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Irreducible representations (irreps) of $\mathbb{Z}_2^2$-graded supersymmetry algebra of ${\cal N}=2$ are obtained by the method of induced representation and they are used to derive $\mathbb{Z}_2^2$-graded supersymmetric classical actions. The irreps are four dimensional for $ \lambda = 0$ where $ \lambda $ is an eigenvalue of the Casimir element, and eight dimensional for $\lambda \neq 0.$ The eight dimensional irreps reduce to four dimensional ones only when $\lambda$ and an eigenvalue of Hamiltonian satisfy a particular relation. The reduced four dimensional irreps are used to define $\mathbb{Z}_2^2$-graded supersymmetry transformations and two types of classical actions invariant under the transformations are presented. It is shown that one of the Noether charges vanishes if all the variables of specific $\mathbb{Z}_2^2$-degree are auxiliary., Comment: 17 pages, no figures

We investigate superfield formulation of the minimal $\mathbb{Z}_2^2$-supersymmetry. It is shown that the integrability on $\mathbb{Z}_2^2$-superspace guarantees the invariance of action. Then we present two superfields which carry distinct irreducible representations of the $\mathbb{Z}_2^2$-supersymmetry algebra. One of them gives integrable Lagrangian and the other does not. We also show that integrable superfields with different $\mathbb{Z}_2^2$-degree also carry irreducible representations and they give invariant actions. To perform this analysis, the representation theory of the minimal $\mathbb{Z}_2^2$-supersymmetry algebra is studied in some detail., Comment: 18 pages, no figure

In the present paper we study the representations of the Jacobi algebra. More concretely, we define, analogously to the case of semi-simple Lie algebras, the Verma modules over the Jacobi algebra ${\cal G}_2$. We study their reducibility and give explicit construction of the reducible Verma modules exhibiting the corresponding singular vectors. Using this information we give a complete classification of the reducible Verma modules. More than this we exhibit their interrelation of embeddings between these modules. These embeddings are illustrated by diagrams of the embedding patterns so that each reducible Verma module appears in one such diagram., Comment: 41pages, many figures

Multi-layered materials provide fascinating platforms to realize various functional properties, possibly leading to future electronic devices controlled by external fields. In particular, layered magnets coupled with conducting layers have been extensively studied recently for possible control of their transport properties via the spin structure. Successful control of quantum-transport properties in the materials with antiferromagnetic (AFM) layers, so-called natural spin-valve structure, has been reported for the Dirac Fermion and topological/axion materials. However, a bulk crystal in which magnetic and superconducting layers are alternately stacked has not been realized until now, and the search for functional properties in it is an interesting yet unexplored field in material science. Here, we discover superconductivity providing such an ideal platform in EuSn2As2 with the van der Waals stacking of magnetic Eu layers and superconducting Sn-As layers, and present the first demonstration of a natural spin-valve effect on the superconducting current. Below the superconducting transition temperature (Tc), the electrical resistivity becomes zero in the in-plane direction. In contrast, it, surprisingly, remains finite down to the lowest temperature in the out-of-plane direction, mostly due to the structure of intrinsic magnetic Josephson junctions in EuSn2As2. The magnetic order of the Eu layers (or natural spin-valve) is observed to be extremely soft, allowing one to easy control of the out-of-plane to in-plane resistivities ratio from 1 to infinity by weak external magnetic fields. The concept of multi-functional materials with stacked magnetic-superconducting layers will open a new pathway to develop novel spintronic devices with magnetically controllable superconductivity., Comment: 17 pages, 4 figures

It is shown that the ${\cal N}=1$ supersymmetric quantum mechanics (SQM) can be extended to a $\mathbb{Z}_2^n$-graded superalgebra. This is done by presenting quantum mechanical models which realize, with the aid of Clifford gamma matrices, the $\mathbb{Z}_2^n$-graded Poincar\'e algebra in one-dimensional spacetime. Reflecting the fact that the $\mathbb{Z}_2^n$-graded Poincar\'e algebra has a number of central elements, a sequence of models defining the $\mathbb{Z}_2^n$-graded version of SQM are provided for a given value of $n.$ In a model of the sequence, the central elements having the same $\mathbb{Z}_2^n$-degree are realized as dependent or independent operators. It is observed that as use the Clifford algebra of lager dimension, more central elements are realized as independent operators., Comment: 19 pages, no figure

In the recent paper, Bruce and Duplij introduced a double-graded version of supersymmetric quantum mechanics (SQM). It is an extension of Lie superalgebraic nature of ${\cal N}=1$ SQM to a $\mathbb{Z}_2^2$-graded superalgebra. In this work, we propose an extension of Bruce-Duplij model to higher values of $\cal N.$ Furthermore, it is shown that our construction of double-graded SQM is a special case of the method which converts a given Lie superalgebra to a $\mathbb{Z}_2^2$-graded superalgebra. By employing this method one may convert a model of superconformal mechanics to its double-graded version. The simplest example of ${\cal N}=1$ double-graded superconformal mechanics is studied in some detail., Comment: 15 pages, 2 figures, new reference, error corrections

We present a relationship between the Calogero-Moser particles confined in harmonic oscillator potentials and a representation theory of the infinite dimensional Lie algebra which is a semi-direct sum of Virasoro algebra and its module. More precisely, it is a correspondence of excited states of the model and singular vectors in Verma modules over the algebra. This is found by a free field realization of the time evolution operator of the model. We investigate the Verma modules and some explicit example of singular vectors are given., Comment: Eq.(27) is incorrect which will alter the main result

We study the three-dimensional (3D) compact U(1) lattice gauge theory coupled with $N$-flavor Higgs fields by means of the Monte Carlo simulations. This model is relevant to multi-component superconductors, antiferromagnetic spin systems in easy plane, inflational cosmology, etc. It is known that there is no phase transition in the N=1 model. For N=2, we found that the system has a second-order phase transition line $\tilde{c}_1(c_2)$ in the $c_2$(gauge coupling)$-c_1$(Higgs coupling) plane, which separates the confinement phase and the Higgs phase. Numerical results suggest that the phase transition belongs to the universality class of the 3D XY model as the previous works by Babaev et al. and Smiseth et al. suggested. For N=3, we found that there exists a critical line similar to that in the N=2 model, but the critical line is separated into two parts; one for $c_2 < c_{2{\rm tc}}=2.4\pm 0.1$ with first-order transitions, and the other for $ c_{2{\rm tc}} < c_2$ with second-order transitions, indicating the existence of a tricritical point. We verified that similar phase diagram appears for the N=4 and N=5 systems. We also studied the case of anistropic Higgs coupling in the N=3 model and found that there appear two second-order phase transitions or a single second-order transition and a crossover depending on the values of the anisotropic Higgs couplings. This result indicates that an "enhancement" of phase transition occurs when multiple phase transitions coincide at a certain point in the parameter space., Comment: Long version of arXiv:0704.1323, New results added