In this paper, we examine the p-adic Ising model with an external field on a Cayley tree. Specifically, we consider weakly periodic p-adic generalized Gibbs measures, which generalize the concept of periodic Gibbs measures. In particular, we investigate weakly periodic Gibbs measures associated with normal subgroups of the group representation of a Cayley tree. We characterize weakly periodic p-adic generalized Gibbs measures corresponding to normal divisors of index two for the considered model on a Cayley tree of order two, where the prime p > 3. Our results demonstrate that if p=1 (mod 4), there exists five unbounded weakly periodic p-adic generalized Gibbs measures. Conversely, if p=3 (mod 4), there exists three unbounded weakly periodic p-adic generalized Gibbs measures (GGMs). This confirms the occurrence of both a phase transition and a quasi-phase transition in the model for all p > 3. Furthermore, we analyze the dynamical system associated with these weakly periodic measures on Q4p. We show that the dynamical system possesses three attractive and two repelling fixed points when p=1 (mod 4), whereas it has three attractive fixed points when p=3 (mod 4).
In 1910, Hendrik Antoon Lorentz delved into the enigmatic Laplace eigenvalue equation, also known as the Helmholtz equation, pondering to what extent the geometry in which one solves the equation can be recovered from knowledge of the eigenvalues. Lorentz, inspired by physical and musical analogies, conjectured a fundamental relationship between eigenvalues, domain volume, and dimensionality. While his conjecture initially seemed insurmountable, Hermann Weyl’s groundbreaking proof in 1912 illuminated the deep connection between eigenvalues and geometric properties. Over the ensuing 113 years, mathematicians and physicists have continued to decipher the intricate interplay between eigenvalues and geometry. From Weyl’s law to Milnor’s example of isospectral non-isometric flat tori, and Kac’s inspiring question about hearing the shape of a drum, the field has witnessed remarkable progress, uncovering spectral invariants and advancing our understanding of geometric properties discernible through eigenvalues. We present an overview of this field amenable to both physicists and mathematicians.
We prove that wave operators of scattering theory for fourth-order Schr odinger operators H= triangle(2)+V(x) on R-2 with real potentialsV(x) such that < x(3)V(x)is an element of L4/3(R2) and < x > 10+epsilon V(x)is an element of L1(R-2) for an epsilon >0,< x >= (1 +|x|2)1/2, are bounded in Lp(R-2) for all 1< p (-1)u(x)is an element of L infinity(R-2) and if positive eigenvalues areabsent fromH. This reducesLp-mapping properties of functions f(H) of H to those of Fourier multipliers f(triangle(2)).
In this paper, we investigate spectral isoperimetric inequalities for the Robin Laplacian with or without a magnetic field in smooth, simply connected planar domains. By revisiting known results and introducing new geometric bounds, we explore the interplay between the geometry of the domain, the magnetic field intensity, and the Robin parameter. Specifically, we analyze the behavior of the lowest eigenvalue under area or perimeter constraints, highlighting cases where the disk ceases to be the optimizer. Through a combination of theoretical proofs and numerical computations, we demonstrate oscillations in the spectral isoperimetric inequality as the magnetic field varies, particularly for ellipses. Additionally, we formulate new geometric bounds for the Robin Laplacian without a magnetic field and extend quantitative reverse Faber-Krahn inequalities to ellipses with Neumann boundary conditions.
In this work, we investigate rogue waves in the integrable nonlocal pair-transition-coupled nonlinear Schr & ouml;dinger equation, a & Pscr;& Tscr;-symmetric and integrable Hamiltonian model. Building on the generalized Darboux transformation with variable separation, we observe intriguing phenomena: Coexistence of dark-bright rogue waves, displacement-parameter-induced alterations in wave peak count, destructive collapsed wave pairs, and ordered circular clusters. The key to these phenomena lies in the combination of linear solutions and displacement parameters. Finally, we investigated the modulation instability conditions of this system. These results have potential applications in applied sciences such as nonlinear optics and oceanic engineering.
We revisit and extend results by Ueltschi [Random loop representations for quantum spin systems, J. Math. Phys. 54(8) (2013) 083301] on the application of reflection positivity to loop models with theta is an element of N->= 2. By exploiting additional flexibility in the method, we prove the existence of long loops over a broader range of parameters u and theta, and establish new lower bounds for connection probabilities and the critical parameter beta(c). Our results are compared with recent numerical simulations, providing further insight into the phase diagram of quantum spin systems.
In this paper, we prove necessary and sufficient conditions for the Schrodinger operators to have zero-energy bound states at the threshold of the essential spectrum such that they have bounded kth moment. This result is an extension of the results published in D. Hundertmark, M. Jex and M. Lange [Forum Math. Sigma 11 (2023)].
In this paper, we investigate a two-variable p-adic dynamical system associated with the Potts model with an external field on the Cayley tree of order two. We analyze the fixed points of the corresponding operator and identify four saddle points. Based on these fixed points, we construct four new translation-invariant p-adic quasi Gibbs measures for the Potts model. Furthermore, we establish new sufficient conditions for the existence of a phase transition in the model by examining the boundedness of these measures.
We give a short review of our construction of a higher-loop perturbative invariant of framed 3-manifolds, generalizing the perturbative Chern-Simons invariant of Witten-Axelrod-Singer, associated to an acyclic flat connection, to an invariant given by the integral of a certain "Chern-Simons volume form" over a smooth closed component of the moduli space of flat connections.
In the practice of physics model building, the process of renormalization, resummation, and anomaly cancellation is to incrementally repair initially ill-defined Lagrangian quantum field theories by a successive choice of partial fixes. Impressive as this is, one would rather have concisely defined complete theories to begin with, and understand these choices as emergent from fundamental principles. As an instructive example, we recall renormalization choices for Wilson loop observables in abelian Chern-Simons theory. Then we show that these emerge in a novel non-Lagrangian topological completion of 5D Maxwell-Chern-Simons QFT, by means of proper flux quantization in 2-Cohomotopy. This result is a modest cousin, with applications to topologically ordered quantum materials, of the more ambitious completion of 11D supergravity by flux quantization in 4-Cohomotopy ("Hypothesis H").
The theory of optimal transport of probability measures has wide-ranging applications across a number of different fields, including concentration of measure, machine learning, Markov chains, and economics. The generalisation of optimal transport tools from probability measures to quantum states has shown great promise over the last few years, particularly in the development of the theory of Wasserstein-style distances and divergences between quantum states. Such distances have already led to a broad range of developments in the quantum setting such as functional inequalities, convergence of solutions in many-body physics, improvements to quantum generative adversarial networks, and more. However, the literature in this field is quite scattered, with very few links between different works and no real consensus on a `true' quantum Wasserstein distance. The aim of this review is to bring these works together under one roof and give a full overview of the state of the art in the development of quantum Wasserstein distances. We also present a variety of open problems and unexplored avenues in the field, and examine the future directions of this promising line of research. This review is written for those interested in quantum optimal transport in coming from both the fields of classical optimal transport and of quantum information theory, and as a resource for those working in one area of quantum optimal transport interested in how existing work may relate to their own.
This paper deals with a generalization of the superadiabatic projectors method. In a general framework, the well-known superadiabatic projectors are constructed and accurately described in the case of rank one, when a remarkable factorization occurs. We apply these ideas to spectral theory and we explain how our abstract results allow to recover or improve recent results about the semiclassical magnetic Laplacian.
In this paper, we consider a class of Unitary Quantum Walks on arbitrary graphs, parameterized by a family of scattering matrices. These Scattering Quantum Walks model the discrete dynamics of a system on the graph’s edges, where a scattering process at the vertices is governed by the scattering matrices assigned to each vertex. We show the Scattering Quantum Walks encompass several known Quantum Walks. We further introduce two classes of Scattering Open Quantum Walks on arbitrary graphs, also parameterized by scattering matrices: one defined on the edges, the other on the vertices of the graph. We show these walks give rise to proper quantum channels and describe their main spectral and dynamical properties, relating them to naturally associated classical Markov chains.
Matrix field theory is a combinatorially non-local field theory which has recently been found to be a non-trivial but solvable QFT example. To generalize such non-perturbative structures to other models, a more combinatorial understanding of Dyson-Schwinger equations and their solutions is of high interest. To this end we consider combinatorial Dyson-Schwinger equations manifestly relying on the Hopf-algebraic structure of perturbative renormalization. We find that these equations are fully compatible with renormalization, relying only on the superficially divergent diagrams which are planar ribbon graphs, i.e. decompleted dual combinatorial maps. Still, they are of a similar kind as in realistic models of local QFT, featuring in particular an infinite number of primitive diagrams as well as graph-dependent combinatorial factors.
Continual Lie algebras are infinite dimensional generalizations of ordinary Lie with continuous set of roots. In this paper we establish a general relation between chain complexes and continual Lie algebras. First, by defining and appropriate product among a chain complex spaces we endow it with the structure of an exterior differential graded algebra $\mathcal C$. The natural orthogonality condition with respect to a product bring about to $\mathcal C$ the structure of a graded algebra with differential relations and compatibility conditions for elements of chain complex spaces. By considering the union of all relations for independent elements in the tree graph structure of the system, we prove the main result of this paper: a chain complex brings about the structure of a continual Lie algebra with the root space determined by parameters for the complex. This provides a new source of examples of continual Lie algebras. Finally, as an example, we consider the case of Cech-de Rham complex associated to a foliation on a smooth manifold. In a particular case of this chain complex, we derive explicitly the commutation relations for corresponding continual Lie algebra.
In this paper, we obtain a new representation of Christoffel–Darboux formula for random matrix with external source and constrained eigenvalues. In order to simplify the Christoffel–Darboux formula, we define two types of orthogonal polynomials, and find out the close relation between Riemann–Hilbert problem and orthogonal polynomials. Furthermore, the Christoffel–Darboux formula can be expressed as a solution of the Riemann–Hilbert problem.
We show that equality of R & eacute;nyi's or Tsallis' entropies for two faithful normal states on a semifinite von Neumann algebra yields the existence of a & lowast;-isomorphism of the algebras generated by these states. Moreover, it is shown that this isomorphism can be extended to a normal completely positive map on the algebra. A similar result is shown for equality of the Segal or R & eacute;nyi or Tsallis entropies of some perturbed states of the initial states on a finite von Neumann algebra.
This paper is devoted to characterize a generalized Z-tensor. In the second part of this paper, we describe the behavior of the associated vector field of a generalized Z-flat spacetime in some special cases. It has been investigated that such a spacetime turns into special spacetimes when the associated scalars have different structures. In the third part, generalized Z-flat spacetime in f(R,G)-gravity is examined.
Using Liu's modular invariance method and its odd-dimensional extension by Han-Yu, we establish new Witten rigidity theorems for the elliptic genus of twisted Dirac operators on even-dimensional spin manifolds and twisted Toeplitz operators on odd-dimensional spin manifolds with circle actions.
In this paper, we have established the general relationship between 3D & Nscr; = 2 gauge theories and quantum spin chains [X.-M. Ding and T. Zhang, Bethe/Gauge correspondence for ABCDEFG-type 3d gauge theories, J. High Energy Phys. 04 (2023) 036, arXiv:2303.03102], known as the Bethe/Gauge correspondence. In this paper, we propose to derive the correspondence with alternative setting for a different boundary conditions, and clarify the distinctions between the two approaches. We argue that these differences originate from a form of Langlands duality between Lie algebras of types B-N and C-N.
