From orthosymplectic structure to super topological matter

Abstract

Topological supermatter is given by ordinary topological matter constrained by supersymmetry or graded supergroups such as OSP(2N|2N). Using results on super oscillators and lattice QFT${}_d$, we construct a super tight binding model on hypercubic super lattice with supercharge $\mathbf{Q}={\sum }_{\mathbf{k}}{\hat{F}}_{\mathbf{k}}.{\mathbf{q}}_{\mathbf{k}}.{\hat{B}}_{\mathbf{k}}$. We first show that the algebraic triplet $(\Omega ,G,J)$ of super oscillators can be derived from the $OSp(2N|2N)$ supergroup containing the symplectic $Sp(2N)$ and the orthogonal $SO(2N)$ as even subgroups. Then, we apply the obtained result on super oscillating matter to super bands and investigate its topological obstructions protected by TPC symmetries. We also give a classification of the Bose/Fermi coupling matrix ${\mathbf{q}}_{\mathbf{k}}$ in terms of subgroups of $OSp(2N|2N)$\ and show that there are $2P_N$ (partition of $N$) classes ${\mathbf{q}}_{\mathbf{k}}$ given by unitary subgroups of $U\left(2\right)\times U\left(N\right)$. Other features are also given.

LaTeX, 74 pages, 7 figures

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