# The phase transitions between Z _{n} × Z _{n} bosonic topological phases in 1 + 1D, and a constraint on the central charge for the critical points between bosonic symmetry protected topological phases

## Abstract

The study of continuous phase transitions triggered by spontaneous symmetry breaking has brought revolutionary ideas to physics. Recently, through the discovery of symmetry protected topological phases, it is realized that continuous quantum phase transition can also occur between states with the same symmetry but different topology. Here in this paper we study a specific class of such phase transitions in 1+1 dimensions – the phase transition between bosonic topological phases protected by Z _{n} × Z _{n}. We find in all cases the critical point possesses two gap opening relevant operators: one leads to a Landau-forbidden symmetry breaking phase transition and the other to the topological phase transition. We also obtained a constraint on the central charge for general phase transitions between symmetry protected bosonic topological phases in 1+1D.

- Authors:

- Univ. of California, Berkeley, CA (United States). Dept. of Physics
- SLAC National Accelerator Lab., Menlo Park, CA (United States). Stanford Institute for Materials and Energy Science (SIMES)
- Univ. of California, Berkeley, CA (United States). Dept. of Physics; Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Materials Sciences Division

- Publication Date:

- Research Org.:
- SLAC National Accelerator Lab., Menlo Park, CA (United States); Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)

- OSTI Identifier:
- 1349311

- Alternate Identifier(s):
- OSTI ID: 1369421; OSTI ID: 1413729

- Grant/Contract Number:
- AC02-05CH11231; AC02-76SF00515

- Resource Type:
- Journal Article: Published Article

- Journal Name:
- Nuclear Physics. B

- Additional Journal Information:
- Journal Volume: 919; Journal Issue: C; Journal ID: ISSN 0550-3213

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS

### Citation Formats

```
Tsui, Lokman, Huang, Yen-Ta, Jiang, Hong-Chen, and Lee, Dung-Hai.
```*The phase transitions between Zn × Zn bosonic topological phases in 1 + 1D, and a constraint on the central charge for the critical points between bosonic symmetry protected topological phases*. United States: N. p., 2017.
Web. doi:10.1016/j.nuclphysb.2017.03.021.

```
Tsui, Lokman, Huang, Yen-Ta, Jiang, Hong-Chen, & Lee, Dung-Hai.
```*The phase transitions between Zn × Zn bosonic topological phases in 1 + 1D, and a constraint on the central charge for the critical points between bosonic symmetry protected topological phases*. United States. doi:10.1016/j.nuclphysb.2017.03.021.

```
Tsui, Lokman, Huang, Yen-Ta, Jiang, Hong-Chen, and Lee, Dung-Hai. Mon .
"The phase transitions between Zn × Zn bosonic topological phases in 1 + 1D, and a constraint on the central charge for the critical points between bosonic symmetry protected topological phases". United States.
doi:10.1016/j.nuclphysb.2017.03.021.
```

```
@article{osti_1349311,
```

title = {The phase transitions between Zn × Zn bosonic topological phases in 1 + 1D, and a constraint on the central charge for the critical points between bosonic symmetry protected topological phases},

author = {Tsui, Lokman and Huang, Yen-Ta and Jiang, Hong-Chen and Lee, Dung-Hai},

abstractNote = {The study of continuous phase transitions triggered by spontaneous symmetry breaking has brought revolutionary ideas to physics. Recently, through the discovery of symmetry protected topological phases, it is realized that continuous quantum phase transition can also occur between states with the same symmetry but different topology. Here in this paper we study a specific class of such phase transitions in 1+1 dimensions – the phase transition between bosonic topological phases protected by Zn × Zn. We find in all cases the critical point possesses two gap opening relevant operators: one leads to a Landau-forbidden symmetry breaking phase transition and the other to the topological phase transition. We also obtained a constraint on the central charge for general phase transitions between symmetry protected bosonic topological phases in 1+1D.},

doi = {10.1016/j.nuclphysb.2017.03.021},

journal = {Nuclear Physics. B},

number = C,

volume = 919,

place = {United States},

year = {Mon Mar 27 00:00:00 EDT 2017},

month = {Mon Mar 27 00:00:00 EDT 2017}

}

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