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

The scaling of the largest eigenvalue lambda(0) 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 lambda(0) similar to B N-C. 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 gamma and beta, which describe the scaling of the peak of the momentum distribution with N and the natural orbital corresponding to lambda(0), respectively, and we derive the scaling relation gamma + 2 beta = C. Taking as a specific case the power-law potential V(x) alpha x(2n), we give analytical formulas for gamma and beta 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.