# ${\mathbb{Z}}_{n}$ superconductivity of composite bosons and the $7/3$ fractional quantum Hall effect

## Abstract

The topological p-wave pairing of composite fermions, believed to be responsible for the 5/2 frac- tional quantum Hall effect (FQHE), has generated much exciting physics. Motivated by the parton theory of the FQHE, we consider the possibility of a new kind of emergent “superconductivity” in the 1/3 FQHE, which involves condensation of clusters of n composite bosons. From a microscopic perspective, the state is described by the $$n\bar{n}$$111 parton wave function P_{LLL}Φ_{n}Φ$$*\atop{n}$$Φ$$3\atop{1}$$, where Φ_{n} is the wave function of the integer quantum Hall state with n filled Landau levels and P_{LLL} is the lowest- Landau-level projection operator. It represents a Zn superconductor of composite bosons, because the factor Φ$$3\atop{1}$$ ~ Πj_{j} - z_{k})^{3}, where zj = x_{j} - iy_{j} is the coordinate of the jth electron, binds three vortices to electrons to convert them into composite bosons, which then condense into the Zn 2 superconducting state |Φ_{n}|^{2}. From a field theoretical perspective, this state can be understood by starting with the usual Laughlin theory and gauging a Z_{n} subgroup of the U(1) charge conservation symmetry. We find from detailed quantitative calculations that the 2$$\bar{2}$$111 and 3$$\bar{3}$$111 states are at least as plausible as the Laughlin wave function for the exact Coulomb ground state at filling ν = 7/3, suggesting that this physics is possibly relevant for the 7/3 FQHE. The Z_{n} order leads to several observable consequences, including quasiparticles with fractionally quantized charges of magnitude e/(3n) and the existence of multiple neutral collective modes. It is interesting that the FQHE may be a promising venue for the realization of exotic Z_{n} superconductivity.

- Authors:

- Publication Date:

- Research Org.:
- Pennsylvania State Univ., University Park, PA (United States)

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

- OSTI Identifier:
- 1605634

- Alternate Identifier(s):
- OSTI ID: 1606396

- Grant/Contract Number:
- SC0005042

- Resource Type:
- Published Article

- Journal Name:
- Physical Review Research

- Additional Journal Information:
- Journal Name: Physical Review Research Journal Volume: 2 Journal Issue: 1; Journal ID: ISSN 2643-1564

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; topological superconductivity; composite bosons; fractional quantum Hall effect

### Citation Formats

```
Balram, Ajit C., Jain, J. K., and Barkeshli, Maissam. Z n superconductivity of composite bosons and the 7 / 3 fractional quantum Hall effect. United States: N. p., 2020.
Web. doi:10.1103/PhysRevResearch.2.013349.
```

```
Balram, Ajit C., Jain, J. K., & Barkeshli, Maissam. Z n superconductivity of composite bosons and the 7 / 3 fractional quantum Hall effect. United States. doi:https://doi.org/10.1103/PhysRevResearch.2.013349
```

```
Balram, Ajit C., Jain, J. K., and Barkeshli, Maissam. Fri .
"Z n superconductivity of composite bosons and the 7 / 3 fractional quantum Hall effect". United States. doi:https://doi.org/10.1103/PhysRevResearch.2.013349.
```

```
@article{osti_1605634,
```

title = {Z n superconductivity of composite bosons and the 7 / 3 fractional quantum Hall effect},

author = {Balram, Ajit C. and Jain, J. K. and Barkeshli, Maissam},

abstractNote = {The topological p-wave pairing of composite fermions, believed to be responsible for the 5/2 frac- tional quantum Hall effect (FQHE), has generated much exciting physics. Motivated by the parton theory of the FQHE, we consider the possibility of a new kind of emergent “superconductivity” in the 1/3 FQHE, which involves condensation of clusters of n composite bosons. From a microscopic perspective, the state is described by the $n\bar{n}$111 parton wave function PLLLΦnΦ$*\atop{n}$Φ$3\atop{1}$, where Φn is the wave function of the integer quantum Hall state with n filled Landau levels and PLLL is the lowest- Landau-level projection operator. It represents a Zn superconductor of composite bosons, because the factor Φ$3\atop{1}$ ~ Πjj - zk)3, where zj = xj - iyj is the coordinate of the jth electron, binds three vortices to electrons to convert them into composite bosons, which then condense into the Zn 2 superconducting state |Φn|2. From a field theoretical perspective, this state can be understood by starting with the usual Laughlin theory and gauging a Zn subgroup of the U(1) charge conservation symmetry. We find from detailed quantitative calculations that the 2$\bar{2}$111 and 3$\bar{3}$111 states are at least as plausible as the Laughlin wave function for the exact Coulomb ground state at filling ν = 7/3, suggesting that this physics is possibly relevant for the 7/3 FQHE. The Zn order leads to several observable consequences, including quasiparticles with fractionally quantized charges of magnitude e/(3n) and the existence of multiple neutral collective modes. It is interesting that the FQHE may be a promising venue for the realization of exotic Zn superconductivity.},

doi = {10.1103/PhysRevResearch.2.013349},

journal = {Physical Review Research},

number = 1,

volume = 2,

place = {United States},

year = {2020},

month = {3}

}

DOI: https://doi.org/10.1103/PhysRevResearch.2.013349

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