ゲストプロフェッサーによるセミナー
場所；中百舌鳥キャンパス A13-306 (セミナー室 B)
プログラム:
Title: Search for solvable potentials by extensions of supersymmetric quantum mechanics
Title: Beyond the tenfold way: 13 associative Z2*Z2-graded superdivision algebras
Title: First quantization of braided Majorana férmions
************** Abstracts ********************************************
Inoue
The necessary and sufficient conditions for a potential to be solvable in one-dimensional quantum mechanical systems are not known yet. Clarifying this will clarify what the solvability of the potential problem is. Most of the known solvable potentials have the property called shape invariance. Therefore, I start with a brief review that the shape invariance is a sufficient condition for solvability. To search for conditions for solvability, it will be helpful to find more solvable potentials. In order to find new solvable potential, I will introduce a combination of para-supersymmetric quantum mechanics and multi-fold supersymmetric quantum mechanics, which are extensions of supersymmetric quantum mechanics that I have focused on in this research.
Kuznetsova
The notion of the tenfold way is related to the classification of Hamiltonians for topological insulators and superconductors. Surprisingly, the tenfold way classification is in a strong relation to two different mathematical structures: Cartan's classification of symmetric spaces and 10 classes of super-division algebras.
In the talk I present the further step: a classification of Z2*Z2-graded division algebras. The classification is done in the alphabetic presentation technique. We obtain 13 new classes of Z2*Z2-graded superdivision algebras. The relation to topological insulators and superconductors is the next step in this direction.
Toppan
Majorana fermions and their braiding properties started being intensively investigated after Kitaev’s proposal to use them for encoding logical operations of a topological quantum computer which offers protection from decoherence. In this talk a quantization framework to construct braided multi-particle Majorana fermions is introduced. A single Majorana fermion is given by a Z_2-graded qubit. The multiparticle sectors are expressed by a graded Hopf algebra endowed with a braided tensor product. A parameter t encodes the braiding property. A nontrivial braiding is essential to produce Hilbert spaces described by qubits, qutrits, etc. Truncations of the Hilbert spaces appear at certain roots of unity for t; these roots of unity are organized into “levels” which specify the maximal number of allowed Majorana fermions in a multiparticle sector.
(Based on FT, NPB 980 (2022) 115834; arXiv:2203.01776)
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