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

## Abstract

A famous example of gauge/gravity duality is the result that the entropy density of strongly coupled $${\cal N}=4$$ SYM in four dimensions for large N is exactly 3/4 of the Stefan-Boltzmann limit. In this work, I revisit the massless O(N) model in 2+1 dimensions, which is analytically solvable at finite temperature $T$ for all couplings $$\lambda$$ in the large N limit. I find that the entropy density monotonically decreases from the Stefan-Boltzmann limit at $$\lambda=0$$ to exactly 4/5 of the Stefan-Boltzmann limit at $$\lambda=\infty$$. Calculating the retarded energy-momentum tensor correlator in the scalar channel at $$\lambda=\infty$$, I find that it has two logarithmic branch cuts originating at $$\omega=\pm 4 T \ln \frac{1+\sqrt{5}}{2}$$, but no singularities in the whole complex frequency plane. I show that the ratio 4/5 and the location of the branch points both are universal within a large class of bosonic CFTs in 2+1 dimensions.

- Authors:

- Publication Date:

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

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

- OSTI Identifier:
- 1526328

- Alternate Identifier(s):
- OSTI ID: 1593875

- Grant/Contract Number:
- SC0017905

- Resource Type:
- Published Article

- Journal Name:
- Physical Review Letters

- Additional Journal Information:
- Journal Name: Physical Review Letters Journal Volume: 122 Journal Issue: 23; Journal ID: ISSN 0031-9007

- Publisher:
- American Physical Society

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS

### Citation Formats

```
Romatschke, Paul. Finite-Temperature Conformal Field Theory Results for All Couplings: O ( N ) Model in 2 + 1 Dimensions. United States: N. p., 2019.
Web. https://doi.org/10.1103/PhysRevLett.122.231603.
```

```
Romatschke, Paul. Finite-Temperature Conformal Field Theory Results for All Couplings: O ( N ) Model in 2 + 1 Dimensions. United States. https://doi.org/10.1103/PhysRevLett.122.231603
```

```
Romatschke, Paul. Fri .
"Finite-Temperature Conformal Field Theory Results for All Couplings: O ( N ) Model in 2 + 1 Dimensions". United States. https://doi.org/10.1103/PhysRevLett.122.231603.
```

```
@article{osti_1526328,
```

title = {Finite-Temperature Conformal Field Theory Results for All Couplings: O ( N ) Model in 2 + 1 Dimensions},

author = {Romatschke, Paul},

abstractNote = {A famous example of gauge/gravity duality is the result that the entropy density of strongly coupled ${\cal N}=4$ SYM in four dimensions for large N is exactly 3/4 of the Stefan-Boltzmann limit. In this work, I revisit the massless O(N) model in 2+1 dimensions, which is analytically solvable at finite temperature $T$ for all couplings $\lambda$ in the large N limit. I find that the entropy density monotonically decreases from the Stefan-Boltzmann limit at $\lambda=0$ to exactly 4/5 of the Stefan-Boltzmann limit at $\lambda=\infty$. Calculating the retarded energy-momentum tensor correlator in the scalar channel at $\lambda=\infty$, I find that it has two logarithmic branch cuts originating at $\omega=\pm 4 T \ln \frac{1+\sqrt{5}}{2}$, but no singularities in the whole complex frequency plane. I show that the ratio 4/5 and the location of the branch points both are universal within a large class of bosonic CFTs in 2+1 dimensions.},

doi = {10.1103/PhysRevLett.122.231603},

journal = {Physical Review Letters},

number = 23,

volume = 122,

place = {United States},

year = {2019},

month = {6}

}

https://doi.org/10.1103/PhysRevLett.122.231603

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Works referencing / citing this record:

##
Large charge at large N

journal, December 2019

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- Journal of High Energy Physics, Vol. 2019, Issue 12

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

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