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Upper bound on the ground state energy of the two-component charged Bose gas with arbitrary masses

Andreas Schön, Jan Philip Solovej

Published Jun 29, 2022• 0 citations
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Abstract

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 $2\times 2$-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.

DOI
10.4171/90-2/41

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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.
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.
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
Using a Jastrow trial wave function and the hypernetted-chain integral equation, as well as Monte Carlo computations, to calculate energy expectation values, we have obtained a variational upper bound to the ground-state energy of the charged Bose gas over an extensive range of densities. In the high-density region our data are in good agreement with previous weak-coupling results. In the low-density (strong-coupling) limit, comparison of our fluid energies with the self-consistent phonon calculations of Glyde and Keech in the solid allow us to locate the Wigner transition for charged bosons at rs=139{r}_{s}=139rs​=139, where rs{r}_{s}rs​ is the usual density parameter. We propose a simple and accurate equation of state based on our numerical data. Furthermore, we find that the Bose condensate fraction drops from a value of about 80% at rs=2{r}_{s}=2rs​=2, to a small but finite value (\ensuremath{\sim}1%) at crystallization. Finally we use our results for the charged Bose gas to derive a heuristic equation of state of the electron gas in the strong-coupling limit (${r}_{s}\ensuremath{\gg}1$).