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.

arXiv comment14 pages, 5 figures

References in corpus (43)

• One dimensional Bosons: From Condensed Matter Systems to Ultracold Gases
• Quantum Inverse Scattering Method and Correlation Functions
• Realization of an Excited, Strongly-Correlated Quantum Gas Phase
• Pinning quantum phase transition for a Luttinger liquid of strongly interacting bosons
• Conformal Field Theory for Inhomogeneous One-dimensional Quantum Systems: the Example of Non-Interacting Fermi Gases
• An introduction to integrable techniques for one-dimensional quantum systems
• Exact coherent states of a harmonically confined Tonks-Girardeau gas
• Ground state properties of a one-dimensional condensate of hard core bosons in a harmonic trap
• Beyond Tonks-Girardeau: strongly correlated regime in quasi-one-dimensional Bose gases
• Equilibration of a Tonks-Girardeau gas following a trap release
• Finite one dimensional impenetrable Bose systems: Occupation numbers
• Many-body solitons in a one-dimensional condensate of hard core bosons
• Universal properties of hard-core bosons confined on one-dimensional lattices
• Evidence for the super Tonks-Girardeau gas
• Momentum distribution dynamics of a Tonks-Girardeau gas: Bragg reflections of a quantum many-body wavepacket
• Critical behavior and scaling in trapped systems
• Universal contact for a Tonks-Girardeau gas at finite temperature
• Mott Transition for Strongly-Interacting 1D Bosons in a Shallow Periodic Potential
• Trap-size scaling in confined particle systems at quantum transitions
• Quantum critical behavior and trap-size scaling of trapped bosons in a one-dimensional optical lattice
• One-dimensional Bose gas in optical lattices of arbitrary strength
• Local Correlations in the Super Tonks-Girardeau Gas
• The Inhomogeneous Gaussian Free Field, with application to ground state correlations of trapped 1d Bose gases
• Interference of a thermal Tonks gas on a ring
• Painlevé transcendent evaluations of finite system density matrices for 1d impenetrable Bosons
• One-particle density matrix of trapped one-dimensional impenetrable bosons from conformal invariance
• Momentum distribution for a one-dimensional trapped gas of hard-core bosons
• Metastable criticality and the super Tonks-Girardeau gas
• Super Tonks-Girardeau state in an attractive one-dimensional dipolar gas
• Tan's contact of a harmonically trapped one-dimensional Bose gas: strong-coupling expansion and conjectural approach at arbitrary interactions
• Anderson localization of a Tonks-Girardeau gas in potentials with controlled disorder
• Decay of correlations and absence of superfluidity in the disordered Tonks-Girardeau gas
• One-body reduced density matrix of trapped impenetrable anyons in one dimension
• Ground and excited states of spinor Fermi gases in tight waveguides and the Lieb-Liniger-Heisenberg model
• Ground state properties of hard-core anyons in a harmonic potential
• The pinning quantum phase transition in a Tonks Girardeau gas: diagnostics by ground state fidelity and the Loschmidt echo
• Random matrix averages and the impenetrable Bose gas in Dirichlet and Neumann boundary conditions
• Deviations from Off-Diagonal Long-Range Order in One-Dimensional Quantum Systems
• Dynamical depinning of a Tonks Girardeau gas
• Optical lattices as a tool to study defect-induced superfluidity
• Signatures of the single particle mobility edge in the ground state properties of Tonks-Girardeau and non-interacting Fermi gases in a bichromatic potential
• Tan's contact scaling behaviour for trapped Lieb-Liniger bosons: from two to many
• Ground state properties of a Tonks-Girardeau Gas in a periodic potential

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• Finite Temperature Off-Diagonal Long-Range Order for Interacting Bosons
• One-particle density matrix and momentum distribution of the out-of-equilibrium 1D Tonks-Girardeau gas: Analytical results at large $N$
• One-body dynamical correlation function of Lieb-Liniger model at finite temperature
• Fermionization of a Few-Body Bose System Immersed into a Bose-Einstein Condensate
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