# 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. doi: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. doi: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. doi: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}

}

DOI: 10.1103/PhysRevD.100.073009

Works referenced in this record:

##
Coupling constant dependence in the thermodynamics of N = 4 supersymmetric Yang-Mills theory

journal, November 1998

- Gubser, Steven S.; Klebanov, Igor R.; Tseytlin, Arkady A.
- Nuclear Physics B, Vol. 534, Issue 1-2

##
Is the Composite Fermion a Dirac Particle?

journal, September 2015

- Son, Dam Thanh
- Physical Review X, Vol. 5, Issue 3

##
Perturbative results on QCD3 at finite temperature

journal, July 1982

- D'Hoker, Eric
- Nuclear Physics B, Vol. 201, Issue 3

##
Chiral-symmetry breaking in three-dimensional electrodynamics

journal, May 1984

- Pisarski, Robert D.
- Physical Review D, Vol. 29, Issue 10

##
Critical Behavior in (2+1)-Dimensional QED

journal, June 1988

- Appelquist, Thomas; Nash, Daniel; Wijewardhana, L. C. R.
- Physical Review Letters, Vol. 60, Issue 25

##
More Abelian dualities in 2 + 1 dimensions

journal, January 2017

- Karch, Andreas; Robinson, Brandon; Tong, David
- Journal of High Energy Physics, Vol. 2017, Issue 1

##
On C J and C T in conformal QED

journal, August 2016

- Giombi, Simone; Tarnopolsky, Grigory; Klebanov, Igor R.
- Journal of High Energy Physics, Vol. 2016, Issue 8

##
Shear Viscosity of Strongly Coupled $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4\mathrm{}$ Supersymmetric Yang-Mills Plasma

journal, August 2001

- Policastro, G.; Son, D. T.; Starinets, A. O.
- Physical Review Letters, Vol. 87, Issue 8

##
${\mathrm{QED}}_{3}$ theory of pairing pseudogap in cuprates: From $d$ -wave superconductor to antiferromagnet via an algebraic Fermi liquid

journal, August 2002

- Franz, M.; Tešanović, Z.; Vafek, O.
- Physical Review B, Vol. 66, Issue 5

##
No evidence for bilinear condensate in parity-invariant three-dimensional QED with massless fermions

journal, February 2016

- Karthik, Nikhil; Narayanan, Rajamani
- Physical Review D, Vol. 93, Issue 4

##
Massless three-dimensional QED with explicit fermions

journal, April 2003

- Lee, Dean; Maris, Pieter
- Physical Review D, Vol. 67, Issue 7

##
Simple non-perturbative resummation schemes beyond mean-field: case study for scalar ϕ4 theory in 1+1 dimensions

journal, March 2019

- Romatschke, Paul
- Journal of High Energy Physics, Vol. 2019, Issue 3

##
The large $N$ limit of superconformal field theories and supergravity

journal, January 1998

- Maldacena, Juan
- Advances in Theoretical and Mathematical Physics, Vol. 2, Issue 2

##
Nonperturbative beta function in three-dimensional electrodynamics

journal, July 2014

- Raviv, Ohad; Shamir, Yigal; Svetitsky, Benjamin
- Physical Review D, Vol. 90, Issue 1

##
Foam diagram summation at finite temperature

journal, August 1998

- Drummond, I. T.; Horgan, R. R.; Landshoff, P. V.
- Nuclear Physics B, Vol. 524, Issue 3

##
Effective coupling constant of plasmons

journal, September 2019

- Carrington, Margaret E.; Mrówczyński, Stanisław
- Physical Review D, Vol. 100, Issue 5

##
Infrared problem in the thermodynamics of the Yang-Mills gas

journal, November 1980

- Linde, A. D.
- Physics Letters B, Vol. 96, Issue 3-4

##
Quantum criticality of U(1) gauge theories with fermionic and bosonic matter in two spatial dimensions

journal, April 2008

- Kaul, Ribhu K.; Sachdev, Subir
- Physical Review B, Vol. 77, Issue 15

##
Advances in perturbative thermal field theory

journal, February 2004

- Kraemmer, U.; Rebhan, A.
- Reports on Progress in Physics, Vol. 67, Issue 3

##
Finite-Temperature Conformal Field Theory Results for All Couplings: $\mathrm{O}(N)$ Model in $2+1$ Dimensions

journal, June 2019

- Romatschke, Paul
- Physical Review Letters, Vol. 122, Issue 23

##
Supergravity and the large $N$ limit of theories with sixteen supercharges

journal, July 1998

- Itzhaki, Nissan; Maldacena, Juan M.; Sonnenschein, Jacob
- Physical Review D, Vol. 58, Issue 4

##
Particle-Vortex Duality from 3D Bosonization

journal, September 2016

- Karch, Andreas; Tong, David
- Physical Review X, Vol. 6, Issue 3

##
PHASE STRUCTURE OF COMPACT LATTICE QED _{3} WITH MASSLESS FERMIONS

journal, December 1993

- Azcoiti, Vicente; Luo, Xiang-Qian
- Modern Physics Letters A, Vol. 08, Issue 38

##
Non-compact QED3 with Nf⩾2

journal, November 2002

- Hands, S. J.; Kogut, J. B.; Strouthos, C. G.
- Nuclear Physics B, Vol. 645, Issue 1-2

##
Analytic transport from weak to strong coupling in the $O(N)$ model

journal, September 2019

- Romatschke, Paul
- Physical Review D, Vol. 100, Issue 5

##
Pressure of Hot QCD at Large *N* _{f}

journal, October 2002

- Moore, Guy D.
- Journal of High Energy Physics, Vol. 2002, Issue 10

##
Comment on and erratum to ``Pressure of hot QCD at large *N* _{ f } ''

journal, January 2003

- Ipp, Andreas; Moore, Guy D.; Rebhan, Anton
- Journal of High Energy Physics, Vol. 2003, Issue 01