Mathematical Physics
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 [1] arXiv:2406.13089 [pdf, html, other]

Title: Uniqueness and CLT for the Ground State of the Disordered MonomerDimer Model on $\mathbb{Z}^{d}$Subjects: Mathematical Physics (mathph); Probability (math.PR)
We prove that the disordered monomerdimer model does not admit infinite volume incongruent ground states in $\mathbb{Z}^d$ which can be obtained as a limit of finite volume ground states. Furthermore, we also prove that these ground states are stable under perturbation of the weights in a precise sense.
As an application, we obtain a CLT for the ground state weight for a growing sequence of tori. Our motivation stems from a similar and long standing open question for the short range EdwardsAnderson spin glass model.  [2] arXiv:2406.13503 [pdf, html, other]

Title: Integrable $\mathbb{Z}_2^2$graded Extensions of the Liouville and SinhGordon TheoriesComments: 25 pagesSubjects: Mathematical Physics (mathph); High Energy Physics  Theory (hepth); Exactly Solvable and Integrable Systems (nlin.SI)
In this paper we present a general framework to construct integrable $\mathbb{Z}_2^2$graded extensions of classical, twodimensional Toda and conformal affine Toda theories. The scheme is applied to define the extended Liouville and SinhGordon models; they are based on $\mathbb{Z}_2^2$graded color Lie algebras and their fields satisfy a parabosonic statististics. The mathematical tools here introduced are the $\mathbb{Z}_2^2$graded covariant extensions of the Lax pair formalism and of the Polyakov's soldering procedure. The $\mathbb{Z}_2^2$graded SinhGordon model is derived from an affine $\mathbb{Z}_2^2$graded color Lie algebra, mimicking a procedure originally introduced by BabelonBonora to derive the ordinary SinhGordon model. The color Lie algebras under considerations are: the $6$generator $\mathbb{Z}_2^2$graded $sl_2$, the $\mathbb{Z}_2^2$graded affine ${\widehat{sl_2}}$ algebra with two central extensions, the $\mathbb{Z}_2^2$graded Virasoro algebra obtained from a Hamiltonian reduction.
New submissions for Friday, 21 June 2024 (showing 2 of 2 entries )
 [3] arXiv:2406.12855 (crosslist from math.GM) [pdf, other]

Title: Moving frame and spin field representations of submanifolds in flat spaceComments: 20 pages, 1 figureSubjects: General Mathematics (math.GM); General Relativity and Quantum Cosmology (grqc); High Energy Physics  Theory (hepth); Mathematical Physics (mathph)
We introduce a spin field approach, that is compatible with the Cartan moving frame method, to describe the submanifold in a flat space. In fact, we consider a kind of spin field $\psi$, that satisfies a Killing spin field equation (analogous to a Killing spinor equation) written in terms of the Clifford algebra, and we use the spin field to locally rotate the orthonormal basis $\{\hat{e}_\mathtt{I}\}$. Then, the deformed orthonormal frame $\{\tilde{\psi}\hat{e}_\mathtt{I}\psi\}$ can be seen as the moving frame of a submanifold. We find some solutions to the Killing spin field equation and demonstrate an explicit example. Using the product of the spin fields, one can easily generate a new immersion submanifold, and this technique should be useful for studies in geometry and physics. Through the spin field, we find a linear relation between the connection and the extrinsic curvature of the submanifold. We propose a conjecture that any solution of the Killing spin field equation can be locally written as the product of the solutions we find.
 [4] arXiv:2406.12868 (crosslist from math.SP) [pdf, other]

Title: Triple products of eigenfunctions and spectral geometryComments: 6 pages, no figuresSubjects: Spectral Theory (math.SP); Mathematical Physics (mathph); Functional Analysis (math.FA)
Using elementary techniques from Geometric Analysis, Partial Differential Equations, and Abelian $C^*$ Algebras, we uncover a novel, yet familiar, global geometric invariant  namely the indexed set of integrals of triple products of eigenfunctions of the LaplaceBeltrami operator, to precisely characterize which isospectral closed Riemannian manifolds are isometric.
 [5] arXiv:2406.12895 (crosslist from qbio.NC) [pdf, html, other]

Title: Temporal Complexity of a HopfieldType Neural Model in Random and ScaleFree GraphsSubjects: Neurons and Cognition (qbio.NC); Disordered Systems and Neural Networks (condmat.disnn); Mathematical Physics (mathph); Numerical Analysis (math.NA); Adaptation and SelfOrganizing Systems (nlin.AO)
The Hopfield network model and its generalizations were introduced as a model of associative, or contentaddressable, memory. They were widely investigated both as a unsupervised learning method in artificial intelligence and as a model of biological neural dynamics in computational neuroscience. The complexity features of biological neural networks are attracting the interest of scientific community since the last two decades. More recently, concepts and tools borrowed from complex network theory were applied to artificial neural networks and learning, thus focusing on the topological aspects. However, the temporal structure is also a crucial property displayed by biological neural networks and investigated in the framework of systems displaying complex intermittency. The IntermittencyDriven Complexity (IDC) approach indeed focuses on the metastability of selforganized states, whose signature is a powerdecay in the interevent time distribution or a scaling behavior in the related eventdriven diffusion processes. The investigation of IDC in neural dynamics and its relationship with network topology is still in its early stages. In this work we present the preliminary results of a IDC analysis carried out on a bioinspired Hopfieldtype neural network comparing two different connectivities, i.e., scalefree vs. random network topology. We found that random networks can trigger complexity features similar to that of scalefree networks, even if with some differences and for different parameter values, in particular for different noise levels.
 [6] arXiv:2406.12962 (crosslist from condmat.strel) [pdf, html, other]

Title: Gauging modulated symmetries: KramersWannier dualities and noninvertible reflectionsComments: 67 pagesSubjects: Strongly Correlated Electrons (condmat.strel); High Energy Physics  Theory (hepth); Mathematical Physics (mathph); Quantum Physics (quantph)
Modulated symmetries are internal symmetries that act in a nonuniform, spatially modulated way and are generalizations of, for example, dipole symmetries. In this paper, we systematically study the gauging of finite Abelian modulated symmetries in ${1+1}$ dimensions. Working with local Hamiltonians of spin chains, we explore the dual symmetries after gauging and their potential new spatial modulations. We establish sufficient conditions for the existence of an isomorphism between the modulated symmetries and their dual, naturally implemented by lattice reflections. For instance, in systems of prime qudits, translation invariance guarantees this isomorphism. For nonprime qudits, we show using techniques from ring theory that this isomorphism can also exist, although it is not guaranteed by lattice translation symmetry alone. From this isomorphism, we identify new KramersWannier dualities and construct related noninvertible reflection symmetry operators using sequential quantum circuits. Notably, this noninvertible reflection symmetry exists even when the system lacks ordinary reflection symmetry. Throughout the paper, we illustrate these results using various simple toy models.
 [7] arXiv:2406.12973 (crosslist from quantph) [pdf, html, other]

Title: Tomography of clock signals using the simplest possible referenceComments: 7 pages + 10 pages appendix, 1 figureSubjects: Quantum Physics (quantph); Mathematical Physics (mathph)
We show that finite physical clocks always have wellbehaved signals, namely that every waitingtime distribution generated by a physical process on a system of finite size is guaranteed to be bounded by a decay envelope. Following this consideration, we show that one can reconstruct the distribution using only operationally available information, namely, that of the ordering of the ticks of one clock with the respect to those of another clock (which we call the reference), and that the simplest possible reference clock  a Poisson process  suffices.
 [8] arXiv:2406.13033 (crosslist from math.CO) [pdf, html, other]

Title: Arrival of information at a target set in a networkSubjects: Combinatorics (math.CO); Mathematical Physics (mathph); Dynamical Systems (math.DS)
We consider labelings of a finite regular tree by a finite alphabet subject to restrictions specified by a nonnegative transition matrix, propose an algorithm for determining whether the set of possible configurations on the last row of the tree is independent of the symbol at the root, and prove that the algorithm succeeds in a bounded number of steps, provided that the dimension of the tree is greater than or equal to the maximum row sum of the transition matrix. (The question was motivated by calculation of topological pressure on trees and is an extension of the idea of primitivity for nonnegative matrices.)
 [9] arXiv:2406.13047 (crosslist from condmat.suprcon) [pdf, html, other]

Title: Symplectic Representation of the GinzburgLandau TheoryComments: 9 pages, 1 figuresSubjects: Superconductivity (condmat.suprcon); Mathematical Physics (mathph)
In this work, the GinzburgLandau theory is represented on a symplectic manifold with a phase space content. The order parameter is defined by a quasiprobability amplitude, which gives rise to a quasiprobability distribution function, i.e., a Wignertype function. The starting point is the thermal group representation of Euclidean symmetries and gauge symmetry. Wellknown basic results on the behavior of a superconductor are rederived, providing the consistency of representation. The critical superconducting current density is determined and its usual behavior is inferred. The negativety factor associated with the quasidistribution function is analyzed, providing information about the nonclassicality nature of the superconductor state in the region closest to the edge of the superconducting material.
 [10] arXiv:2406.13071 (crosslist from condmat.statmech) [pdf, html, other]

Title: Structural analysis of Gibbs states and metastates in shortrange classical spin glasses: indecomposable metastates, dynamicallyfrozen states, and metasymmetryComments: 69 pages (apologies)Subjects: Statistical Mechanics (condmat.statmech); Disordered Systems and Neural Networks (condmat.disnn); Mathematical Physics (mathph)
We consider shortrange classical spin glasses, or other disordered systems, consisting of Ising spins. For a lowtemperature Gibbs state in infinite size in such a system, for given random bonds, it is controversial whether its decomposition into pure states will be trivial or nontrivial. We undertake a general study of the overall structure of this problem, based on metastates, which are essential to prove the existence of a thermodynamic limit. A metastate is a probability distribution on Gibbs states, for given disorder, that satisfies certain covariance properties. First, we prove that any metastate can be decomposed as a mixture of indecomposable metastates, and that all Gibbs states drawn from an indecomposable metastate are alike macroscopically. Next, we consider stochastic stability of a metastate under random perturbations of the disorder, and prove that any metastate is stochastically stable. Dynamicallyfrozen states play a role in the analysis of Gibbs states drawn from a metastate, either as states or as parts of states. Using a mapping into real Hilbert space, we prove results about Gibbs states, and classify them into six types. Any indecomposable metastate has a compact symmetry group, though it may be trivial; we call this a metasymmetry. Metastateaverage states are studied, and can be related to states arising dynamically at long times after a quench from high temperature, under some conditions. Many features that are permitted by general results are already present in replica symmetry breaking (RSB). Our results are for cases both with and without spinflip symmetry of the Hamiltonian and, technically, we use mixed $p$spininteraction models.
 [11] arXiv:2406.13120 (crosslist from math.RT) [pdf, html, other]

Title: A different approach to positive traces on generalized qWeyl algebrasComments: 6 pages, slightly different version to appear in the Proceedings of the 15th International Workshop "Lie Theory and Its Applications in Physics" (LT15), 1925 June 2023, Varna, BulgariaSubjects: Representation Theory (math.RT); High Energy Physics  Theory (hepth); Mathematical Physics (mathph)
Positive twisted traces are mathematical objects that could be useful in computing certain parameters of superconformal field theories. The case when $\mathcal{A}$ is a $q$Weyl algebra and $\rho$ is a certain antilinear automorphism of $\mathcal{A}$ was considered in arXiv:2105.12652. Here we consider more general choices of $\rho$. In particular, we show that for $\rho$ corresponding to a standard Schur index of a fourdimensional gauge theory a positive trace is unique.
 [12] arXiv:2406.13245 (crosslist from condmat.disnn) [pdf, html, other]

Title: Free energy equivalence between meanfield models and nonsparsely diluted meanfield modelsComments: 9 pages, 0 figureSubjects: Disordered Systems and Neural Networks (condmat.disnn); Statistical Mechanics (condmat.statmech); Mathematical Physics (mathph); Probability (math.PR)
We studied nonsparsely diluted meanfield models that differ from sparsely diluted meanfield models, such as the VianaBray model. We prove that the free energy of nonsparsely diluted meanfield models coincides exactly with that of the corresponding meanfield models with different parameters in ferromagnetic and spinglass models composed of any discrete spin $S$ in the thermodynamic limit. Our results are a broad generalization of the results of a previous study [Bovier and Gayrard, J. Stat. Phys. 72, 643 (1993)], where the densely diluted meanfield ferromagnetic Ising model (diluted CurieWeiss model) was analyzed rigorously, and it was proven that its free energy was exactly equivalent to that of the corresponding meanfield model (CurieWeiss model) with different parameters.
 [13] arXiv:2406.13423 (crosslist from nlin.SI) [pdf, html, other]

Title: Lagrangian multiform structure of discrete and semidiscrete KP systemsComments: 25 pages, 1 figureSubjects: Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (mathph)
A variational structure for the potential AKP system is established using the novel formalism of a Lagrangian multiforms. The structure comprises not only the fully discrete equation on the 3D lattice, but also its semidiscrete variants including several differentialdifference equations asssociated with, and compatible with, the partial difference equation. To this end, an overview is given of the various (discrete and semidiscrete) variants of the KP system, and their associated Lax representations, including a novel `generating PDE' for the KP hierarchy. The exterior derivative of the Lagrangian 3form for the lattice potential KP equation is shown to exhibit a doublezero structure, which implies the corresponding generalised EulerLagrange equations. Alongside the 3form structures, we develop a variational formulation of the corresponding Lax systems via the square eigenfunction representation arising from the relevant direct linearization scheme.
 [14] arXiv:2406.13438 (crosslist from math.RT) [pdf, other]

Title: Computing the center of a fusion categorySubjects: Representation Theory (math.RT); Mathematical Physics (mathph); Category Theory (math.CT); Quantum Algebra (math.QA)
We present a (Las Vegas) algorithm for explicitly computing the simple objects of the categorical (Drinfeld) center of a spherical fusion category. Our approach is based on decomposing the images of simple objects under the induction functor from the category to its center. We have implemented this algorithm in a generalpurpose software framework TensorCategories.jl for tensor categories that we develop within the opensource computer algebra system OSCAR. While the required computations are still too heavy to investigate standard examples whose center is not yet known up to some equivalence, our algorithm has the advantage of determining explicit halfbraidings for the simple central objects, avoiding abstract equivalences and the like. Furthermore, it also works over not necessarily algebraically closed fields, and this yields new explicit examples of nonsplit modular categories.
 [15] arXiv:2406.13459 (crosslist from nlin.SI) [pdf, html, other]

Title: The RiemannHilbert approach for the nonlocal derivative nonlinear Schr\"odinger equation with nonzero boundary conditionsSubjects: Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (mathph)
In this paper, the nonlocal reverse spacetime derivative nonlinear Schrödinger equation under nonzero boundary conditions is investigated using the RiemannHilbert (RH) approach. The direct scattering problem focuses on the analyticity, symmetries, and asymptotic behaviors of the Jost eigenfunctions and scattering matrix functions, leading to the construction of the corresponding RH problem. Then, in the inverse scattering problem, the Plemelj formula is employed to solve the RH problem. So the reconstruction formula, trace formulae, $\theta$ condition, and exact expression of the singlepole and doublepole solutions are obtained. Furthermore, darkdark solitons, brightdark solitons, and breather solutions of the reverse spacetime derivative nonlinear Schrödinger equation are presented along with their dynamic behaviors summarized through graphical simulation.
 [16] arXiv:2406.13528 (crosslist from math.CO) [pdf, html, other]

Title: Enumeration of maps with tight boundaries and the Zhukovsky transformationComments: 62 pages, 7 figuresSubjects: Combinatorics (math.CO); Mathematical Physics (mathph); Probability (math.PR)
We consider maps with tight boundaries, i.e. maps whose boundaries have minimal length in their homotopy class, and discuss the properties of their generating functions $T^{(g)}_{\ell_1,\ldots,\ell_n}$ for fixed genus $g$ and prescribed boundary lengths $\ell_1,\ldots,\ell_n$, with a control on the degrees of inner faces. We find that these series appear as coefficients in the expansion of $\omega^{(g)}_n(z_1,\ldots,z_n)$, a fundamental quantity in the EynardOrantin theory of topological recursion, thereby providing a combinatorial interpretation of the Zhukovsky transformation used in this context. This interpretation results from the socalled trumpet decomposition of maps with arbitrary boundaries. In the planar bipartite case, we obtain a fully explicit formula for $T^{(0)}_{2\ell_1,\ldots,2\ell_n}$ from the ColletFusy formula. We also find recursion relations satisfied by $T^{(g)}_{\ell_1,\ldots,\ell_n}$, which consist in adding an extra tight boundary, keeping the genus $g$ fixed. Building on a result of Norbury and Scott, we show that $T^{(g)}_{\ell_1,\ldots,\ell_n}$ is equal to a paritydependent quasipolynomial in $\ell_1^2,\ldots,\ell_n^2$ times a simple power of the basic generating function $R$. In passing, we provide a bijective derivation in the case $(g,n)=(0,3)$, generalizing a recent construction of ours to the non bipartite case.
 [17] arXiv:2406.13624 (crosslist from hepth) [pdf, other]

Title: Generalized $ \widetilde{W} $ algebrasComments: 47 pages, 2 figuresSubjects: High Energy Physics  Theory (hepth); Mathematical Physics (mathph)
Recently, a new generalized family of infinitedimensional $ \widetilde{W} $ algebras, each associated with a particular element of a commutative subalgebra of the $ W_{1+\infty} $ algebra, was described. This paper provides a comprehensive account of the aforementioned association, accompanied by the requisite proofs and illustrative examples. This approach allows a derivation of Ward identities for selected WLZZ matrix models and the expansion of corresponding $ W $operators in terms of an infinite set of variables $ p_k $.
 [18] arXiv:2406.13661 (crosslist from cs.LG) [pdf, html, other]

Title: Hitchhiker's guide on EnergyBased Models: a comprehensive review on the relation with other generative models, sampling and statistical physicsDavide Carbone (1 and 2) ((1) Dipartimento di Scienze Matematiche, Politecnico di Torino, Torino, Italy, (2) INFN, Sezione di Torino, Torino, Italy)Subjects: Machine Learning (cs.LG); Mathematical Physics (mathph); Applied Physics (physics.appph); Data Analysis, Statistics and Probability (physics.dataan)
EnergyBased Models (EBMs) have emerged as a powerful framework in the realm of generative modeling, offering a unique perspective that aligns closely with principles of statistical mechanics. This review aims to provide physicists with a comprehensive understanding of EBMs, delineating their connection to other generative models such as Generative Adversarial Networks (GANs), Variational Autoencoders (VAEs), and Normalizing Flows. We explore the sampling techniques crucial for EBMs, including Markov Chain Monte Carlo (MCMC) methods, and draw parallels between EBM concepts and statistical mechanics, highlighting the significance of energy functions and partition functions. Furthermore, we delve into stateoftheart training methodologies for EBMs, covering recent advancements and their implications for enhanced model performance and efficiency. This review is designed to clarify the often complex interconnections between these models, which can be challenging due to the diverse communities working on the topic.
 [19] arXiv:2406.13680 (crosslist from physics.fludyn) [pdf, html, other]

Title: Effects of settling on inertial particle slip velocity statistics in wall bounded flowsSubjects: Fluid Dynamics (physics.fludyn); Mathematical Physics (mathph); Atmospheric and Oceanic Physics (physics.aoph)
Developing reduced order models for the transport of solid particles in turbulence typically requires a statistical description of the particleturbulence interactions. In this work, we utilize a statistical framework to derive continuum equations for the moments of the slip velocity of inertial settling Lagrangian particles in a turbulent boundary layer. Using coupled EulerianLagrangian direct numerical simulations, we then identify the dominant mechanisms controlling the slip velocity variance, and find that for a range of St+, Sv+, and Re, the slip variance is primarily controlled by local differences between the "seen" variance and the particle velocity variance, while terms appearing due to the inhomogeneity of the turbulence are subleading until Sv+ becomes large. We also consider several comparative metrics to assess the relative magnitudes of the fluctuating slip velocity and the mean slip velocity, and we find that the vertical mean slip increases rapidly with Sv+, rendering the variance relatively small  an effect found to be most substantial for Sv+>1. Finally, we compare the results to a model of the acceleration variance Berk and Coletti (2021) based the concept of a response function described in Csanady (1963), highlighting the role of the crossing trajectories mechanism. We find that while there is good agreement for low Sv+, systematic errors remain, possibly due to implicit nonlocal effects arising from rapid particle settling and inhomogeneous turbulence. We conclude with a discussion of the implications of this work for modeling the transport of coarse dust grains in the atmospheric surface layer.
 [20] arXiv:2406.13701 (crosslist from nlin.PS) [pdf, html, other]

Title: Windwave interaction in finite depth: linear and nonlinear approaches, blowup and soliton breaking in finite time, integrability perspectivesSubjects: Pattern Formation and Solitons (nlin.PS); Mathematical Physics (mathph)
This work is an analytical investigation of the evolution of surface water waves in Miles and Jeffreys theories of wind wave interaction in water of finite depth. The present review is divided into two major parts. The first corresponds to the surface water waves in a linear regime and its nonlinear extensions. In this part, Miles theory of wave amplification by wind is extended to the case of finite depth. The dispersion relation provides a wave growth rate depending on depth. Our theoretical results are in good agreement with the data from the Australian Shallow Water Experiment and the data from the Lake George experiment. In the second part of this study, Jeffreys theory of wave amplification by wind is extended to the case of finite depth, where the SerreGreenNaghdi is derived. We find the solitary wave solution of the system, with an increasing amplitude under the action of the wind. This continuous increase in amplitude leads to the soliton breaking and blowup of the surface wave in finite time. The theoretical blowup time is calculated based on actual experimental data. By applying an appropriate perturbation method, the SGN equation yields Korteweg de Vries Burger equation (KdVB). We show that the continuous transfer of energy from wind to water results in the growth of the KdVB soliton amplitude, velocity, acceleration, and energy over time while its effective wavelength decreases. This phenomenon differs from the classical results of Jeffreys approach due to finite depth. Again, blowup and breaking occur in finite time. These times are calculated and expressed for soliton and windappropriate parameters and values. These values are measurable in usual experimental facilities. The kinematics of the breaking is studied, and a detailed analysis of the breaking time is conducted using various criteria. Finally, some integrability perspectives are presented.
 [21] arXiv:2406.13767 (crosslist from physics.fludyn) [pdf, html, other]

Title: A fully observercovariant formulation of the fluid dynamics of simple fluids: derivation, simple examples and a generalized OrrSommerfeld equationSubjects: Fluid Dynamics (physics.fludyn); Mathematical Physics (mathph)
We present a formalism to describe the motion of a fluid fully which is fully covariant with respect to arbitrary observers. To achieve fully covariance, we write prognostic equations for quantities that belong to the graded exterior algebra of the cotangent bundle of the manifold occupied by the fluid. With the new formalism, we consider problems of stability, and we derive a generalization of the OrrSommerfeld equation that describes the evolution of perturbations relative to an arbitrary observer. The latter is applied to cases where the observer is the Lagrangian observer comoving with the background flow.
 [22] arXiv:2406.13773 (crosslist from math.AP) [pdf, other]

Title: A rigorous approach to pattern formation for isotropic isoperimetric problems with competing nonlocal interactionsSubjects: Analysis of PDEs (math.AP); Mathematical Physics (mathph)
We introduce a rigorous approach to the study of the symmetry breaking and pattern formation phenomenon for isotropic functionals with local/nonlocal interactions in competition.
We consider a general class of nonlocal variational problems in dimension $d\geq 2$, in which an isotropic surface term favouring pure phases competes with an isotropic nonlocal term with power law kernel favouring alternation between different phases.
Close to the critical regime in which the two terms are of the same order, we give a rigorous proof of the conjectured structure of global minimizers, in the shape of domains with flat boundary (e.g., stripes or lamellae).
The natural framework in which our approach is set and developed is the one of calculus of variations and geometric measure theory.
Among others, we detect a nonlocal curvaturetype quantity which is controlled by the energy functional and whose finiteness implies flatness for sufficiently regular boundaries.
The power of decay of the considered kernels at infinity is $p\geq d+3$ and it is related to pattern formation in synthetic antiferromagnets. The decay $p=d+3$ is optimal to get the flatness of regular boundaries of finite energy in the critical regime.  [23] arXiv:2406.13823 (crosslist from quantph) [pdf, other]

Title: Inevitable Negativity: Additivity Commands Negative Quantum Channel EntropyComments: 16 pages (main text) + 21 pages (appendix), 6 figures, comments are welcomeSubjects: Quantum Physics (quantph); Mathematical Physics (mathph)
Quantum channels represent a broad spectrum of operations crucial to quantum information theory, encompassing everything from the transmission of quantum information to the manipulation of various resources. In the domain of states, the concept of majorization serves as a fundamental tool for comparing the uncertainty inherent in both classical and quantum systems. This paper establishes a rigorous framework for assessing the uncertainty in both classical and quantum channels. By employing a specific class of superchannels, we introduce and elucidate three distinct approaches to channel majorization: constructive, axiomatic, and operational. Intriguingly, these methodologies converge to a consistent ordering. This convergence not only provides a robust basis for defining entropy functions for channels but also clarifies the interpretation of entropy in this broader context. Most notably, our findings reveal that any viable entropy function for quantum channels must assume negative values, thereby challenging traditional notions of entropy.
 [24] arXiv:2406.14063 (crosslist from math.AP) [pdf, html, other]

Title: Global counterexamples to uniqueness for a Calder\'on problem with $C^k$ conductivitiesSubjects: Analysis of PDEs (math.AP); Mathematical Physics (mathph); Spectral Theory (math.SP)
Let $\Omega \subset R^n$, $n \geq 3$, be a fixed smooth bounded domain, and let $\gamma$ be a smooth conductivity in $\overline{\Omega}$. Consider a nonzero frequency $\lambda_0$ which does not belong to the Dirichlet spectrum of $L_\gamma = {\rm div} (\gamma \nabla \cdot)$. Then, for all $k \geq 1$, there exists an infinite number of pairs of nonisometric $C^k$ conductivities $(\gamma_1, \gamma_2)$ on $\overline{\Omega}$, which are close to $\gamma$ such that the associated DN maps at frequency $\lambda_0$ satisfy \begin{equation*}
\Lambda_{\gamma_1,\lambda_0} = \Lambda_{\gamma_2,\lambda_0}. \end{equation*}  [25] arXiv:2406.14153 (crosslist from quantph) [pdf, html, other]

Title: On random classical marginal problems with applications to quantum information theorySubjects: Quantum Physics (quantph); Mathematical Physics (mathph); Probability (math.PR)
In this paper, we study random instances of the classical marginal problem. We encode the problem in a graph, where the vertices have assigned fixed binary probability distributions, and edges have assigned random bivariate distributions having the incident vertex distributions as marginals. We provide estimates on the probability that a joint distribution on the graph exists, having the bivariate edge distributions as marginals. Our study is motivated by Fine's theorem in quantum mechanics. We study in great detail the graphs corresponding to CHSH and BellWigner scenarios providing rations of volumes between the local and nonsignaling polytopes.
 [26] arXiv:2406.14229 (crosslist from physics.fludyn) [pdf, html, other]

Title: Approaches to conservative Smoothed Particle Hydrodynamics with entropySubjects: Fluid Dynamics (physics.fludyn); Mathematical Physics (mathph)
Smoothed particle hydrodynamics (SPH) is typically used for barotropic fluids, where the pressure depends only on the local mass density. Here, we show how to incorporate the entropy into the SPH, so that the pressure can also depend on the temperature, while keeping the growth of the total entropy, conservation of the total energy, and symplecticity of the reversible part of the SPH equations. The SPH system of ordinary differential equations with entropy is derived by means of the Poisson reduction and the LagrangeEuler transformation. We present several approaches towards SPH with entropy, which are then illustrated on systems with discontinuities, on adiabatic and nonadiabatic expansion, and on the RayleighBeenard convection without the Boussinesq approximation. Finally, we show how to model hyperbolic heat conduction within the SPH, extending the SPH variables with not only entropy but also a heatfluxrelated vector field.
 [27] arXiv:2406.14248 (crosslist from condmat.statmech) [pdf, html, other]

Title: Starving Random WalksComments: 21 pages, 5 figures. Contribution to the book "The Mathematics of Movement: an Interdisciplinary Approach to Mutual Challenges in Animal Ecology and Cell Biology" edited by Luca Giuggioli and Philip MainiSubjects: Statistical Mechanics (condmat.statmech); Mathematical Physics (mathph)
In this chapter, we review recent results on the starving random walk (RW) problem, a minimal model for resourcelimited exploration. Initially, each lattice site contains a single food unit, which is consumed upon visitation by the RW. The RW starves whenever it has not found any food unit within the previous $\mathcal{S}$ steps. To address this problem, the key observable corresponds to the intervisit time $\tau_k$ defined as the time elapsed between the finding of the $k^\text{th}$ and the $(k+1)^\text{th}$ food unit. By characterizing the maximum $M_n$ of the intervisit times $\tau_0,\dots,\tau_{n1}$, we will see how to obtain the number $N_\mathcal{S}$ of food units collected at starvation, as well as the lifetime $T_\mathcal{S}$ of the starving RW.
 [28] arXiv:2406.14320 (crosslist from hepth) [pdf, html, other]

Title: Anyon condensation in mixedstate topological orderComments: 52 pages, 14 figuresSubjects: High Energy Physics  Theory (hepth); Strongly Correlated Electrons (condmat.strel); Mathematical Physics (mathph); Category Theory (math.CT); Quantum Physics (quantph)
We discuss anyon condensation in mixedstate topological order. The phases were recently conjectured to be classified by premodular fusion categories. Just like anyon condensation in purestate topological order, a bootstrap analysis shows condensable anyons are given by connected étale algebras. We explain how to perform generic anyon condensation including noninvertible anyons and successive condensations. Interestingly, some condensations lead to purestate topological orders. We clarify when this happens. We also compute topological invariants of equivalence classes.
 [29] arXiv:2406.14327 (crosslist from condmat.statmech) [pdf, html, other]

Title: Application of Haldane's statistical correlation theory in classical systemsSubjects: Statistical Mechanics (condmat.statmech); Mathematical Physics (mathph); Data Analysis, Statistics and Probability (physics.dataan); Quantum Physics (quantph)
This letter investigates the application of Haldane's statistical correlation theory in classical systems. A modified statistical correlation theory has been proposed by including nonlinearity into the original theory of Haldane. It is shown that indistinguishability can be introduced as a form of external statistical correlation into distinguishable systems. It is proved that this modified statistical correlation theory can be used to derive classical fractional exclusion statistics (CFES) using maximum entropy methods for a selfcorrelating system. An extended nonlinear correlation model based on power series expansion is also proposed, which can produce various intermediate statistical models.
 [30] arXiv:2406.14414 (crosslist from math.AG) [pdf, html, other]

Title: Normal forms for ordinary differential operators, IComments: 67 pSubjects: Algebraic Geometry (math.AG); Mathematical Physics (mathph); Quantum Algebra (math.QA); Rings and Algebras (math.RA)
In this paper we develop the generalised Schur theory offered in the recent paper by the second author in dimension one case, and apply it to obtain two applications in different directions of algebra/algebraic geometry.
The first application is a new explicit parametrisation of torsion free rank one sheaves on projective irreducible curves with vanishing cohomology groups.
The second application is a commutativity criterion for operators in the Weyl algebra or, more generally, in the ring of ordinary differential operators, which we prove in the case when operators have a normal form with the restriction top line (for details see Introduction).
Both applications are obtained with the help of normal forms. Namely, considering the ring of ordinary differential operators $D_1=K[[x]][\partial ]$ as a subring of a certain complete noncommutative ring $\hat{D}_1^{sym}$, the normal forms of differential operators mentioned here are obtained after conjugation by some invertible operator ("Schur operator"), calculated using one of the operators in a ring. Normal forms of commuting operators are polynomials with constant coefficients in the differentiation, integration and shift operators, which have a finite order in each variable, and can be effectively calculated for any given commuting operators.  [31] arXiv:2406.14533 (crosslist from math.CO) [pdf, other]

Title: Local symmetries in partially ordered setsComments: 33 pages, 5 figures, 3 tablesSubjects: Combinatorics (math.CO); General Relativity and Quantum Cosmology (grqc); Mathematical Physics (mathph)
Partially ordered sets (posets) have a universal appearance as an abstract structure in many areas of mathematics. Though, even their explicit enumeration remains unknown in general, and only the counts of all partial orders on sets of up to 16 unlabelled elements have been calculated to date, see sequence A000112 in the OEIS.
In this work, we study automorphisms of posets in order to formulate a classification by local symmetries. These symmetries give rise to a division operation on the set of all posets and lead us to the construction of symmetry classes that are easier to characterise and enumerate. Additionally to the enumeration of symmetry classes, I derive polynomial expressions that count certain subsets of posets with a large number of layers (a large height). As an application in physics, I investigate local symmetries (or rather their lack of) in causal sets, which are discrete spacetime models used as a candidate framework for quantum gravity.  [32] arXiv:2406.14547 (crosslist from math.SG) [pdf, html, other]

Title: A Mathematical Definition of Path Integrals on Symplectic ManifoldsComments: 20 pagesSubjects: Symplectic Geometry (math.SG); High Energy Physics  Theory (hepth); Mathematical Physics (mathph); Quantum Algebra (math.QA)
We give a mathematical definition of some path integrals, emphasizing those relevant to the quantization of symplectic manifolds (and more generally, Poisson manifolds) $\unicode{x2013}$ in particular, the coherent state path integral. We show that Kähler manifolds provide many computable examples.
Cross submissions for Friday, 21 June 2024 (showing 30 of 30 entries )
 [33] arXiv:2203.16441 (replaced) [pdf, html, other]

Title: Mixed state representability of entropydensity pairsSubjects: Mathematical Physics (mathph); Quantum Physics (quantph)
We show the representability of densityentropy pairs with canonical and grandcanonical states, and we provide bounds on the kinetic energy of the representing states.
 [34] arXiv:2305.06237 (replaced) [pdf, html, other]

Title: Weyl laws for interacting particlesComments: 37 pagesSubjects: Mathematical Physics (mathph); Spectral Theory (math.SP)
We study grandcanonical interacting fermionic systems in the meanfield regime, in a trapping potential. We provide the first order term of integrated and pointwise Weyl laws, but in the case with interaction. More precisely, we prove the convergence of the densities of the grandcanonical HartreeFock ground state to the ThomasFermi ground state in the semiclassical limit $\hbar\to 0$. For the proof, we write the grandcanonical version of the results of Fournais, Lewin and Solovej (Calc. Var. Partial Differ. Equ., 2018) and of Conlon (Commun. Math. Phys., 1983).
 [35] arXiv:2306.07110 (replaced) [pdf, html, other]

Title: Invariant measures on padic Lie groups: the padic quaternion algebra and the Haar integral on the padic rotation groupsComments: 49 pages, minor changesJournalref: Lett. Math. Phys. 114, 78 (2024)Subjects: Mathematical Physics (mathph); Functional Analysis (math.FA); Number Theory (math.NT)
We provide a general expression of the Haar measure $$ that is, the essentially unique translationinvariant measure $$ on a $p$adic Lie group. We then argue that this measure can be regarded as the measure naturally induced by the invariant volume form on the group, as it happens for a standard Lie group over the reals. As an important application, we next consider the problem of determining the Haar measure on the $p$adic special orthogonal groups in dimension two, three and four (for every prime number $p$). In particular, the Haar measure on $\mathrm{SO}(2,\mathbb{Q}_p)$ is obtained by a direct application of our general formula. As for $\mathrm{SO}(3,\mathbb{Q}_p)$ and $\mathrm{SO}(4,\mathbb{Q}_p)$, instead, we show that Haar integrals on these two groups can conveniently be lifted to Haar integrals on certain $p$adic Lie groups from which the special orthogonal groups are obtained as quotients. This construction involves a suitable quaternion algebra over the field $\mathbb{Q}_p$ and is reminiscent of the quaternionic realization of the real rotation groups. Our results should pave the way to the development of harmonic analysis on the $p$adic special orthogonal groups, with potential applications in $p$adic quantum mechanics and in the recently proposed $p$adic quantum information theory.
 [36] arXiv:2312.16482 (replaced) [pdf, html, other]

Title: CwikelLiebRozenblum type inequalities for HardySchr\"odinger operatorComments: 15 pages, final version to appear in J. Math. Pures ApplSubjects: Mathematical Physics (mathph); Functional Analysis (math.FA); Spectral Theory (math.SP)
We prove a CwikelLiebRozenblum type inequality for the number of negative eigenvalues of the HardySchrödinger operator $\Delta  (d2)^2/(4x^2) W(x)$ on $L^2(\mathbb{R}^d)$. The bound is given in terms of a weighted $L^{d/2}$norm of $W$ which is sharp in both large and small coupling regimes. We also obtain a similar bound for the fractional Laplacian.
 [37] arXiv:2401.07449 (replaced) [pdf, html, other]

Title: Fock space: A bridge between Fredholm index and the quantum Hall effectComments: Revision with extra example in Section 10. 38 pages, 1 figureSubjects: Mathematical Physics (mathph); Mesoscale and Nanoscale Physics (condmat.meshall); Functional Analysis (math.FA); Quantum Algebra (math.QA)
We compute the quantized Hall conductance at various Landau levels by using the classic trace. The computations reduce to the single elementary one for the lowest Landau level. By using the theories of HeltonHoweCareyPincus, and Toeplitz operators on the classic Fock space and higher Fock spaces, the Hall conductance is naturally identified with a Fredholm index. This brings new mathematical insights to the extraordinary precision of quantization observed in quantum Hall measurements.
 [38] arXiv:2401.14298 (replaced) [pdf, html, other]

Title: Characterising the Haar measure on the $p$adic rotation groups via inverse limits of measure spacesComments: 42 pages; to appear in Expositiones MathematicaeSubjects: Mathematical Physics (mathph); Functional Analysis (math.FA); Group Theory (math.GR); Number Theory (math.NT)
We determine the Haar measure on the compact $p$adic special orthogonal groups of rotations $\mathrm{SO}(d)_p$ in dimension $d=2,3$, by exploiting the machinery of inverse limits of measure spaces, for every prime $p>2$. We characterise $\mathrm{SO}(d)_p$ as inverse limits of finite groups, of which we provide parametrisations and orders, together with an equivalent description through a multivariable Hensel lifting. Supplying these finite groups with their normalised counting measures, we get an inverse family of Haar measure spaces for each $\mathrm{SO}(d)_p$. Finally, we constructively prove the existence of the socalled inverse limit measure of these inverse families, which is explicitly computable, and prove that it gives the Haar measure on $\mathrm{SO}(d)_p$. Our results pave the way towards the study of the irreducible projective unitary representations of the $p$adic rotation groups, with potential applications to the recently proposed $p$adic quantum information theory.
 [39] arXiv:2403.09345 (replaced) [pdf, html, other]

Title: ClassicalQuantum correspondence in Lindblad evolutionComments: Main article by Jeffrey Galkowski and Maciej Zworski with an appendix by Zhen Huang and Maciej Zworski  new appendix with numerical experiments and a new section providing estimates for the HilbertSchmidt norm under Lindblad evolutionSubjects: Mathematical Physics (mathph); Analysis of PDEs (math.AP); Quantum Physics (quantph)
We show that for the Lindblad evolution defined using (at most) quadratically growing classical Hamiltonians and (at most) linearly growing classical jump functions (quantized into jump operators assumed to satisfy certain ellipticity conditions and modeling interaction with a larger system), the evolution of a quantum observable remains close to the classical FokkerPlanck evolution in the HilbertSchmidt norm for times vastly exceeding the Ehrenfest time (the limit of such agreement with no jump operators). The time scale is the same as in the recent papers by HernándezRanardRiedel but the statement and methods are different. The appendix presents numerical experiments illustrating the classical/quantum correspondence in Lindblad evolution and comparing it to the mathematical results.
 [40] arXiv:2405.05388 (replaced) [pdf, other]

Title: The Asymptotic Behavior of the Mayer Series Coefficients for a Dimer Gas on a Rectangular LatticeComments: 11 pages, ALL NEW and COMELYSubjects: Mathematical Physics (mathph); Statistical Mechanics (condmat.statmech)
This paper continues and complements the research in the earlier version of this paper (essentially Part 3 and Part 4 herein.) We now assume as suggested at the end of Part 4 that $b(n)$ is asymptotically of the form in eq.(A1).
\begin{equation}\label{A1} \tag{A1}
b(n) \sim \exp{( k_{1} n + k_{0} ln(n) + \frac{k_{1}}{n} + \frac{k_{2}}{n^{2}}...)} \end{equation} Using the details of the six approximations used in Part 3, corresponding to the six values of $r$, $ 1 \leq r \leq 6$, in dimensions $d$ equal $2$. $3$, $5$, $11$ and $20$ we find in Part 1 an approximate value for the right side of eq.(A1) keeping the three terms in the exponent in $k_{1}$, $k_{0}$, and $k_{1}$. In the range $5 \leq n \leq 20$ the two sides of eq.(A1) may be said to agree roughly to 5 parts per 100. ( With an appropriate choice of a constant of proportionality. )
In Part 2 an approximation is found keeping the term in $k_{2}$ also that may be said to agree to 5 parts in 1000 in the range $8 \leq n \leq 20$. Not only is it amazing that the relation in eq.(A1) seems to hold, it is equally amazing that it is so accurate for such small values of $n$.  [41] arXiv:2406.04182 (replaced) [pdf, html, other]

Title: Minimal Walgebras with nonadmissible levels and intermediate Lie algebrasComments: 23 pagesSubjects: Mathematical Physics (mathph); High Energy Physics  Theory (hepth); Quantum Algebra (math.QA)
In \cite{Kawasetsu:2018irs}, Kawasetsu proved that the simple Walgebra associated with a minimal nilpotent element $W_{k}(\mathfrak{g},f_\theta)$ is rational and $C_2$cofinite for $\mathfrak{g}=D_4,E_6,E_7,E_8$ with nonadmissible level $k=h^\vee/6$. In this paper, we study ${W}_{k}(\mathfrak{g},f_\theta)$ algebra for $\mathfrak{g}=E_6,E_7,E_8$ with nonadmissible level $k=h^\vee/6+1$. We determine all irreducible (Ramond twisted) modules, compute their characters and find coset constructions and Hecke operator interpretations. These Walgebras are closely related to intermediate Lie algebras and intermediate vertex subalgebras.
 [42] arXiv:2203.04362 (replaced) [pdf, html, other]

Title: The Sobolev Wavefront Set of the Causal Propagator in Finite RegularitySubjects: Analysis of PDEs (math.AP); Mathematical Physics (mathph)
Given a globally hyperbolic spacetime $M=\mathbb{R}\times \Sigma$ of dimension four and regularity $C^\tau$, we estimate the Sobolev wavefront set of the causal propagator $K_G$ of the KleinGordon operator. In the smooth case, the propagator satisfies $WF'(K_G)=C$, where $C\subset T^*(M\times M)$ consists of those points $(\tilde{x},\tilde{\xi},\tilde{y},\tilde{\eta})$ such that $\tilde{\xi},\tilde{\eta}$ are cotangent to a null geodesic $\gamma$ at $\tilde{x}$ resp. $\tilde{y}$ and parallel transports of each other along $\gamma$.
We show that for $\tau>2$, $WF'^{2+\tau{\epsilon}}(K_G)\subset C$ for every ${\epsilon}>0$. Furthermore, in regularity $C^{\tau+2}$ with $\tau>2$, $C\subset WF'^{\frac{1}{2}}(K_G)\subset WF'^{\tau\epsilon}(K_G)\subset C$ holds for $0<\epsilon<\tau+\frac{1}{2}$.
In the ultrastatic case with $\Sigma$ compact, we show $WF'^{\frac{3}{2}+\tau\epsilon}(K_G)\subset C$ for $\epsilon >0$ and $\tau>2$ and $WF'^{\frac{3}{2}+\tau\epsilon}(K_G)= C$ for $\tau>3$ and $\epsilon<\tau3$. Moreover, we show that the global regularity of the propagator $K_G$ is $H^{\frac{1}{2}\epsilon}_{loc}(M\times M)$ as in the smooth case.  [43] arXiv:2204.01426 (replaced) [pdf, html, other]

Title: Empirical adequacy of the time operator canonically conjugate to a Hamiltonian generating translationsComments: Accepted version. The mathematical results are unchanged, but I added substantial explanations of the physical significance. Phys. Scr. (2024)Subjects: Quantum Physics (quantph); Mathematical Physics (mathph); History and Philosophy of Physics (physics.histph)
To admit a canonically conjugate time operator, the Hamiltonian has to be a generator of translations (like the momentum operator generates translations in space), so its spectrum must be unbounded. But the Hamiltonian governing our world is thought to be bounded from below. Also, judging by the number of fields and parameters of the Standard Model, the Hamiltonian seems much more complicated.
In this article I give examples of worlds governed by Hamiltonians generating translations. They can be expressed as a partial derivative operator just like the momentum operator, but when expressed in function of other observables they can exhibit any level of complexity. The examples include any quantum world realizing a standard ideal measurement, any quantum world containing a clock or a free massless fermion, the quantum representation of any deterministic timereversible dynamical system without time loops, and any quantum world that cannot return to a past state.
Such worlds are as sophisticated as our world, but they admit a time operator. I show that, despite having unbounded Hamiltonian, they do not decay to infinite negative energy any more than any quantum or classical world. Since two such quantum systems of the same Hilbert space dimension are unitarily equivalent even if the physical content of their observables is very different, they are concrete counterexamples to Hilbert Space Fundamentalism (HSF). Taking the observables into account removes the ambiguity of HSF and the clock ambiguity problem attributed to the PageWootters formalism, also caused by assuming HSF. These results provide additional motivations to restore the spacetime symmetry in the formulation of Quantum Mechanics and for the PageWootters formalism.  [44] arXiv:2204.04493 (replaced) [pdf, html, other]

Title: Entanglementinvertible channelsComments: 38 pages, many diagrams. Rev 4: Final versionJournalref: J. Math. Phys. 65, 062203 (2024)Subjects: Quantum Physics (quantph); Mathematical Physics (mathph); Operator Algebras (math.OA)
In a wellknown result [Werner2001], Werner classified all tight quantum teleportation and dense coding schemes, showing that they correspond to unitary error bases. Here tightness is a certain dimensional restriction: the quantum system to be teleported and the entangled resource must be of dimension d, and the measurement must have d^2 outcomes.
In this work we generalise this classification so as to remove the dimensional restriction altogether, thereby resolving an open problem raised in that work. In fact, we classify not just teleportation and dense coding schemes, but entanglementreversible channels. These are channels between finitedimensional C*algebras which are reversible with the aid of an entangled resource state, generalising ordinary reversibility of a channel.
In Werner's classification, a bijective correspondence between tight teleportation and dense coding schemes was shown: swapping Alice and Bob's operations turns a teleportation scheme into a dense coding scheme and vice versa. We observe that this property generalises ordinary invertibility of a channel; we call it entanglementinvertibility. We show that entanglementinvertible channels are precisely the quantum bijections previously studied in the setting of quantum combinatorics [Musto2018], which are classified in terms of the representation theory of the quantum permutation group.  [45] arXiv:2209.02027 (replaced) [pdf, html, other]

Title: Emergence of quantum dynamics from chaos: The case of prequantum cat mapsComments: v2: We clarify some statements and simplify the arguments by restricting to the case of cat maps in checkerboard form. 24 pagesSubjects: Dynamical Systems (math.DS); Mathematical Physics (mathph); Spectral Theory (math.SP)
Faure and Tsujii have recently proposed a novel quantization procedure, named natural quantization, for smooth symplectic Anosov diffeomorphisms. Their method starts with prequantization, which is also the first step of geometric quantization as proposed by KostantSouriauKirillov, and then relies on the RuellePollicott spectrum of the prequantum transfer operator, which they show to have a particular band structure. The appeal of this new quantization scheme resides in its naturalness: the quantum behavior appears dynamically in the classical correlation functions of the prequantum transfer operator. In this paper, we explicitly work out the case of cat maps on the $2n$dimensional torus, showing in particular that the outcome is equivalent to that of the usual Weyl quantization. We also provide a concrete construction of all the prequantum cat maps.
 [46] arXiv:2210.15047 (replaced) [pdf, other]

Title: A stochastic analysis of subcritical Euclidean fermionic field theoriesComments: The errors in some proofs have been corrected and some typos have been fixed. Accepted for publication in Annals of ProbabilitySubjects: Probability (math.PR); Mathematical Physics (mathph)
Building on previous work on the stochastic analysis for Grassmann random variables, we introduce a forwardbackward stochastic differential equation (FBSDE) which provides a stochastic quantisation of Grassmann measures. Our method is inspired by the socalled continuous renormalisation group, but avoids the technical difficulties encountered in the direct study of the flow equation for the effective potentials. As an application, we construct a family of weakly coupled subcritical Euclidean fermionic field theories and prove exponential decay of correlations.
 [47] arXiv:2306.07766 (replaced) [pdf, html, other]

Title: Consistency of eightdimensional supergravities: Anomalies, Lattices and CountertermsComments: 37 pages. Comments and suggestions are welcomedSubjects: High Energy Physics  Theory (hepth); Mathematical Physics (mathph)
We reexamine the question of quantum consistency of supergravities in eight dimensions. Theories with 16 supercharges suffer from the anomalies under the action of its discrete modular groups. In minimally supersymmetric theory coupled to YangMills multiples of rank $l$ with the moduli space given by $\text{SO}(2,l)/ (\text{U}(1) \times \text{SO}(l))$, the existence of a counterterm together with the requirement that its poles and zeros correspond to the gauge symmetry enhancement imposes nontrivial constraints on the lattice. The counterterms needed for anomaly cancellation for all cases, that are believed to lead to consistent theories of quantum gravity ($l = 2,10,18$), are discussed.
 [48] arXiv:2308.06158 (replaced) [pdf, html, other]

Title: Infinitesimal Modular Group: $q$Deformed $\mathfrak{sl}_2$ and Witt AlgebraJournalref: SIGMA 20 (2024), 053, 16 pagesSubjects: Quantum Algebra (math.QA); Mathematical Physics (mathph)
We describe new $q$deformations of the 3dimensional Heisenberg algebra, the simple Lie algebra $\mathfrak{sl}_2$ and the Witt algebra. They are constructed through a realization as differential operators. These operators are related to the modular group and $q$deformed rational numbers defined by MorierGenoud and Ovsienko and lead to $q$deformed Möbius transformations acting on the hyperbolic plane.
 [49] arXiv:2310.09082 (replaced) [pdf, html, other]

Title: From Maximum of Intervisit Times to Starving Random WalksComments: 6 pages, 3 figures + 16 pages, 11 figuresJournalref: Phys. Rev. Lett. 132, 127101 (2024)Subjects: Statistical Mechanics (condmat.statmech); Mathematical Physics (mathph); Data Analysis, Statistics and Probability (physics.dataan)
Very recently, a fundamental observable has been introduced and analyzed to quantify the exploration of random walks: the time $\tau_k$ required for a random walk to find a site that it never visited previously, when the walk has already visited $k$ distinct sites. Here, we tackle the natural issue of the statistics of $M_n$, the longest duration out of $\tau_0,\dots,\tau_{n1}$. This problem belongs to the active field of extreme value statistics, with the difficulty that the random variables $\tau_k$ are both correlated and nonidentically distributed. Beyond this fundamental aspect, we show that the asymptotic determination of the statistics of $M_n$ finds explicit applications in foraging theory and allows us to solve the open $d$dimensional starving random walk problem, in which each site of a lattice initially contains one food unit, consumed upon visit by the random walker, which can travel $\mathcal{S}$ steps without food before starving. Processes of diverse nature, including regular diffusion, anomalous diffusion, and diffusion in disordered media and fractals, share common properties within the same universality classes.
 [50] arXiv:2312.14885 (replaced) [pdf, html, other]

Title: Full Record Statistics of 1d Random WalksComments: 16 pages, 5 figuresJournalref: Phys. Rev. E 109, 064101 (2024)Subjects: Statistical Mechanics (condmat.statmech); Mathematical Physics (mathph); Data Analysis, Statistics and Probability (physics.dataan)
We develop a comprehensive framework for analyzing full record statistics, covering record counts $M(t_1), M(t_2), \ldots$, and their corresponding attainment times $T_{M(t_1)}, T_{M(t_2)}, \ldots$, as well as the intervals until the next record. From this multipletime distribution, we derive general expressions for various observables related to record dynamics, including the conditional number of records given the number observed at a previous time and the conditional time required to reach the current record, given the occurrence time of the previous one. Our formalism is exemplified by a variety of stochastic processes, including biased nearestneighbor random walks, asymmetric runandtumble dynamics, and random walks with stochastic resetting.
 [51] arXiv:2401.09908 (replaced) [pdf, html, other]

Title: Algorithm for differential equations for Feynman integrals in general dimensionsComments: 47 pages. v2: Clarifications and comments added. Version to appear in Letters in Mathematical Physics. Results for differential operators are on the repository : this https URLSubjects: High Energy Physics  Theory (hepth); High Energy Physics  Phenomenology (hepph); Mathematical Physics (mathph)
We present an algorithm for determining the minimal order differential equations associated to a given Feynman integral in dimensional or analytic regularisation. The algorithm is an extension of the GriffithsDwork pole reduction adapted to the case of twisted differential forms. In dimensional regularisation, we demonstrate the applicability of this algorithm by explicitly providing the inhomogeneous differential equations for the multiloop twopoint sunset integrals: up to 20 loops for the equal mass case, the generic mass case at two and threeloop orders. Additionally, we derive the differential operators for various infrareddivergent twoloop graphs. In the analytic regularisation case, we apply our algorithm for deriving a system of partial differential equations for regulated Witten diagrams, which arise in the evaluation of cosmological correlators of conformally coupled $\phi^4$ theory in fourdimensional de Sitter space.
 [52] arXiv:2403.06377 (replaced) [pdf, html, other]

Title: Adiabatic versus instantaneous transitions from a harmonic oscillator to an inverted oscillatorComments: 18 pages, 6 figures, 101 references, published in: A. Dodonov and C. C. H. Ribeiro (Eds.), Proceedings of the Second International Workshop on Quantum Nonstationary Systems (LF Editorial, Sao Paulo, 2024, ISBN 9786555634464), Chapter 2, pp. 2142; this https URL. arXiv admin note: text overlap with arXiv:2303.08299Subjects: Quantum Physics (quantph); Mathematical Physics (mathph)
We have obtained explicit analytical formulas for the mean energy and its variance (characterizing the energy fluctuations) of a quantum harmonic oscillator with timedependent frequency in the adiabatic regimes after the frequency passes through zero. The behavior of energy turns out to be quite different in two cases: when the frequency remains real and when it becomes imaginary. In the first case, the mean energy always increases when the frequency returns to its initial value, and the increment coefficient is determined by the exponent in the power law of the frequency crossing zero. On the other hand, if the frequency becomes imaginary, the absolute value of mean energy increases exponentially, even in the adiabatic regime, unless the Hamiltonian becomes time independent. Small corrections to the leading terms of simple adiabatic approximate formulas are crucial in this case, due to the unstable nature of the motion.
 [53] arXiv:2403.06391 (replaced) [pdf, html, other]

Title: Towards verifications of Krylov complexityComments: typos and errors are corrected, LaTeX 29pages, no figureJournalref: Progress of Theoretical and Experimental Physics, Volume 2024, Issue 6, June 2024, 063A01,Subjects: Quantum Physics (quantph); High Energy Physics  Theory (hepth); Mathematical Physics (mathph)
Krylov complexity is considered to provide a measure of the growth of operators evolving under Hamiltonian dynamics. The main strategy is the analysis of the structure of Krylov subspace $\mathcal{K}_M(\mathcal{H},\eta)$ spanned by the multiple applications of the Liouville operator $\mathcal{L}$ defined by the commutator in terms of a Hamiltonian $\mathcal{H}$, $\mathcal{L}:=[\mathcal{H},\cdot]$ acting on an operator $\eta$, $\mathcal{K}_M(\mathcal{H},\eta)=\text{span}\{\eta,\mathcal{L}\eta,\ldots,\mathcal{L}^{M1}\eta\}$. For a given inner product $(\cdot,\cdot)$ of the operators, the orthonormal basis $\{\mathcal{O}_n\}$ is constructed from $\mathcal{O}_0=\eta/\sqrt{(\eta,\eta)}$ by Lanczos algorithm. The moments $\mu_m=(\mathcal{O}_0,\mathcal{L}^m\mathcal{O}_0)$ are closely related to the important data $\{b_n\}$ called Lanczos coefficients. I present the exact and explicit expressions of the moments $\{\mu_m\}$ for 16 quantum mechanical systems which are {\em exactly solvable both in the Schrödinger and Heisenberg pictures}. The operator $\eta$ is the variable of the eigenpolynomials. Among them six systems show a clear sign of `noncomplexity' as vanishing higher Lanczos coefficients $b_m=0$, $m\ge3$.
 [54] arXiv:2405.11630 (replaced) [pdf, other]

Title: General Christoffel Perturbations for Mixed Multiple Orthogonal PolynomialsComments: 32 pages. Some minor typos corrected in the second versionSubjects: Classical Analysis and ODEs (math.CA); Mathematical Physics (mathph); Spectral Theory (math.SP)
Performing both right and left multiplication operations using general regular matrix polynomials, which need not be monic and may possess leading coefficients of arbitrary rank, on a rectangular matrix of measures associated with mixed multiple orthogonal polynomials, reveals corresponding Christoffel formulas. These formulas express the perturbed mixed multiple orthogonal polynomials in relation to the original ones. Utilizing the divisibility theorem for matrix polynomials, we establish a criterion for the existence of perturbed orthogonality, expressed through the noncancellation of certain $\tau$ determinants.
 [55] arXiv:2405.18806 (replaced) [pdf, html, other]

Title: Propagation of Waves from Finite Sources Arranged in Line Segments within an Infinite Triangular LatticeComments: 23 Pages, 15 Figures, 2 Tables. This revision comprises several enhancements: the modification and refinement of Figure 1, the incorporation of additional information pertaining to the description of the proposed method and the application of the stated problem within the introduction section, and an expanded reference listSubjects: Analysis of PDEs (math.AP); Mathematical Physics (mathph); Classical Analysis and ODEs (math.CA)
This paper examines the propagation of time harmonic waves through a twodimensional triangular lattice with sources located on line segments. Specifically, we investigate the discrete Helmholtz equation with a wavenumber $k \in \left( 0,2\sqrt{2} \right)$, where input data is prescribed on finite rows or columns of lattice sites. We focus on two main questions: the efficacy of the numerical methods employed in evaluating the Green's function, and the necessity of the cone condition. Consistent with a continuum theory, we employ the notion of radiating solution and establish a unique solvability result and Green's representation formula using difference potentials. Finally, we propose a numerical computation method and demonstrate its efficiency through examples related to the propagation problems in the lefthanded 2D inductorcapacitor metamaterial.
 [56] arXiv:2406.04310 (replaced) [pdf, html, other]

Title: Neural Networks Assisted MetropolisHastings for Bayesian Estimation of Critical Exponent on Elliptic Black Hole Solution in 4D Using Quantum Perturbation TheoryComments: V2: 3 extra figures for loss functions on Gaussian proposal distributions are added. Section 4 is modified. 37 pages, 14 figuresSubjects: High Energy Physics  Theory (hepth); General Relativity and Quantum Cosmology (grqc); Mathematical Physics (mathph); Quantum Physics (quantph)
It is wellknown that the critical gravitational collapse produces continuous selfsimilar solutions characterized by the Choptuik critical exponent, $\gamma$. We examine the solutions in the domains of the linear perturbation equations, considering the numerical measurement errors. Specifically, we study quantum perturbation theory for the fourdimensional Einsteinaxiondilaton system of the elliptic class of $\text{SL}(2,\mathbb{R})$ transformations. We develop a novel artificial neural networkassisted MetropolisHastings algorithm based on quantum perturbation theory to find the distribution of the critical exponent in a Bayesian framework. Unlike existing methods, this new probabilistic approach identifies the available deterministic solution and explores the range of physically distinguishable critical exponents that may arise due to numerical measurement errors.
 [57] arXiv:2406.05842 (replaced) [pdf, html, other]

Title: Replica symmetry breaking in spin glasses in the replicafree Keldysh formalismComments: 17 pages, 4 figures, submitted versionSubjects: Disordered Systems and Neural Networks (condmat.disnn); Statistical Mechanics (condmat.statmech); Strongly Correlated Electrons (condmat.strel); Mathematical Physics (mathph); Quantum Physics (quantph)
At asymptotically late times ultrametricity can emerge from the persistent slow aging dynamics of the glass phase. We show that this suffices to recover the breaking of replica symmetry in meanfield spin glasses from the late time limit of the time evolution using the Keldysh path integral. This provides an alternative approach to replica symmetry breaking by connecting it rigorously to the dynamic formulation. Stationary spin glasses are thereby understood to spontaneously break thermal symmetry, or the KuboMartinSchwinger relation of a state in global thermal equilibrium. We demonstrate our general statements for the spherical quantum $p$spin model and the quantum SherringtonKirkpatrick model in the presence of transverse and longitudinal fields. In doing so, we also derive their dynamical GinzburgLandau effective Keldysh actions starting from microscopic quantum models.