# Static and Dynamic Properties of a Few Spin 1/2 Interacting Fermions Trapped in a Harmonic Potential

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Approach

#### 2.1. The Hamiltonian of the System

#### 2.2. Non-Interacting and Infinite Interaction Limits

#### 2.2.1. Non-Interacting Case: Fermi Gas

#### 2.2.2. Infinite Interaction Case

#### 2.3. Second Quantization

#### 2.3.1. Creation and Annihilation Operators

#### 2.3.2. Fock Space

#### 2.3.3. Operators in Second Quantization

#### 2.3.4. The Hamiltonian in Second Quantization

## 3. Numerical Methods

#### 3.1. Direct Diagonalization

#### Basis Truncation

#### 3.2. The Two-Body Matrix Elements of the Interaction

#### 3.3. A Benchmark for the Two-Particle Case

#### 3.3.1. Theoretical Spectrum for Two Particles

#### 3.3.2. Comparison of Analytical and Numerical Results

## 4. Ground State Properties

#### 4.1. Energy and Virial Theorem

#### 4.2. One Body Density Matrix

#### 4.2.1. Density Profile

#### 4.2.2. Natural Orbits

## 5. Low-Energy Excited States

#### 5.1. Energy Spectrum

#### 5.2. Spin Determination

## 6. Dynamical Excitation

#### 6.1. Sudden Change in the Trap Frequency: Breathing Mode

#### 6.1.1. Dynamic Structure Function

#### 6.1.2. Sum Rules

#### 6.2. Interaction Quench

#### 6.2.1. Time Evolution of the Perturbed System

#### 6.2.2. Central Density Oscillations

## 7. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Derivation of the Virial Theorem

## Appendix B. Evaluation of the One-Body Matrix Elements

- (a)
- The m-th derivative of a Hermite polynomial,$${H}_{n}^{\left(m\right)}={2}^{m}\frac{m!}{(n-m)!}{H}_{n-m}\left(x\right)\phantom{\rule{0.166667em}{0ex}},$$
- (b)
- The recurrence relation,$${H}_{n+1}\left(x\right)=2x{H}_{n}\left(x\right)-2n{H}_{n-1}\left(x\right)\phantom{\rule{0.166667em}{0ex}},$$
- (c)
- The orthogonality of the Hermite polynomials$${\int}_{-\infty}^{\infty}{H}_{m}\left(x\right){H}_{n}\left(x\right){e}^{-{x}^{2}}={\delta}_{m,n}{2}^{n}n!\sqrt{\pi}\phantom{\rule{0.166667em}{0ex}},$$
- (d)
- The n-th power of x expressed in terms of Hermite polynomials$${x}^{n}=\frac{n!}{{2}^{n}}\sum _{m=0}^{n/2}\frac{1}{m!(n-2m)!}{H}_{n-2m}\phantom{\rule{0.166667em}{0ex}},$$
- (e)
- The product of two Hermite polynomials as a function of the sum of Hermite polynomials$${H}_{m}\left(x\right){H}_{n}\left(x\right)={2}^{n}n!\sum _{r=0}^{n}\frac{m!}{(n-r)!(m-n+r)!}\frac{{H}_{m-n+2r}\left(x\right)}{{2}^{r}r!},n\le m\phantom{\rule{0.166667em}{0ex}}.$$

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**Figure 1.**Energy spectrum of the two-particle system as a function of the interaction strength. The calculations are performed using 100 single-particle modes (green dots). The analytical spectrum (dashed red line) is obtained with Equation (43) for the energy of the relative system and adding after the possible energies of the center of mass. Energy given in harmonic oscillator units. The energies obtained via Equation (43) correspond to even-parity states, the corresponding odd-parity states do not depend on the interaction, and result in horizontal lines in close agreement with the numerical ones.

**Figure 2.**Differences between the numerical results and the exact ones as a function of the number of single particle modes, for two values of the interaction strength, $g=1$ and $g=5$, for the ground (g.s) and first excited (ex.) state. The lines are fits of the type ${C}_{1}/{N}_{M}^{1/2}+{C}_{2}/{N}_{M}$ to the calculated points. Both axes are in a logarithmic scale.

**Figure 3.**Ground-state energy as a function of the interaction strength for different number of particles and spin configurations. The calculations have been performed by using 100, 50, 40 and 30 single-particle modes to construct the many-body basis for N = 2,3,4 and 5 particles respectively. The black squares are the corresponding results reported in Ref. [54].

**Figure 4.**Different energy contributions to the total ground-state energy, as a function of the interaction strength for different number of particles and total third-spin component. The total energy and the fulfilment of the virial theorem is also shown (Equation (47)). The legend is common to all graphics.

**Figure 5.**Density profiles of the ground state for different number of particles and spin projections. The density profiles are shown for various values of the interaction strength. The legend is common to all panels.

**Figure 6.**Densities of the spin up (${\rho}_{\uparrow}$, left panels) and spin down (${\rho}_{\downarrow}$, right panels) particles in the ground state. The density profiles are shown for various values of the interaction strength and for different number of particles and spin projection. The legend is common to all panels. The corresponding total densities are reported in Figure 5.

**Figure 7.**The largest 10 eigenvalues of the one-body density matrix (OBDM) of the ground state of systems with different number of particles and spin configurations as a function of the interaction strength.

**Figure 8.**Low-energy spectrum for several number of particles and spin configurations as a function of the interaction strength. Notice that the excitation energies are measured with respect to the ground-state energy of the non-interacting system.

**Figure 9.**Low-energy spectrum of the fermionic system with four particles as a function of the interaction strength. The states are labelled according to their total spin, as explained in the text.

**Figure 10.**Dynamic structure function of a mono-polar excitation for the case of two fermions. The total spin of the ground state and of all excited states connected to it through the mono-polar excitation, are zero. The different panels correspond to four different values of interaction strength. From the non-interacting case, $g=0$ to a strongly interacting one $g=15$.

**Figure 11.**Dynamic structure function of a mono-polar excitation for the case of four particles. The total spin of the ground state and of all excited states connected to it through the mono-polar excitation, are zero. The different panels correspond to four different values of interaction strength. From the non-interacting case, $g=0$ to a strongly interacting one $g=15$.

**Figure 12.**Values of the three energy momenta ${M}_{-1}$, ${M}_{1}$ and ${M}_{3}$ for the cases of two and four particles as a function of the interaction strength. The calculations done using the explicit value of the dynamic structure function, Equation (57) are represented by dots and the line is the value computed using the sum rules, Equation (59). For ${M}_{-1}$ the results from Equations (59) and (60) are indistinguishable on the figure.

**Figure 13.**Ratios of the sum rules $\sqrt{{M}_{1}/{M}_{-1}}$ and $\sqrt{{M}_{3}/{M}_{1}}$ as a function of the interaction strength for several number of particles and spin configurations. The purple + signs correspond to the excitation energy of the main peak of the dynamic structure function.

**Figure 14.**Projection of the initial ground state for $g=0$ into the eigenstates of $g=1$ and $g=5$ as a function of its energy, for a two particles systems with zero total spin. The y axis is in logarithmic scale.

**Figure 15.**Frequency analysis of the time dependence of the central density after an interaction quench from $g=0$ to $g=1$. The average density has been subtracted. We consider the cases $N=2$ (solid lines) and $N=4$ (dot-dashed lines). In the inset we depict the first oscillations of the signal used to compute the Fourier analysis shown in the main panels. For the Fourier analysis we used a time interval, t = 500. All magnitudes in the figure are in harmonic oscillator units.

**Table 1.**Number of single-particle states used in the construction of the many-body basis states in the second column. The number of many-body basis states, with and without energy restriction, are shown in the third and fourth columns, respectively.

Number of Single-Particle States | Number of Many-Body Basis States | ||
---|---|---|---|

with Energy Restriction | without Restriction | ||

2 particles, M = 0 | 100 | 5050 | 10,000 |

3 particles, M = 1/2 | 50 | 10,725 | 61,250 |

4 particles, M = 0 | 40 | 30,800 | 608,400 |

4 particles, M = 1 | 40 | 19,530 | 395,200 |

5 particles, M = 1/2 | 30 | 22,923 | 1,766,100 |

5 particles, M = 3/2 | 30 | 11,349 | 822,150 |

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**MDPI and ACS Style**

Rojo-Francàs, A.; Polls, A.; Juliá-Díaz, B.
Static and Dynamic Properties of a Few Spin 1/2 Interacting Fermions Trapped in a Harmonic Potential. *Mathematics* **2020**, *8*, 1196.
https://doi.org/10.3390/math8071196

**AMA Style**

Rojo-Francàs A, Polls A, Juliá-Díaz B.
Static and Dynamic Properties of a Few Spin 1/2 Interacting Fermions Trapped in a Harmonic Potential. *Mathematics*. 2020; 8(7):1196.
https://doi.org/10.3390/math8071196

**Chicago/Turabian Style**

Rojo-Francàs, Abel, Artur Polls, and Bruno Juliá-Díaz.
2020. "Static and Dynamic Properties of a Few Spin 1/2 Interacting Fermions Trapped in a Harmonic Potential" *Mathematics* 8, no. 7: 1196.
https://doi.org/10.3390/math8071196