, the natural Hamiltonian evolution equation associated to the Ginzburg–Landau energy.

Glass encapsulation of molecular-doped epitaxial graphene for quantum resistance metrology

The large Landau energy spacing, stemming from the linear energy-momentum dispersion of quasi-particles in graphene, allows an efficient realization of the quantum Hall effect at a small density of charge carriers. Promising scalable epitaxial graphene on silicon carbide (SiC), however, requires molecular doping, which is generally unstable under ambient conditions, to compensate for electron transfer from the SiC substrate. Here, we employed classical glass encapsulation common in organic electronics to passivate molecular-doped epitaxial graphene against water and oxygen molecules in air. We have investigated the stability of Hall quantization in a glass-encapsulated device for almost 1 year. The Hall quantization is maintained above a threshold magnetic field within 2 nΩ Ω−1 smaller than the measurement uncertainty of 3.5 nΩ Ω−1 through multiple thermal cycles for almost 1 year, while the ordinary unencapsulated device in air distinctly shows a relative deviation larger than 0.05% from the nominal quantized Hall resistance in 1 month.

Open charm mesons and charmonia in magnetized strange hadronic matter * *A.J.C.S acknowledges the support towards this work from the Department of Science and Technology, Government of India, via an INSPIRE fellowship (INSPIRE Code IF170745). AM acknowledges financial support from Department of Science and Technology (DST), Government of India (CRG/2018/002226)

We investigate the in-medium masses of open charm mesons (D( , ), ( , ), ( , )) and charmonium states ( , , , , ) in strongly magnetized isospin asymmetric strange hadronic matter using a chiral effective model. In the presence of a magnetic field, the number and scalar densities of charged baryons have contributions from Landau energy levels. The mass modifications of open charm mesons result from their interactions with nucleons, hyperons, and the scalar fields (the non-strange field σ, strange field ζ, and isovector field δ) in the presence of a magnetic field. The mass modifications of the charmonium states result from the modification of gluon condensates in a medium simulated by the variation in the dilaton field (χ) in the chiral effective model. The effects of finite quark masses are also incorporated in the trace of the energy-momentum tensor in quantum chromodynamics to investigate the mass shifts of charmonium states. The in-medium masses of open charm mesons and charmonia are observed to decrease with an increase in baryon density. The charged , , , and mesons have additional positive mass shifts due to Landau quantization in the presence of a magnetic field. The effects of the strangeness fraction are observed to be more dominant for mesons compared with D mesons. The mass shifts of charmonia are observed to be larger in hyperonic media compared with nuclear media when the effect of the finite quark mass term is neglected. These medium mass modifications can have observable consequences on the production of the open charm mesons and charmonia in high-energy asymmetric heavy-ion collision experiments.

Quantum Hall effect in q-formalism based on Fermi gas model

To the best of our knowledge, for the first time, the quantum Hall effect (QHE) is considered in thermal non-equilibrium conditions (TNEC) via [Formula: see text]-algebra. The [Formula: see text]-algebra is a topological tool that provides a controlling [Formula: see text]-parameter to approximate theoretical results to the laboratory ones. We obtain the QHE spectrum and [Formula: see text]-deformed Hall potential (or Hall potential in TNEC) via the density probability current. Because of the Pauli Exclusion Principle, we model the QHE using relativistic Fermi electron gas and the [Formula: see text]-deformed Fermi–Dirac distribution. We find the dependence of deformed Hall potential to temperature and magnetic field for different geometric [Formula: see text]-regions. Further, we geometrically prove that the magnitude of the Hall potentials depends on the [Formula: see text]-regions (i.e. [Formula: see text]). We find that for some [Formula: see text]-amounts, the Landau energy gap on graphene-based materials can be explained in the room-temperature quantum Hall effect (RTQHE). Our method may be applied to analyze new electronic devices based on graphene in TNEQ.

A provably efficient monotonic-decreasing algorithm for shape optimization in Stokes flows by phase-field approaches

In this work, we study shape optimization problems in the Stokes flows. By phase-field approaches, the resulted total objective function consists of the dissipation energy of the fluids and the Ginzburg–Landau energy functional as a regularizing term for the generated diffusive interface, together with a Lagrangian multiplier for volume constraint. An efficient decoupled scheme is proposed to implement by the gradient flow approach to decrease the objective function. In each loop, we first update the velocity field by solving the Stokes equation with the phase field variable given in the previous iteration, which is followed by updating the phase field variable by solving an Allen–Cahn-type equation using a stabilized scheme. We then take the cut-off post-processing for the phase-field variable to constrain its value in [0,1]. In the last step of each loop, the Lagrangian parameter is updated with an appropriate artificial time step. We rigorously prove that the proposed scheme permits an unconditionally monotonic-decreasing property. To enhance the overall efficiency of the algorithm, in each loop we update the phase field variable and Lagrangian parameter several time steps but update the velocity field only one time. Numerical results for various shape optimizations are presented to validate the effectiveness of our numerical scheme.

Upsilon decay widths in magnetized asymmetric nuclear matter

The in-medium partial decay widths of [Formula: see text] in magnetized asymmetric nuclear matter are studied using a field theoretic model for composite hadrons with quark (and antiquark) constituents. [Formula: see text] is the lowest bottomonium state which can decay to [Formula: see text] in vacuum. The medium modifications of the decay widths of [Formula: see text] to [Formula: see text] pair in magnetized matter arise due to the mass modifications of the decaying [Formula: see text] as well as of the produced [Formula: see text] and [Formula: see text] mesons. The in-medium masses of the open bottom meson in magnetized nuclear matter are computed from their interactions with the nucleons and the scalar mesons within a chiral effective model. The mass modification of the [Formula: see text] arises due to the medium modification of a scalar dilaton field, which is introduced in the model to simulate the gluon condensates of QCD. The effects of the anomalous magnetic moments for the proton and neutron are taken into consideration in the present investigation. The presence of the external magnetic field is observed to lead to different mass modifications within the [Formula: see text] as well as the [Formula: see text] doublets, even in isospin symmetric nuclear matter. This is due to the difference in the interactions of the proton and the neutron to the electromagnetic field. The charged [Formula: see text] mesons have additional contributions from the Landau energy levels, leading to positive shifts in their masses in the presence of a magnetic field. In the presence of an external magnetic field, there are contributions to the masses of the B, [Formula: see text] mesons and [Formula: see text] state (longitudinal component) due to the pseudoscalar meson–vector meson (PV) mixing ([Formula: see text], [Formula: see text] and [Formula: see text] mixings), which are also considered in this study. The PV mixing effects are observed to be the dominant contributions to the mass shifts of these mesons, which lead to appreciable modifications in the decay widths of [Formula: see text] to the neutral ([Formula: see text]) and the charged ([Formula: see text]) pairs in the presence of a magnetic field. These should have observable consequence in the production of open bottom mesons and bottomonium states at LHC and RHIC, where huge magnetic fields are produced in ultra-relativistic peripheral heavy-ion collisions.

Magnetic field induced hair structure in the charmonium gluon dissociation

We study the electromagnetic field effect on charmonium gluon dissociation in quark-gluon plasma. With the effective Hamiltonian derived from QCD multipole expansion under an external electromagnetic field, we first solve the two-body Schr\"odinger equation for a pair of charm quarks with mean field potentials for color and electromagnetic interactions and obtain the charmonium binding energies and wave functions, and then calculate the gluon dissociation cross section and decay width by taking the color electric and magnetic dipole interactions as perturbations above the mean field and employing Fermi's golden rule. Considering the charmonium deformation in a magnetic field, the discrete Landau energy levels make the dissociation cross section grow hair, and the electric dipole channel is significantly changed, especially for the $P$-wave states ${\ensuremath{\chi}}_{c0}$ and ${\ensuremath{\chi}}_{c\ifmmode\pm\else\textpm\fi{}}$. From our numerical calculation, the magnetic field strength $eB=5{m}_{\ensuremath{\pi}}^{2}$ already changes the gluon dissociation strongly, which may indicate measurable effects in high-energy nuclear collisions.

Isolated and hybrid bilayer graphene quantum rings

Using the continuum model, we investigate the electronic properties of two types of bilayer graphene (BLG) quantum ring (QR) geometries: (i) an isolated BLG QR and (ii) a monolayer graphene (MLG) with a QR put on top of an infinite graphene sheet (hybrid BLG QR). Solving the Dirac-Weyl equation in the presence of a perpendicular magnetic field and applying the infinite-mass boundary condition at the ring boundaries, we obtain analytical results for the energy levels and corresponding wave spinors for both structures. In the case of isolated BLG QR, we observe a sizeable and magnetically tunable band gap which agrees with the tight-binding transport simulations. Our analytical results also show the intervalley symmetry $ E^K_e (m) = -E^{K'}_h(m) $ between the electron (e) and hole (h) states ($ m $ being the angular momentum quantum number) for the energy spectrum of the isolated BLG QR. The presence of interface boundary in a hybrid BLG QR modifies drastically the energy levels as compared to that of an isolated BLG QR. Its energy levels are tunable from MLG dot, to isolated BLG QR, and to MLG Landau energy levels as magnetic field is varied. Our predictions can be verified experimentally using different techniques such as by magnetotransport measurements.

Unconditionally energy stable schemes for fluid-based topology optimization

We present first- and second-order unconditionally energy stable schemes for fluid-based topology optimization problems. Our objective functional composes of five terms including mechanical property, Ginzburg–Landau energy, two penalized terms for solid, and the volume constraint. We consider the steady-state Stokes equation in the fluid domain and Darcy flow through porous medium. By coupling a Stokes type equation and the Allen–Cahn equation, we obtain the evolutionary equation for the fluid-based topology optimization. We use the backward Euler method and the Crank–Nicolson method to discretize the coupling system. The first- and second-order accurate schemes are presented correspondingly. We prove that our proposed schemes are unconditionally energy stable. The preconditioned conjugate gradient method is applied to solve the system. Several numerical tests are performed to verify the efficiency and accuracy of our schemes.

Relativistic electron dynamics in magnetic fields with low-degree of field nonlinearity

Abstract We study the relativistic bound state dynamics of electrons in the presence of electromagnetic fields within the framework of a novel class of nonlinear Dirac wave equations. In 2 + 1 space–time dimensions, we address a specific interacting electron model in a homogeneous magnetic field providing the allowed nonlinear ground-state Landau energy levels. We concentrate on the regime of laboratory high magnetic fields, and thus, comment on the relevance of this approach to study 2D electron systems.

Averaging of magnetic fields and applications

In this paper, we estimate the magnetic Laplacian energy norm in appropriate planar domains under a weak regularity hypothesis on the magnetic field. Our main contribution is an averaging estimate, valid in small cells, allowing us to pass from non-uniform to uniform magnetic fields. As a matter of application, we derive new upper and lower bounds of the lowest eigenvalue of the Dirichlet Laplacian which match in the regime of large magnetic field intensity. Furthermore, our averaging technique allows us to estimate the nonlinear Ginzburg–Landau energy, and as a byproduct, yields a non-Gaussian trial state for the Dirichlet magnetic Laplacian.

Magnetic Ginzburg–Landau energy with a periodic rapidly oscillating and diluted pinning term

Magnetic Ginzburg–Landau energy with a periodic rapidly oscillating and diluted pinning term

A Cahn–Hilliard–Biot system and its generalized gradient flow structure

In this work, we propose a new model for flow through deformable porous media, where the solid material has two phases with distinct material properties. The two phases of the porous material evolve according to a generalized Ginzburg–Landau energy functional, with additional impact from both elastic and fluid effects, and the coupling between flow and deformation is governed by Biot’s theory. This results in a three-way coupled system which can be seen as an extension of the Cahn–Larché equations with the inclusion of a fluid flowing through the medium. The model covers essential coupling terms for several relevant applications, including solid tumor growth, biogrout, and wood growth simulation. Moreover, we show that this coupled set of equations follow a generalized gradient flow framework. This opens a toolbox of analysis and solvers which can be used for further study of the model. Additionally, we provide a numerical example showing the impact of the flow on the solid phase evolution in comparison to the Cahn–Larché system.

Boundary conditions manipulation of polar vortex domains in BiFeO3 membranes via phase-field simulations

Polar vortex domains have recently become an emergent research field due to the abundant physical phenomena and potential applications in high-density memories. Here, we explore the mechanisms of creating polar vortex domains in the BiFeO3 (BFO) membranes subjected to different boundary conditions using phase-field simulations. A major difference is that the vortex in membrane can be stabilized even under short-circuit electrical boundary conditions compared to vortex in other systems, such as thin film or superlattice. We found that (a) the formation of polar vortex domains at the membrane interior under bending is mainly driven by the reduction of elastic energy under short-circuit boundary condition, and the vortex chirality (namely, clockwise and counterclockwise) could be identified by n-shape and u-shape bending; (b) in the unbent open-circuit BFO membrane case, exotic trapezoid-shaped vortex nanodomains form at the terminations of 109° domain walls (DWs) and partially charged 71° DWs, which is driven by the local depolarization field and the interplay among electrostatic, elastic, and gradient and Landau energies. We also examine Kittel’s law by establishing the dependence of vortex periods on the membrane thickness. These results give further understanding of the effect of boundary conditions on the formation of polar vortex domains, guiding experimental designs of vortex-based high-density memories.

Electron dynamics in noncommutative geometry with magnetic field and Zitterbewegung phenomenon

Starting from a gauge-invariant Dirac Hamiltonian with noncommutativity of space sector in the presence of an external uniform magnetic field, the resulting Dirac equation has been solved for electrons and its corresponding Zitterbewegung (ZBW) phenomenon has been studied. The corresponding energy spectrum is shown to be different from previous studies wherein the non-gauge-invariant Dirac Hamiltonian has been used. The effects of noncommutativity alter the amplitude as well as the cyclotron and the ZBW frequencies of the average velocity of charge carriers. This result is contrary to previous studies wherein there was no magnetic field, and hence, neither the amplitude nor the frequency of the motion was affected. Moreover, all of the ZBW frequencies of the Landau energy levels appear in the results. Also, in weak magnetic fields, we have calculated the average velocity of charge carriers for two initially localized spin-up and spin-down cases. The plotted trajectories reveal difference between these two cases. In addition, the ZBW phenomenon has been shown that manifests itself as a circular motion whose direction is spin dependent while accompanied by the cyclotron motion.

On a nonlocal Cahn–Hilliard model permitting sharp interfaces

A nonlocal Cahn–Hilliard model with a non-smooth potential of double-well obstacle type that promotes sharp interfaces in the solution is presented. To capture long-range interactions between particles, a nonlocal Ginzburg–Landau energy functional is defined which recovers the classical (local) model as the extent of nonlocal interactions vanish. In contrast to the local Cahn–Hilliard problem that always leads to diffuse interfaces, the proposed nonlocal model can lead to a strict separation into pure phases of the substance. Here, the lack of smoothness of the potential is essential to guarantee the aforementioned sharp-interface property. Mathematically, this introduces additional inequality constraints that, in a weak formulation, lead to a coupled system of variational inequalities which at each time instance can be restated as a constrained optimization problem. We prove the well-posedness and regularity of the semi-discrete and continuous in time weak solutions, and derive the conditions under which pure phases are admitted. Moreover, we develop discretizations of the problem based on finite element methods and implicit–explicit time-stepping methods that can be realized efficiently. Finally, we illustrate our theoretical findings through several numerical experiments in one and two spatial dimensions that highlight the differences in features of local and nonlocal solutions and also the sharp interface properties of the nonlocal model.

A Variational Singular Perturbation Problem Motivated by Ericksen’s Model for Nematic Liquid Crystals

We study the asymptotic behavior, when \(\varepsilon \rightarrow 0\), of the minimizers \(\{u_\varepsilon \}_{\varepsilon >0}\) for the energy $$\begin{aligned} E_\varepsilon (u)=\int _{\Omega }\Big (|\nabla u|^2+\big (\frac{1}{\varepsilon ^2}-1\big )|\nabla |u||^2\Big ), \end{aligned}$$over the class of maps \(u\in H^1(\Omega ,{{\mathbb {R}}}^2)\) satisfying the boundary condition \(u=g\) on \(\partial \Omega \), where \(\Omega \) is a smooth, bounded and simply connected domain in \({{\mathbb {R}}}^2\) and \(g:\partial \Omega \rightarrow S^1\) is a smooth boundary data of degree \(D\ge 1\). The motivation comes from a simplified version of the Ericksen model for nematic liquid crystals with variable degree of orientation. We prove convergence (up to a subsequence) of \(\{u_\varepsilon \}\) towards a singular \(S^1\)–valued harmonic map \(u_*\), a result that resembles the one obtained in Bethuel et al. (Ginzburg–Landau Vortices, Birkhäuser, 1994) for an analogous problem for the Ginzburg–Landau energy. There are however two striking differences between our result and the one involving the Ginzburg–Landau energy. First, in our problem, the singular limit \(u_*\) may have singularities of, degree strictly larger than one. Second, we find that the principle of “equipartition” holds for the energy of the minimizers, i.e., the contributions of the two terms in \(E_\varepsilon (u_\varepsilon )\) are essentially equal.