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In this work we investigate the decorated domain wall construction in bosonic group-cohomology symmetry-protected topological (SPT) phases and related quantum anomalies in bosonic topological phases. We first show that a general decorated domain wall construction can be described mathematically as an Atiyah-Hirzebruch spectral sequence, where the terms on the $E_2$ page correspond to decorations by lower-dimensional SPT states at domain wall junctions. For bosonic group-cohomology SPT phases, the spectral sequence becomes the Lyndon-Hochschild-Serre (LHS) spectral sequence for ordinary group cohomology. We then discuss the physical interpretations of the differentials in the spectral sequence, particularly in the context of anomalous SPT phases and symmetry-enriched gauge theories. As the main technical result, we obtain a full description of the LHS spectral sequence concretely at the cochain level. The explicit formulae are then applied to explain Lieb-Schultz-Mattis theorems for SPT phases, and also derive a new LSM theorem for easy-plane spin model in a $pi$ flux lattice. We also revisit the classifications of symmetry-enriched 2D and 3D Abelian gauge theories using our results.
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We study the ground-state entanglement of gapped domain walls between topologically ordered systems in two spatial dimensions. We derive a universal correction to the ground-state entanglement entropy, which is equal to the logarithm of the total quantum dimension of a set of superselection sectors localized on the domain wall. This expression is derived from the recently proposed entanglement bootstrap method.
Matrix Product States (MPSs) provide a powerful framework to study and classify gapped quantum phases --symmetry-protected topological (SPT) phases in particular--defined in one dimensional lattices. On the other hand, it is natural to expect that gapped quantum phases in the limit of zero correlation length are described by topological quantum field theories (TFTs or TQFTs). In this paper, for (1+1)-dimensional bosonic SPT phases protected by symmetry $G$, we bridge their descriptions in terms of MPSs, and those in terms of $G$-equivariant TFTs. In particular, for various topological invariants (SPT invariants) constructed previously using MPSs, we provide derivations from the point of view of (1+1) TFTs. We also discuss the connection between boundary degrees of freedom, which appear when one introduces a physical boundary in SPT phases, and open TFTs, which are TFTs defined on spacetimes with boundaries.
Given a (2+1)D fermionic topological order and a symmetry fractionalization class for a global symmetry group $G$, we show how to construct a (3+1)D topologically invariant path integral for a fermionic $G$ symmetry-protected topological state ($G$-FSPT) in terms of an exact combinatorial state sum. This provides a general way to compute anomalies in (2+1)D fermionic symmetry-enriched topological states of matter. Equivalently, our construction provides an exact (3+1)D combinatorial state sum for a path integral of any FSPT that admits a symmetry-preserving gapped boundary, including the (3+1)D topological insulators and superconductors in class AII, AIII, DIII, and CII that arise in the free fermion classification. Our construction uses the fermionic topological order (characterized by a super-modular tensor category) and symmetry fractionalization data to define a (3+1)D path integral for a bosonic theory that hosts a non-trivial emergent fermionic particle, and then condenses the fermion by summing over closed 3-form $mathbb{Z}_2$ background gauge fields. This procedure involves a number of non-trivial higher-form anomalies associated with Fermi statistics and fractional quantum numbers that need to be appropriately canceled off with a Grassmann integral that depends on a generalized spin structure. We show how our construction reproduces the $mathbb{Z}_{16}$ anomaly indicator for time-reversal symmetric topological superconductors with ${bf T}^2 = (-1)^F$. Mathematically, with standard technical assumptions, this implies that our construction gives a combinatorial state sum on a triangulated 4-manifold that can distinguish all $mathbb{Z}_{16}$ $mathrm{Pin}^+$ smooth bordism classes. As such, it contains the topological information encoded in the eta invariant of the pin$^+$ Dirac operator, thus giving an example of a state sum TQFT that can distinguish exotic smooth structure.
We describe topologically ordered and fracton ordered states on novel geometries which do not have an underlying manifold structure. Using tree graphs such as the $k$-coordinated Bethe lattice ${cal B}(k)$ and a hypertree called the $(k,n)$-hyper-Bethe lattice ${cal HB}(k,n)$ consisting of $k$-coordinated hyperlinks (defined by $n$ sites), we construct multidimensional arboreal arenas such as ${cal B}(k_1) square {cal B}(k_2)$ by the notion of a graph Cartesian product $square$. We study various quantum systems such as the ${mathbb Z}_2$ gauge theory, generalized quantum Ising models (GQIM), the fractonic X-cube model, and related X-cube gauge theory defined on these arenas. Even the simplest ${mathbb Z}_2$ gauge theory on a 2d arboreal arena is fractonic -- the monopole excitation is immobile. The X-cube model on a 3d arboreal arena is fully fractonic, all multipoles are rendered immobile. We obtain variational ground state phase diagrams of these gauge theories. Further, we find an intriguing class of dualities in arboreal arenas as illustrated by the ${mathbb Z}_2$ gauge theory defined on ${cal B}(k_1) square {cal B}(k_2)$ being dual to a GQIM defined on ${cal HB}(2,k_1) square {cal HB}(2,k_2)$. Finally, we discuss different classes of topological and fracton orders on arboreal arenas. We find three distinct classes of arboreal toric code orders on 2d arboreal arenas, those that occur on ${cal B}(2) square {cal B}(2)$, ${cal B}(k) square {cal B}(2), k >2$, and ${cal B}(k_1) square {cal B}(k_2)$, $k_1,k_2>2$. Likewise, four classes of X-cube fracton orders are found in 3d arboreal arenas -- those on ${cal B}(2)square{cal B}(2)square {cal B}(2)$, ${cal B}(k) square {cal B}(2)square {cal B}(2), k>2$, ${cal B}(k_1) square {cal B}(k_2) square {cal B}(2), k_1,k_2 >2$, and ${cal B}(k_1) square {cal B}(k_2) square {cal B}(k_3), k_1,k_2,k_3 >2$.
We review the dimensional reduction procedure in the group cohomology classification of bosonic SPT phases with finite abelian unitary symmetry group. We then extend this to include general reductions of arbitrary dimensions and also extend the procedure to fermionic SPT phases described by the Gu-Wen super-cohomology model. We then show that we can define topological invariants as partition functions on certain closed orientable/spin manifolds equipped with a flat connection. The invariants are able to distinguish all phases described within the respective models. Finally, we establish a connection to invariants obtained from braiding statistics of the corresponding gauged theories.
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