# Non-Abelian fermionization and fractional quantum Hall transitions

## Abstract

There has been a recent surge of interest in dualities relating theories of Chern-Simons gauge fields coupled to either bosons or fermions within the condensed matter community, particularly in the context of topological insulators and the half-filled Landau level. Here, we study the application of one such duality to the long-standing problem of quantum Hall interplateaux transitions. The key motivating experimental observations are the anomalously large value of the correlation length There has been a recent surge of interest in dualities relating theories of Chern-Simons gauge fields coupled to either bosons or fermions within the condensed matter community, particularly in the context of topological insulators and the half-filled Landau level. Here, we study the application of one such duality to the long-standing problem of quantum Hall inter-plateaux transitions. The key motivating experimental observations are the anomalously large value of the correlation length exponent $$\nu \approx 2.3$$ and that $$\nu$$ is observed to be super-universal, i.e., the same in the vicinity of distinct critical points [S.L. Sondhi et al., Rev. Mod. Phys. \textbf{69}, 315 (1997)]. Duality motivates effective descriptions for a fractional quantum Hall plateau transition involving a Chern-Simons field with $$U(N_c)$$ gauge group coupled to $$N_f = 1$$ fermion. We study one class of theories in a controlled limit where $$N_f \gg N_c$$ and calculate $$\nu$$ to leading non-trivial order in the absence of disorder. Although these theories do not yield an anomalously large exponent $$\nu$$ within the large $$N_f \gg N_c$$ expansion, they do offer a new parameter space of theories that is apparently different from prior works involving abelian Chern-Simons gauge fields [X.-G. Wen and Y.-S. Wu, Phys. Rev. Lett. \textbf{70}, 1501 (1993); W. Chen. et al., Phys. Rev. B. \textbf{48}, 13749 (1993)].

- Authors:

- Cornell Univ., Ithaca, NY (United States)
- Univ. of California, Riverside, CA (United States)

- Publication Date:

- Research Org.:
- Cornell Univ., Ithaca, NY (United States)

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

- OSTI Identifier:
- 1423635

- Alternate Identifier(s):
- OSTI ID: 1419905

- Grant/Contract Number:
- SC0010313

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Physical Review B

- Additional Journal Information:
- Journal Volume: 97; Journal Issue: 8; Journal ID: ISSN 2469-9950

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY

### Citation Formats

```
Hui, Aaron, Mulligan, Michael, and Kim, Eun-Ah.
```*Non-Abelian fermionization and fractional quantum Hall transitions*. United States: N. p., 2018.
Web. doi:10.1103/PhysRevB.97.085112.

```
Hui, Aaron, Mulligan, Michael, & Kim, Eun-Ah.
```*Non-Abelian fermionization and fractional quantum Hall transitions*. United States. doi:10.1103/PhysRevB.97.085112.

```
Hui, Aaron, Mulligan, Michael, and Kim, Eun-Ah. Thu .
"Non-Abelian fermionization and fractional quantum Hall transitions". United States. doi:10.1103/PhysRevB.97.085112. https://www.osti.gov/servlets/purl/1423635.
```

```
@article{osti_1423635,
```

title = {Non-Abelian fermionization and fractional quantum Hall transitions},

author = {Hui, Aaron and Mulligan, Michael and Kim, Eun-Ah},

abstractNote = {There has been a recent surge of interest in dualities relating theories of Chern-Simons gauge fields coupled to either bosons or fermions within the condensed matter community, particularly in the context of topological insulators and the half-filled Landau level. Here, we study the application of one such duality to the long-standing problem of quantum Hall interplateaux transitions. The key motivating experimental observations are the anomalously large value of the correlation length There has been a recent surge of interest in dualities relating theories of Chern-Simons gauge fields coupled to either bosons or fermions within the condensed matter community, particularly in the context of topological insulators and the half-filled Landau level. Here, we study the application of one such duality to the long-standing problem of quantum Hall inter-plateaux transitions. The key motivating experimental observations are the anomalously large value of the correlation length exponent $\nu \approx 2.3$ and that $\nu$ is observed to be super-universal, i.e., the same in the vicinity of distinct critical points [S.L. Sondhi et al., Rev. Mod. Phys. \textbf{69}, 315 (1997)]. Duality motivates effective descriptions for a fractional quantum Hall plateau transition involving a Chern-Simons field with $U(N_c)$ gauge group coupled to $N_f = 1$ fermion. We study one class of theories in a controlled limit where $N_f \gg N_c$ and calculate $\nu$ to leading non-trivial order in the absence of disorder. Although these theories do not yield an anomalously large exponent $\nu$ within the large $N_f \gg N_c$ expansion, they do offer a new parameter space of theories that is apparently different from prior works involving abelian Chern-Simons gauge fields [X.-G. Wen and Y.-S. Wu, Phys. Rev. Lett. \textbf{70}, 1501 (1993); W. Chen. et al., Phys. Rev. B. \textbf{48}, 13749 (1993)].},

doi = {10.1103/PhysRevB.97.085112},

journal = {Physical Review B},

number = 8,

volume = 97,

place = {United States},

year = {Thu Feb 08 00:00:00 EST 2018},

month = {Thu Feb 08 00:00:00 EST 2018}

}

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