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International Journal of Modern Physics B

OFF-DIAGONAL LONG RANGE ORDER OF THE REDUCED DENSITY MATRICES FOR A ONE-DIMENSIONAL SOLVABLE MODEL OF FERMIONS

Fu‐Cho Pu, Bao‐Heng Zhao

Published Dec 1, 1988• 0 citations
Publisher

Abstract

A one-dimensional solvable model of fermions is mapped to the XXX ferromagnetic chain. At T = 0, it is shown that there is off-diagonal long range order for the reduced density matrix ρ2 in the model. In momentum space the single particle state pairs are coherently occupied. The momenta of the particles in each pair are equal but with opposite directions.

DOI
10.1142/s021797928800127x

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We discuss new possibilities for off-diagonal long range order (ODLRO) in spin chains involving operators which add or delete sites from the chain. For the one-dimensional, periodic Heisenberg, inverse square exchange, and XY\mathrm{XY}XY models we give numerical evidence for the hidden ODLRO conjectured by Anderson. A connection to the singlet pair correlations in one-dimensional models of interacting electrons is made and briefly discussed.
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One-dimensional spinor gases with strong $\ensuremath{\delta}$ interaction fermionize and form a spin chain. The spatial degrees of freedom of this atom chain can be described by a mapping to spinless noninteracting fermions and the spin degrees of freedom are described by a spin-chain model with nearest-neighbor interactions. Here, we compute momentum and occupation-number distributions of up to 16 strongly interacting spinor fermions and bosons as a function of their spin imbalance, the strength of an externally applied magnetic field gradient, the length of their spin, and for different excited states of the multiplet. We show that the ground-state momentum distributions resemble those of the corresponding noninteracting systems, apart from flat background distributions, which extend to high momenta. Moreover, we show that the spin order of the spin chain---in particular antiferromagnetic spin order---may be deduced from the momentum and occupation-number distributions of the system. Finally, we present efficient numerical methods for the calculation of the single-particle densities and one-body density matrix elements and of the local exchange coefficients of the spin chain for large systems containing more than 20 strongly interacting particles in arbitrary confining potentials.
Strongly interacting one-dimensional (1D) Bose--Fermi mixtures form a tunable XXZXXZXXZ spin chain. Within the spin-chain model developed here, all properties of these systems can be calculated from states representing the ordering of the bosons and fermions within the atom chain and from the ground-state wave function of spinless noninteracting fermions. We validate the model by means of an exact diagonalization of the full few-body Hamiltonian in the strongly interacting regime. Using the model, we explore the phase diagram of the atom chain as a function of the boson-boson (BB) and boson-fermion (BF) interaction strengths and calculate the densities, momentum distributions, and trap-level occupancies for up to 17 particles. In particular, we find antiferromagnetic (AFM) and ferromagnetic (FM) order and a demixing of the bosons and fermions in certain interaction regimes. We find, however, no demixing for equally strong BB and BF interactions, in agreement with earlier calculations that combined the Bethe ansatz with a local-density approximation.