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arXiv:1201.6019

Ground-state Properties of Tonks-Girardeau Gas in One Dimensional Periodic Potential

Zhao-Liang, Wang, An-Min, Wang, Xuechao Li

Published Jan 29, 2012• 0 citations
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Abstract

The relations among the occupation number of the lowest natural orbital (ONLNO), momentum distributions (MD) and off-diagonal long-range element (ODLRE) of the reduced single-particle density matrix (RSPDM) are studied while Tonks-Girardeau gas in one dimensional periodic potential is in the ground state. For $N$-body systems of large enough, RSPDM and its lowest natural orbital do not vary with $N$ in overlapped areas in commensurate and incommensurate cases correspondingly. In commensurate case, the ODLRE is exponential attenuation with $N$, which results in that the ONLNO and MD are invariant with $N$. While in contrast, in incommensurate case, the off-diagonal elements are inversely proportional to $\sqrt{N}$, which results in the different behavior of the ONLNO and MD.

DOI
10.48550/arxiv.1201.6019

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The relations among the occupation number of the lowest natural orbital (ONLNO), momentum distributions (MD) and off-diagonal long-range element (ODLRE) of the reduced single-particle density matrix (RSPDM) are studied while Tonks-Girardeau gas in one dimensional periodic potential is in the ground state. For NNN-body systems of large enough, RSPDM and its lowest natural orbital do not vary with NNN in overlapped areas in commensurate and incommensurate cases correspondingly. In commensurate case, the ODLRE is exponential attenuation with NNN, which results in that the ONLNO and MD are invariant with NNN. While in contrast, in incommensurate case, the off-diagonal elements are inversely proportional to N\sqrt{N}N​, which results in the different behavior of the ONLNO and MD.
The relations among the occupation number of the lowest natural orbital, momentum distributions and off-diagonal elements of the reduced single-particle density matrix (RSPDM) are studied while a Bose gas in periodic potentials is in the ground state. For large particle number systems, RSPDM and its lowest natural orbital of NNN-body system are the same as that of (N+1)(N+1)(N+1)-body system in overlapped areas in both commensurate and incommensurate cases. For large NNN in commensurate case, the occupation number of the lowest natural orbital and momentum distributions (MD) are invariant with NNN that results from the exponential attenuation with NNN of the off-diagonal elements. While in contrast, in incommensurate case, the off-diagonal elements are inversely proportional to N\sqrt{N}N​, which result in the different behavior of the occupation number of the lowest natural orbital and MD. There are no Bose-Einstein condensation (BEC) for periodic trapped TG gas in both commensurate and incommensurate cases.
In this paper, we investigate the ground-state properties of a bosonic Tonks-Girardeau (TG) gas confined in a one-dimensional periodic potential. The single-particle reduced density matrix is computed numerically for systems up to N=265N=265N=265 bosons. Scaling analysis of the occupation number of the lowest orbital shows that there are no Bose-Einstein condensation (BEC) for the periodically trapped TG gas in both commensurate and incommensurate cases. We find that, in the commensurate case, the scaling exponents of the occupation number of the lowest orbital, the amplitude of the lowest orbital and the zero-momentum peak height with the particle numbers are 0, $\ensuremath{-}0.5$ and 1, respectively, while in the incommensurate case, they are 0.5, $\ensuremath{-}0.5$, and 1.5, respectively. These exponents are related to each other in a universal relation.
We study the reduced single-particle density matrix (RSPDM), the momentum distribution, and natural orbitals and their occupancies of dark ``soliton'' (DS) states in a Tonks-Girardeau gas. DS states are specially tailored excited many-body eigenstates, which have a dark solitonic notch in their single-particle density. The momentum distribution of DS states has a characteristic shape with two sharp spikes. We find that the two spikes arise due to the high degree of correlation observed within the RSPDM between the mirror points (xxx and $\ensuremath{-}x$) with respect to the dark notch at x=0x=0x=0; the correlations oscillate rather than decay as the points xxx and $\ensuremath{-}x$ are being separated.
The scaling of the largest eigenvalue ${\ensuremath{\lambda}}_{0}$ of the one-body density matrix of a system with respect to its particle number NNN defines an exponent C\mathcal{C}C and a coefficient B\mathcal{B}B via the asymptotic relation ${\ensuremath{\lambda}}_{0}\ensuremath{\sim}\mathcal{B}\phantom{\rule{0.16em}{0ex}}{N}^{\mathcal{C}}$. The case C=1\mathcal{C}=1C=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\mathcal{C}=1/2C=1/2. Here we investigate the inhomogeneous case, initially addressing the behavior of C\mathcal{C}C in the presence of a general external trapping potential VVV. We argue that the value C=1/2\mathcal{C}=1/2C=1/2 characterizes the hard-core system independently of the nature of the potential VVV. We then define the exponents $\ensuremath{\gamma}$ and $\ensuremath{\beta}$, which describe the scaling of the peak of the momentum distribution with NNN and the natural orbital corresponding to ${\ensuremath{\lambda}}_{0}$, respectively, and we derive the scaling relation $\ensuremath{\gamma}+2\ensuremath{\beta}=\mathcal{C}$. Taking as a specific case the power-law potential $V(x)\ensuremath{\propto}{x}^{2n}$, we give analytical formulas for $\ensuremath{\gamma}$ and $\ensuremath{\beta}$ as functions of nnn. Analytical predictions for the coefficient B\mathcal{B}B are also obtained. These formulas are derived by exploiting a recent field theoretical formulation and checked against numerical results. The agreement is excellent.
We calculate the reduced single-particle density matrix (RSPDM), momentum distribution, natural orbitals and their occupancies, for a strongly repulsive one-dimensional Bose-Fermi mixture in a double-well potential with a large central barrier. We assume that all particles have the same mass, and fermions are spin polarized. For mesoscopic systems, we find that the ground-state properties qualitatively differ for mixtures with even number of particles (both odd-odd and even-even mixtures) in comparison to mixtures with odd particle numbers (odd-even and even-odd mixtures). For even mixtures the momentum distribution is smooth, whereas the momentum distribution of odd mixtures possesses distinct modulations; the differences are observed also in the off-diagonal correlations of the RSPDM, and in the occupancies of natural orbitals. The calculation is based on a formula which enables efficient calculation of the RSPDM for mesoscopic mixtures in various potentials.
In one-dimensional (1D) quantum gases, the momentum distribution (MD) of the atoms is a standard experimental observable, routinely measured in various experimental setups. The MD is sensitive to correlations, and it is notoriously hard to compute theoretically for large numbers of atoms NNN, which often prevents direct comparison with experimental data. Here we report significant progress on this problem for the 1D Tonks-Girardeau (TG) gas in the asymptotic limit of large NNN, at zero temperature and driven out of equilibrium by a quench of the confining potential. We find an exact analytical formula for the one-particle density matrix $\ensuremath{\langle}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Psi}}}}^{\ifmmode\dagger\else\textdagger\fi{}}(x)\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Psi}}}({x}^{\ensuremath{'}})\ensuremath{\rangle}$ of the out-of-equilibrium TG gas in the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, valid on distances $|x\ensuremath{-}{x}^{\ensuremath{'}}|$ much larger than the interparticle distance. By comparing with time-dependent Bose-Fermi mapping numerics, we demonstrate that our analytical formula can be used to compute the out-of-equilibrium MD with great accuracy for a wide range of momenta (except in the tails of the distribution at very large momenta). For a quench from a double-well potential to a single harmonic well, which mimics a ``quantum Newton cradle'' setup, our method predicts the periodic formation of peculiar, multiply peaked, momentum distributions.