2018/09/05 by Andrea Colcelli, A. Colcelli, Jacopo Viti +2 · 17 citations
Mathematics · Physics and Astronomy · #Cold Atom Physics and Bose-Einstein Condensates #Exponent #Lambda #Mathematical physics #Mathematics #Order (exchange) #Physics #Quantum many-body systems #Quantum mechanics #Quantum, superfluid, helium dynamics #Scaling #cond-mat.quant-gas #cond-mat.stat-mech
paper · pdf · doi:10.1103/physreva.98.063633
published in Physical Review A 98(6) (American Physical Society) · 14 pages, 5 figures
arxiv created 2018/09/05 · openalex created_date 2018/09/27 · openalex publication_date 2018/12/26 · arxiv updated 2018/12/31 · openalex updated_date 2026/08/05
The scaling of the largest eigenvalue \ensuremathλ0 of the one-body density matrix of a system with respect to its particle number N defines an exponent C and a coefficient B via the asymptotic relation \ensuremathλ0\ensuremath∼B\phantom\rule0.16em0exNC. The case C=1 corresponds to off-diagonal long-range order. For a one-dimensional homogeneous Tonks-Girardeau gas, a well-known result also confirmed by bosonization gives instead C=1/2. Here we investigate the inhomogeneous case, initially addressing the behavior of C in the presence of a general external trapping potential V. We argue that the value C=1/2 characterizes the hard-core system independently of the nature of the potential V. We then define the exponents \ensuremathγ and \ensuremathβ, which describe the scaling of the peak of the momentum distribution with N and the natural orbital corresponding to \ensuremathλ0, respectively, and we derive the scaling relation \ensuremathγ+2\ensuremathβ=C. Taking as a specific case the power-law potential V(x)\ensuremath∝x2n, we give analytical formulas for \ensuremathγ and \ensuremathβ as functions of n. Analytical predictions for the coefficient B are also obtained. These formulas are derived by exploiting a recent field theoretical formulation and checked against numerical results. The agreement is excellent.