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arXiv:0906.0536 · Physical Review A

Quantum entanglement of particles on a ring with fractional statistics

Hongli Guo, Yajiang Hao, Shu Chen

Published Nov 25, 2009• 23 citations
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Abstract

In this paper we investigate the von Neumann entropy in the ground state of one-dimensional anyonic systems with the repulsive interaction. Based on the Bethe-ansatz solutions, the entanglement properties for the arbitrary statistical parameter $(0\ensuremath{\le}\ensuremath{\kappa}\ensuremath{\le}1)$ are obtained from the one-particle reduced density matrix in the full interacting regime. It is shown that the entanglement entropy increases with the increase in the interaction strength and statistical parameter. The statistic parameter affects the entanglement properties from two aspects: renormalizing the effective interaction strength and introducing an additional anyonic phase. We also evaluate the entanglement entropy of hard-core anyons for different statistical parameters in order to clarify the effect solely induced by the anyonic phase.

DOI
10.1103/physreva.80.052332

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In this paper we investigate the von Neumann entropy in the ground state of one-dimensional anyonic systems with the repulsive interaction. Based on the Bethe-ansatz solutions, the entanglement properties for the arbitrary statistical parameter (0\ensuremath≤\ensuremathκ\ensuremath≤1)(0\ensuremath{\le}\ensuremath{\kappa}\ensuremath{\le}1)(0\ensuremath≤\ensuremathκ\ensuremath≤1) are obtained from the one-particle reduced density matrix in the full interacting regime. It is shown that the entanglement entropy increases with the increase in the interaction strength and statistical parameter. The statistic parameter affects the entanglement properties from two aspects: renormalizing the effective interaction strength and introducing an additional anyonic phase. We also evaluate the entanglement entropy of hard-core anyons for different statistical parameters in order to clarify the effect solely induced by the anyonic phase.
We study the one-particle von Neumann entropy of a system of N hard-core anyons on a ring. The entropy is found to have a clear dependence on the anyonic parameter which characterizes the generalized fractional statistics described by the anyons. This confirms that the entanglement is a valuable quantity for investigating topological properties of quantum states. We derive the generalization to anyonic statistics of the Lenard formula for the one-particle density matrix of N hard-core bosons in the large N limit and extend our results by a numerical analysis of the entanglement entropy, providing additional insight into the problem under consideration.
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 kkk 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 ccc. 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.
We present a Bethe-ansatz-solvable dissipative model of one-dimensional hard-core anyons under dephasing, where the particles obey fractional statistics. We establish an exact mapping between the Liouvillian and a non-Hermitian Hubbard model with special twisted boundary conditions. With the help of string hypothesis, we rigorously calculate the finite-size corrections and the Liouvillian gap in terms of the statistical parameter, total particle number, lattice size, and dissipation strength. The statistical parameter breaks the $\ensuremath{\eta}$-pairing symmetry of the system, partially lifts out the degeneracy of the spectrum, and contributes to the Liouvillian gap in finite-size corrections. We also derive the general form of the equation of motion for expectation values of the 2w2w2w-point correlator of hard-core anyons and present a proposal for experimentally measuring the correlation function. Our results shed light on the Liouvillians of exactly solvable dissipative models of anyonic particles that obey the fractional statistics.
We study dynamical correlations of interacting hard-core anyons on a one-dimensional lattice at infinite temperature. This is a setting in which the many-body spectrum is independent of the statistical phase θ , while dynamical correlators remain sensitive to θ through nonlocal Jordan-Wigner strings. We compute single-particle Green's functions, spectral functions, and density-density correlators, thereby separating the effects of fractional statistics on one-body coherence from those on density transport in a maximally mixed ensemble. In the noninteracting case V = 0 , high-temperature averaging leads to inversion-symmetric Green's functions for all θ despite the presence of anyonic strings. Finite nearest-neighbor interactions V generate, however, a pronounced left-right asymmetry in the Green's functions for 0 < θ < π , with the strongest chirality appearing at intermediate couplings V ∼ J where interactions and hopping compete most effectively. In this regime, the Green's function decays exponentially in time with a statistical-angle-dependent decay rate. At strong coupling, the dynamics crosses over to an atomic-limit regime in which the dependence on θ is reduced. Here, the Green's function decays universally as t − 1 and the corresponding spectral function displays a three-band structure. In contrast, density-density correlations are insensitive to statistics and recover the known infinite-temperature transport regimes of the XXZ chain, including ballistic, superdiffusive, and diffusive behaviors. These results identify dynamical correlation functions as direct probes of fractional statistics in high-entropy quantum systems.
Using the fractional statistical properties of so-called anyonic particles, we present solutions of the Schr\"odinger equation for up to six strongly interacting particles in one-dimensional confinement that interpolate the usual bosonic and fermionic limits. These solutions are exact to linear order in the inverse coupling strength of the zero-range interaction of our model. Specifically, we consider two-component mixtures of anyons and use these to eludicate the mixing-demixing properties of both balanced and imbalanced systems. Importantly, we demonstrate that the degree of demixing depends sensitively on the external trap in which the particles are confined. We also show how one may in principle probe the statistical parameter of an anyonic system by injection a strongly interacting impurity and doing spectral or tunneling measurements.
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.