Mathematical Physics
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Showing new listings for Tuesday, 26 November 2024
- [1] arXiv:2411.15188 [pdf, other]
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Title: Approximability of Poisson structures for the 4-vertex model, and the higher-spin XXX chain, and Yang-Baxter algebrasComments: 72 pages. An informal discussion of the work is at this https URL, while presentations on related topics are at this https URL, this https URL, this https URL, this https URL, this https URL, this https URL, this https URLSubjects: Mathematical Physics (math-ph); Probability (math.PR); Exactly Solvable and Integrable Systems (nlin.SI)
We implement the quantum inverse scattering method for the 4-vertex model. In comparison to previous works of the author which examined the 6-vertex, and 20-vertex, models, the 4-vertex model exhibits different characteristics, ranging from L-operators expressed in terms of projectors and Pauli matrices to algebraic and combinatorial properties, including Poisson structure and boxed plane partitions. With far fewer computations with an L-operator provided for the 4-vertex model by Bogoliubov in 2007, in comparison to those for L-operators of the 6, and 20, vertex models, from lower order expansions of the transfer matrix we derive a system of relations from the structure of operators that can be leveraged for studying characteristics of the higher-spin XXX chain in the weak finite volume limit. In comparison to quantum inverse scattering methods for the 6, and 20, vertex models which can be used to further study integrability, and exact solvability, an adaptation of such an approach for the 4-vertex model can be used to approximate, asymptotically in the weak finite volume limit, sixteen brackets which generate the Poisson structure. From explicit relations for operators of the 4-vertex transfer matrix, we conclude by discussing corresponding aspects of the Yang-Baxter algebra, which is closely related to the operators obtained from products of L-operators for approximating the transfer, and quantum monodromy, matrices. The structure of computations from L-operators of the 4-vertex model directly transfers to L-operators of the higher-spin XXX chain, revealing a similar structure of another Yang-Baxter algebra of interest.
- [2] arXiv:2411.15496 [pdf, html, other]
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Title: Legendre transformations of a class of generalized Frobenius manifolds and the associated integrable hierarchiesComments: 52 pagesSubjects: Mathematical Physics (math-ph); Differential Geometry (math.DG); Exactly Solvable and Integrable Systems (nlin.SI)
For two generalized Frobenius manifolds related by a Legendre-type transformation, we show that the associated integrable hierarchies of hydrodynamic type, which are called the Legendre-extended Principal Hierarchies, are related by a certain linear reciprocal transformation; we also show, under the semisimplicity condition, that the topological deformations of these Legendre-extended Principal Hierarchies are related by the same linear reciprocal transformation.
- [3] arXiv:2411.15635 [pdf, html, other]
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Title: Computing marginal eigenvalue distributions for the Gaussian and Laguerre orthogonal ensemblesComments: 25 pages, 2 figures, 5 ancillary Mathematica filesSubjects: Mathematical Physics (math-ph); Statistics Theory (math.ST)
The Gaussian and Laguerre orthogonal ensembles are fundamental to random matrix theory, and the marginal eigenvalue distributions are basic observable quantities. Notwithstanding a long history, a formulation providing high precision numerical evaluations for $N$ large enough to probe asymptotic regimes, has not been provided. An exception is for the largest eigenvalue, where there is a formalism due to Chiani which uses a combination of the Pfaffian structure underlying the ensembles, and a recursive computation of the matrix elements. We augment this strategy by introducing a generating function for the conditioned gap probabilities. A finite Fourier series approach is then used to extract the sequence of marginal eigenvalue distributions as a linear combination of Pfaffians, with the latter then evaluated using an efficient numerical procedure available in the literature. Applications are given to illustrating various asymptotic formulas, local central limit theorems, and central limit theorems, as well as to probing finite size corrections. Further, our data indicates that the mean values of the marginal distributions interlace with the zeros of the Hermite polynomial (Gaussian ensemble) and a Laguerre polynomial (Laguerre ensemble).
- [4] arXiv:2411.16040 [pdf, html, other]
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Title: Rota-Baxter operators on crossed modules of Lie groups and categorical solutions of the Yang-Baxter equationSubjects: Mathematical Physics (math-ph); Group Theory (math.GR); Rings and Algebras (math.RA)
In this paper, we construct a categorical solution $(\huaC, R)$ of the Yang-Baxter equation, i.e. $\huaC$ is a small category and $R: \huaC\times\huaC\lon\huaC\times\huaC$ is an invertible functor satisfying $$ (R\times\Id_\huaC)(\Id_\huaC\times R)(R\times\Id_\huaC)=(\Id_\huaC\times R)(R\times\Id_\huaC)(\Id_\huaC\times R),
$$ where $\huaC\times\huaC$ is the product category. First, the notion of Rota-Baxter operators on crossed modules of Lie groups is defined and its various properties are established. Then, we use Rota-Baxter operators on crossed modules of Lie groups to construct categorical solutions of the Yang-Baxter equation. We also study the Rota-Baxter operators on crossed modules of Lie algebras which are infinitesimals of Rota-Baxter operators on crossed modules of Lie groups, they can give connections on manifolds. Finally, we study the integration of Rota-Baxter operators on crossed modules of Lie algebras and the differentials of Rota-Baxter operators on crossed modules of Lie groups. - [5] arXiv:2411.16182 [pdf, html, other]
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Title: Resonant signal reversal in a waveguide connected to a resonatorComments: 6 pages, 1 figureSubjects: Mathematical Physics (math-ph)
It has been proven that when connecting two infinite semi-cylinders or waveguides with a finite cylinder or resonator at a certain frequency, it is possible to transmit a signal almost completely from one semi-cylinder to another. In this case, the reflected field is arbitrarily small. A very simple technique based on the expansion of the solution in a Fourier series in cylinders and matching the series for the signal and its derivatives in the conjugation boundaries of cylinders of different radii is used for the proof. The main feature of this method is its elementary nature, which allows for a certain class of boundaries to establish resonant scattering effects.
- [6] arXiv:2411.16274 [pdf, html, other]
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Title: Out-of-Time Ordered Correlator for a Chaotic Many-Body Quantum SystemSubjects: Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech); Quantum Physics (quant-ph)
Using the parametric representation of a chaotic many-body
quantum system derived earlier, we calculate explicitly the
large-time dependence and asymptotic value of the out-of-time
correlator (OTOC) of that system. The dependence on time $t$ is
determined by $\Delta t / \hbar$. Here $\Delta$ is the energy
correlation width within which the Bohigas-Giannoni-Schmit
conjecture applies. We conjecture that $\Delta$ is universally
related to the leading Ljapunov coefficient of the corresponding
classical system by $\Delta = \hbar \lambda_{\max}$. Then the
large-time behavior of OTOC is given by the dimensionless parameter
$\lambda_{\max} t$. - [7] arXiv:2411.16401 [pdf, html, other]
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Title: On finite-temperature Fredholm determinantsComments: 28 pages; 2 figuresSubjects: Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech); Exactly Solvable and Integrable Systems (nlin.SI)
We consider finite-temperature deformation of the sine kernel Fredholm determinants acting on the closed contours. These types of expressions usually appear as static two-point correlation functions in the models of free fermions and can be equivalently presented in terms of Toeplitz determinants. The corresponding symbol, or the phase shift, is related to the temperature weight. We present an elementary way to obtain large-distance asymptotic behavior even when the phase shift has a non-zero winding number. It is done by deforming the original kernel to the so-called effective form factors kernel that has a completely solvable matrix Riemann-Hilbert problem. This allows us to find explicitly the resolvent and address the subleading corrections. We recover Szego, Hartwig and Fisher, and Borodin-Okounkov asymptotic formulas.
New submissions (showing 7 of 7 entries)
- [8] arXiv:2406.01567 (cross-list from hep-th) [pdf, html, other]
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Title: Modularity in $d > 2$ free conformal field theoryComments: 45 pages; v2: reference added and statements improved; v3: Add references and some discussionsSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
We derive new closed form expressions for the partition functions of free conformally-coupled scalars on $S^{2D-1}\times S^1$ which resum the exact high-temperature expansion. The derivation relies on an identification of the partition functions, analytically continued in chemical potentials and temperature, with multiple elliptic Gamma functions. These functions satisfy interesting modular properties, which we use to arrive at our expressions. We describe a geometric interpretation of the modular properties of multiple elliptic Gamma functions in the context of superconformal field theory. Based on this, we suggest a geometric interpretation of the modular property in the context of the free scalar CFT in even dimensions and comment on extensions to odd dimensions and free fermions.
- [9] arXiv:2411.15171 (cross-list from physics.flu-dyn) [pdf, html, other]
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Title: A Hamiltonian set-up for 4-layer density stratified Euler fluidsComments: arXiv admin note: text overlap with arXiv:2105.12851Subjects: Fluid Dynamics (physics.flu-dyn); Mathematical Physics (math-ph); Geophysics (physics.geo-ph)
By means of the Hamiltonian approach to two-dimensional wave motions in heterogeneous fluids proposed by Benjamin, we derive a natural Hamiltonian structure for ideal fluids, density stratified in four homogenous layers, constrained in a channel of fixed total height and infinite lateral length. We derive the Hamiltonian and the equations of motion in the dispersionless long-wave limit, restricting ourselves to the so-called Boussinesq approximation. The existence of special symmetric solutions, which generalize to the four-layer case the ones obtained in the paper for the three-layer case, is examined.
- [10] arXiv:2411.15184 (cross-list from nlin.PS) [pdf, html, other]
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Title: Two-Crested Stokes WavesSubjects: Pattern Formation and Solitons (nlin.PS); Mathematical Physics (math-ph)
We study two-crested traveling Stokes waves on the surface of an ideal fluid with infinite depth. Following Chen and Saffman (1980), we refer to these waves as class $\mathrm{II}$ Stokes waves. The class $\mathrm{II}$ waves are found from bifurcations from the primary branch of Stokes waves away from the flat surface. These waves are strongly nonlinear, and are disconnected from small-amplitude solutions. Distinct class $\mathrm{II}$ bifurcations are found to occur in the first two oscillations of the velocity versus steepness diagram. The bifurcations in distinct oscillations are not connected via a continuous family of class $\mathrm{II}$ waves. We follow the first two families of class $\mathrm{II}$ waves, which we refer to as the secondary branch (that is primary class $\mathrm{II}$ branch), and the tertiary branch (that is secondary class $\mathrm{II}$ branch). Similar to Stokes waves, the class $\mathrm{II}$ waves follow through a sequence of oscillations in velocity as their steepness rises, and indicate the existence of limiting class $\mathrm{II}$ Stokes waves characterized by a $120$ degree angle at every other wave crest.
- [11] arXiv:2411.15343 (cross-list from hep-lat) [pdf, html, other]
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Title: Galerkin Formulation of Path Integrals in Lattice Field TheorySubjects: High Energy Physics - Lattice (hep-lat); Mathematical Physics (math-ph); Numerical Analysis (math.NA)
We present a mathematical framework for Galerkin formulations of path integrals in lattice field theory. The framework is based on using the degrees of freedom associated to a Galerkin discretization as the fundamental lattice variables. We formulate standard concepts in lattice field theory, such as the partition function and correlation functions, in terms of the degrees of freedom. For example, using continuous finite element spaces, we show that the two-point spatial correlation function can be defined between any two points on the domain (as opposed to at just lattice sites) and furthermore, that this satisfies a weak propagator (or Green's function) identity, in analogy to the continuum case. Furthermore, this framework leads naturally to higher-order formulations of lattice field theories by considering higher-order finite element spaces for the Galerkin discretization. We consider analytical and numerical examples of scalar field theory to investigate how increasing the order of piecewise polynomial finite element spaces affect the approximation of lattice observables. Finally, we sketch an outline of this Galerkin framework in the context of gauge field theories.
- [12] arXiv:2411.15406 (cross-list from math.AP) [pdf, html, other]
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Title: Uniform-in-Time Estimates on the Size of Chaos for Interacting Particle SystemsComments: 25 pagesSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Probability (math.PR)
For any weakly interacting particle system with bounded kernel, we give uniform-in-time estimates of the $L^2$ norm of correlation functions, provided that the diffusion coefficient is large enough. When the condition on the kernels is more restrictive, we can remove the dependence of the lower bound for diffusion coefficient on the initial data and estimate the size of chaos in a weaker sense. Based on these estimates, we may study fluctuation around the mean-field limit.
- [13] arXiv:2411.15522 (cross-list from math.AP) [pdf, html, other]
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Title: On the magnetic Dirichlet to Neumann operator on the disk -- strong diamagnetism and strong magnetic field limit--Comments: 25 pagesSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Spectral Theory (math.SP)
Inspired by a paper by T. Chakradhar, K. Gittins, G. Habib and N. Peyerimhoff, we analyze their conjecture that the ground state energy of the magnetic Dirichlet-to-Neumann operator on the disk tends to $+\infty$ as the magnetic field tends to $+\infty$. This is an important step towards the analysis of the curvature effect in the case of general domains in $\mathbb R^2$.
- [14] arXiv:2411.15536 (cross-list from quant-ph) [pdf, html, other]
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Title: Four-Qubit CHSH GamesComments: 17 pages, 11 figures, 7 tablesSubjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
In this paper, the CHSH quantum game is extended to four players. This is achieved by exploring all possible 4-variable Boolean functions to identify those that yield a game scenario with a quantum advantage using a specific entangled state. Notably, two new four-player quantum games are presented. In one game, the optimal quantum strategy is achieved when players share a $GHZ$-state, breaking the traditional 10\% gain observed in 2 and 3 qubit CHSH games and achieving a 22.5\% gap. In the other game, players gain a greater advantage using a $W$-state as their quantum resource. Quantum games with other four-qubit entangled states are also explored. To demonstrate the results, these game scenarios are implemented on an online quantum computer, and the advantage of the respective quantum resource for each game is experimentally verified.
- [15] arXiv:2411.15599 (cross-list from nlin.SI) [pdf, html, other]
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Title: The generalized Darboux matrices with the same poles and their applicationsSubjects: Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (math-ph)
Darboux transformation plays a key role in constructing explicit closed-form solutions of completely integrable systems. This paper provides an algebraic construction of generalized Darboux matrices with the same poles for the $2\times2$ Lax pair, in which the coefficient matrices are polynomials of spectral parameter. The first-order monic Darboux matrix is constructed explicitly and its classification theorem is presented. Then by using the solutions of the corresponding adjoint Lax pair, the $n$-order monic Darboux matrix and its inverse, both sharing the same unique pole, are derived explicitly. Further, a theorem is proposed to describe the invariance of Darboux matrix regarding pole distributions in Darboux matrix and its inverse. Finally, a unified theorem is offered to construct formal Darboux transformation in general form. All Darboux matrices expressible as the product of $n$ first-order monic Darboux matrices can be constructed in this way. The nonlocal focusing NLS equation, the focusing NLS equation and the Kaup-Boussinesq equation are taken as examples to illustrate the application of these Darboux transformations.
- [16] arXiv:2411.15608 (cross-list from hep-th) [pdf, html, other]
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Title: Condensation of Magnetic Fluxes and Landscape of QCD VacuumComments: 10 pagesSubjects: High Energy Physics - Theory (hep-th); High Energy Physics - Phenomenology (hep-ph); Mathematical Physics (math-ph)
I discuss new non-perturbative solutions of the sourceless Yang-Mills equation representing the superposition of oppositely oriented chromomagnetic flux tubes (vortices) similar in their form to a lattice of superposed Abrikosov-Nielsen-Olesen chromomagnetic vortices. These solutions represent highly degenerate classical vacua of the Yang Mills theory that are separated by potential barriers and are forming a complicated potential landscape of the QCD vacuum. It is suggested that the solutions describe a lattice of dense chromomagnetic vortices representing a dual analog of the Cooper pairs condensate in a superconductor.
- [17] arXiv:2411.15830 (cross-list from math.PR) [pdf, html, other]
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Title: Deformations of biorthogonal ensembles and universalityComments: 32 pagesSubjects: Probability (math.PR); Mathematical Physics (math-ph); Classical Analysis and ODEs (math.CA)
We consider a large class of deformations of continuous and discrete biorthogonal ensembles and investigate their behavior in the limit of a large number of particles. We provide sufficient conditions to ensure that if a biorthogonal ensemble converges to a (universal) limiting process, then the deformed biorthogonal ensemble converges to a deformed version of the same limiting process. To construct the deformed version of the limiting process, we rely on a procedure of marking and conditioning. Our approach is based on an analysis of the probability generating functionals of the ensembles and is conceptually different from the traditional approach via correlation kernels. Thanks to this method, our sufficient conditions are rather mild and do not rely on much regularity of the original ensemble and of the deformation. As a consequence of our results, we obtain probabilistic interpretations of several Painlevé-type kernels that have been constructed in the literature, as deformations of classical sine and Airy point processes.
- [18] arXiv:2411.15886 (cross-list from math.AP) [pdf, other]
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Title: Low-Regularity Local Well-Posedness for the Elastic Wave SystemComments: 110 pagesSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph)
We study the elastic wave system in three spatial dimensions. For admissible harmonic elastic materials, we prove a desired low-regularity local well-posedness result for the corresponding elastic wave equations. For such materials, we can split the dynamics into the divergence-part and the curl-part, and each part satisfies a distinct coupled quasilinear wave system with respect to different acoustical metrics. Our main result is that the Sobolev norm $H^{3+}$ of the divergence-part (the faster-wave part) and the $H^{4+}$ of the curl-part (the slower-wave part) can be controlled in terms of initial data for short times. We note that the Sobolev norm assumption $H^{3+}$ is optimal for the divergence-part. This marks the first favorable low-regularity local well-posedness result for a wave system with multiple wave speeds.
- [19] arXiv:2411.15906 (cross-list from math.SP) [pdf, html, other]
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Title: Convergence of supercell and superspace methods for computing spectra of quasiperiodic operatorsSubjects: Spectral Theory (math.SP); Mathematical Physics (math-ph); Analysis of PDEs (math.AP)
We study the convergence of two of the most widely used and intuitive approaches for computing the spectra of differential operators with quasiperiodic coefficients: the supercell method and the superspace method. In both cases, Floquet-Bloch theory for periodic operators can be used to compute approximations to the spectrum. We illustrate our results with examples of Schrödinger and Helmholtz operators.
- [20] arXiv:2411.15954 (cross-list from math.PR) [pdf, other]
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Title: A gradient model for the Bernstein polynomial basisSubjects: Probability (math.PR); Mathematical Physics (math-ph); Cellular Automata and Lattice Gases (nlin.CG)
We introduce and study a symmetric, gradient exclusion process, in the class of non-cooperative kinetically constrained lattice gases, modelling a non-linear diffusivity where mass transport is constrained by the local density not being too small or too large. Maintaining the gradient property is the main technical challenge. The resulting model enjoys of properties in common with the Bernstein polynomial basis, and is associated with the diffusion coefficient $D_{n,k}(\rho)=\binom{n+k}{k}\rho^n(1-\rho)^k$, for $n,k$ arbitrary natural numbers. The dynamics generalizes the Porous Media Model, and we show, via the entropy method, the hydrodynamic limit for the empirical measure associated with a perturbed, irreducible version of the process. The hydrodynamic equation is proved to be a Generalized Porous Media Equation.
- [21] arXiv:2411.15975 (cross-list from physics.hist-ph) [pdf, html, other]
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Title: Effective Theory Building and Manifold LearningComments: 33 pagesSubjects: History and Philosophy of Physics (physics.hist-ph); High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
Manifold learning and effective model building are generally viewed as fundamentally different types of procedure. After all, in one we build a simplified model of the data, in the other, we construct a simplified model of the another model. Nonetheless, I argue that certain kinds of high-dimensional effective model building, and effective field theory construction in quantum field theory, can be viewed as special cases of manifold learning. I argue that this helps to shed light on all of these techniques. First, it suggests that the effective model building procedure depends upon a certain kind of algorithmic compressibility requirement. All three approaches assume that real-world systems exhibit certain redundancies, due to regularities. The use of these regularities to build simplified models is essential for scientific progress in many different domains.
- [22] arXiv:2411.15977 (cross-list from math.QA) [pdf, html, other]
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Title: A quantum space of Euclidean linesComments: 22 pages, no figuresSubjects: Quantum Algebra (math.QA); Mathematical Physics (math-ph); Symplectic Geometry (math.SG)
This article presents a differential groupoid with ``coaction'' of the groupoid underlying the
Quantum Euclidean Group (i.e. its $C^*$-algebra is the $C^*$-algebra of this quantum group).
The dual of the Lie algebroid is a Poisson manifold that can be identified with the space of oriented
lines in Euclidean space equipped with a Poisson action of the Poisson-Lie Euclidean group. - [23] arXiv:2411.16013 (cross-list from math.AP) [pdf, html, other]
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Title: Stochastic Analysis and White Noise Calculus of Nonlinear Wave Equations with Application to Laser Propagation and GenerationSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Functional Analysis (math.FA); Probability (math.PR); Quantum Algebra (math.QA)
In this paper we study a large class of nonlinear stochastic wave equations that arise in laser generation models and models for propagation in random media in a unified mathematical framework. Continuous and pulse-wave propagation models, free electron laser generation models, as well as laser-plasma interaction models have been cast in a convenient and unified abstract framework as semilinear evolution equations in a Hilbert space to enable stochastic analysis. We formulate Ito calculus and white noise calculus methods of treating stochastic terms and prove existence and uniqueness of mild solutions.
- [24] arXiv:2411.16022 (cross-list from math.CA) [pdf, other]
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Title: General Geronimus Perturbations for Mixed Multiple Orthogonal PolynomialsComments: 40 pagesSubjects: Classical Analysis and ODEs (math.CA); Mathematical Physics (math-ph)
General Geronimus transformations, defined by regular matrix polynomials that are neither required to be monic nor restricted by the rank of their leading coefficients, are applied through both right and left multiplication to a rectangular matrix of measures associated with mixed multiple orthogonal polynomials. These transformations produce Christoffel-type formulas that establish relationships between the perturbed and original polynomials. Moreover, it is proven that the existence of Geronimus-perturbed orthogonality is equivalent to the non-cancellation of certain $\tau$-determinants. The effect of these transformations on the Markov-Stieltjes matrix functions is also determined. As a case study, we examine the Jacobi-Piñeiro orthogonal polynomials with three weights.
- [25] arXiv:2411.16033 (cross-list from hep-th) [pdf, html, other]
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Title: Generative AI for Brane Configurations, Tropical Coamoeba and 4d N=1 Quiver Gauge TheoriesComments: 21 pages, 8 figures, 1 tableSubjects: High Energy Physics - Theory (hep-th); Machine Learning (cs.LG); Mathematical Physics (math-ph); Algebraic Geometry (math.AG)
We introduce a generative AI model to obtain Type IIB brane configurations that realize toric phases of a family of 4d N=1 supersymmetric gauge theories. These 4d N=1 quiver gauge theories are worldvolume theories of a D3-brane probing a toric Calabi-Yau 3-fold. The Type IIB brane configurations that realize this family of 4d N=1 theories are known as brane tilings and are given by the tropical coamoeba projection of the mirror curve associated with the toric Calabi-Yau 3-fold. The shape of the mirror curve and its coamoeba projection, as well as the corresponding Type IIB brane configuration and the toric phase of the 4d N=1 theory, all depend on the complex structure moduli parameterizing the mirror curve. We train a generative AI model, a conditional variational autoencoder (CVAE), that takes a choice of complex structure moduli as input and generates the corresponding tropical coamoeba. This enables us not only to obtain a high-resolution representation of the entire phase space for a family of brane tilings corresponding to the same toric Calabi-Yau 3-fold, but also to continuously track the movements of the mirror curve and individual branes in the corresponding Type IIB brane configurations during phase transitions associated with Seiberg duality.
- [26] arXiv:2411.16058 (cross-list from math.PR) [pdf, html, other]
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Title: Gaussian deconvolution on $\mathbb R^d$ with application to self-repellent Brownian motionComments: 13 pagesSubjects: Probability (math.PR); Mathematical Physics (math-ph)
We consider the convolution equation $(\delta - J) * G = g$ on $\mathbb R^d$, $d>2$, where $\delta$ is the Dirac delta function and $J,g$ are given functions. We provide conditions on $J, g$ that ensure the deconvolution $G(x)$ to decay as $( x \cdot \Sigma^{-1} x)^{-(d-2)/2}$ for large $|x|$, where $\Sigma$ is a positive-definite diagonal matrix. This extends a recent deconvolution theorem on $\mathbb Z^d$ proved by the author and Slade to the possibly anisotropic, continuum setting while maintaining its simplicity. Our motivation comes from studies of statistical mechanical models on $\mathbb R^d$ based on the lace expansion. As an example, we apply our theorem to a self-repellent Brownian motion in dimensions $d>4$, proving its critical two-point function to decay as $|x|^{-(d-2)}$, like the Green function of the Laplace operator $\Delta$.
- [27] arXiv:2411.16071 (cross-list from math.CA) [pdf, html, other]
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Title: Long time evolution of the H\'enon-Heiles system for small energyComments: 22 pages, 6 figuresSubjects: Classical Analysis and ODEs (math.CA); Mathematical Physics (math-ph); Dynamical Systems (math.DS)
The Hénon-Heiles system, initially introduced as a simplified model of galactic dynamics, has become a paradigmatic example in the study of nonlinear systems. Despite its simplicity, it exhibits remarkably rich dynamical behavior, including the interplay between regular and chaotic orbital dynamics, resonances, and stochastic regions in phase space, which have inspired extensive research in nonlinear dynamics. In this work, we investigate the system's solutions at small energy levels, deriving asymptotic constants of motion that remain valid over remarkably long timescales -- far exceeding the range of validity of conventional perturbation techniques. Our approach leverages the system's inherent two-scale dynamics, employing a novel analytical framework to uncover these long-lived invariants. The derived formulas exhibit excellent agreement with numerical simulations, providing a deeper understanding of the system's long-term behavior.
- [28] arXiv:2411.16090 (cross-list from math.AP) [pdf, html, other]
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Title: The final state problem for the nonlinear Schrodinger equation in dimensions 1, 2 and 3Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph)
In this article we consider the defocusing nonlinear Schrödinger equation, with time-dependent potential, in space dimensions $n=1, 2$ and $3$, with nonlinearity $|u|^{p-1} u$, $p$ an odd integer, satisfying $p \geq 5$ in dimension $1$, $p \geq 3$ in dimension $2$ and $p=3$ in dimension $3$. We also allow a metric perturbation, assumed to be compactly supported in spacetime, and nontrapping. We work with module regularity spaces, which are defined by regularity of order $k \geq 2$ under the action of certain vector fields generating symmetries of the free Schrödinger equation. We solve the large data final state problem, with final state in a module regularity space, and show convergence of the solution to the final state.
- [29] arXiv:2411.16109 (cross-list from math.AP) [pdf, html, other]
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Title: Existence and uniqueness of the solution of a mixed problem for a parabolic equation under nonconventional boundary conditionsComments: 19 pagesJournal-ref: Journal of Physical Mathematics & its Applications, 2024, Vol. 2(4): 1-10Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph)
In this study, we investigate a mixed problem linked to a second-order parabolic equation, characterized by temporal dependencies and variable~coefficients, and constrained by non-local, non-self-adjoint boundary conditions. By defining precise conditions on the input data, we establish the unique solvability of the problem through a synthesis of the residue and contour integral methods. Moreover, our research yields an explicit analytical solution, facilitating the direct resolution of the stated problem.
- [30] arXiv:2411.16176 (cross-list from hep-th) [pdf, html, other]
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Title: Yangian Form-alism for Planar Gauge TheoriesComments: 27 pagesSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
In this article, we reconsider the formulation of Yangian symmetry for planar N=4 supersymmetric Yang-Mills theory, and we investigate to what extent this symmetry lifts to the beta/gamma-deformation of the model. We first apply cohomology of variational forms towards a thorough derivation of the invariance statement for the undeformed action from covariance of the equations of motion under the Yangian algebra. We then apply a twist deformation to these statements paying particular attention to cyclicity aspects. We find that the equations of motion remain covariant while invariance of the action only holds for the Yangian subalgebra that is uncharged under the twist.
- [31] arXiv:2411.16272 (cross-list from math.QA) [pdf, html, other]
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Title: Twisted vertex algebra modules for irregular connections: A case studySubjects: Quantum Algebra (math.QA); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
A vertex algebra with an action of a group $G$ comes with a notion of $g$-twisted modules, forming a $G$-crossed braided tensor category. For a Lie group $G$, one might instead wish for a notion of $(\mathrm{d}+A)$-twisted modules for any $\mathfrak{g}$-connection on the formal punctured disc. For connections with a regular singularity, this reduces to $g$-twisted modules, where $g$ is the monodromy around the puncture. The case of an irregular singularity is much richer and involved, and we are not aware that it has appeared in vertex algebra language. The present article is intended to spark such a treatment, by providing a list of expectations and an explicit worked-through example with interesting applications.
Concretely, we consider the vertex super algebra of symplectic fermions, or equivalently the triplet vertex algebra $\mathcal{W}_p(\mathfrak{sl}_2)$ for $p=2$, and study its twisted module with respect to irregular $\mathfrak{sl}_2$-connections. We first determine the category of representations, depending on the formal type of the connection. Then we prove that a Sugawara type construction gives a Virasoro action and we prove that as Virasoro modules our representations are direct sums of Whittaker modules.
Conformal field theory with irregular singularities resp. wild ramification appear in the context of geometric Langlands correspondence, and in particular in work by Witten [Wit08]. It has also been studied for example in the context of Gaudin models [FFTL10] and in the context of AGT correspondence [GT12]. Our original motivation comes from semiclassical limits of the generalized quantum Langlands kernel, which fibres over the space of connections [FL24], similar to the affine Lie algebra at critical level. Our present article now describes, in the smallest case, the - [32] arXiv:2411.16324 (cross-list from math.NA) [pdf, html, other]
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Title: Parameter Error Analysis for the 3D Modified Leray-alpha Model: Analytical and Numerical ApproachesComments: 27 pages, 12 figures. arXiv admin note: text overlap with arXiv:2409.03042Subjects: Numerical Analysis (math.NA); Mathematical Physics (math-ph); Analysis of PDEs (math.AP)
In this study, we conduct a parameter error analysis for the 3D modified Leray-$\alpha$ model using both analytical and numerical approaches. We first prove the global well-posedness and continuous dependence of initial data for the assimilated system. Furthermore, given sufficient conditions on the physical parameters and norms of the true solution, we demonstrate that the true solution can be recovered from the approximation solution, with an error determined by the discrepancy between the true and approximating parameters. Numerical simulations are provided to validate the convergence criteria.
- [33] arXiv:2411.16432 (cross-list from math.RT) [pdf, html, other]
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Title: Langlands Duality and Invariant Differential OperatorsComments: 21 pages, 2 figures. arXiv admin note: text overlap with arXiv:1311.7557, arXiv:1208.0409, arXiv:1412.8038Subjects: Representation Theory (math.RT); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Quantum Algebra (math.QA)
Langlands duality is one of the most influential topics in mathematical research. It has many different appearances and influential subtopics. Yet there is a topic that until now seems unrelated to the Langlands program. That is the topic of invariant differential operators. That is strange since both items are deeply rooted in Harish-Chandra's representation theory of semisimple Lie groups. In this paper we start building the bridge between the two programs.
- [34] arXiv:2411.16452 (cross-list from math.PR) [pdf, html, other]
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Title: Interface scaling limit for the critical planar Ising model perturbed by a magnetic fieldSubjects: Probability (math.PR); Mathematical Physics (math-ph)
We prove that the interface separating $+1$ and $-1$ spins in the critical planar Ising model with Dobrushin boundary conditions perturbed by an external magnetic field has a scaling limit. This result holds when the Ising model is defined on a bounded and simply connected subgraph of $\delta \mathbb{Z}^2$, with $\delta >0$. We show that if the scaling of the external field is of order $\delta^{15/8}$, then, as $\delta \to 0$, the interface converges in law to a random curve whose law is conformally covariant and absolutely continuous with respect to SLE$_3$. This limiting law is a massive version of SLE$_3$ in the sense of Makarov and Smirnov and we give an explicit expression for its Radon-Nikodym derivative with respect to SLE$_3$. We also prove that if the scaling of the external field is of order $\delta^{15/8}g_1(\delta)$ with $g_1(\delta)\to 0$, then the interface converges in law to SLE$_3$. In contrast, we show that if the scaling of the external field is of order $\delta^{15/8}g_2(\delta)$ with $g_2(\delta) \to \infty$, then the interface degenerates to a boundary arc.
- [35] arXiv:2411.16494 (cross-list from math.SP) [pdf, html, other]
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Title: The relativistic rotated harmonic oscillatorComments: 3 figuresSubjects: Spectral Theory (math.SP); Mathematical Physics (math-ph)
We introduce a relativistic version of the non-self-adjoint operator obtained by a dilation analytic transformation of the quantum harmonic oscillator. While the spectrum is real and discrete, we show that the eigenfunctions do not form a basis and that the pseudospectra are highly non-trivial.
- [36] arXiv:2411.16517 (cross-list from hep-th) [pdf, html, other]
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Title: A basic triad in Macdonald theoryComments: 9 pagesSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Quantum Algebra (math.QA)
Within the context of wavefunctions of integrable many-body systems, rational multivariable Baker-Akhiezer (BA) functions were introduced by O. Chalykh, M. Feigin and A. Veselov and, in the case of the trigonometric Ruijsenaars-Schneider system, can be associated with a reduction of the Macdonald symmetric polynomials at $t=q^{-m}$ with integer partition labels substituted by arbitrary complex numbers. A parallel attempt to describe wavefunctions of the bispectral trigonometric Ruijsenaars-Schneider problem was made by M. Noumi and J. Shiraishi who proposed a power series that reduces to the Macdonald polynomials at particular values of parameters. It turns out that this power series also reduces to the BA functions at $t=q^{-m}$, as we demonstrate in this letter. This makes the Macdonald polynomials, the BA functions and the Noumi-Shiraishi (NS) series a closely tied {\it triad} of objects, which have very different definitions, but are straightforwardly related with each other. In particular, theory of the BA functions provides a nice system of simple linear equations, while the NS functions provide a nice way to represent the multivariable BA function explicitly with arbitrary number of variables.
- [37] arXiv:2411.16536 (cross-list from math.AP) [pdf, other]
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Title: A priori bounds for the dynamic fractional $\Phi^4$ model on $\mathbb{T}^3$ in the full subcritical regimeSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Probability (math.PR)
We show a priori bounds for the dynamic fractional $\Phi^4$ model on $\mathbb{T}^3$ in the full subcritical regime using the framework of Hairer's regularity structures theory. Assuming the model bounds our estimates imply global existence of solutions and existence of an invariant measure. We extend the method developed for the usual heat operator by Chandra, Moinat and Weber [CMW23] to the fractional heat operator, thereby treating a more physically relevant model. A key ingredient in this work is the development of localised multilevel Schauder estimates for the fractional heat operator which is not covered by Hairer's original work. Furthermore, the algebraic arguments from [CMW23] are streamlined significantly.
- [38] arXiv:2411.16572 (cross-list from math.PR) [pdf, html, other]
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Title: Optimal decay of eigenvector overlap for non-Hermitian random matricesComments: 39 pagesSubjects: Probability (math.PR); Mathematical Physics (math-ph)
We consider the standard overlap $\mathcal{O}_{ij}: =\langle \mathbf{r}_j, \mathbf{r}_i\rangle\langle \mathbf{l}_j, \mathbf{l}_i\rangle$ of any bi-orthogonal family of left and right eigenvectors of a large random matrix $X$ with centred i.i.d. entries and we prove that it decays as an inverse second power of the distance between the corresponding eigenvalues. This extends similar results for the complex Gaussian ensemble from Bourgade and Dubach [arXiv:1801.01219], as well as Benaych-Georges and Zeitouni [arXiv:1806.06806], to any i.i.d. matrix ensemble in both symmetry classes. As a main tool, we prove a two-resolvent local law for the Hermitisation of $X$ uniformly in the spectrum with optimal decay rate and optimal dependence on the density near the spectral edge.
- [39] arXiv:2411.16631 (cross-list from math.DS) [pdf, html, other]
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Title: A note on integrabiliy of Hamiltonian systems on the co-adjoint Lie groupoidsComments: 15 pagesSubjects: Dynamical Systems (math.DS); Mathematical Physics (math-ph)
As we said in our previous work [4], the main idea of our research is to introduce a class of Lie groupoids by means of co-adjoint representation of a Lie groupoid on its isotropy Lie algebroid, which we called coadjoint Lie groupoids. In this paper, we will examine the relationship between structural mappings of the Lie algebroid associated to Lie groupoid and co-adjoint Lie algebroid. Also, we try to construct and define integrabiliy of Hamiltonian system on the co-adjoint Lie groupoids. In addition, we show that co-adjoint Lie groupoid associated to a symplectic Lie groupoid is a symplectic Lie groupoid.
- [40] arXiv:2411.16640 (cross-list from math.OC) [pdf, html, other]
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Title: Optimal control problems on the co-adjoint Lie groupoidsComments: 17 pagesSubjects: Optimization and Control (math.OC); Mathematical Physics (math-ph)
In this work we study the invariant optimal control problem on Lie groupoids. We show that any invariant optimal control problem on a Lie groupoid reduces to its co-adjoint Lie algebroid. In the final section of the paper, we present an illustrative example.
Cross submissions (showing 33 of 33 entries)
- [41] arXiv:2010.07723 (replaced) [pdf, html, other]
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Title: Airy kernel determinant solutions to the KdV equation and integro-differential Painlev\'e equationsComments: 44 pages. V3: Remark 1.2 correctedJournal-ref: Commun. Math. Phys. 386, 1107-1153 (2021)Subjects: Mathematical Physics (math-ph); Classical Analysis and ODEs (math.CA); Probability (math.PR)
We study a family of unbounded solutions to the Korteweg-de Vries equation which can be constructed as log-derivatives of deformed Airy kernel Fredholm determinants, and which are connected to an integro-differential version of the second Painlevé equation. The initial data of the Korteweg-de Vries solutions are well-defined for $x>0$, but not for $x<0$, where the solutions behave like $\frac{x}{2t}$ as $t\to 0$, and hence would be well-defined as solutions of the cylindrical Korteweg-de Vries equation. We provide uniform asymptotics in $x$ as $t\to 0$; for $x>0$ they involve an integro-differential analogue of the Painlevé V equation. A special case of our results yields improved estimates for the {tails} of the narrow wedge solution to the Kardar-Parisi-Zhang equation.
- [42] arXiv:2307.12773 (replaced) [pdf, html, other]
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Title: Violation of Ferromagnetic Ordering of Energy Levels in Spin Rings by Weak Paramagnetism of the SingletComments: Reduced the length, 14 pages, 4 figuresSubjects: Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech)
Sutherland considered the spin-$1/2$ Heisenberg ferromagnetic spin ring for all total spin sectors. He discovered that there is an instability at total spin $0$. The total spin 1 sector has a higher energy groundstate than the groundstate among spin singlets. He called this ``weak paramagnetism.''
Some parts of Sutherland's analysis were obscure. There was a later reconsideration by Dhar and Shastry, who showed that Bloch wall states give a good approximation to the lowest energy eigenstates in each momentum sector. Unfortunately, their ansatz demonstrates no weak paramagnetism.
The question resurfaced due to a conjecture called ``ferromagnetic ordering of energy levels,'' which Sutherland's weak paramagnetism falsifies. We show that Sutherland's finding is numerically validated for spin rings up to size $L=20$. We also show Dhar and Shastry's approximation is demonstrably inexact at total spin $0$, for theoretical reasons. We finally show that the single mode approximation together with a symmetry of the spin singlet can explain weak paramagnetism, heuristically. - [43] arXiv:2406.07717 (replaced) [pdf, html, other]
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Title: Bifurcation analysis of figure-eight choreography in the three-body problem based on crystallographic point groupsSubjects: Mathematical Physics (math-ph)
The bifurcation of figure-eight choreography is analyzed by its symmetry group based on the variational principle of the action. The irreducible representations determine the symmetry and the dimension of the Lyapunov-Schmidt reduced action, which yields four types of bifurcations in the sequence of the bifurcation cascade. Type 1 bifurcation, represented by trivial representation, bifurcates two solutions. Type 2, by non-trivial one-dimensional representation, bifurcates two congruent solutions. Type 3 and 4, by two-dimensional irreducible representations, bifurcate two sets of three and six congruent solutions, respectively. We analyze numerical bifurcation solutions previously published and four new ones: non-symmetric choreographic solution of type 2, non-planar solution of type 2, $y$-axis symmetric solution of type 3, and non-symmetric solution of type 4.
- [44] arXiv:2406.17479 (replaced) [pdf, html, other]
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Title: A representation-theoretical approach to higher-dimensional Lie-Hamilton systems: The symplectic Lie algebra $\mathfrak{sp}(4,\mathbb{R})$Comments: 44 pages. Some typos and misprints have been corrected. Several comments and an appendix have been addedJournal-ref: Commun. Nonlinear Sci. Numer. Simulat. 141 (2025) 108452Subjects: Mathematical Physics (math-ph); Dynamical Systems (math.DS); Exactly Solvable and Integrable Systems (nlin.SI)
A new procedure for the construction of higher-dimensional Lie-Hamilton systems is proposed. This method is based on techniques belonging to the representation theory of Lie algebras and their realization by vector fields. The notion of intrinsic Lie-Hamilton system is defined, and a sufficiency criterion for this property given. Novel four-dimensional Lie-Hamilton systems arising from the fundamental representation of the symplectic Lie algebra $\mathfrak{sp}(4,\mathbb{R})$ are obtained and proved to be intrinsic. Two distinguished subalgebras, the two-photon Lie algebra $\mathfrak{h}_{6}$ and the Lorentz Lie algebra $\mathfrak{so}(1,3)$, are also considered in detail. As applications, coupled time-dependent systems which generalize the Bateman oscillator and the one-dimensional Caldirola-Kanai models are constructed, as well as systems depending on a time-dependent electromagnetic field and generalized coupled oscillators. A superposition rule for these systems, exhibiting interesting symmetry properties, is obtained using the coalgebra method.
- [45] arXiv:2407.01500 (replaced) [pdf, html, other]
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Title: Lie-Hamilton systems on Riemannian and Lorentzian spaces from conformal transformations and some of their applicationsComments: 41 pages, 3 figures. Minor correctionsJournal-ref: J. Phys. A: Math. Theor. 57 (2024) 485203Subjects: Mathematical Physics (math-ph); Dynamical Systems (math.DS); Exactly Solvable and Integrable Systems (nlin.SI)
We propose a generalization of two classes of Lie-Hamilton systems on the Euclidean plane to two-dimensional curved spaces, leading to novel Lie-Hamilton systems on Riemannian spaces (flat $2$-torus, product of hyperbolic lines, sphere and hyperbolic plane), pseudo-Riemannian spaces (anti-de Sitter, de Sitter, and Minkowski spacetimes), as well as to semi-Riemannian spaces (Newtonian or non-relativistic spacetimes). The vector fields, Hamiltonian functions, symplectic form and constants of the motion of the Euclidean classes are recovered by a contraction process. The construction is based on the structure of certain subalgebras of the so-called conformal algebras of the two-dimensional Cayley-Klein spaces. These curved Lie-Hamilton classes allow us to generalize naturally the Riccati, Kummer-Schwarz and Ermakov equations on the Euclidean plane to curved spaces, covering both the Riemannian and Lorentzian possibilities, and where the curvature can be considered as an integrable deformation parameter of the initial Euclidean system.
- [46] arXiv:2410.16747 (replaced) [pdf, html, other]
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Title: Quantum dispersionless KdV hierarchy revisitedSubjects: Mathematical Physics (math-ph); Combinatorics (math.CO); Quantum Algebra (math.QA)
We quantize Hamiltonian structures with hydrodynamic leading terms using the Heisenberg vertex algebra. As an application, we construct the quantum dispersionless KdV hierarchy via a non-associative Weyl quantization procedure and compute the corresponding eigenvalue problem.
- [47] arXiv:2411.02626 (replaced) [pdf, html, other]
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Title: On classical aspects of Bose-Einstein condensationComments: 47 pagesSubjects: Mathematical Physics (math-ph)
Berezin and Weyl quantization are renown procedures for mapping classical, commutative Poisson algebras of observables to their non-commutative, quantum counterparts. The latter is famous for its use on Weyl algebras, while the former is more appropriate for continuous functions decaying at infinity. In this work, we define a variant of the Berezin quantization map, which acts on the classical Weyl algebra $\mathcal{W}(E,0)$ and constitutes a positive \textit{strict deformation quantization}. We use this map as a mathematical tool to compare classical and quantum thermal equilibrium states for a boson gas by computing the classical limit of the latter.
For this scope, we first define a purely algebraic notion of KMS states for the classical Weyl algebra and verify that in the finite volume setting there is only one possible KMS state, which can be interpreted as the Fourier transform of a Gibbs measure on some Hilbert space. Subsequently, we perform a thermodynamic limit and show that the limit points of the finite volume classical KMS state manifest condensation in the zero mode, similarly to what happens in the standard formulation of Bose-Einstein condensation. Lastly, we prove that there exist sequences of quantum KMS states for the infinite volume Bose gas, that converge weak-$^*$ to classical KMS states. Moreover, as the different thermal phases are preserved by this limit, it is demonstrated that a quantum condensate is mapped to a classical one. - [48] arXiv:2411.07093 (replaced) [pdf, html, other]
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Title: Generalized Airy polynomials, Hankel determinants and asymptoticsComments: 27 pagesSubjects: Mathematical Physics (math-ph)
We further study the orthogonal polynomials with respect to the generalized Airy weight based on the work of Clarkson and Jordaan [{\em J. Phys. A: Math. Theor.} {\bf 54} ({2021}) {185202}]. We prove the ladder operator equations and associated compatibility conditions for orthogonal polynomials with respect to a general Laguerre-type weight of the form $w(x)=x^\lambda w_0(x),\;\lambda>-1, x\in\mathbb{R}^+$. By applying them to the generalized Airy polynomials, we are able to derive a discrete system for the recurrence coefficients. Combining with the Toda evolution, we establish the relation between the recurrence coefficients, the sub-leading coefficient of the monic generalized Airy polynomials and the associated Hankel determinant. Using Dyson's Coulomb fluid approach and with the aid of the discrete system for the recurrence coefficients, we obtain the large $n$ asymptotic expansions for the recurrence coefficients and the sub-leading coefficient of the monic generalized Airy polynomials. The large $n$ asymptotic expansion (including the constant term) of the Hankel determinant has been derived by using a recent result in the literature. The long-time asymptotics of these quantities have also been discussed explicitly.
- [49] arXiv:2201.12110 (replaced) [pdf, html, other]
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Title: Dynamical Landauer's principle: Quantifying information transmission by thermodynamicsComments: 5+3 pages & 3 figures. With new title and new resultsSubjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Energy transfer and information transmission are two fundamental aspects of nature. They are seemingly unrelated, while recent findings suggest that a deep connection between them is to be discovered. This amounts to asking: Can we phrase the processes of transmitting classical bits equivalently as specific energy-transmitting tasks, thereby uncovering foundational links between them? We answer this question positively by showing that, for a broad class of classical communication tasks, a quantum dynamics' ability to transmit $n$ bits of classical information is equivalent to its ability to transmit $n$ units of energy in a thermodynamic task. This finding not only provides an analytical correspondence between information transmission and energy extraction tasks, but also quantifies classical communication by thermodynamics. Furthermore, our findings uncover the dynamical version of Landauer's principle, showing the strong link between transmitting information and energy. In the asymptotic regime, our results further provide thermodynamic meanings for the well-known Holevo-Schumacher-Westmoreland Theorem in quantum communication theory.
- [50] arXiv:2209.07803 (replaced) [pdf, html, other]
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Title: Generalized gravitational fields and well-posedness of the Boussinesq systems on non-compact Riemannian ManifoldsComments: 28 pagesSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Differential Geometry (math.DG); Dynamical Systems (math.DS); Functional Analysis (math.FA)
We study the global existence, uniqueness and exponential stability of mild solutions to the Boussinesq systems equipped with a generalized gravitational field on the framework of non-compact Riemannian manifolds. We work on some manifolds satisfying some bounded and negative conditions on curvature tensors. We consider a couple of Stokes and heat semigroups associated with the corresponding linear system which provides a vectorial matrix semigoup. By using dispersive and smoothing estimates of the vectorial matrix semigroup we establish the global-in-time existence and uniqueness of mild solutions for linear systems. Next, we can pass from the linear system to the semilinear systems to obtain the well-posedness by using fixed point arguments. Moreover, we will prove the exponential stability of such solutions by using Gronwall's inequality.
- [51] arXiv:2211.13176 (replaced) [pdf, html, other]
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Title: A Chiral ${\Lambda}$-$\mathfrak{bms}_4$ Symmetry of AdS$_4$ GravityComments: 34 pages. v3: Accepted for publication in Nuclear Phys.B. References added, minor changes in Introduction and Conclusions. v2: The title has been changed to reflect better the content of the paper. The manuscript has been revised to add the section on the derivation of $\mathcal{W}$-algebra from AdS$_4$ gravity. Elaborated introduction. References addedSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
Generalising the chiral boundary conditions of $\mathbb{R}^{1,3}$ gravity for AdS$_4$ gravity, we derive chiral locally AdS$_4$ solutions in the Newman-Unti gauge consistent with a variational principle whose asymptotic symmetry algebra we show, to be an infinite-dimensional chiral extension of $\mathfrak{so}(2,3)$. This symmetry algebra coincides with the chiral $\mathfrak{bms}_4$ algebra in the flat space limit. We posit this symmetry algebra as the chiral version of recently discovered $\Lambda$-$\mathfrak{bms}_4$ algebra. We postulate line integral charges from the bulk AdS$_4$ gravity corresponding to this chiral symmetry algebra and show that the charges obey the semi-classical limit of a $\mathcal{W}$-algebra that includes a level $\kappa$ Kac-Moody $\mathfrak{sl}(2,\mathbb{R})$ current algebra. Furthermore, using the standard tools of $2d$ CFT, we derive the quantum version of this $\mathcal{W}$-algebra which may be denoted by $\mathcal{W}(2;(3/2)^2,1^3)$.
- [52] arXiv:2303.14629 (replaced) [pdf, html, other]
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Title: Average entropy and asymptoticsComments: Appeared in J. Korean Math. Soc. 2024Subjects: Differential Geometry (math.DG); Mathematical Physics (math-ph)
We determine the $N\to \infty$ asymptotics of the expected value of entanglement entropy in $H_{1,N}\otimes H_{2,N}$, where $H_{1,N}$ and $H_{2,N}$ are the spaces of holomorphic sections of the $N$-th tensor powers of hermitian ample line bundles on compact complex manifolds.
- [53] arXiv:2303.15338 (replaced) [pdf, other]
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Title: Decay and non-decay for the massless Vlasov equation on subextremal and extremal Reissner-Nordstr\"om black holesJournal-ref: Arch Rational Mech Anal 248, 118 (2024)Subjects: General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph); Analysis of PDEs (math.AP); Differential Geometry (math.DG)
We study the massless Vlasov equation on the exterior of the subextremal and extremal Reissner-Nordström spacetimes. We prove that moments decay at an exponential rate in the subextremal case and at a polynomial rate in the extremal case. This polynomial rate is shown to be sharp along the event horizon. In the extremal case we show that transversal derivatives of certain components of the energy momentum tensor do not decay along the event horizon if the solution and its first time derivative are initially supported on a neighbourhood of the event horizon. The non-decay of transversal derivatives in the extremal case is compared to the work of Aretakis on instability for the wave equation. Unlike Aretakis' results for the wave equation, which exploit a hierarchy of conservation laws, our proof is based entirely on a quantitative analysis of the geodesic flow and conservation laws do not feature in the present work.
- [54] arXiv:2304.14375 (replaced) [pdf, html, other]
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Title: High moments of the SHE in the clustering regimesComments: 44 pages, 7 figures. Updated to match the published versionJournal-ref: Journal of Functional Analysis 288(1), 2025, 110675Subjects: Probability (math.PR); Mathematical Physics (math-ph)
We analyze the high moments of the Stochastic Heat Equation (SHE) via a transformation to the attractive Brownian Particles (BPs), which are Brownian motions interacting via pairwise attractive drift. In those scaling regimes where the particles tend to cluster, we prove a Large Deviation Principle (LDP) for the empirical measure of the attractive BPs. Under what we call the 1-to-$n$ initial-terminal condition, we characterize the unique minimizer of the rate function and relate the minimizer to the spacetime limit shapes of the Kardar--Parisi--Zhang (KPZ) equation in the upper tails. The results of this paper are used in the companion paper Lin and Tsai (2023) to prove an $n$-point, upper-tail LDP for the KPZ equation and to characterize the corresponding spacetime limit shape.
- [55] arXiv:2309.05492 (replaced) [pdf, html, other]
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Title: Majorana fermion induced power-law scaling in the violations of the Wiedemann-Franz lawComments: 15 pages, 4 figures, 1 table, revised version accepted for publication in Journal of Applied PhysicsJournal-ref: Journal of Applied Physics (2024)Subjects: Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Strongly Correlated Electrons (cond-mat.str-el); Mathematical Physics (math-ph); Popular Physics (physics.pop-ph); Quantum Physics (quant-ph)
Violation of the Wiedemann-Franz law in a 2D topological insulator due to Majorana bound states is studied via the Lorenz ratio in the single-particle picture. We study the scaling of the Lorenz ratio in the presence and absence of Majorana bound states with inelastic scattering modeled using a Buttiker voltage-temperature probe. We compare our results with that seen in a quantum dot junction in the Luttinger liquid picture operating in the topological Kondo regime. We explore the scaling of the Lorentz ratio in our setup when either phase and momentum relaxation or phase relaxation is present. This scaling differs from that predicted by the Luttinger liquid picture for both uncoupled and coupled Majorana cases.
- [56] arXiv:2309.08461 (replaced) [pdf, html, other]
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Title: Combed Trisection Diagrams and Non-Semisimple 4-Manifold InvariantsComments: 62 pages, many figures and diagrams; v3: text expanded, typos correctedSubjects: Quantum Algebra (math.QA); Mathematical Physics (math-ph); Geometric Topology (math.GT)
Given a triple $H$ of (possibly non-semisimple) Hopf algebras equipped with pairings satisfying a set of properties, we describe a construction of an associated smooth, scalar invariant $\tau_H(X,\pi)$ of a simply connected, compact, oriented $4$-manifold $X$ and an open book $\pi$ on its boundary. This invariant generalizes an earlier semisimple version and is calculated using a trisection diagram $T$ for $X$ and a certain type of combing of the trisection surface. We explain a general calculation of this invariant for a family of exotic 4-manifolds with boundary called Stein nuclei, introduced by Yasui. After investigating many low-dimensional Hopf algebras up to dimension 11, we have not been able to find non-semisimple Hopf triples that satisfy the criteria for our invariant. Nonetheless, appropriate Hopf triples may exist outside the scope of our explorations.
- [57] arXiv:2310.08554 (replaced) [pdf, other]
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Title: Generalized symmetries in singularity-free nonlinear $\sigma$ models and their disordered phasesComments: 13+12 pages, 1+2 figures. v2: published versionJournal-ref: Phys. Rev. B 110, 195149 (2024)Subjects: Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
We study the nonlinear $\sigma$-model in ${(d+1)}$-dimensional spacetime with connected target space $K$ and show that, at energy scales below singular field configurations (such as vortices), it has an emergent non-invertible higher symmetry. The symmetry defects of the emergent symmetry are described by the $d$-representations of a discrete $d$-group $\mathbb{G}^{(d)}$ (i.e. the emergent symmetry is the dual of the invertible $d$-group $\mathbb{G}^{(d)}$ symmetry). The $d$-group $\mathbb{G}^{(d)}$ is determined such that its classifying space $B\mathbb{G}^{(d)}$ is given by the $d$-th Postnikov stage of $K$. In $(2+1)$D and for finite $\mathbb{G}^{(2)}$, this symmetry is always holo-equivalent to an invertible ${0}$-form (ordinary) symmetry with potential 't Hooft anomaly. The singularity-free disordered phase of the nonlinear $\sigma$-model spontaneously breaks this symmetry, and when $\mathbb{G}^{(d)}$ is finite, it is described by the deconfined phase of $\mathbb{G}^{(d)}$ higher gauge theory. We consider examples of such disordered phases. We focus on a singularity-free $S^2$ nonlinear $\sigma$-model in ${(3+1)}$D and show that it has an emergent non-invertible higher symmetry. As a result, its disordered phase is described by axion electrodynamics and has two gapless modes corresponding to a photon and a massless axion. Notably, this non-perturbative result is different from the results obtained using the $S^N$ and $\mathbb{C}P^{N-1}$ nonlinear $\sigma$-models in the large-$N$ limit.
- [58] arXiv:2311.15621 (replaced) [pdf, html, other]
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Title: Operator-state correspondence in simple current extended conformal field theories: Toward a general understanding of chiral conformal field theories and topological ordersComments: A figure is added, and discussions on a system with higher space-time dimensions are added (v2). Literature on recent mathematical and theoretical studies are added, and the detailed explanations of the technique are added. Typos are corrected (v3)Subjects: High Energy Physics - Theory (hep-th); Statistical Mechanics (cond-mat.stat-mech); Strongly Correlated Electrons (cond-mat.str-el); Mathematical Physics (math-ph)
In this work, we revisit the operator-state correspondence in the Majorana conformal field theory (CFT) with emphasis on its semion representation. Whereas the semion representation (or $Z_{2}$ extension of the chiral Ising CFT) gives a concise ``abelian" (or invertible) representation in the level of fusion rule and quantum states, there exists subtlety when considering the chiral multipoint correlation function. In this sense, the operator-state correspondence in the semion sector of the fermionic theory inevitably contains difficulty coming from its anomalous conformal dimension $1/16$ as a $Z_{2}$ symmetry operator. By analyzing the asymptotic behaviors of the existing correlation functions, we propose a nontrivial correspondence between the chiral conformal blocks and bulk correlation functions containing both order and disorder fields. One can generalize this understanding to $Z_{N}$ models or fractional supersymmetric models (in which there exist long-standing open problems). We expect this may improve our understanding of the simple current extension of CFT which can appear commonly in the studies of topologically ordered systems.
- [59] arXiv:2311.18706 (replaced) [pdf, html, other]
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Title: Certified algorithms for equilibrium states of local quantum HamiltoniansJournal-ref: Nature Communications 15, 7394 (2024)Subjects: Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)
Predicting observables in equilibrium states is a central yet notoriously hard question in quantum many-body systems. In the physically relevant thermodynamic limit, certain mathematical formulations of this task have even been shown to result in undecidable problems. Using a finite-size scaling of algorithms devised for finite systems often fails due to the lack of certified convergence bounds for this limit. In this work, we design certified algorithms for computing expectation values of observables in the equilibrium states of local quantum Hamiltonians, both at zero and positive temperature. Importantly, our algorithms output rigorous lower and upper bounds on these values. This allows us to show that expectation values of local observables can be approximated in finite time, contrasting related undecidability results. When the Hamiltonian is commuting on a 2-dimensional lattice, we prove fast convergence of the hierarchy at high temperature and as a result for a desired precision $\varepsilon$, local observables can be approximated by a convex optimization program of quasi-polynomial size in $1/\varepsilon$.
- [60] arXiv:2404.03492 (replaced) [pdf, html, other]
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Title: Three perspectives on entropy dynamics in a non-Hermitian two-state systemComments: 12 pages, 6 figuresJournal-ref: Phys. Scr. 99, 125234 (2024)Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
A comparative study of entropy dynamics as an indicator of physical behavior in an open two-state system with balanced gain and loss is presented. We distinguish the perspective taken in utilizing the conventional framework of Hermitian-adjoint states from an approach that is based on biorthogonal-adjoint states and a third case based on an isospectral mapping. In this it is demonstrated that their differences are rooted in the treatment of the environmental coupling mode. For unbroken $\mathcal{PT}$ symmetry of the system, a notable characteristic feature of the perspective taken is the presence or absence of purity oscillations, with an associated entropy revival. The description of the system is then continued from its $\mathcal{PT}$-symmetric pseudo-Hermitian phase into the regime of spontaneously broken symmetry, in the latter two approaches through a non-analytic operator-based continuation, yielding a Lindblad master equation based on the $\mathcal{PT}$ charge operator $\mathcal{C}$. This phase transition indicates a general connection between the pseudo-Hermitian closed-system and the Lindbladian open-system formalism through a spontaneous breakdown of the underlying physical reflection symmetry.
- [61] arXiv:2405.02018 (replaced) [pdf, html, other]
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Title: Time-of-arrival distributions for continuous quantum systems and application to quantum backflowComments: 13 pages, 2 Figures, 1 Table. This new version contains a general formula for the current of a superposition of two waves and applications to a superposition of two Gaussian wave packetsJournal-ref: Phys. Rev. A 110, 052217 (2024)Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Using standard results from statistics, we show that for any continuous quantum system (Gaussian or otherwise) and any observable $\widehat{A}$ (position or otherwise), the distribution $\pi_{a}\left(t\right)$ of time measurement at a fixed state $a$ can be inferred from the distribution $\rho_{t}\left( a\right)$ of a state measurement at a fixed time $t$ via the transformation $\pi_{a}(t) \propto \left\vert \frac{\partial }{\partial t} \int_{-\infty }^a \rho_t(u) du \right\vert$. This finding suggests that the answer to the long-lasting time-of-arrival problem is in fact secretly hidden within the Born rule, and therefore does not require the introduction of a time operator or a commitment to a specific (e.g., Bohmian) ontology. The generality and versatility of the result are illustrated by applications to the time-of-arrival at a given location for a free particle in a superposed state and to the time required to reach a given velocity for a free-falling quantum particle. Our approach also offers a potentially promising new avenue toward the design of an experimental protocol for the yet-to-be-performed observation of the phenomenon of quantum backflow.
- [62] arXiv:2405.08083 (replaced) [pdf, html, other]
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Title: 5d 2-Chern-Simons theory and 3d integrable field theoriesComments: 29 pages; v2: Minor updates; v3: final version to appear in Communications in Mathematical PhysicsSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
The $4$-dimensional semi-holomorphic Chern-Simons theory of Costello and Yamazaki provides a gauge-theoretic origin for the Lax connection of $2$-dimensional integrable field theories. The purpose of this paper is to extend this framework to the setting of $3$-dimensional integrable field theories by considering a $5$-dimensional semi-holomorphic higher Chern-Simons theory for a higher connection $(A,B)$ on $\mathbb{R}^3 \times \mathbb{C}P^1$. The input data for this theory are the choice of a meromorphic $1$-form $\omega$ on $\mathbb{C}P^1$ and a strict Lie $2$-group with cyclic structure on its underlying Lie $2$-algebra. Integrable field theories on $\mathbb{R}^3$ are constructed by imposing suitable boundary conditions on the connection $(A,B)$ at the $3$-dimensional defects located at the poles of $\omega$ and choosing certain admissible meromorphic solutions of the bulk equations of motion. The latter provides a natural notion of higher Lax connection for $3$-dimensional integrable field theories, including a $2$-form component $B$ which can be integrated over Cauchy surfaces to produce conserved charges. As a first application of this approach, we show how to construct a generalization of Ward's $(2+1)$-dimensional integrable chiral model from a suitable choice of data in the $5$-dimensional theory.
- [63] arXiv:2405.19215 (replaced) [pdf, html, other]
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Title: Two dimensional potential theory with a view towards vortex motion: Energy, capacity and Green functionsComments: 104 pages, 6 figuresSubjects: Complex Variables (math.CV); Mathematical Physics (math-ph); Fluid Dynamics (physics.flu-dyn)
The paper reviews some parts of classical potential theory with applications to two dimensional fluid dynamics, in particular vortex motion. Energy and forces within a system of point vortices are similar to those for point charges when the vortices are kept fixed, but the dynamics is different in the case of free vortices. Starting from Bernoulli's equation we derive these laws. Letting the number of vortices tend to infinity leads in the limit to considerations of capacity, harmonic measure and many other notions in potential theory. In particular various kinds of Green functions have a central role in the paper, where we make a difference between electrostatic and hydrodynamic Green function.
We also consider the corresponding concepts in the case of closed Riemann surfaces provided with a metric. From a canonically defined monopole Green function we rederive much of the classical theory of harmonic and analytic forms. In the final section of the paper we return to the planar case, then reappearing in form of a symmetric Riemann surface, the Schottky double. Bergman kernels, electrostatic and hydrodynamic, come up naturally, and associated to the Green function the is a certain Robin function which is important for vortex motion and which also relates to capacity functions in classical potential theory. - [64] arXiv:2406.18409 (replaced) [pdf, html, other]
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Title: Gauged Skyrme analogue of Chern-PontryaginComments: Detailed calculation of Skyrme--Chern-Pontryagin densities in 2,3,4,5 dimensionsSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
An analogue of the Chern-Pontryagin density for $SO(D)$ gauged $O(D+1)$ Skyrme systems, referred to as Skyrme--Chern-Pontryagin (SCP) densities is known for dimensions $D=2,3,4$. Since these are defined only through a prescription, it is necessary to extend the realisation to higher $D$, which is carried out here for $D=5$. The construction of SCP densities in $D=2,3,4,5$ is presented here in a unified pattern with the aim of pointing out to the possible extrapolation to all dimensions.
- [65] arXiv:2408.09983 (replaced) [pdf, other]
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Title: Integrability of Generalised Skew-Symmetric Replicator Equations via Graph EmbeddingsComments: 38 pages, 17 figuresSubjects: Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (math-ph); Dynamical Systems (math.DS)
It is known that there is a one-to-one mapping between oriented directed graphs and zero-sum replicator dynamics (Lotka-Volterra equations) and that furthermore these dynamics are Hamiltonian in an appropriately defined nonlinear Poisson bracket. In this paper, we investigate the problem of determining whether these dynamics are Liouville-Arnold integrable, building on prior work graph in graph decloning by Evripidou et al. [J. Phys. A., 55:325201, 2022] and graph embedding by Paik and Griffin [Phys. Rev. E. 107(5): L052202, 2024]. Using the embedding procedure from Paik and Griffin, we show (with certain caveats) that when a graph producing integrable dynamics is embedded in another graph producing integrable dynamics, the resulting graph structure also produces integrable dynamics. We also construct a new family of graph structures that produces integrable dynamics that does not arise either from embeddings or decloning. We use these results, along with numerical methods, to classify the dynamics generated by almost all oriented directed graphs on six vertices, with three hold-out graphs that generate integrable dynamics and are not part of a natural taxonomy arising from known families and graph operations. These hold-out graphs suggest more structure is available to be found. Moreover, the work suggests that oriented directed graphs leading to integrable dynamics may be classifiable in an analogous way to the classification of finite simple groups, creating the possibility that there is a deep connection between integrable dynamics and combinatorial structures in graphs.
- [66] arXiv:2408.13342 (replaced) [pdf, html, other]
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Title: Teleparallel Geometry with Spherical Symmetry: The diagonal and proper framesComments: 29 pagesJournal-ref: JCAP11(2024)033Subjects: General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph); Differential Geometry (math.DG)
We present the proper co-frame and its corresponding (diagonal) co-frame/spin connection pair for spherically symmetric geometries which can be used as an initial ansatz in any theory of teleparallel gravity. The Lorentz transformation facilitating the move from one co-frame to the other is also presented in factored form. The factored form also illustrates the nature of the two degrees of freedom found in the spin connection. The choice of coordinates in restricting the number of arbitrary functions is also presented. Beginning with a thorough presentation of teleparallel gravity using the metric affine gauge theory approach and concentrating on f(T) teleparallel gravity, we express the field equations in the diagonal co-frame. We argue that the choice of diagonal co-frame may be more advantageous over the proper co-frame choice. Finally, assuming one additional symmetry, we restrict ourselves to the Kantowski-Sachs tele-parallel geometries, and determine some solutions.
- [67] arXiv:2409.04571 (replaced) [pdf, html, other]
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Title: Statistics for Differential Topological Properties between Data Sets with an Application to Reservoir ComputersComments: 17 pages, 14 figuresSubjects: Chaotic Dynamics (nlin.CD); Mathematical Physics (math-ph); Data Analysis, Statistics and Probability (physics.data-an)
It is common for researchers to record long, multiple time series from experiments or calculations. But sometimes there are no good models for the systems or no applicable mathematical theorems that can tell us when there are basic relationships between subsets of the time series data such as continuity, differentiability, embeddings, etc. The data is often higher dimensional and simple plotting will not guide us. At that point fitting the data to polynomials, Fourier series, etc. becomes uncertain. Even at the simplest level, having data that shows there is a function between the data subsets is useful and a negative answer means that more particular data fitting or analysis will be suspect and probably fail. We show here statistics that test time series subsets for basic mathematical properties and relations between them that not only indicate when more specific analyses are safe to do, but whether the systems are operating correctly. We apply these statistics to examples from reservoir computing where an important property of reservoir computers is that the reservoir system establishes an embedding of the drive system in order to make any other calculations with the reservoir computer successful.
- [68] arXiv:2409.07381 (replaced) [pdf, html, other]
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Title: A Lie algebraic pattern behind logarithmic CFTsComments: 31 pages, minor corrections including title changes and addition of referencesSubjects: Representation Theory (math.RT); Mathematical Physics (math-ph)
We introduce a new concept named shift system. This is a purely Lie algebraic setting to develop the geometric representation theory of Feigin-Tipunin construction of logarithmic conformal field theories. After reformulating the discussion in the second author's past works under this new setting, as an application, we extend almost all the main results of these works to the (multiplet) principal W-algebra at positive integer level associated with a simple Lie algebra $\mathfrak{g}$ and Lie superalgebra $\mathfrak{osp}(1|2n)$, respectively. This paper also contains an appendix by Myungbo Shim on the relationship between Feigin-Tipunin construction and recent quantum field theories.
- [69] arXiv:2409.17031 (replaced) [pdf, html, other]
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Title: Null geodesics around a black hole with weakly coupled global monopole chargeComments: 16 pages, 11 figuresSubjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Astrophysical Phenomena (astro-ph.HE); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
In this paper, we study an asymptotically flat black hole spacetime with weakly nonminimally coupled monopole charge. We analytically and numerically investigate light ray propagation around such a black hole by employing the common Lagrangian formalism. Our analysis encompasses both radial and angular geodesics, for which we present analytical solutions in terms of incomplete Lauricella hypergeometric functions. Additionally, we explore the impact of the coupling constant on geodesic motion. Based on observations from the Event Horizon Telescope, we constrain the black hole parameters, resulting in a coupling constant range of $-0.5\lesssim \alpha\lesssim 0.5$. Throughout our analysis, we simulate all possible trajectories and, where necessary, perform numerical inversion of the included integrals.
- [70] arXiv:2410.15201 (replaced) [pdf, html, other]
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Title: Learning the Rolling Penny DynamicsSubjects: Dynamical Systems (math.DS); Machine Learning (cs.LG); Mathematical Physics (math-ph)
We consider learning the dynamics of a typical nonholonomic system -- the rolling penny. A nonholonomic system is a system subject to nonholonomic constraints. Unlike a holonomic constraints, a nonholonomic constraint does not define a submanifold on the configuration space. Therefore, the inverse problem of finding the constraints has to involve the tangent space. This paper discusses how to learn the dynamics, as well as the constraints for such a system, given the data set of discrete trajectories on the tangent bundle $TQ$.
- [71] arXiv:2410.19039 (replaced) [pdf, html, other]
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Title: Quantum State Tomography of Photonic Qubits with Realistic Coherent Light SourcesJournal-ref: Quantum Inf. Comput. Vol. 24 (No. 1), 31-39 (2024)Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph); Computational Physics (physics.comp-ph); Optics (physics.optics)
Quantum state tomography (QST) is an essential technique for characterizing quantum states. However, practical implementations of QST are significantly challenged by factors such as shot noise, attenuation, and Raman scattering, especially when photonic qubits are transmitted through optical fibers alongside classical signals. In this paper, we present a numerical framework to simulate and evaluate the efficiency of QST under these realistic conditions. The results reveal how the efficiency of QST is influenced by the power of the classical signal. By analyzing the fidelity of reconstructed states, we provide insights into the limitations and potential improvements for QST in noisy environments.
- [72] arXiv:2411.12267 (replaced) [pdf, html, other]
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Title: Cost of controllability of the Burgers' equation linearized at a steady shock in the vanishing viscosity limitSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph)
We consider the one-dimensional Burgers' equation linearized at a stationary shock, and investigate its null-controllability cost with a control at the left endpoint. We give an upper and a lower bound on the control time required for this cost to remain bounded in the vanishing viscosity limit, as well as a rough description of an admissible control. The proof relies on complex analysis and adapts methods previously used to tackle the same issue with a constant transport term.
- [73] arXiv:2411.14302 (replaced) [pdf, html, other]
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Title: Electrodynamics of Vortices in Quasi-2D Scalar Bose-Einstein CondensatesComments: 15 pages, 1 figureSubjects: Quantum Gases (cond-mat.quant-gas); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph); Plasma Physics (physics.plasm-ph); Quantum Physics (quant-ph)
In two spatial dimensions, vortex-vortex interactions approximately vary with the logarithm of the inter-vortex distance, making it possible to describe an ensemble of vortices as a Coulomb gas. We introduce a duality between vortices in a quasi-two-dimensional (quasi-2D) scalar Bose-Einstein condensates (BEC) and effective Maxwell's electrodynamics. Specifically, we address the general scenario of inhomogeneous, time-dependent BEC number density with dissipation or rotation. Starting from the Gross-Pitaevskii equation (GPE), which describes the mean-field dynamics of a quasi-2D scalar BEC without dissipation, we show how to map vortices in a quasi-2D scalar BEC to 2D electrodynamics beyond the point-vortex approximation, even when dissipation is present or in a rotating system. The physical meaning of this duality is discussed.