A novel Bogomol’nyi–Prasad–Sommerfield (BPS) bound for the Gross–Pitaevskii equations in two spatial dimensions is presented herein. The energy can be bounded from below in terms of the combination of two boundary terms, one related to the vorticity (but “dressed” by the condensate profile) and the second to the “skewness” of the configurations. The bound is saturated by configurations that satisfy a system of two first-order partial differential equations. When such a BPS system is satisfied, the Gross–Pitaevskii equations are also satisfied. The analytic solutions of this BPS system in the present manuscript represent configurations with fractional vorticity living in an annulus. Using these techniques, we present the first analytic examples of this kind. The hydrodynamic interpretation of the BPS system is discussed, and the implications of these results are outlined.

Chile

Fabrizio Canfora

Czechia

Pablo Pais

A Bogomol’nyi–Prasad–Sommerfield bound with a first-order system in the 2$\textit{D}$ Gross–Pitaevskii equation

The (generalized) Gross-Pitaevskii equation (GPE) for a complex scalar field in two spatial dimensions is analyzed. It is shown that there is an infinite family of self-interaction potentials which admit Bogomol'nyi-Prasad-Sommerfield (BPS) bounds together with the corresponding first-order BPS systems. For each member of this family, the solutions of the first-order BPS systems are automatically solutions of the corresponding second-order generalized GPE. The simplest topologically non-trivial solutions of these first-order BPS systems describe configurations with quantized fractional vorticity. The corresponding fraction is related to the degree of non-linearity. The case in which the self-interaction potential is of order six (namely <math altimg='si1.svg'><mo stretchy='false'>|</mo><mi mathvariant='normal'>&#936;</mi><msup><mrow><mo stretchy='false'>|</mo></mrow><mrow><mn>6</mn></mrow></msup></math>, which is a relevant theory both in relativistic quantum field theories in <math altimg='si2.svg'><mo stretchy='false'>(</mo><mn>2</mn><mo linebreak='badbreak' linebreakstyle='after'>+</mo><mn>1</mn><mo stretchy='false'>)</mo></math> dimensions in connection with the quantum Hall effect as well as in the theory of the supersolids) is analyzed in detail. Such formalism can also be extended to the case of quantum mixtures with multi-component GPEs. The relationship between these techniques and supersymmetry will be discussed. In particular, despite several common features, we will show that there are multi-component GPEs that are not supersymmetric (at least, not in the standard sense) and possess a BPS system of the above type.

Fractional vorticity, Bogomol'nyi-Prasad-Sommerfield systems and complex structures for the (generalized) spinor Gross-Pitaevskii equations

It was recently shown that, taking into account the granular structure of graphene lattice, the Dirac-like dynamics of its quasiparticles resists beyond the lowest energy approximation. This can be described in terms of new phase-space variables, <math><mo>(</mo><mover><mrow><mi>X</mi></mrow><mrow><mo>→</mo></mrow></mover><mo>,</mo><mover><mrow><mi>P</mi></mrow><mrow><mo>→</mo></mrow></mover><mo>)</mo></math>, that enjoy generalized Heisenberg algebras. In this letter, we add to that picture the important case of noncommuting <math><mover><mrow><mi>X</mi></mrow><mrow><mo>→</mo></mrow></mover></math>, for which <math><mo>[</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msup><mo>,</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>j</mi></mrow></msup><mo>]</mo><mo>=</mo><mi>i</mi><mspace width="0.2em"></mspace><msup><mrow><mi>θ</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msup></math> and we find that <math><msup><mrow><mi>θ</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msup><mo>=</mo><msup><mrow><mi>ℓ</mi></mrow><mrow><mn>2</mn></mrow></msup><mspace width="0.2em"></mspace><msup><mrow><mi>ϵ</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msup></math>, with ℓ the lattice spacing. We close by giving both the general recipe and a possible specific kinematic setup for the practical implementation of this approach to test noncommutative theories in tabletop analog experiments on graphene.

Alfredo Iorio

Austria

Boris Ivetić

Turning graphene into a lab for noncommutativity

By incorporating two gauge connections, transgression forms provide a generalization of Chern-Simons actions that are genuinely gauge-invariant on bounded manifolds. In this work, we show that, when defined on a manifold with a boundary, the Hamiltonian formulation of a transgression field theory can be consistently carried out without the need to implement regularizing boundary terms at the level of first-class constraints. By considering boundary variations of the relevant functionals in the Poisson brackets, the surface integral in the very definition of a transgression action can be translated into boundary contributions in the generators of gauge transformations and diffeomorphisms. This prescription systematically leads to the corresponding surface charges of the theory, reducing to the general expression for conserved charges in (higher-dimensional) Chern-Simons theories when one of the gauge connections in the transgression form is set to zero.

Poland

Patricio Salgado-Rebolledo

Aldo Vera

A note on the Hamiltonian structure of transgression forms

A remarkable exact mapping, valid for low-enough energy scales and close to a sharp boundary distribution of hadronic matter, from the (<math><mrow><mn>3</mn><mo>+</mo><mn>1</mn></mrow></math>)-dimensional Skyrme model to the sine-Gordon theory in (<math><mn>1</mn><mo>+</mo><mn>1</mn></math>) dimensions in the attractive regime is explicitly constructed. Besides the intrinsic theoretical interest to be able to describe the prototype of nonintegrable theories (namely, quantum chromodynamics in the infrared regime) in terms of the prototype of integrable relativistic field theories (namely, sine-Gordon theory in (<math><mn>1</mn><mo>+</mo><mn>1</mn></math>) dimensions), we will show that this mapping can be extremely useful to analyze both equilibrium and out-of-equilibrium features of baryonic distributions in a cavity.

Marcela Lagos

Exact mapping from the <math><mrow><mo>(</mo><mrow><mn>3</mn><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></math>-dimensional Skyrme model to the <math><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></math>-dimensional sine-Gordon theory and some applications

Beginning with a simple set of planar equations, we discuss novel realizations of the Pais-Uhlenbeck oscillator in various contexts. First, due to the bi-Hamiltonian character of this model, we develop a Hamiltonian approach for the Eisenhart-Duval lift of the related dynamics. We apply this approach to the previously worked example of a circularly polarized periodic gravitational wave. Then, we present our further results. Firstly, we show that the transverse dynamics of the Lukash plane wave and a complete gravitational wave pulse can also lead to the Pais-Uhlenbeck oscillator. We express the related Carroll Killing vectors in terms of the Pais-Uhlenbeck frequencies and derive extra integrals of motion from the conformal Newton-Hooke symmetry. In addition, we find that the <math><mn>3</mn><mo>+</mo><mn>1</mn></math> dimensional Penning trap can be canonically mapped to the 6th order Pais-Uhlenbeck oscillator. We also carry the problem to the non-commutative plane. Lastly, we discuss other examples like the motion of a charged particle under electromagnetic field created with double copy.

Türkiye

Mahmut Elbistan

Krzysztof Andrzejewski

Various disguises of the Pais-Uhlenbeck oscillator

We employ path integral methods to calculate the Casimir energy and force densities in a chiral extension of QED. Manifestly gauge invariant perfect electromagnetic boundary conditions, a natural generalization of perfect electric and perfect magnetic conditions, are implemented directly in the action by the usage of auxiliary fields. The chiral properties of the vacuum are modelled using a background θ field, and we introduce techniques to efficiently calculate the path integral in this chiral medium. The flexibility of our method allows us to naturally obtain results for a variety of configurations, and where comparison is possible our results are in perfect agreement with existing literature. Among these are multiple situations where a repulsive Casimir force is possible.

Belgium

David Dudal

Thomas Oosthuyse

Italy

Luigi Rosa

The Casimir effect in chiral media using path integral techniques

We perform the BFV-BRST quantization of the fourth-order Pais-Uhlenbeck oscillator (PUO) for the first time. We show that although the PUO is not naturally constrained in the sense of Dirac-Bergmann, it is possible to profit from the introduction of suitable constraints in phase space in order to obtain a proper BRST invariant quantum system. Starting from its second-class constrained system description, we use the BFFT formalism to obtain first-class constraints as gauge symmetry generators. After the Abelianization of the constraints, we obtain the conserved BRST charge, the corresponding BRST transformations and proceed further to the BFV functional quantization of the model. We further construct appropriate finite field dependent BRST transformation to establish the interconnections between different BRST invariant effective theories of PUO in different gauges. Our approach sheds light on the open problem of the quantization of general higher derivative quantum field theories.

India

Bhabani Prasad Mandal

Vipul Kumar Pandey

Brazil

Ronaldo Thibes

BFV quantization and BRST symmetries of the gauge invariant fourth-order Pais-Uhlenbeck oscillator

Two Lagrangian formulations for describing the damped harmonic oscillator have been introduced by Bateman. For these models we construct higher derivative generalizations which enjoy the l-conformal Newton-Hooke symmetry. It is shown that the dynamics of generalized systems corresponds to the damped Pais-Uhlenbeck oscillator for a particular choice of its frequencies.

Russia

Ivan Masterov

-conformal Newton-Hooke symmetry of the damped Pais-Uhlenbeck oscillator

The “Bargmann framework” allows us to prove the equivalence of the Pais-Uhlenbeck oscillator with a spin-zero particle in a circularly polarized periodic gravitational wave. Pushing forward the symmetries (which include Carroll symmetry) of the latter, we derive conserved charges for the Pais-Uhlenbeck oscillator.