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arXiv:0805.1988 · Physical Review A

Ground-state properties of one-dimensional anyon gases

Yajiang Hao, Yunbo Zhang, Shu Chen

Published Aug 26, 2008• 62 citations
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Abstract

We investigate the ground state of the one-dimensional interacting anyonic system based on the exact Bethe ansatz solution for arbitrary coupling constant $(0\ensuremath{\leqslant}c\ensuremath{\leqslant}\ensuremath{\infty})$ and statistics parameter $(0\ensuremath{\leqslant}\ensuremath{\kappa}\ensuremath{\leqslant}\ensuremath{\pi})$. It is shown that the density of state in quasimomentum $k$ space and the ground-state energy are determined by the renormalized coupling constant ${c}^{\ensuremath{'}}$. The effect induced by the statistics parameter $\ensuremath{\kappa}$ exhibits in the momentum distribution in two aspects: Besides the effect of renormalized coupling, the anyonic statistics results in the nonsymmetric momentum distribution when the statistics parameter $\ensuremath{\kappa}$ deviates from 0 (Bose statistics) and $\ensuremath{\pi}$ (Fermi statistics) for any coupling constant $c$. The momentum distribution evolves from a Bose distribution to a Fermi one as $\ensuremath{\kappa}$ varies from 0 to $\ensuremath{\pi}$. The asymmetric momentum distribution comes from the contribution of the imaginary part of the nondiagonal element of reduced density matrix, which is an odd function of $\ensuremath{\kappa}$. The peak at positive momentum will shift to negative momentum if $\ensuremath{\kappa}$ is negative.

DOI
10.1103/physreva.78.023631

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We investigate the ground state of the one-dimensional interacting anyonic system based on the exact Bethe ansatz solution for arbitrary coupling constant (0\ensuremath⩽c\ensuremath⩽\ensuremath∞)(0\ensuremath{\leqslant}c\ensuremath{\leqslant}\ensuremath{\infty})(0\ensuremath⩽c\ensuremath⩽\ensuremath∞) and statistics parameter (0\ensuremath⩽\ensuremathκ\ensuremath⩽\ensuremathπ)(0\ensuremath{\leqslant}\ensuremath{\kappa}\ensuremath{\leqslant}\ensuremath{\pi})(0\ensuremath⩽\ensuremathκ\ensuremath⩽\ensuremathπ). It is shown that the density of state in quasimomentum kkk space and the ground-state energy are determined by the renormalized coupling constant c\ensuremath′{c}^{\ensuremath{'}}c\ensuremath′. The effect induced by the statistics parameter \ensuremathκ\ensuremath{\kappa}\ensuremathκ exhibits in the momentum distribution in two aspects: Besides the effect of renormalized coupling, the anyonic statistics results in the nonsymmetric momentum distribution when the statistics parameter \ensuremathκ\ensuremath{\kappa}\ensuremathκ deviates from 0 (Bose statistics) and \ensuremathπ\ensuremath{\pi}\ensuremathπ (Fermi statistics) for any coupling constant ccc. The momentum distribution evolves from a Bose distribution to a Fermi one as \ensuremathκ\ensuremath{\kappa}\ensuremathκ varies from 0 to \ensuremathπ\ensuremath{\pi}\ensuremathπ. The asymmetric momentum distribution comes from the contribution of the imaginary part of the nondiagonal element of reduced density matrix, which is an odd function of \ensuremathκ\ensuremath{\kappa}\ensuremathκ. The peak at positive momentum will shift to negative momentum if \ensuremathκ\ensuremath{\kappa}\ensuremathκ is negative.
The low-energy properties of the one-dimensional anyon gas with a delta-function interaction are discussed in the context of its Bethe ansatz solution. It is found that the anyonic statistical parameter and the dynamical coupling constant induce Haldane exclusion statistics interpolating between bosons and fermions. Moreover, the anyonic parameter may trigger statistics beyond Fermi statistics for which the exclusion parameter alpha is greater than one. The Tonks-Girardeau and the weak coupling limits are discussed in detail. The results support the universal role of alpha in the dispersion relations.
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We study a one-dimensional system of interacting anyons with short-range interactions under external confinement. This model, which describes anyons with zero-range sss-wave and ppp-wave scattering interactions in the strong interaction regime, continuously interpolates between spin-polarized fermions with ppp-wave interactions and free bosons. At zero temperature, the correlation functions decay exponentially with distance, with oscillations governed by the statistics parameter. The decay rate is maximal for ppp-wave fermions and decreases monotonically as the statistics parameter approaches the bosonic limit, where it vanishes. The momentum distribution is asymmetric, a hallmark of one-dimensional anyons, and takes the form of a shifted Lorentzian with universal power-law tails, ${lim}_{k\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}}n(k)\ensuremath{\sim}C/{k}^{2}$. We prove analytically that, following release from a harmonic trap, the asymptotic momentum distribution converges to that of free bosons in the same trap, a phenomenon known as dynamical bosonization. We also establish the universality of the ground-state nnn-particle reduced density matrices: their natural occupations are independent of the confining potential, while the associated natural nnn-point functions for different confinements are related through a simple analytical transformation. In particular, for the one-particle reduced density matrix, we derive exact expressions for both the natural occupations and the natural orbitals at arbitrary particle number. These results extend and unify earlier partial findings for ppp-wave fermions, and they provide a clear conceptual explanation of the double degeneracy observed in their spectrum.
We study the nonequilibrium quench dynamics of a one-dimensional anyonic gas. We focus on the integrable anyonic Lieb-Liniger model and consider the quench from noninteracting to hard-core anyons. We study the dynamics of the local properties of the system. By means of integrability-based methods, we compute analytically the one-body density matrix and the density-density correlation function at all times after the quench and in particular at infinite time. Our results show that the system evolves from an initial state where the local momentum distribution function is nonsymmetric to a steady state where it becomes symmetric. Furthermore, while the initial momentum distribution functions (and the equilibrium ones) explicitly depend on the anyonic parameter, the final ones do not. This is reminiscent of the dynamical fermionization observed in the context of free expansions after release from a confining trap.
We point out that the momentum distribution is not a proper observable for a system of anyons in two dimensions. In view of anyons as Wilczek's composite charged flux tubes, this is a consequence of the fact that the orthogonal components of the kinetic momentum operator do not commute at the position of a flux tube, and thus cannot be diagonalized in the same basis. As a substitute for the momentum distribution of an anyonic (spatially localized) state, we propose to use the asymptotic single-particle density after the expansion of anyons in free space from the state. This definition is identical to the standard one when the statistical parameter approaches that for bosons or fermions. Exact examples which underpin our proposal are shown. They reveal that the quasimomentum distribution can be used to identify anyonic statistics in standard time-of-flight measurements.
We present a systematic study of the Green functions of a one-dimensional gas of impenetrable anyons. We show that the one-particle density matrix is the determinant of a Toeplitz matrix whose large N asymptotic is given by the Fisher–Hartwig conjecture. We provide a careful numerical analysis of this determinant for general values of the anyonic parameter, showing in full details the crossover between bosons and fermions and the reorganization of the singularities of the momentum distribution function. We show that the one-particle density matrix satisfies a Painlevé VI differential equation that is then used to derive the small distance and large momentum expansions. We find that the first non-vanishing term in this expansion is always k −4 , that is proved to be true for all couplings in the Lieb–Liniger anyonic gas and that can be traced back to the presence of a delta function interaction in the Hamiltonian.