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Chinese Physics Letters

Ground State Energy of Unitary Fermion Gas with the Thomson Problem Approach

陈继胜

Published Jan 1, 2007• 1 citation
OpenAlex

Abstract

尺寸更少通用 coellicient x i 与 T = 在非相对论的限制为理想的非交往的定义单一的费米子精力密度的比率到那 0。古典 Thomson 问题作为一只 nonperturbative 量许多身体手臂被拿包括低精力探讨地面状态精力非线性的量 fluctuation/correlationeffects。与相对论的迪拉克一起，连续统保持了理论形式主义，为在两有限温度的强烈交往的限制费米子功能的精力密度的简明表示和密度被获得。经分解，通用因素被计算是 x i = 4/9。精力差距是三角洲 =(5 )(18 ) k (f)(2 )/(2m ) 。

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尺寸更少通用 coellicient x i 与 T = 在非相对论的限制为理想的非交往的定义单一的费米子精力密度的比率到那 0。古典 Thomson 问题作为一只 nonperturbative 量许多身体手臂被拿包括低精力探讨地面状态精力非线性的量 fluctuation/correlationeffects。与相对论的迪拉克一起，连续统保持了理论形式主义，为在两有限温度的强烈交往的限制费米子功能的精力密度的简明表示和密度被获得。经分解，通用因素被计算是 x i = 4/9。精力差距是三角洲 =(5 )(18 ) k (f)(2 )/(2m ) 。
The dimensionless universal coefficient ξ defines the ratio of the unitary fermions energy density to that for the ideal non-interacting ones in the non-relativistic limit with T = 0. The classical Thomson problem is taken as a nonperturbative quantum many-body arm to address the ground state energy including the low energy nonlinear quantum fluctuation/correlation effects. With the relativistic Dirac continuum field theory formalism, the concise expression for the energy density functional of the strongly interacting limit fermions at both finite temperature and density is obtained. Analytically, the universal factor is calculated to be ξ = 4/9. The energy gap is Δ = (5/18) k f 2 /(2 m ).
We update the ground-state energy ratio of unitary Fermi gas to noninteracting Fermi gas $(\ensuremath{\xi})$ from the ϵϵϵ expansion by including the next-to-next-to-leading-order (NNLO) term near two spatial dimensions. Interpolations of the NNLO ϵϵϵ expansions around four and two spatial dimensions with the use of Pad\'e approximants give $\ensuremath{\xi}\ensuremath{\approx}0.360\ifmmode\pm\else\textpm\fi{}0.020$ in three dimensions with the uncertainty due to different interpolation functions. This value is consistent with the previous interpolations of the NLO ϵϵϵ expansions $\ensuremath{\xi}\ensuremath{\approx}0.377\ifmmode\pm\else\textpm\fi{}0.014$ in spite of the large NNLO corrections.
Recent developments in the physics of low density trapped gases make it worthwhile to verify old, well known results that, while plausible, were based on perturbation theory and assumptions about pseudopotentials. We use and extend recently developed techniques to give a rigorous derivation of the asymptotic formula for the ground state energy of a dilute gas of NNN fermions interacting with a short-range, positive potential of scattering length aaa. For spin 1/2 fermions, this is E∼E0+(ℏ2/2m)2πNρaE \sim E^0 + (\hbar^2/2m) 2 πN ρaE∼E0+(ℏ2/2m)2πNρa, where E0E^0E0 is the energy of the non-interacting system and ρρρ is the density.
Recent developments in the physics of low-density trapped gases make it worthwhile to verify old, well-known results that, while plausible, were based on perturbation theory and assumptions about pseudopotentials. We use and extend recently developed techniques to give a rigorous derivation of the asymptotic formula for the ground-state energy of a dilute gas of NNN fermions interacting with a short-range, positive potential of scattering length aaa. For spin-1∕21∕21∕2 fermions, this is $E\ensuremath{\sim}{E}^{0}+({\ensuremath{\hbar}}^{2}∕2m)2\ensuremath{\pi}N\ensuremath{\varrho}a$, where E0{E}^{0}E0 is the energy of the noninteracting system and $\ensuremath{\varrho}$ is the density. A similar formula holds in two dimensions (2D), with $\ensuremath{\varrho}a$ replaced by $\ensuremath{\varrho}∕\ensuremath{\mid}\mathrm{ln}(\ensuremath{\varrho}{a}^{2})\ensuremath{\mid}$. Obviously this 2D energy is not the expectation value of a density-independent pseudopotential.
Foldy, and later Girardeau, calculated the ground-state energy of a charged Bose gas at high density. We rederive the common first term obtained by these authors by using a nonperturbation method developed previously. Our aims are: (i) to establish the validity of this common result, which has not been proved; (ii) to establish the validity and usefulness of our nonperturbation method. We also show that our method will give the correct functional dependence of the ground-state energy on the density at low density, although the exact coefficient must await a numerical computation.