Symmetry-protected topological (SPT) phases (e.g., the well known topological insulators) are a class of energetically gapped condensed matter systems that exhibit interesting topological properties only in the presence of certain global symmetries. While a great understanding of noninteracting fermionic SPT phases has been achieved recently, interacting SPT phases, in particular those that exist only in strongly interacting systems, are much less understood in general. Here, the authors study strongly interacting fermionic SPT phases in two spatial dimensions with finite Abelian unitary symmetries and provide a potentially complete classification of them.
Symmetry-protected topological (SPT) phases (e.g., the well known topological insulators) are a class of energetically gapped condensed matter systems that exhibit interesting topological properties only in the presence of certain global symmetries. While a great understanding of noninteracting fermionic SPT phases has been achieved recently, interacting SPT phases, in particular those that exist only in strongly interacting systems, are much less understood in general. Here, the authors study strongly interacting fermionic SPT phases in two spatial dimensions with finite Abelian unitary symmetries and provide a potentially complete classification of them.
Symmetry protected topological (SPT) phases are a minimal generalization of the concept of topological insulators to interacting systems. In this paper, we describe the classification and properties of such phases for three-dimensional (3D) electronic systems with a number of different symmetries. For symmetries representative of all classes in the famous 10-fold way of free-fermion topological insulators/superconductors, we determine the stability to interactions. By combining with results on bosonic SPT phases, we obtain a classification of electronic 3D SPT phases for these symmetries. In cases with a normal U(1) subgroup we show that this classification is complete. We describe the nontrivial surface and bulk properties of these states. In particular, we discuss interesting correlated surface states that are not captured in a free-fermion description. We show that in many, but not all, cases, the surface can be gapped while preserving symmetry if it develops intrinsic topological order.
The construction and classification of symmetry-protected topological (SPT) phases in interacting bosonic and fermionic systems have been intensively studied in the past few years. Very recently, a complete classification and construction of space-group SPT phases were also proposed for interacting bosonic systems. In this Rapid Communication, we attempt to generalize this classification and construction scheme systematically into interacting fermion systems. In particular, we construct and classify point-group SPT phases for two-dimensional (2D) interacting fermion systems via lower-dimensional block-state decorations. We discover several intriguing fermionic SPT states that can only be realized in interacting fermion systems (i.e., not in free-fermion or bosonic SPT systems). Moreover, we also verify the recently conjectured crystalline equivalence principle for 2D interacting fermion systems. Finally, the potential experimental realization of these different classes of point-group SPT phases in 2D correlated superconductors is addressed.
It has been shown that the symmetry-protected topological (SPT) phases with finite Abelian symmetries can be described by Chern-Simons field theory. We propose a topological response theory to uniquely identify the SPT orders, which allows us to obtain a systematic scheme to classify bosonic SPT phases with any finite Abelian symmetry group. We point out that even for finite Abelian symmetry, there exist bosonic SPT phases beyond the current Chern-Simons theory framework. We also apply the theory to fermionic SPT phases with Zm symmetry and find the classification of SPT phases depends on the parity of m: for even m there are 2m classes, m out of which is intrinsically fermionic SPT phases and cannot be realized in any bosonic system. Finally we propose a classification scheme of fermionic SPT phases for any finite, Abelian symmetry.
We study the classification of interacting fermionic symmetry-protected topological (SPT) phases with both rotation symmetry and Abelian internal symmetries in one, two, and three dimensions. By working out this classification, on the one hand, we demonstrate the recently proposed correspondence principle between crystalline topological phases and those with internal symmetries through explicit block-state constructions. We find that for the precise correspondence to hold it is necessary to change the central extension structure of the symmetry group by the Z2 fermion parity. On the other hand, we uncover new classes of intrinsically fermionic SPT phases that are only enabled by interactions, both in 2D and 3D. Several 3D Majorana-type fermionic SPT phases are identified. Moreover, several new instances of Lieb-Schultz-Mattis-type theorems for Majorana-type fermionic SPT phases are obtained and we discuss their interpretations from the perspective of bulk-boundary correspondence.
In this thesis, we study gapped topological phases of matter in systems with strong inter-particle interaction. They are challenging to analyze theoretically, because interaction not only gives rise to a plethora of phases that are otherwise absent, but also renders methods used to analyze non-interacting systems inadequate. By now, people have had a relatively systematic understanding of topological orders in two spatial dimensions. However, less is known about the higher dimensional cases. In Chapter 2, we will explore three dimensional long-range entangled topological orders in the framework of Walker-Wang models, which are a class of exactly solvable models for three-dimensional topological phases that are not known previously to be able to capture these phases. We find that they can represent a class of twisted discrete gauge theories, which were discovered using a different formalism. Meanwhile, a systematic theory of bosonic symmetry protected topological (SPT) phases in all spatial dimensions have been developed based on group cohomology. A generalization of the theory to group supercohomology has been proposed to classify and characterize fermionic SPT phases in all dimensions. However, it can only handle cases where the symmetry group of the system is a product of discrete unitary symmetries. Furthermore, the classification is known to be incomplete for certain symmetries. In Chapter 3, we will construct an exactly solvable model for the two-dimensional time-reversal-invariant topological superconductors, which could be valuable as a first attempt to a systematic understanding of strongly interacting fermionic SPT phases with anti-unitary symmetries in terms of exactly solvable models. In Chapter 4, we will propose an alternative classification of fermionic SPT phases using the spin cobordism theory, which hopefully can capture all the phases missing in the supercohomology classification. We test this proposal in the case of fermionic SPT phases with Z2 symmetry, where Z2 is either time-reversal or an internal symmetry. We find that cobordism classification correctly describes all known fermionic SPT phases in space dimensions less than or equal to 3.
We present the correspondence between symmetry-protected topological (SPT) phases and their anomalous boundary states, based on examples in various spacetime dimensions. Through the study of the effect of interactions on these SPT phases, we discovered a new formalism of quantum anomalies, associated with discrete spacetime (such as time-reversal and spatial reflection) symmetries in particular, to classify distinct interacting topological phases. An example is the Z2 classification of the (2+1)d topological insulator protected by charge U(1) and time-reversal (or CP) symmetries, which can be deduced by the form of the global U(1) gauge anomalies on its edge theories defined on closed unorientable manifolds. In this case, the nontrivial phase (in free systems) is robust against electron interactions. Another example is the (3+1)d topological superconductor protected by only time-reversal or reflection symmetry. For this system, we identified the bulk phase by studying the global gravitational anomalies of the surface theories formulated on unorientable spacetime manifolds, and also discussed its connection to the collapse of the non-interacting classification by an integer Z to Z16, in the presence of interactions. We also revisit the problem of gauging a discrete internal symmetry in theories of chiral (Weyl) fermions in 3+1 dimensions – which have not been fully understood so far – from the perspective of fermionic SPT phases in 4+1 dimensions. Comparing with the previous results, we give a complete answer for the anomalies constraints on the discrete symmetry, as our approach is based on purely geometrical considerations, namely, our assumption is more fundamental and general. Furthermore, our result also provides an understanding of gapped states of fermions with anomalous discrete symmetries, and we present a model, based on weak coupling, for constructing these anomalous gapped states.