# The Inverse-Square Interaction Phase Diagram: Unitarity in the Bosonic Ground State

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## Abstract

**:**

## 1. Introduction

## 2. Homogeneous Calogero–Sutherland Model

## 3. Bogoliubov Theory

## 4. Quantum Monte Carlo Approach

#### 4.1. Long-Wavelength Part of Wave Function

#### 4.2. Short-Range Part of Wave Function

#### 4.3. Guiding Wave Function

## 5. Jellium Model

#### 5.1. Mean-Field Contribution

#### 5.2. Direct Summation

#### 5.3. Smooth Version of the Long-Range Potential

## 6. Classical Limit

#### 6.1. Equilibrium Energy

#### 6.2. Harmonic Approximation

#### 6.3. Excitation Spectrum

## 7. Numerical Results

#### 7.1. Thermodynamic Properties

#### 7.2. Quantum Phase Transition

#### 7.3. Coherence and Structural Properties

#### 7.4. Excitation Spectrum and Plasmons

#### 7.5. Critical Parameters

#### 7.6. Universal Scaling Properties

## 8. Considerations for Experimental Realization

## 9. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DMC | diffusion Monte Carlo |

HA | Harmonic approximation |

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**Figure 1.**Bertsch parameter $\xi \left(Q\right)$ extracted from the ground-state energy E scaled by $NQ$ as a function of Q on a semi-logarithmic scale. Symbols are Monte Carlo results, solids lines are fits; dashed lines are asymptotic expansions. Including, solid circles, total energy in the gas phase; open circles, total energy in the solid phase; diamonds, potential energy; squares, kinetic energy; dashed lines, Bogoliubov theory prediction valid for small Q for the total (20), kinetic (54), and potential (53) energies; dash-dotted line, large Q limit of a classical fcc crystal (40) solid lines, fits given by Equation (46) for the total energy, Equation (51) for the potential energy and Equation (52) for the kinetic energy. The arrow shows the position of the gas-solid phase transition ${Q}^{\ast}$. The data points represent the energy of a gas for $Q<{Q}^{\ast}$ and of a solid otherwise. Note that the jellium contribution is subtracted from the total and potential energies resulting in a negative energy even if the interaction is entirely repulsive.

**Figure 2.**The difference of the energy per particle between the gas and solid phases as a function of Q. Symbols report Monte Carlo data, line shows the linear fit. The point where the difference is equal to zero corresponds to the location of the phase transition.

**Figure 3.**Condensate fraction as a function of Q. Symbols report Monte Carlo data; dashed line shows the Bogoliubov theory prediction given by Equation (19).

**Figure 4.**Pair distribution function in the gas phase for different values of $Q=0.1;1;10;100$ (from top to bottom). Solid lines, Monte Carlo results; dashed lines, short-range expansion ${\left|r\right|}^{2\lambda}$ with the coefficient of proportionality fixed by a fitting procedure (see Equation (30)). Note that the point with $Q=100$ corresponds to a metastable state, as the ground state in this regime is a solid.

**Figure 5.**Static structure factor in the gas phase for different values of $Q=0.1;1;10;100$ (from top to bottom). Symbols, DMC data; dashed lines, ${k}^{3/2}$ low momentum behavior with the coefficient of proportionality taken from Bogoliubov theory, Equation (47).

**Figure 6.**Upper bound for the excitation spectrum $E\left(k\right)$ on a double logarithmic scale as provided by Feynman relation (48) by using the data presented in Figure 5. Symbols, Monte Carlo results; solid thick lines, ${k}^{1/2}$ plasmon relation; dashed line, Bogoliubov excitation spectrum (18); thin solid line, free particle ${\hslash}^{2}{k}^{2}/2m$ energy. Units of ${\hslash}^{2}{n}^{2/3}/m$ are used for the energy and ${n}^{1/3}$ for the momentum.

**Table 1.**Literature overview for the typical values of the Lindemann parameter $\gamma $ (solid phase), height of the peak of the static structure factor ${S}_{peak}\left(k\right)$ and condensate fraction ${N}_{0}/N$ (gas phase) at the critical point of zero-temperature gas(liquid)-solid phase transition.

Reference | Authors | $\mathit{\gamma}$ | ${\mathit{S}}_{\mathbf{peak}}$ (k) | ${\mathit{N}}_{0}/\mathit{N}$ | Dim. | Interaction | Method |
---|---|---|---|---|---|---|---|

— | present work | 0.24(1) | 1.63(5) | 0.008(2) | 3D | $1/{r}^{2}$ | DMC |

[77] | C. Cazorla et al. | 0.26(1) | — | — | 3D | ${}^{4}$He | DMC |

[78] | S. A. Vitiello et al. | 0.23(1) | 1.55 | — | 3D | ${}^{4}$He | GFMC |

[79] | S. A. Vitiello et al. | 0.23(1) | — | — | 3D | ${}^{4}$He | VMC |

[74] | D. Ceperley et al. | 0.28(2) | — | — | 3D | Yukawa | DMC |

[80] | A. R. Denton et al. | 0.27(1) | — | — | 3D | hard core | DFT |

[81] | M. D. Jones and D. M. Ceperley | 0.27(1) | — | — | 3D | electrons | PIMC |

[82] | D. Ceperley | 0.29(1) | — | — | 2D | electrons | VMC |

[83] | P. A. Whitlock et al. | 0.254(2) | 1.70(2) | — | 2D | ${}^{4}$He | GFMC |

[84] | L. Xing | 0.279(1) | 1.54(2) | — | 2D | hard core | DMC |

[85] | W. R. Magro and D. M. Ceperley | 0.245(15) | — | — | 2D | Yukawa | DMC |

[86] | W. R. Magro and D. M. Ceperley | 0.24(1) | — | — | 2D | 1/r charges | DMC |

[87] | G. E. Astrakharchik et al. | 0.230(6) | 1.70(3) | 0.014(2) | 2D | dipoles | DMC |

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## Share and Cite

**MDPI and ACS Style**

Astrakharchik, G.E.; Kryuchkov, P.S.; Kurbakov, I.L.; Lozovik, Y.E.
The Inverse-Square Interaction Phase Diagram: Unitarity in the Bosonic Ground State. *Crystals* **2018**, *8*, 246.
https://doi.org/10.3390/cryst8060246

**AMA Style**

Astrakharchik GE, Kryuchkov PS, Kurbakov IL, Lozovik YE.
The Inverse-Square Interaction Phase Diagram: Unitarity in the Bosonic Ground State. *Crystals*. 2018; 8(6):246.
https://doi.org/10.3390/cryst8060246

**Chicago/Turabian Style**

Astrakharchik, Grigori E., P. S. Kryuchkov, I. L. Kurbakov, and Yu. E. Lozovik.
2018. "The Inverse-Square Interaction Phase Diagram: Unitarity in the Bosonic Ground State" *Crystals* 8, no. 6: 246.
https://doi.org/10.3390/cryst8060246