The scaling of the largest eigenvalue λ0 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 λ0∼BNC. 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 γ and β, which describe the scaling of the peak of the momentum distribution with N and the natural orbital corresponding to λ0, respectively, and we derive the scaling relation γ+2β=C. Taking as a specific case the power-law potential V(x)2n, we give analytical formulas for γ and β 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.

Universal off-diagonal long-range-order behavior for a trapped Tonks-Girardeau gas / Colcelli, Andrea; Viti, J.; Mussardo, G.; Trombettoni, A.. - In: PHYSICAL REVIEW A. - ISSN 2469-9926. - 98:6(2018), pp. 1-12. [10.1103/PhysRevA.98.063633]

Universal off-diagonal long-range-order behavior for a trapped Tonks-Girardeau gas

Colcelli, Andrea;Viti, J.;Mussardo, G.;Trombettoni, A.
2018

Abstract

The scaling of the largest eigenvalue λ0 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 λ0∼BNC. 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 γ and β, which describe the scaling of the peak of the momentum distribution with N and the natural orbital corresponding to λ0, respectively, and we derive the scaling relation γ+2β=C. Taking as a specific case the power-law potential V(x)2n, we give analytical formulas for γ and β 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.
98
6
1
12
063633
https://journals.aps.org/pra/abstract/10.1103/PhysRevA.98.063633
https://arxiv.org/abs/1809.01592
Colcelli, Andrea; Viti, J.; Mussardo, G.; Trombettoni, A.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11767/88158
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