After briefly recalling the quantum entanglement-based view of topological phases of matter in order to outline the general context, we give an overview of different approaches to the classification problem of topological insulators and superconductors of non-interacting Fermions. In particular, we review in some detail general symmetry aspects of the "ten-fold way" which forms the foundation of the classification, and put different approaches to the classification in relationship with each other. We end by briefly mentioning some of the results obtained on the effect of interactions, mainly in three spatial dimensions.
After briefly recalling the quantum entanglement-based view of topological phases of matter in order to outline the general context, we give an overview of different approaches to the classification problem of topological insulators and superconductors of non-interacting Fermions. In particular, we review in some detail general symmetry aspects of the "ten-fold way" which forms the foundation of the classification, and put different approaches to the classification in relationship with each other. We end by briefly mentioning some of the results obtained on the effect of interactions, mainly in three spatial dimensions.
The “ten-fold way” underlying the classification of nonin- teracting topological insulators and superconductors maps the infrastructure of the “civilized” part of the topological world. With interactions, we step into the wild. The main challenges that stem from the presence of interactions can be organized into three categories. The first set of problems is generated by the lack of a general classification of interacting topological phases (i.e., the interacting analog of the “ten-fold way”), which could be used as the basis for a systematic understanding of these quantum states. Moreover, it is not even clear what type of mathematical structure should underlie such a classification. Nonetheless, significant progress has been made in the past few years involving partial classifications of certain types of interacting topological states based on tensor categories [ 512 ], group cohomology theory [ 98 , 226 ], cobordism [ 276 ], and other “scary mathematical beasts.”
T he “ten-fold way” underlying the classification of non-interacting topological insulators and superconductors maps the infras- tructure of the “civilized” part of the topological world. With interactions, we step into the wild. The main challenges that stem from the presence of interactions can be organized into three categories. The first set of problems is generated by the lack of a general classification of interacting topological phases (i.e., the interacting analog of the “ten-fold way”), which could be used as the basis for a systematic understanding of these quantum states. Moreover, it is not even clear what type of mathematical structure should underlie such a classification. Nonetheless, significant progress has been made in the past few years involving partial classifications of certain types of interacting topological states based on tensor categories [429], group cohomology theory [82, 200], cobordism [246], and other “scary mathematical beasts.”
These lecture notes explain the classification of some simple fermionic topological phases of matter in a pedestrian manner, with an aim to be maximally pedagogical = doing things in excruciating detail. We focus on a many-body perspective, even if many of the models we work with are non-interacting. We start out with symmetry protected topological (SPT) phases of free fermions that are protected by U(1) symmetry = topological insulators. We then look at fermion topological phases that don’t even need a symmetry = topological superconductors, and explain how their classification changes in presence of spinless time-reversal symmetry. We close by perturbatively checking which of the 1D topological phases we had found are stable to interactions.
We give a brief overview on interacting symmetry-protected topological (SPT) phases. In particular, we will focus on various cases where the classification of non-interacting fermionic systems breaks down in the presence of interactions. The examples include topological (and trivial) superconductors in one, two and three spatial dimensions, protected by various symmetries such as time-reversal, spin parity, and reflection symmetries. Common to these examples is the ‘collapse’ of the non-interacting classification, where an integer classification (‘’) at non-interacting level reduces to or in the presence of interactions. We make use of these examples to illustrate various approaches to interacting SPT phases.
We give a brief overview on interacting symmetry-protected topological (SPT) phases. In particular, we will focus on various cases where the classification of non-interacting fermionic systems breaks down in the presence of interactions. The examples include topological (and trivial) superconductors in one, two and three spatial dimensions, protected by various symmetries such as time-reversal, spin parity, and reflection symmetries. Common to these examples is the ‘collapse’ of the non-interacting classification, where an integer classification (‘ ’) at non-interacting level reduces to or in the presence of interactions. We make use of these examples to illustrate various approaches to interacting SPT phases.
We study aspects of gapped phases of matter, focusing on their classification, including the group law under stacking, and their relation to topological quantum field theories (TQFT). In one spatial dimension, it is well-known that Matrix Product States (MPS) efficiently approximate ground states of gapped systems; by showing that these states arise naturally in 1 + 1-dimensional lattice TQFT, which in turn are closely related to continuum TQFT, we provide a concrete connection between ground states of lattice systems and TQFT in 1 + 1 dimensions. We generalize this to systems with symmetries and fermions, and obtain a classification and group law for the stacking of 1 + 1-dimensional symmetry-protected topological phases. Further, we study the effect of turning on/off interactions for the classification: the phase classification of a given symmetry class of Hamiltonians can be different depending on whether we allow interactions or not, and in low dimensions we provide some concrete formulas relating the phases under the non-interacting classification and those under the interacting classification. Lastly, we study the phases of the 2 + 1-dimensional topological superconductor, and show that for all 16 phases braiding statistics of vortices, which determine the underlying TQFT, can be obtained by stacking layers of the basic p + ip superconductor.