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arXiv:cond-mat/9712138 · Physical Review Letters

Ground State Energy of the Low Density Bose Gas

Élliott H. Lieb, Jakob Yngvason

Published Mar 23, 1998• 181 citations
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Abstract

Now that the properties of low temperature Bose gases at low density, $\ensuremath{\rho}$, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors four to five decades ago. One of these is that the leading term in the energy/particle is $2\ensuremath{\pi}{\ensuremath{\Elzxh}}^{2}\ensuremath{\rho}a/m$, where $a$ is the scattering length. Owing to the delicate and peculiar nature of bosonic correlations, four decades of research have failed to establish this plausible formula rigorously. The only known lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct bound is proved here.

DOI
10.1103/physrevlett.80.2504

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Now that the properties of low temperature Bose gases at low density, \ensuremathρ\ensuremath{\rho}\ensuremathρ, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors four to five decades ago. One of these is that the leading term in the energy/particle is 2\ensuremathπ\ensuremath\Elzxh2\ensuremathρa/m2\ensuremath{\pi}{\ensuremath{\Elzxh}}^{2}\ensuremath{\rho}a/m2\ensuremathπ\ensuremath\Elzxh2\ensuremathρa/m, where aaa is the scattering length. Owing to the delicate and peculiar nature of bosonic correlations, four decades of research have failed to establish this plausible formula rigorously. The only known lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct bound is proved here.
Now that the low temperature properties of quantum-mechanical many-body systems (bosons) at low density, ρρρ, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4-5 decades ago. For systems with repulsive (i.e. positive) interaction potentials the experimental low temperature state and the ground state are effectively synonymous -- and this fact is used in all modeling. In such cases, the leading term in the energy/particle is 2πℏ2aρ/m2π\hbar^2 a ρ/m2πℏ2aρ/m where aaa is the scattering length of the two-body potential. Owing to the delicate and peculiar nature of bosonic correlations (such as the strange N7/5N^{7/5}N7/5 law for charged bosons), four decades of research failed to establish this plausible formula rigorously. The only previous lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct asymptotic formula has recently been obtained by us and this work will be presented. The reason behind the mathematical difficulties will be emphasized. A different formula, postulated as late as 1971 by Schick, holds in two-dimensions and this, too, will be shown to be correct. With the aid of the methodology developed to prove the lower bound for the homogeneous gas, two other problems have been successfully addressed. One is the proof by us that the Gross-Pitaevskii equation correctly describes the ground state in the `traps' actually used in the experiments. For this system it is also possible to prove complete Bose condensation, as we have shown. Another topic is a proof that Foldy's 1961 theory of a high density Bose gas of charged particles correctly describes its ground state energy.
According to a formula that was put forward many decades ago the ground state energy per particle of an interacting, dilute Bose gas at density ρρρ is 2πℏ2ρa/m2π\hbar^2ρa/m2πℏ2ρa/m to leading order in ρa3≪1ρa^3\ll 1ρa3≪1, where aaa is the scattering length of the interaction potential and mmm the particle mass. This result, which is important for the theoretical description of current experiments on Bose-Einstein condensation, has recently been established rigorously for the first time. We give here an account of the proof that applies to nonnegative, spherically symmetric potentials decreasing faster than 1/r31/r^31/r3 at infinity.
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.
For a dilute system of non-relativistic bosons interacting through a positive, compactly supported, L1L^1L1-potential vvv with scattering length aaa we prove that the ground state energy density satisfies the bound e(ρ)≥4πaρ2(1+12815πρa3+o(ρa3))e(\rho) \ge 4\pi a \rho^2(1 + \frac{128}{15\sqrt{\pi}} \sqrt{\rho a^3} + o(\sqrt{\rho a^3}))e(ρ)≥4πaρ2(1+15π​128​ρa3​+o(ρa3​)), thereby proving the Lee-Huang-Yang formula for the energy 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 1/2 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.