Exactly soluble local bosonic cocycle models, statistical transmutation, and simplest time-reversal symmetric topological orders in 3+1D

TL;DR

This work develops a versatile, exactly soluble framework of local bosonic lattice models built from space‑time cochains and cocycles to realize a broad family of topological orders, including SPTs and SETs, in 2+1D and 3+1D. By organizing models around Z_n 1‑ and 2‑cocycle twists (and non‑abelian generalizations), the authors demonstrate emergent gauge theories (notably Dijkgraaf–Witten types) and phenomena such as emergent fermions, Kramer doublets, and symmetry fractionalization on defects, with explicit calculations of partition functions and ground state degeneracies on diverse manifolds. They establish dualities, dimension reductions, and detailed classifications of time‑reversal symmetric orders, providing a broad catalog of root families and explicit realizations beyond prior DW/Walker–Wang frameworks. The results illuminate how boundary anomalies, braiding statistics, and symmetry fractionalization intertwine in higher dimensions, with potential implications for constructing lattice realizations of exotic topological phases and for cobordism-based fermionic SPT classification.

Abstract

We propose a generic construction of exactly soluble \emph{local bosonic models} that realize various topological orders with gappable boundaries. In particular, we construct an exactly soluble bosonic model that realizes a 3+1D $Z_2$ gauge theory with emergent fermionic Kramer doublet. We show that the emergence of such a fermion will cause the nucleation of certain topological excitations in space-time without pin$^+$ structure. The exactly soluble model also leads to a statistical transmutation in 3+1D. In addition, we construct exactly soluble bosonic models that realize 2 types of time-reversal symmetry enriched $Z_2$-topological orders in 2+1D, and 20 types of simplest time-reversal symmetry enriched topological (SET) orders which have only one non-trivial point-like and string-like topological excitations. Many physical properties of those topological states are calculated using the exactly soluble models. We find that some time-reversal SET orders have point-like excitations that carry Kramer doublet -- a fractionalized time-reversal symmetry. We also find that some $Z_2$ SET orders have string-like excitations that carry anomalous (non-on-site) $Z_2$ symmetry, which can be viewed as a fractionalization of $Z_2$ symmetry on strings. Our construction is based on cochains and cocycles in algebraic topology, which is very versatile. In principle, it can also realize emergent topological field theory beyond the twisted gauge theory.

Figures (12)

• Figure 1: (Color online) Three identical$Z_3$-symmetry twist defects (blue triangles) on a torus. The red line are the symmetry twist line. A symmetry twist defect is an end of symmetry twist line.
• Figure 2: (Color online) A 1-cochain $a$ has a value $1$ on the red links: $a_{ik}=a_{jk}= 1$ and a value $0$ on other links: $a_{ij}=a_{kl}=0$. $\dd a$ is non-zero on the shaded triangles: $(\dd a)_{jkl} = a_{jk} + a_{kl} - a_{jl}$. For such 1-cohain, we also have $a\cup a=0$. So when viewed as a $\Z_2$-valued cochain, $\cB_2 a \neq a\cup a$ mod 2.
• Figure 3: (Color online) A 1-cochain $a$ has a value $1$ on the red links, Another 1-cochain $a'$ has a value $1$ on the blue links. On the left, $a\cup a'$ is non-zero on the shade triangles: $(a\cup a')_{ijl}=a_{ij}a'_{jl}=1$, while on the right, $a'\cup a$ is zero. Thus $a\cup a'+a'\cup a$ is not a coboundary.
• Figure 4: (Color online) A particle-hole tunneling process is a process where we create a particle-hole pair, move the particle around a non-contractible loop, and then annihilate the particle and the hole. The GSD on a $d$-dimensional space $S^1\times S^{d-1}$ is generated by the particle-hole tunneling process described by the blue loop. Thus, each degenerate ground state correspond to a type of particle, and GSD$(S^1\times S^{d-1}) =$ number of types of point-like excitations. Similarly, GSD$(S^{k}\times S^{d-k}) =$ number of types of $(k-1)$-dimensional excitations.
• Figure 5: (Color online) Two branched simplices with opposite orientations. (a) A branched simplex with positive orientation and (b) a branched simplex with negative orientation.