Search

A class of left bialgebroids whose underlying algebra A ♯ H is a smash product of a bialgebra H with a braided commutative Yetter–Drinfeld H -algebra A has recently been studied in relation to models of field theories on noncommutative spaces. In Borowiec and Pachoł (2017 J. Phys. A: Math. Theor. 50 055205) a proof has been presented that the bialgebroid A F ♯ H F where HF and AF are the twists of H and A by a Drinfeld 2-cocycle F = ∑ F 1 ⊗ F 2 is isomorphic to the twist of bialgebroid A ♯ H by the bialgebroid 2-cocycle ∑ 1 ♯ F 1 ⊗ 1 ♯ F 2 induced by F. They assume H is quasitriangular, which is reasonable for many physical applications. However the proof and the entire paper take for granted that the coaction and the prebraiding are both given by special formulas involving the R-matrix. There are counterexamples of Yetter–Drinfeld modules over quasitriangular Hopf algebras which are not of this special form. Nevertheless, the main result essentially survives. We present a proof with a general coaction and the correct prebraiding, and even without the assumption of quasitriangularity. [ABSTRACT FROM AUTHOR]

The authors replies to the comment made by Yang and Zhou (2024 J. Phys. A: Math. Theor.) on his 2015 paper entitled 'Critical points of Potts and O(N) models from eigenvalue identities in periodic Temperley–Lieb algebras' (Jacobsen 2015 J. Phys. A: Math. Theor. 48 454003). [ABSTRACT FROM AUTHOR]

In this paper, we present the construction of all nonstandard integrable systems in magnetic fields whose integrals have leading order structure corresponding to the case (i) of theorem 1 in Marchesiello and Šnobl (2022 J. Phys. A: Math. Theor. 55 145203). We find that the resulting systems can be written as one family with several parameters. For certain limits of these parameters the system belongs to intersections with already known standard systems separating in Cartesian and / or cylindrical coordinates and the number of independent integrals of motion increases, thus the system becomes minimally superintegrable. These results generalize the particular example presented in section 3 of Marchesiello and Šnobl (2022 J. Phys. A: Math. Theor. 55 145203). [ABSTRACT FROM AUTHOR]

It is well-known that any Lennard–Jones type potential energy must have a periodic ground state given by a triangular lattice in dimension 2. In this paper, we describe a computer-assisted method that rigorously shows such global minimality result among 2-dimensional lattices once the exponents of the potential have been fixed. The method is applied to the widely used classical (12 , 6) Lennard–Jones potential, which is the main result of this work. Furthermore, a new bound on the inverse density (i.e. the co-volume) for which the triangular lattice is minimal is derived, improving those found in (Bétermin and Zhang 2015 Commun. Contemp. Math. 17 1450049) and (Bétermin 2016 SIAM J. Math. Anal. 48 3236–3269). The same results are also shown to hold for other exponents as additional examples and a new conjecture implying the global optimality of a triangular lattice for any parameters is stated. [ABSTRACT FROM AUTHOR]

Let us say that a three-qubit state u 000|000âź© + u 001|001âź© + â‹Ż + u 111|111âź© is real if all its amplitudes u rst are real numbers. We will prove that for every real three-qubit state | Ď• âź©, there exist three angles θ 0, θ 1 and θ 2 such that R y (θ 2) ⊗ R y (θ 1) ⊗ R y (θ 0)| Ď• âź© is a three-qubit of the form λ 1|000âź© + λ 2|011âź© + λ 3|101âź© + λ 4|110âź© + λ 5|111âź© with the λ i real numbers. In contrast with the general case, the case of three-qubits with complex amplitudes, we proved that for three qubit states, the dimension of the real entanglement space (the space obtained by identifying real qubit states with local orthogonal gates, instead of local unitary gates) is 4 and in this paper we find four linearly independent polynomial invariants of degree 4 which are not possible to find for the different Schmidt representations of three qubit states. See (AcĂ-n et al 2000 Phys. Rev. Lett. 85 1560; AcĂ-n et al 2001 J. Phys. A: Math. Gen. 34 6725; Carteret et al 2000 J. Math. Phys. 41 7932; Sudbery 2001 J. Phys. A: Math. Gen. 34 643). [ABSTRACT FROM AUTHOR]

This paper studies the conductance on the universal homology covering space Z of 2D orbifolds in a strong magnetic field, thereby removing the rationality constraint on the magnetic field in earlier works (Avron et al 1994 Phys. Rev. Lett. 73 3255–3257; Mathai and Wilkin 2019 Lett. Math. Phys. 109 2473–2484; Prieto 2006 Commun. Math. Phys. 265 373–396) in the literature. We consider a natural Landau Hamiltonian on Z and study its spectrum which we prove consists of a finite number of low-lying isolated points and calculate the von Neumann degree of the associated holomorphic spectral orbibundles when the magnetic field B is large, and obtain fractional quantum numbers as the conductance. [ABSTRACT FROM AUTHOR]

Godrčche, in Comment on 'Fluctuation dominated phase ordering at a mixed order transition' [2021 J. Phys. A: Math. Theor. 54 038001], has commented on our recent paper Fluctuation dominated phase ordering at a mixed order transition (2019 J. Phys. A: Math. Theor. 52 254001). This comment concerns the prefactor of the cusp-like small-argument singularity of the scaled spin-spin correlation function at criticality. We remark that the approach used in our paper is adequate for computing the cusp exponent, which is what is really needed to establish fluctuation-dominated phase ordering. Computing the precise value of the prefactor of the cusp singularity is irrelevant for this purpose, or for the physics behind the relation between the mixed order phase transition and the fluctuation-dominated phase ordering—the understanding of which was the main purpose of our paper. [ABSTRACT FROM AUTHOR]

The damped and parametrically driven nonlinear Dirac equation with arbitrary nonlinearity parameter is analyzed, when the external force is periodic in space and given by , both numerically and in a variational approximation using five collective coordinates (time dependent shape parameters of the wave function). Our variational approximation satisfies exactly the low-order moment equations. Because of competition between the spatial period of the external force , and the soliton width ls, which is a function of the nonlinearity as well as the initial frequency of the solitary wave, there is a transition (at fixed ) from trapped to unbound behavior of the soliton, which depends on the parameters r and K of the external force and the nonlinearity parameter . We previously studied this phenomena when (Quintero et al 2019 J. Phys. A: Math. Theor. 52 285201) where we showed that for the soliton oscillates in an effective potential, while for it moves uniformly as a free particle. In this paper we focus on the dependence of the transition from oscillatory to particle behavior and explicitly compare the curves of the transition regime found in the collective coordinate approximation as a function of r and K when at fixed value of the frequency . Since the solitary wave gets narrower for fixed as a function of , we expect and indeed find that the regime where the solitary wave is trapped is extended as we increase . [ABSTRACT FROM AUTHOR]

We complete the derivation of the sausage model NLIE by giving a proof of the crucial relation (3.24) of the original paper based on the analytic properties of Q and . [ABSTRACT FROM AUTHOR]

We study the Laurent property, the irreducibility and co-primeness of discrete integrable and non-integrable equations. First we study a discrete integrable equation related to the Somos-4 sequence, and also a non-integrable equation as a comparison. We prove that the conditions of irreducibility and co-primeness hold only in the integrable case. Next, we generalize our previous results on the singularities of the discrete Korteweg–de Vries (dKdV) equation. In our previous paper (Kanki et al 2014 J. Phys. A: Math. Theor.47 065201) we described the singularity confinement test (one of the integrability criteria) using the Laurent property, and the irreducibility, and co-primeness of the terms in the bilinear dKdV equation, in which we only considered simplified boundary conditions. This restriction was needed to obtain simple (monomial) relations between the bilinear form and the nonlinear form of the dKdV equation. In this paper, we prove the co-primeness of the terms in the nonlinear dKdV equation for general initial conditions and boundary conditions, by using the localization of Laurent rings and the interchange of the axes. We assert that co-primeness of the terms can be used as a new integrability criterion, which is a mathematical re-interpretation of the confinement of singularities in the case of discrete equations. [ABSTRACT FROM AUTHOR]

The Witten–Dijkgraaf–Verlinde–Verlinde (or WDVV) equations, as one would expect from an integrable system, has many symmetries, both continuous and discrete. One class—the so-called Legendre transformations—were introduced by Dubrovin. They are a discrete set of symmetries between the stronger concept of a Frobenius manifold, and are generated by certain flat vector fields. In this paper this construction is generalized to the case where the vector field (called here the Legendre field) is non-flat but satisfies a certain set of defining equations. One application of this more general theory is to generate the induced symmetry between almost-dual Frobenius manifolds whose underlying Frobenius manifolds are related by a Legendre transformation. This also provides a map between rational and trigonometric solutions of the WDVV equations. [ABSTRACT FROM AUTHOR]

It has been shown that many networks associated with complex systems are small-world (they have both a large local clustering coefficient and a small diameter) and also scale-free (the degrees are distributed according to a power law). Moreover, these networks are very often hierarchical, as they describe the modularity of the systems that are modeled. Most of the studies for complex networks are based on stochastic methods. However, a deterministic method, with an exact determination of the main relevant parameters of the networks, has proven useful. Indeed, this approach complements and enhances the probabilistic and simulation techniques and, therefore, it provides a better understanding of the modeled systems. In this paper we find the radius, diameter, clustering coefficient and degree distribution of a generic family of deterministic hierarchical small-world scale-free networks that has been considered for modeling real-life complex systems. [ABSTRACT FROM AUTHOR]

We show how the derivation of the Derrida–Lebowitz–Speer–Spohn equation can be prolonged to obtain a new equation, generalizing the models obtained in the paper by these authors. We then investigate its properties from both an analytical and numerical perspective. Specifically, a numerical method is presented to approximate solutions of the prolonged equation. Using this method, we investigate the relationship between the solutions of the prolonged equation and the Tracy–Widom GOE distribution. [ABSTRACT FROM AUTHOR]

State sum constructions, such as Kuperberg’s algorithm, give partition functions of physical systems, like lattice gauge theories, in various dimensions by associating local tensors or weights with different parts of a closed triangulated manifold. Here we extend this construction by including matter fields to build partition functions in both two and three space–time dimensions. The matter fields introduce new weights to the vertices and they correspond to Potts spin configurations described by an -module with an inner product. Performing this construction on a triangulated manifold with a boundary we obtain transfer matrices which are decomposed into a product of local operators acting on vertices, links and plaquettes. The vertex and plaquette operators are similar to the ones appearing in the quantum double models (QDMs) of Kitaev. The link operator couples the gauge and the matter fields, and it reduces to the usual interaction terms in known models such as gauge theory with matter fields. The transfer matrices lead to Hamiltonians that are frustration-free and are exactly solvable. According to the choice of the initial input, that of the gauge group and a matter module, we obtain interesting models which have a new kind of ground state degeneracy that depends on the number of equivalence classes in the matter module under gauge action. Some of the models have confined flux excitations in the bulk which become deconfined at the surface. These edge modes are protected by an energy gap provided by the link operator. These properties also appear in ‘confined Walker–Wang’ models which are 3D models having interesting surface states. Apart from the gauge excitations there are also excitations in the matter sector which are immobile and can be thought of as defects like in the Ising model. We only consider bosonic matter fields in this paper. [ABSTRACT FROM AUTHOR]

In this paper we prove that given a weakly almost periodic measure μ supported inside some model set with closed window W, then the strongly almost periodic component and the null weakly almost periodic component μ0 are both supported inside . As a consequence we prove that given any translation bounded measure ω, supported inside some model set, then each of the pure point diffraction spectrum and the continuous diffraction spectrum is either trivial or has a relatively dense support. [ABSTRACT FROM AUTHOR]