We present an algorithmic classification of the irreps of the N -extended onedimensional supersymmetry algebra linearly realized on a finite number of fields. Our work is based on the 1-to-1 [1] correspondence between Weyl-type Clifford algebras (whose irreps are fully classified) and classes of irreps of the N -extended 1D supersymmetry. The complete classification of irreps is presented up to N ≤ 10. The fields of an irrep are accommodated in l different spin states. N = 10 is the minimal value admitting length l > 4 irreps. The classification of length-4 irreps of the N = 12 and real N = 11 extended supersymmetries is also explicitly presented.Tensoring irreps allows us to systematically construct manifestly (N -extended) supersymmetric multi-linear invariants without introducing a superspace formalism. Multi-linear invariants can be constructed both for unconstrained and multilinearly constrained fields. A whole class of off-shell invariant actions are produced in association with each irreducible representation. The explicit example of the N = 8 off-shell action of the (1, 8, 7) multiplet is presented.Tensoring zero-energy irreps leads us to the notion of the fusion algebra of the 1D N -extended supersymmetric vacua.

Most quantum entanglement investigations are focused on two qubits or some finite (small) chain structure, since the infinite chain structure is a considerably cumbersome task. Therefore, the quantum entanglement properties involving an infinite chain structure is quite important, not only because the mathematical calculation is cumbersome but also because real materials are well represented by an infinite chain. Thus, in this paper we consider an entangled diamond chain with Ising and anisotropic Heisenberg (Ising-XXZ) coupling. Two interstitial particles are coupled through Heisenberg coupling or simply two-qubit Heisenberg, which could be responsible for the emergence of entanglement. These two-qubit Heisenberg operators are interacted with two nodal Ising spins. An infinite diamond chain is organized by interstitial-interstitial and nodal-interstitial (dimer-monomer) site couplings. We are able to get the thermal average of the two-qubit operator, called the reduced two-qubit density operator. Since these density operators are spatially separated, we could obtain the concurrence (entanglement) directly in the thermodynamic limit. The thermal entanglement (concurrence) is constructed for different values of the anisotropic Heisenberg parameter, magnetic field and temperature. We also observed the threshold temperature via the parameter of anisotropy, Heisenberg and Ising interaction, external magnetic field, and temperature.

Quaternionic and octonionic realizations of Clifford algebras and spinors are classified and explicitly constructed in terms of recursive formulas. The most general free dynamics in arbitrary signature space-times for both quaternionic and octonionic spinors is presented. In the octonionic case we further provide a systematic list of results and tables expressing, e.g., the relations of the octonionic Clifford algebras with the G 2 cosets over the Lorentz algebras, the identities satisfied by the higher-rank antisymmetric octonionic tensors and so on. Applications of these results range from the classification of octonionic generalized supersymmetries, the construction of octonionic superstrings, as well as the investigations concerning the recently discovered octonionic M-superalgebra and its superconformal extension.

Quantum entanglement is one of the most fascinating types of correlation that can be shared only among quantum systems. The Heisenberg chain is one of the simplest quantum chains which exhibits a reach entanglement feature, due to the Heisenberg interaction is quantum coupling in the spin system. The two particles were coupled trough XYZ coupling or simply called as two-qubit XYZ spin, which are the responsible for the emergence of thermal entanglement. These two-qubit operators are bonded to two nodal Ising spins, and this process is repeated infinitely resulting in a diamond chain structure. We will discuss two-qubit thermal entanglement effect on Ising-XYZ diamond chain structure. The concurrence could be obtained straightforwardly in terms of two-qubit density operator elements, using this result, we study the thermal entanglement, as well as the threshold temperature where entangled state vanishes. The present model displays a quite unusual concurrence behavior, such as, the boundary of two entangled regions becomes a disentangled region, this is intrinsically related to the XY-anisotropy in the Heisenberg coupling. Despite a similar property had been found for only two-qubit, here we show in the case of a diamond chain structure, which reasonably represents real materials. S i S z J J 0 J Figure 1: (Color Online) Schematic representation of Ising-XYZ chain on diamond structure, σa,i and σ i,b are Heisenberg spins, while Si corresponds to Ising spins.

Zero temperature non-plateau magnetization is a peculiar property of a quantum spin chain and it sometimes appears due to different gyromagnetic factors. In this study, we illustrate a quite unusual nonplateau magnetization property driven by XY-anisotropy in an Ising-XYZ diamond chain. Two particles with spin-1/2 are bonded by XYZ coupling and they are responsible for the emergence of non-plateau magnetization. These two quantum operator spins are bonded to two nodal Ising spins and this process is repeated infinitely to yield a diamond chain structure. Due to the nonplateau magnetization property, we focus our discussion on the magnetocaloric effect of this model by presenting the isentropic curves and the Grüneisen parameters, as well as showing the regions where the model exhibits an efficient magnetocaloric effect. Due to the existence of two phases located very close to each other, the strong XY-anisotropy exhibits a particular behavior with a magnetocaloric effect, with a wider interval in the magnetic field, where the magnetocaloric effect is efficient.PACS numbers: