# Ground-State Magnetization in Mixtures of a Few Ultra-Cold Fermions in One-Dimensional Traps

## Abstract

**:**

## 1. Introduction

## 2. The System Studied

## 3. The Method

## 4. Ground-State Magnetization

#### 4.1. Harmonic Confinement

#### 4.2. Deep Double-Well Confinement

#### 4.3. Intermediate Confinements

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Shape of the external potential ${V}_{\xi}\left(x\right)$ (

**left**) and the single-particle spectrum of the corresponding Hamiltonian ${H}_{\xi}$ (

**right**) as functions of the parameter $\xi $ controlling the barrier between the wells. For a sufficiently deep barrier, the single-particle spectrum becomes doubly quasi-degenerated.

**Figure 2.**Ground-state energies in subspaces of given magnetization $\mathcal{M}$ as functions of interaction strength g in harmonic confinement $\xi =0$. The energies $E(\mathcal{N},\mathcal{M})$ are always ordered according to increasing $\left|M\right|$. Therefore, the magnetization of the global ground-state for any external magnetic field B can be easily determined.

**Figure 3.**Phase diagram of the ground-state magnetization for a different number of particles N and different confinements $\xi $. For given values of a magnetic field and an inter-particle interaction, the magnetization $\mathcal{M}$ of the system is well determined (blue and red tints), and it instantly changes when on boundaries (black lines). In the limit of infinite repulsions, only the maximal magnetization can be achieved. Note that, for particular confinements (close to the deep double-well), not all magnetizations are accessible. See the main text for details.

**Figure 4.**Schematic view on a role of the external magnetic field in the magnetization of the ground-state when the system is confined in deep double-well confinement. When the magnetic field crosses some well defined critical value, appropriate particles on opposite wells simultaneously flip their spins to minimize single-particle energy. Therefore, in contrast to harmonic confinement, magnetization changes by 4 quanta. The mechanism described for $N=4$ and $N=6$ particles can be generalized to a larger number of particles in a straightforward manner.

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Sowiński, T.
Ground-State Magnetization in Mixtures of a Few Ultra-Cold Fermions in One-Dimensional Traps. *Condens. Matter* **2018**, *3*, 7.
https://doi.org/10.3390/condmat3010007

**AMA Style**

Sowiński T.
Ground-State Magnetization in Mixtures of a Few Ultra-Cold Fermions in One-Dimensional Traps. *Condensed Matter*. 2018; 3(1):7.
https://doi.org/10.3390/condmat3010007

**Chicago/Turabian Style**

Sowiński, Tomasz.
2018. "Ground-State Magnetization in Mixtures of a Few Ultra-Cold Fermions in One-Dimensional Traps" *Condensed Matter* 3, no. 1: 7.
https://doi.org/10.3390/condmat3010007