# Tan relations in one dimension

## Abstract

We derive exact relations that connect the universal C/k{sup 4}-decay of the momentum distribution at large k with both thermodynamic properties and correlation functions of two-component Fermi gases in one dimension with contact interactions. The relations are analogous to those obtained by Tan in the three-dimensional case and are derived from an operator product expansion of the one- and two-particle density matrix. They extend earlier results by Olshanii and Dunjko (2003) for the bosonic Lieb-Liniger gas. As an application, we calculate the pair distribution function at short distances and the dimensionless contact in the limit of infinite repulsion. The ground state energy approaches a universal constant in this limit, a behavior that also holds in the three-dimensional case. In both one and three dimensions, a Stoner instability to a saturated ferromagnet for repulsive fermions with zero range interactions is ruled out at any finite coupling. - Highlights: > We derive universal relations for the two-component, contact-interacting 1D Fermi gas. > These relations connect the tail of the momentum distribution to thermodynamic properties. > There is no saturated ferromagnetism at finite, repulsive couplings for the 3D model.

- Authors:

- Publication Date:

- OSTI Identifier:
- 21583283

- Resource Type:
- Journal Article

- Journal Name:
- Annals of Physics (New York)

- Additional Journal Information:
- Journal Volume: 326; Journal Issue: 10; Other Information: DOI: 10.1016/j.aop.2011.05.010; PII: S0003-4916(11)00108-4; Copyright (c) 2011 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Journal ID: ISSN 0003-4916

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; CORRELATION FUNCTIONS; COUPLING; DECAY; DENSITY MATRIX; DISTRIBUTION; DISTRIBUTION FUNCTIONS; FERMI GAS; FERMIONS; FERROMAGNETISM; GROUND STATES; INSTABILITY; INTERACTIONS; OPERATOR PRODUCT EXPANSION; THERMODYNAMIC PROPERTIES; THREE-DIMENSIONAL CALCULATIONS; ENERGY LEVELS; FUNCTIONS; MAGNETISM; MATRICES; PHYSICAL PROPERTIES; SERIES EXPANSION

### Citation Formats

```
Barth, Marcus, and Zwerger, Wilhelm.
```*Tan relations in one dimension*. United States: N. p., 2011.
Web. doi:10.1016/j.aop.2011.05.010.

```
Barth, Marcus, & Zwerger, Wilhelm.
```*Tan relations in one dimension*. United States. https://doi.org/10.1016/j.aop.2011.05.010

```
Barth, Marcus, and Zwerger, Wilhelm. Sat .
"Tan relations in one dimension". United States. https://doi.org/10.1016/j.aop.2011.05.010.
```

```
@article{osti_21583283,
```

title = {Tan relations in one dimension},

author = {Barth, Marcus and Zwerger, Wilhelm},

abstractNote = {We derive exact relations that connect the universal C/k{sup 4}-decay of the momentum distribution at large k with both thermodynamic properties and correlation functions of two-component Fermi gases in one dimension with contact interactions. The relations are analogous to those obtained by Tan in the three-dimensional case and are derived from an operator product expansion of the one- and two-particle density matrix. They extend earlier results by Olshanii and Dunjko (2003) for the bosonic Lieb-Liniger gas. As an application, we calculate the pair distribution function at short distances and the dimensionless contact in the limit of infinite repulsion. The ground state energy approaches a universal constant in this limit, a behavior that also holds in the three-dimensional case. In both one and three dimensions, a Stoner instability to a saturated ferromagnet for repulsive fermions with zero range interactions is ruled out at any finite coupling. - Highlights: > We derive universal relations for the two-component, contact-interacting 1D Fermi gas. > These relations connect the tail of the momentum distribution to thermodynamic properties. > There is no saturated ferromagnetism at finite, repulsive couplings for the 3D model.},

doi = {10.1016/j.aop.2011.05.010},

url = {https://www.osti.gov/biblio/21583283},
journal = {Annals of Physics (New York)},

issn = {0003-4916},

number = 10,

volume = 326,

place = {United States},

year = {2011},

month = {10}

}