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GROUND STATE ENERGY OF THE TWO-COMPONENT CHARGED BOSE GAS

Published Jan 1, 2003• 0 citations
OpenAlex

Abstract

Dedicated to Freeman J. Dyson on the occasion of his 80th birthday We continue the study of the two-component charged Bose gas initiated by Dyson in 1967. He showed that the ground state energy for N particles is at least as negative as −CN 7/5 for large N and this power law was verified by a lower bound found by Conlon, Lieb and Yau in 1988. Dyson conjectured that the exact constant C was given by a mean-field minimization problem that used, as input, Foldy’s calculation (using Bogolubov’s 1947 formalism) for the onecomponent gas. Earlier we showed that Foldy’s calculation is exact insofar as a lower bound of his form was obtained. In this paper we do the same thing for Dyson’s conjecture. The two-component case is considerably more difficult because the gas is very non-homogeneous in its ground state. 1

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Dedicated to Freeman J. Dyson on the occasion of his 80th birthday We continue the study of the two-component charged Bose gas initiated by Dyson in 1967. He showed that the ground state energy for N particles is at least as negative as −CN 7/5 for large N and this power law was verified by a lower bound found by Conlon, Lieb and Yau in 1988. Dyson conjectured that the exact constant C was given by a mean-field minimization problem that used, as input, Foldy’s calculation (using Bogolubov’s 1947 formalism) for the onecomponent gas. Earlier we showed that Foldy’s calculation is exact insofar as a lower bound of his form was obtained. In this paper we do the same thing for Dyson’s conjecture. The two-component case is considerably more difficult because the gas is very non-homogeneous in its ground state. 1
The upper bound on the ground state energy of the two-component charged Bose gas derived in [Comm. Math. Phys. 266 (2006), 797–818] is extended to the more general case, where the mass of the positive bosons can be different from the mass of the negative bosons. The analysis in principle is quite similar, however the formerly scalar problem becomes two-dimensional (one dimension for the + and one for the − case). This leads to a non-linear 2× 2 matrix minimization problem. Its minimizer is calculated and results in a non-linear mass-dependent coefficient in the upper bound. These new results agree in the equal-mass case with the formerly known results. It is also discussed how the result here interpolates between the equal mass case and the case where one mass is infinite.
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.
The ground-state energy of the charged Bose gas is calculated by the pair-correlation variational method of Girardeau and Arnowitt. The method is exact in the high-density limit (${r}_{s}\ensuremath{\ll}1$) and provides a variational extrapolation to intermediate densities. The leading terms of the high-density expansion, obtained by iteration of the variational integral equation, are ${u}_{0}=\ensuremath{-}0.804{{r}_{s}}^{\ensuremath{-}\frac{3}{4}}\ensuremath{-}(\frac{1}{8})\mathrm{ln}{r}_{s}+O(1)$, where u0{u}_{0}u0​ is the ground-state energy per particle in Rydbergs and rs{r}_{s}rs​ is the ratio of the mean interparticle spacing to the Bohr radius. The first term was obtained previously by Foldy, but the logarithmic term is new; it is related to screening of the long-range correlations at a distance ${r}_{0}\ensuremath{\sim}{{r}_{s}}^{\ensuremath{-}\frac{1}{2}}{\ensuremath{\rho}}^{\ensuremath{-}\frac{1}{3}}$, in analogy with the logarithmic term in the correlation energy of the electron gas. Results of numerical solutions for the intermediate-density region are presented, ranging up to rs=10{r}_{s}=10rs​=10. On the basis of a comparison with the energy calculated from the known low-density expansion, it is estimated that the transition into Wigner's electron crystal (here a boson crystal) should occur at ${r}_{s}\ensuremath{\sim}5$.
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.
The upper bound on the ground state energy of the two-component charged Bose gas derived in \[Comm. Math. Phys. 266 (2006), 797–818] is extended to the more general case, where the mass of the positive bosons can be different from the mass of the negative bosons. The analysis in principle is quite similar, however the formerly scalar problem becomes two-dimensional (one dimension for the +++ and one for the −-− case). This leads to a non-linear 2×22\times 22×2-matrix minimization problem. Its minimizer is calculated and results in a non-linear mass-dependent coefficient in the upper bound. These new results agree in the equal-mass case with the formerly known results. It is also discussed how the result here interpolates between the equal mass case and the case where one mass is infinite.
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 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.