# Liquid-gas phase transitions and $\mathcal{C}\mathcal{K}$ symmetry in quantum field theories

## Abstract

A general field-theoretic framework for the treatment of liquid-gas phase transitions is developed. Starting from a fundamental four-dimensional field theory at nonzero temperature and density, an effective three-dimensional field theory is derived. The effective field theory has a sign problem at finite density. Although finite density explicitly breaks charge conjugation C , there remains a symmetry under C K , where K is complex conjugation. Here, we consider four models: relativistic fermions, nonrelativistic fermions, static fermions and classical particles. The interactions are via an attractive potential due to scalar field exchange and a repulsive potential due to massive vector exchange. The field-theoretic representation of the partition function is closely related to the equivalence of the sine-Gordon field theory with a classical gas. The thermodynamic behavior is extracted from C K -symmetric complex saddle points of the effective field theory at tree level. In the cases of nonrelativistic fermions and classical particles, we find complex saddle point solutions but no first-order transitions, and neither model has a ground state at tree level. The relativistic and static fermions show a liquid-gas transition at tree level in the effective field theory. The liquid-gas transition, when it occurs, manifests as a first-order line atmore »

- Authors:

- Brookhaven National Lab. (BNL), Upton, NY (United States). RIKEN Research Center
- Washington Univ., St. Louis, MO (United States). Dept. of Physics

- Publication Date:

- Research Org.:
- Brookhaven National Lab. (BNL), Upton, NY (United States). RIKEN Research Center

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)

- OSTI Identifier:
- 1377048

- Report Number(s):
- BNL-114166-2017-JA

Journal ID: ISSN 2470-0010; PRVDAQ; R&D Project: PO-3

- Grant/Contract Number:
- SC00112704

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Volume: 95; Journal Issue: 7; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Riken BNL Research Center

### Citation Formats

```
Nishimura, Hiromichi, Ogilvie, Michael C., and Pangeni, Kamal.
```*Liquid-gas phase transitions and CK symmetry in quantum field theories*. United States: N. p., 2017.
Web. doi:10.1103/PhysRevD.95.076003.

```
Nishimura, Hiromichi, Ogilvie, Michael C., & Pangeni, Kamal.
```*Liquid-gas phase transitions and CK symmetry in quantum field theories*. United States. doi:10.1103/PhysRevD.95.076003.

```
Nishimura, Hiromichi, Ogilvie, Michael C., and Pangeni, Kamal. Tue .
"Liquid-gas phase transitions and CK symmetry in quantum field theories". United States.
doi:10.1103/PhysRevD.95.076003. https://www.osti.gov/servlets/purl/1377048.
```

```
@article{osti_1377048,
```

title = {Liquid-gas phase transitions and CK symmetry in quantum field theories},

author = {Nishimura, Hiromichi and Ogilvie, Michael C. and Pangeni, Kamal},

abstractNote = {A general field-theoretic framework for the treatment of liquid-gas phase transitions is developed. Starting from a fundamental four-dimensional field theory at nonzero temperature and density, an effective three-dimensional field theory is derived. The effective field theory has a sign problem at finite density. Although finite density explicitly breaks charge conjugation C , there remains a symmetry under C K , where K is complex conjugation. Here, we consider four models: relativistic fermions, nonrelativistic fermions, static fermions and classical particles. The interactions are via an attractive potential due to scalar field exchange and a repulsive potential due to massive vector exchange. The field-theoretic representation of the partition function is closely related to the equivalence of the sine-Gordon field theory with a classical gas. The thermodynamic behavior is extracted from C K -symmetric complex saddle points of the effective field theory at tree level. In the cases of nonrelativistic fermions and classical particles, we find complex saddle point solutions but no first-order transitions, and neither model has a ground state at tree level. The relativistic and static fermions show a liquid-gas transition at tree level in the effective field theory. The liquid-gas transition, when it occurs, manifests as a first-order line at low temperature and high density, terminated by a critical end point. The mass matrix controlling the behavior of correlation functions is obtained from fluctuations around the saddle points. Due to the C K symmetry of the models, the eigenvalues of the mass matrix are not always real but can be complex. This then leads to the existence of disorder lines, which mark the boundaries where the eigenvalues go from purely real to complex. The regions where the mass matrix eigenvalues are complex are associated with the critical line. In the case of static fermions, a powerful duality between particles and holes allows for the analytic determination of both the critical line and the disorder lines. Depending on the values of the parameters, either zero, one, or two disorder lines are found. Our numerical results for relativistic fermions give a very similar picture.},

doi = {10.1103/PhysRevD.95.076003},

journal = {Physical Review D},

number = 7,

volume = 95,

place = {United States},

year = {Tue Apr 04 00:00:00 EDT 2017},

month = {Tue Apr 04 00:00:00 EDT 2017}

}