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arXiv:1602.06816 · Physical Review A

Momentum distributions and numerical methods for strongly interacting one-dimensional spinor gases

F. Deuretzbacher, D. Becker, Luis Santos

Published Aug 4, 2016• 42 citations
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Abstract

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.

DOI
10.1103/physreva.94.023606

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One-dimensional spinor gases with strong \ensuremathδ\ensuremath{\delta}\ensuremathδ 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.
Quantum many-body physics has been studied for many decades. A lot of intriguing phenomena have been observed and theories have been developed. Yet many problems remain unsolved, which is largely due to a lack of general efficient classical computation method. One-dimensional (1D) systems have also drawn much attention over the past few decades. One reason is that many unique strongly correlated quantum phenomena only appear in low dimensions. Another reason is that many exact results can be obtained in 1D for cross benchmarking, such as Bethe Ansatz, Bosonization, Bose-Fermi mapping, etc. In addition, many numerical methods, such as Matrix-Product-State based and Monte Carlo methods, work most efficiently in 1D. In recent years, 1D quantum gases have been realized in many cold atom labs, providing experimental motivation for their studies. One of the most mysterious assumptions about a system of identical quantum particles is that the wavefunction must be symmetric (for bosons) or anti-symmetric (for fermions). This means that no two fermions can occupy the same state, while multiple occupancy is allowed for bosons. Bosons and fermions are therefore generally very different. But in 1D this distinction could become rather subtle, provided that the bosonic multiple occupancy is suppressed, which can happen if strong repulsion exists between them. However, when considering spin degrees of freedom, things become more complicated. In this work, we develop a generalized Bose-Fermi mapping theory, under which, the charge degrees of freedom is mapped to a spinless fermions, while the spin degrees of freedom to an effective spin chain. This mapping works for arbitrary spin, arbitrary trapping potentials, arbitrary spin-dependent interaction potential, and for either spinor bosons or fermions. In the strong interaction limit, the wavefunction of the system is represented by the strong coupling ansatz wavefunction (SCAW), which takes the form of a direct product of spinless fermions wavefunction and spin wavefunction. Using this mapping technique, we study the dynamics and collective modes of the system. Furthermore, we develop a very efficient method to calculate the one-body density matrix, from which we can calculate the momentum distribution of the system.
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.
One of quantum physics' fundamental, but largely unsolved, problems is the computation of the correlation functions in many-body systems. We address this problem in the case of one-dimensional spinor gases with repulsive contact interactions in the presence of a confining potential. We take advantage of the fact that in the strong-coupling limit, the wave function factorizes with the charge degrees of freedom expressed as a Slater determinant of spinless fermions and the spin sector described by a spin chain of Sutherland type with exchange coefficients that depend only on the trapping potential. This factorization is also present in the expressions for the correlation functions. Still, analytical and numerical investigations were hindered by the fact that the local exchange coefficients and the charge component of the correlators are expressed as $N\ensuremath{-}1$ multidimensional integrals, with NNN the number of particles, which are notoriously hard to compute using conventional methods. We introduce a different approach to calculating these integrals that is extremely simple, scales polynomially with the number of particles, and is several orders of magnitude faster than the previous methods reported in the literature. This allows us to investigate the static and dynamic properties, temperature dependence, and nonequilibrium dynamics for systems with a larger number of particles than previously considered and discover new phenomena. We show that contrary to natural expectations, the momentum distribution of strongly interacting trapped spinor gases becomes narrower as we increase the temperature and derive simple determinant representations for the correlators in the spin incoherent regime that is valid for both equilibrium and nonequilibrium situations. For homogeneous systems of impenetrable particles at zero temperature, we analytically compute the large distance asymptotics of the correlators, finding that the leading term of the asymptotic expansion is proportional to ${e}^{\ensuremath{-}x{k}_{F}ln\ensuremath{\kappa}/\ensuremath{\pi}}$, with $\ensuremath{\kappa}$ the number of components of the system, which precludes any singularity of the momentum distribution.
We present an exact analytical solution of the fundamental system of quasi-one-dimensional spin-1 bosons with infinite delta repulsion. The eigenfunctions are constructed from the wave functions of noninteracting spinless fermions, based on Girardeau's Fermi-Bose mapping. We show that the spinor bosons behave like a compound of noninteracting spinless fermions and noninteracting distinguishable spins. This duality is especially reflected in the spin densities and the energy spectrum. We find that the momentum distribution of the eigenstates depends on the symmetry of the spin function. Furthermore, we discuss the splitting of the ground state multiplet in the regime of large but finite repulsion.
We show that the wave function in one spatial sector ${x}_{1}<{x}_{2}<\ensuremath{\cdots}<{x}_{N}$ (with xi{x}_{i}xi​ being the coordinate of the ithi\mathrm{th}ith particle) of a one-dimensional spinor gas with contact sss-wave interaction, either bosonic or fermionic, can be mapped to the direct product of the wave function of a spinless Fermi gas with short-range ppp-wave interaction and that of a spin system governed by spin-parity projection operators. Applying this mapping to strongly interacting spinor gases, we obtain a generalized spin-chain model that captures both the static and dynamics properties of the system. Using this spin-chain model, we investigate the breathing-mode frequency and the quench dynamics of strongly interacting, harmonically trapped spinor gases.
We present a general form of the effective spin-chain model for strongly interacting atomic gases with an arbitrary spin in one-dimensional (1D) traps. In particular, for high-spin systems the atoms can collide in multiple scattering channels, and we find that the resulting form of spin-chain model generically follows the same structure as that of the interaction potentials. This is a unified form working for any spin, statistics (Bose or Fermi), and confinement potentials. We adopt the spin-chain model to reveal both the ferromagnetic (FM) and antiferromagnetic (AFM) magnetic orders for strongly interacting spin-1 bosons in 1D traps. We further show that by adding the spin-orbit coupling, the FM and AFM orders can be gradually destroyed and eventually the ground state exhibits universal spin structure and contacts that are independent of the strength of spin-orbit coupling.