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arXiv:2104.06693 · The European Physical Journal Plus

The momentum distribution of two bosons in one dimension with infinite contact repulsion in harmonic trap gets analytical

K. Bencheikh, Luis Miguel Nieto, L. U. Ancarani

Published Jul 1, 2021• 0 citations
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Abstract

Abstract For a harmonically trapped system consisting of two bosons in one spatial dimension with infinite contact repulsion (hard core bosons), we derive an expression for the one-body density matrix $$\rho _\mathrm{B}$$ ρ B in terms of center of mass and relative coordinates of the particles. The deviation from $$\rho _\mathrm{F}$$ ρ F , the density matrix for the two fermions case, can be clearly identified. Moreover, the obtained $$\rho _\mathrm{B}$$ ρ B allows us to derive a closed form expression of the corresponding momentum distribution $$n_\mathrm{B}(p)$$ n B ( p ) . We show how the result deviates from the noninteracting fermionic case, the deviation being associated with the short-range character of the interaction. Mathematically, our analytical momentum distribution is expressed in terms of one and two variables confluent hypergeometric functions. Our formula satisfies the correct normalization and possesses the expected behavior at zero momentum. It also exhibits the high momentum $$1/p^4 $$ 1 / p 4 tail with the appropriate Tan’s coefficient. Numerical results support our findings.

DOI
10.1140/epjp/s13360-021-01671-x

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The one-particle density matrices for hard-core bosons in a one-dimensional harmonic trap are computed numerically for systems with up to 160 bosons. Diagonalization of the density matrix shows that the many-body ground state is not Bose-Einstein condensed. The ground-state occupation, the amplitude of the lowest natural orbital, and the zero momentum peak height scale as powers of the particle number, and the corresponding exponents are related to each other. Close to its diagonal, the density matrix for hard-core bosons is similar to that of noninteracting fermions.
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