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arXiv:2609.11984

Diagonal Pair Coulomb Density Matrix on a Sphere: a numerical study

Riccardo Fantoni

cond-mat.stat-mechcond-mat.str-elphysics.comp-phquant-phSubmitted Sep 7, 2026• Announced Sep 8, 2026• 0 citations
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Abstract

We tabulate, on the diagonal, the $n$th power of the primitive approximation for the pair Coulomb density matrix on a sphere, with up to $n=5$, and we discuss its behavior at particles contact as $n$ grows. Our results show that even on a curved surface quantum statistical physics is fundamentally different from its classical counterpart. If in the latter one needs to artificially introduce a regularization of the divergent attractive Coulomb potential preventing particles contact, for example with a hard core, in the former this is not necessary.

Comments
12 pages, 1 table, 1 figure

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We tabulate, on the diagonal, the nnnth power of the primitive approximation for the pair Coulomb density matrix on a sphere, with up to n=5n=5n=5, and we discuss its behavior at particles contact as nnn grows. Our results show that even on a curved surface quantum statistical physics is fundamentally different from its classical counterpart. If in the latter one needs to artificially introduce a regularization of the divergent attractive Coulomb potential preventing particles contact, for example with a hard core, in the former this is not necessary.
We obtain bounds for all derivatives of the non-relativistic Coulombic one-particle density matrix γ(x,y)γ(x, y)γ(x,y) near the diagonal x=yx = yx=y.
The thermodynamic properties of the hard-sphere Bose gas are calculated from the one- and two-particle density matrices. The one- and two-particle density matrices are evaluated in the Bogoliubov approximation. The off-diagonal elements of the one- and two-particle density matrices are studied for the presence or absence of off-diagonal long-range order in various temperature regions. Off-diagonal long-range order was found in the single-particle density matrix below the Bose-Einstein transition temperature. The two-particle density matrix also exhibited off-diagonal long-range order whenever the single-particle density matrix did.
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