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See detail${\mathbb{Z}}_{2}{\times}{\mathbb{Z}}_{2}$-graded supersymmetry: 2-d sigma models
Bruce, Andrew UL

in Journal of Physics. A, Mathematical and Theoretical (2020), 53(45), 455201

We propose a natural ${\mathbb{Z}}_{2}{\times}{\mathbb{Z}}_{2}$-graded generalisation of d = 2, $\mathcal{N}=\left(1,1\right)$ supersymmetry and construct a ${\mathbb{Z}}_{2}^{2}$-space realisation ... [more ▼]

We propose a natural ${\mathbb{Z}}_{2}{\times}{\mathbb{Z}}_{2}$-graded generalisation of d = 2, $\mathcal{N}=\left(1,1\right)$ supersymmetry and construct a ${\mathbb{Z}}_{2}^{2}$-space realisation thereof. Due to the grading, the supercharges close with respect to, in the classical language, a commutator rather than an anticommutator. This is then used to build classical (linear and non-linear) sigma models that exhibit this novel supersymmetry via mimicking standard superspace methods. The fields in our models are bosons, right-handed and left-handed Majorana–Weyl spinors, and exotic bosons. The bosons commute with all the fields, the spinors belong to different sectors that cross commute rather than anticommute, while the exotic boson anticommute with the spinors. As a particular example of one of the models, we present a 'double-graded' version of supersymmetric sine-Gordon theory. [less ▲]

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See detailOdd connections on supermanifolds: existence and relation with affine connections
Bruce, Andrew UL; Grabowski, Janusz

in Journal of Physics. A, Mathematical and Theoretical (2020), 53(45), 455203

The notion of an odd quasi-connection on a supermanifold, which is loosely an affine connection that carries non-zero Grassmann parity, is examined. Their torsion and curvature are defined, however, in ... [more ▼]

The notion of an odd quasi-connection on a supermanifold, which is loosely an affine connection that carries non-zero Grassmann parity, is examined. Their torsion and curvature are defined, however, in general, they are not tensors. A special class of such generalised connections, referred to as odd connections in this paper, have torsion and curvature tensors. Part of the structure is an odd involution of the tangent bundle of the supermanifold and this puts drastic restrictions on the supermanifolds that admit odd connections. In particular, they must have equal number of even and odd dimensions. Amongst other results, we show that an odd connection is defined, up to an odd tensor field of type (1, 2), by an affine connection and an odd endomorphism of the tangent bundle. Thus, the theory of odd connections and affine connections are not completely separate theories. As an example relevant to physics, it is shown that $\mathcal{N}=1$ super-Minkowski spacetime admits a natural odd connection. [less ▲]

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See detailTemplates and subtemplates of Rössler attractors from a bifurcation diagram
Rosalie, Martin UL

in Journal of Physics. A, Mathematical and Theoretical (2016), 49(31), 315101

We study the bifurcation diagram of the Rössler system. It displays the various dynamical regimes of the system (stable or chaotic) when a parameter is varied. We choose a diagram that exhibits coexisting ... [more ▼]

We study the bifurcation diagram of the Rössler system. It displays the various dynamical regimes of the system (stable or chaotic) when a parameter is varied. We choose a diagram that exhibits coexisting attractors and banded chaos. We use the topological characterization method to study these attractors. Then, we detail how the templates of these attractors are subtemplates of a unique template. Our main result is that only one template describes the topological structure of eight attractors. This leads to a topological partition of the bifurcation diagram that gives the symbolic dynamic of all bifurcation diagram attractors with a unique template. [less ▲]

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See detailDeterminant representation of the domain-wall boundary condition partition function of a Richardson–Gaudin model containing one arbitrary spin
Faribault, Alexandre; Tschirhart, Hugo UL; Muller, Nicolas

in Journal of Physics. A, Mathematical and Theoretical (2016)

In this work we present a determinant expression for the domain-wall boundary condition partition function of rational (XXX) Richardson–Gaudin models which, in addition to N-1 spins 1/2, contains one ... [more ▼]

In this work we present a determinant expression for the domain-wall boundary condition partition function of rational (XXX) Richardson–Gaudin models which, in addition to N-1 spins 1/2, contains one arbitrarily large spin S. The proposed determinant representation is written in terms of a set of variables which, from previous work, are known to define eigenstates of the quantum integrable models belonging to this class as solutions to quadratic Bethe equations. Such a determinant can be useful numerically since systems of quadratic equations are much simpler to solve than the usual highly nonlinear Bethe equations. It can therefore offer significant gains in stability and computation speed. [less ▲]

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See detailAlgebraic Bethe ansätze and eigenvalue-based determinants for Dicke–Jaynes–Cummings–Gaudin quantum integrable models
Tschirhart, Hugo UL; Faribault, Alexandre

in Journal of Physics. A, Mathematical and Theoretical (2014), 47

In this work, we construct an alternative formulation to the traditional Algebraic Bethe ansatz for quantum integrable models derived from a generalised rational Gaudin algebra realised in terms of a ... [more ▼]

In this work, we construct an alternative formulation to the traditional Algebraic Bethe ansatz for quantum integrable models derived from a generalised rational Gaudin algebra realised in terms of a collection of spins 1/2 coupled to a single bosonic mode. The ensemble of resulting models which we call Dicke-Jaynes-Cummings-Gaudin models are particularly relevant for the description of light-matter interaction in the context of quantum optics. Having two distinct ways to write any eigenstate of these models we then combine them in order to write overlaps and form factors of local operators in terms of partition functions with domain wall boundary conditions. We also demonstrate that they can all be written in terms of determinants of matrices whose entries only depend on the eigenvalues of the conserved charges. Since these eigenvalues obey a much simpler set of quadratic Bethe equations, the resulting expressions could then offer important simplifications for the numerical treatment of these models. [less ▲]

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See detailTransfer matrix analysis of one-dimensional majority cellular automata with thermal noise
Lemoy, Rémi UL; Mozeika, Alexander; Seki, Shinnosuke

in Journal of Physics. A, Mathematical and Theoretical (2014), 47(10), 105001-11

Thermal noise in a cellular automaton (CA) refers to a random perturbation in its function which eventually leads the automaton to an equilibrium state controlled by a temperature parameter. We study the ... [more ▼]

Thermal noise in a cellular automaton (CA) refers to a random perturbation in its function which eventually leads the automaton to an equilibrium state controlled by a temperature parameter. We study the one-dimensional majority-3 CA under this model of noise. Without noise, each cell in the automaton decides its next state by majority voting among itself and its left and right neighbour cells. Transfer matrix analysis shows that the automaton always reaches a state in which every cell is in one of its two states with probability 1/2 and thus cannot remember even one bit of information. Numerical experiments, however, support the possibility of reliable computation for a long but finite time. [less ▲]

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