Tenfold Classification for Interacting Fermions and Relation with Homogeneous Spaces

Abstract

Topological classifi cation of topological insulator superconductor, various quantum Hall states are probably the most discussed topic in theoretical quantum condensed matter community as well as many mathematicians especially after Kitaev's periodic table answers the topological classi fication problem for noninteracting fermions, and the details of these classifi cation theories require a lot of K-theory and other advanced topological study. However, there is a parallel method proposed by synder etc all which does the same classi cation but taking the target space for some nonlinearsigmamodel in d-1 dimension. However what happens with interaction is still an open problem and KItaev's spin chain shows that in BDI class under interaction classi cation can change from Z to Z8 which clearly not there in any class. So certainly interacting systems hold much for the surprise. What happens under interaction, the partial answer is given in Ref[1], the authors have been able to compute the Hamiltonian structure for each class for K-body interaction. So naturally, the second question is that can we classify the interacting systems now. Its noted that the time evolution operator for each of these Hamiltonian will be some Homogeneous spaces. In this thesis, we provide a review of the classi cation scheme as well as a hopefully, precise way of doing calculus on Homogeneous space like calculating connection curvature etc. Such that one can do the calculus on this whenever it appears as a target space of the nonlinearsigmamodel. We have also shown that symmetric spaces (which appeared in time evolution operator for the noninteracting case) are homogeneous spaces with additions constraints on their tangent space, such that we can see whenever we end up getting the noninteracting Hamiltonian as well. Finally, we have also given an alternative derivation of getting the Hamiltonian structure via projective representation, it is originally given in (Ref[1]) however at the conclusion we draw a rough connection with Kahler potential which appears on the quantization of arbitrary functional space. We hope that this connection might give a new insight into the connection between Cartan's symmetric spaces and Classi fication of noninteracting Hamiltonian.