# Thermal free energy of large ${N}_{f}$ QED in $2+1$ dimensions from weak to strong coupling

## Abstract

In 2+1 dimensions, QED becomes exactly solvable for all values of the fermion charge e in the limit of many fermions ${N}_{f}$ >> 1. We present results for the free energy density at finite temperature T to next-to-leading-order in large ${N}_{f}$. In the naive large ${N}_{f}$ limit, we uncover an apparently UV-divergent contribution to the vacuum energy at order O(e^{6}N$$^{3}_{f}$$/ε), which we argue would become a finite contribution of order O(e^{6}N$$^{4}_{f}$$) when resumming formally higher-order 1/ ${N}_{f}$ contributions. Still in the limit of large ${N}_{f}$, we find the finite-temperature free energy to be well-behaved for all values of the dimensionless coupling e2Nf/T, and to be bounded from above by the free energy of ${N}_{f}$ free fermions and bounded from below by non-interacting QED3. We invite follow-up studies from finite-temperature lattice gauge theory at large but fixed ${N}_{f}$ to test our results in the regime e^{2} ${N}_{f}$/T >> 1.

- Authors:

- Publication Date:

- Research Org.:
- Univ. of Colorado, Boulder, CO (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Nuclear Physics (NP)

- OSTI Identifier:
- 1571337

- Alternate Identifier(s):
- OSTI ID: 1593888

- Grant/Contract Number:
- SC0017905

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Name: Physical Review D Journal Volume: 100 Journal Issue: 7; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 73 NUCLEAR PHYSICS AND RADIATION PHYSICS; Conformal field theory; Finite temperature field theory; Large-N expansion in field theory; Lower-dimensional field theories; Quantum electrodynamics

### Citation Formats

```
Romatschke, Paul, and Säppi, Matias. Thermal free energy of large N f QED in 2 + 1 dimensions from weak to strong coupling. United States: N. p., 2019.
Web. https://doi.org/10.1103/PhysRevD.100.073009.
```

```
Romatschke, Paul, & Säppi, Matias. Thermal free energy of large N f QED in 2 + 1 dimensions from weak to strong coupling. United States. https://doi.org/10.1103/PhysRevD.100.073009
```

```
Romatschke, Paul, and Säppi, Matias. Tue .
"Thermal free energy of large N f QED in 2 + 1 dimensions from weak to strong coupling". United States. https://doi.org/10.1103/PhysRevD.100.073009.
```

```
@article{osti_1571337,
```

title = {Thermal free energy of large N f QED in 2 + 1 dimensions from weak to strong coupling},

author = {Romatschke, Paul and Säppi, Matias},

abstractNote = {In 2+1 dimensions, QED becomes exactly solvable for all values of the fermion charge e in the limit of many fermions Nf >> 1. We present results for the free energy density at finite temperature T to next-to-leading-order in large Nf. In the naive large Nf limit, we uncover an apparently UV-divergent contribution to the vacuum energy at order O(e6N$^{3}_{f}$/ε), which we argue would become a finite contribution of order O(e6N$^{4}_{f}$) when resumming formally higher-order 1/Nf contributions. Still in the limit of large Nf, we find the finite-temperature free energy to be well-behaved for all values of the dimensionless coupling e2Nf/T, and to be bounded from above by the free energy of Nf free fermions and bounded from below by non-interacting QED3. We invite follow-up studies from finite-temperature lattice gauge theory at large but fixed Nf to test our results in the regime e2Nf/T >> 1.},

doi = {10.1103/PhysRevD.100.073009},

journal = {Physical Review D},

number = 7,

volume = 100,

place = {United States},

year = {2019},

month = {10}

}

https://doi.org/10.1103/PhysRevD.100.073009

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