Abstract Topological crystalline states (TCSs) are short-range entangled states jointly protected by onsite and crystalline symmetries. Here we present a unified scheme for constructing all TCSs, bosonic and fermionic, free and interacting, from real-space building blocks and connectors. Building blocks are lower-dimensional topological states protected by onsite symmetries alone, and connectors are glues that complete the open edges shared by two or multiple building blocks. The resulted assemblies are selected against two physical criteria we call the no-open-edge condition and the bubble equivalence. The scheme is then applied to obtaining the full classification of bosonic TCSs protected by several onsite symmetry groups and each of the 17 wallpaper groups in two dimensions and 230 space groups in three dimensions. We claim that our construction scheme can give the complete set of TCSs for bosons and fermions, and prove the boson case analytically using a spectral-sequence expansion.
Topological crystalline states (TCSs) are short-range entangled states jointly protected by onsite and crystalline symmetries. Here we present a unified scheme for constructing all TCSs, bosonic and fermionic, free and interacting, from real-space building blocks and connectors. Building blocks are lower-dimensional topological states protected by onsite symmetries alone, and connectors are glues that complete the open edges shared by two or multiple building blocks. The resulted assemblies are selected against two physical criteria we call the no-open-edge condition and the bubble equivalence. The scheme is then applied to obtaining the full classification of bosonic TCSs protected by several onsite symmetry groups and each of the 17 wallpaper groups in two dimensions and 230 space groups in three dimensions. We claim that our construction scheme can give the complete set of TCSs for bosons and fermions, and prove the boson case analytically using a spectral-sequence expansion.
We present a scheme to explicitly construct and classify general topological states jointly protected by an onsite symmetry group and a spatial symmetry group. We show that all these symmetry-protected topological states can be adiabatically deformed into a special class of states we call topological crystals. A topological crystal in, for example, three dimensions is a real-space assembly of finite-sized pieces of topological states in one and two dimensions protected by the local symmetry group alone, arranged in a configuration invariant under the spatial group and glued together such that there is no open edge or end. As a demonstration of principle, we explicitly enumerate all inequivalent topological crystals for noninteracting time-reversal symmetric electronic insulators with spin-orbit coupling and any one of the 230 space groups. This enumeration gives topological crystalline insulators a full classification.
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.
Crystalline symmetry and time-reversal symmetry are commonly present in real superconducting materials. However, the topological classification of systems respecting these symmetries, particularly for interacting fermions, remains incomplete. In this work, we systematically classify time-reversal symmetry-protected crystalline topological superconductors in two-dimensional interacting fermionic systems using an explicit real-space construction. Among the resulting phases, we identify intrinsically interacting fermionic topological superconductors, i.e., phases that cannot be realized in either free-fermion or interacting bosonic systems. For spinless fermions with protecting symmetry group
C4×Z2T or
D4×Z2T (plus fermion parity), the intrinsic sector has a
Z4 classification. The corresponding root phases generating this
Z4 classification admit a transparent real-space construction in terms of decorated 1D blocks. These blocks are 1D fermionic symmetry-protected topological (FSPT) phases, realizable as double Majorana chains. We further find the corresponding
Z4 spinless intrinsic phases for wallpaper groups
p4,
p4m, and
p4g. We also find an additional
Z2 intrinsically interacting phase for spinless fermions with wallpaper group
pm, which is absent with the corresponding point-group symmetry alone. Moreover, these intrinsic phases naturally give rise to higher-order FSPT phases that support corner zero modes. Finally, we verify the crystalline equivalence principle for generic 2D interacting FSPT systems with both crystalline and internal symmetries.
The construction and classification of crystalline symmetry protected topological (SPT) phases in interacting bosonic and fermionic systems have been intensively studied in the past few years. Crystalline SPT phases are not only of conceptual importance, but also provide us great opportunities towards experimental realization since space group symmetries naturally exist for any realistic material. In this paper, we systematically classify the crystalline topological superconductors (TSC) and topological insulators (TI) in 2D interacting fermionic systems by using an explicit real-space construction. In particular, we discover an intriguing fermionic crystalline topological superconductor that can only be realized in interacting fermionic systems (i.e., not in free-fermion or interacting bosonic systems). Moreover, we also verify the recently conjectured crystalline equivalence principle for generic 2D interacting fermionic systems.
We develop a theoretical framework for the classification and construction of symmetry protected topological (SPT) phases, which are a special class of zero-temperature phases of strongly interacting gapped quantum many-body systems that exhibit topological properties. The framework unifies various proposals for the classification of SPT phases, including the group (super-)cohomology proposal, the (spin-)cobordism proposal, the Freed-Hopkins proposal, and the Kitaev proposal. The power of the framework is demonstrated in a number of applications: (1) the classification and construction of 3D fermionic SPT phases in Wigner-Dyson classes A and AII with glide symmetry, (2) the classification and construction of 3D bosonic SPT phases with space-group symmetries for all 230 space groups, (3) the derivation of a Mayer-Vietoris sequence relating the classification of SPT phases with and without reflection symmetry, and (4) an interpretation of the structure of general crystalline SPT phases via the Atiyah-Hirzebruch spectral sequence. The framework is based on the idea that short-range entangled states form what are known as Ω-spectra in the sense of algebraic topology, and models the classification of SPT phases by what are called generalized cohomology theories.