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

Abstract: 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 $\mathcal{C}$ and a coefficient $\mathcal{B}$ via the asymptotic relation $\lambda_0 \sim \mathcal{B}\,N^{\mathcal{C}}$. The case $\mathcal{C}=1$ 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 $\mathcal{C}=1/2$. Here we investigate the inhomogeneous case, initially addressing the behaviour of $\mathcal{C}$ in presence of a general external trapping potential $V$. We argue that the value $\mathcal{C}= 1/2$ characterises the hard-core system independently of the nature of the potential $V$. We then define the exponents $\gamma$ and $\beta$ which describe the scaling with $N$ of the peak of the momentum distribution and the natural orbital corresponding to $\lambda_0$ respectively, and we derive the scaling relation $\gamma + 2\beta= \mathcal{C}$. Taking as a specific case the power-law potential $V(x)\propto x^{2n}$, we give analytical formulas for $\gamma$ and $\beta$ as functions of $n$. Analytical predictions for the coefficient $\mathcal{B}$ are also obtained. These formulas are derived exploiting a recent field theoretical formulation and checked against numerical results. The agreement is excellent.