# Tunneling topological vacua via extended operators: (Spin-)TQFT spectra and boundary deconfinement in various dimensions

## Abstract

Distinct quantum vacua of topologically ordered states can be tunneled into each other via extended operators. The possible applications include condensed matter and quantum cosmology. We present a straightforward approach to calculate the partition function on various manifolds and ground state degeneracy (GSD), mainly based on continuum/cochain topological quantum field theories (TQFTs), in any dimension. This information can be related to the counting of extended operators of bosonic/fermionic TQFTs. On the lattice scale, anyonic particles/strings live at the ends of line/surface operators. Certain systems in different dimensions are related to each other through dimensional reduction schemes, analogous to (de)categorification. Examples include spin TQFTs derived from gauging the interacting fermionic symmetry-protected topological states (with fermion $$\mathbb Z$$$f\atop{2}$$) of parity symmetry groups $$\mathbb Z{_4}$$×$$\mathbb Z{_2}$$ and ($$\mathbb Z{_4}$$) ^{2} in 3+1D, also $$\mathbb Z{_2}$$ and ($$\mathbb Z{_2}$$) ^{2} in 2+1D. Gauging the last three cases begets non-Abelian spin TQFTs (fermionic topological order). We consider situations where a TQFT lives on (1) a closed spacetime or (2) a spacetime with a boundary, such that the bulk and boundary are fully gapped and short- or long-range entangled (SRE/LRE). Anyonic excitations can be deconfined on the boundary. We introduce new exotic topological interfaces on which neither particle nor string excitations alone condense, but only fuzzy-composite objects of extended operators can end (e.g., a string-like composite object formed by a set of particles can end on a special 2+1D boundary of 3+1D bulk). We explore the relations between group extension constructions and partially breaking constructions (e.g., 0-form/higher-form/“composite” breaking) of topological boundaries, after gauging. We comment on the implications of entanglement entropy for some such LRE systems.

- Authors:

- Institute for Advanced Study, Princeton, NJ (United States); Harvard Univ., Cambridge, MA (United States)
- Institute for Advanced Study, Princeton, NJ (United States)
- Princeton Univ., NJ (United States)
- Univ. of Science and Technology of China, Hefei (China)
- Harvard Univ., Cambridge, MA (United States)
- Harvard Univ., Cambridge, MA (United States); Tsinghua Univ., Beijing (China)

- Publication Date:

- Research Org.:
- Inst. for Advanced Study, Princeton, NJ (United States)

- Sponsoring Org.:
- USDOE; National Science Foundation (NSF); US–Israel Binational Science Foundation

- OSTI Identifier:
- 1499898

- Grant/Contract Number:
- SC0009988; PHY-1314311; 11431010; 11571329; PHY-1306313; PHY-0937443; DMS-1308244; DMS-0804454; DMS-1159412

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Progress of Theoretical and Experimental Physics

- Additional Journal Information:
- Journal Volume: 2018; Journal Issue: 5; Journal ID: ISSN 2050-3911

- Publisher:
- Physical Society of Japan

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS

### Citation Formats

```
Wang, Juven, Ohmori, Kantaro, Putrov, Pavel, Zheng, Yunqin, Wan, Zheyan, Guo, Meng, Lin, Hai, Gao, Peng, and Yau, Shing-Tung.
```*Tunneling topological vacua via extended operators: (Spin-)TQFT spectra and boundary deconfinement in various dimensions*. United States: N. p., 2018.
Web. doi:10.1093/ptep/pty051.

```
Wang, Juven, Ohmori, Kantaro, Putrov, Pavel, Zheng, Yunqin, Wan, Zheyan, Guo, Meng, Lin, Hai, Gao, Peng, & Yau, Shing-Tung.
```*Tunneling topological vacua via extended operators: (Spin-)TQFT spectra and boundary deconfinement in various dimensions*. United States. doi:10.1093/ptep/pty051.

```
Wang, Juven, Ohmori, Kantaro, Putrov, Pavel, Zheng, Yunqin, Wan, Zheyan, Guo, Meng, Lin, Hai, Gao, Peng, and Yau, Shing-Tung. Wed .
"Tunneling topological vacua via extended operators: (Spin-)TQFT spectra and boundary deconfinement in various dimensions". United States. doi:10.1093/ptep/pty051. https://www.osti.gov/servlets/purl/1499898.
```

```
@article{osti_1499898,
```

title = {Tunneling topological vacua via extended operators: (Spin-)TQFT spectra and boundary deconfinement in various dimensions},

author = {Wang, Juven and Ohmori, Kantaro and Putrov, Pavel and Zheng, Yunqin and Wan, Zheyan and Guo, Meng and Lin, Hai and Gao, Peng and Yau, Shing-Tung},

abstractNote = {Distinct quantum vacua of topologically ordered states can be tunneled into each other via extended operators. The possible applications include condensed matter and quantum cosmology. We present a straightforward approach to calculate the partition function on various manifolds and ground state degeneracy (GSD), mainly based on continuum/cochain topological quantum field theories (TQFTs), in any dimension. This information can be related to the counting of extended operators of bosonic/fermionic TQFTs. On the lattice scale, anyonic particles/strings live at the ends of line/surface operators. Certain systems in different dimensions are related to each other through dimensional reduction schemes, analogous to (de)categorification. Examples include spin TQFTs derived from gauging the interacting fermionic symmetry-protected topological states (with fermion $\mathbb Z$$f\atop{2}$) of parity symmetry groups $\mathbb Z{_4}$×$\mathbb Z{_2}$ and ($\mathbb Z{_4}$)2 in 3+1D, also $\mathbb Z{_2}$ and ($\mathbb Z{_2}$)2 in 2+1D. Gauging the last three cases begets non-Abelian spin TQFTs (fermionic topological order). We consider situations where a TQFT lives on (1) a closed spacetime or (2) a spacetime with a boundary, such that the bulk and boundary are fully gapped and short- or long-range entangled (SRE/LRE). Anyonic excitations can be deconfined on the boundary. We introduce new exotic topological interfaces on which neither particle nor string excitations alone condense, but only fuzzy-composite objects of extended operators can end (e.g., a string-like composite object formed by a set of particles can end on a special 2+1D boundary of 3+1D bulk). We explore the relations between group extension constructions and partially breaking constructions (e.g., 0-form/higher-form/“composite” breaking) of topological boundaries, after gauging. We comment on the implications of entanglement entropy for some such LRE systems.},

doi = {10.1093/ptep/pty051},

journal = {Progress of Theoretical and Experimental Physics},

issn = {2050-3911},

number = 5,

volume = 2018,

place = {United States},

year = {2018},

month = {5}

}