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arXiv:0808.0479 · Physical Review A

Dynamics of a one-dimensional spinor Bose liquid: A phenomenological approach

Alex Kamenev, L. I. Glazman

Published Jul 16, 2009• 59 citations
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Abstract

The ground state of a spinor Bose liquid is ferromagnetic, while the softest excitation above the ground state is the magnon mode. The dispersion relation of the magnon in a one-dimensional liquid is periodic in the wave number $q$ with the period of $2\ensuremath{\pi}n$, determined by the density $n$ of the liquid. Dynamic correlation functions, such as, e.g., spin-spin correlation function, exhibit power-law singularities at the magnon spectrum, $\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{m}(q,n)$. Without using any specific model of the interparticle interactions, we relate the corresponding exponents to independently measurable quantities $\ensuremath{\partial}{\ensuremath{\omega}}_{m}/\ensuremath{\partial}q$ and $\ensuremath{\partial}{\ensuremath{\omega}}_{m}/\ensuremath{\partial}n$.

DOI
10.1103/physreva.80.011603

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The ground state of a spinor Bose liquid is ferromagnetic, while the softest excitation above the ground state is the magnon mode. The dispersion relation of the magnon in a one-dimensional liquid is periodic in the wave number qqq with the period of 2\ensuremathπn2\ensuremath{\pi}n2\ensuremathπn, determined by the density nnn of the liquid. Dynamic correlation functions, such as, e.g., spin-spin correlation function, exhibit power-law singularities at the magnon spectrum, \ensuremathω\ensuremath→\ensuremathωm(q,n)\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{m}(q,n)\ensuremathω\ensuremath→\ensuremathωm​(q,n). Without using any specific model of the interparticle interactions, we relate the corresponding exponents to independently measurable quantities \ensuremath∂\ensuremathωm/\ensuremath∂q\ensuremath{\partial}{\ensuremath{\omega}}_{m}/\ensuremath{\partial}q\ensuremath∂\ensuremathωm​/\ensuremath∂q and \ensuremath∂\ensuremathωm/\ensuremath∂n\ensuremath{\partial}{\ensuremath{\omega}}_{m}/\ensuremath{\partial}n\ensuremath∂\ensuremathωm​/\ensuremath∂n.
We investigate the propagation of spin excitations in a one-dimensional ferromagnetic Bose gas. While the spectrum of longitudinal spin waves in this system is soundlike, the dispersion of transverse spin excitations is quadratic, making a direct application of the Luttinger liquid theory impossible. By using a combination of different analytic methods we derive the large time asymptotic behavior of the spin-spin dynamical correlation function for strong interparticle repulsion. The result has an unusual structure associated with a crossover from the regime of trapped spin wave to an open regime and does not have analogues in known low-energy universality classes of quantum 1D systems.
The achievement in cooling alkali atomic gases, such as 87^{87}87Rb, 23^{23}23Na and 7^{7}7Li, to quantum degeneracy opens up a way to study magnetism in spinor bosons, because these constituent atoms usually have a hyperfine spin degree of freedom. This article reviews several basic problems related to the ferromagnetic phase transition in spinor atomic Bose gases from a theoretical perspective. After a brief discussion on various possible origins of the ferromagnetic interaction, the phase diagram of the ferromagnetically coupled spinor bosons is investigated. It is found that the ferromagnetic transition occurs always above Bose-Einstein condensation and the Bose condensate is fully polarized. The low-lying collective excitations of the spinor condensate, including spin and density modes, are discussed. The spectrum of the density mode is of the Bogliubov form and the spin wave spectrum has a k2k^2k2-formed dispersion relation at long wavelengths. The spin-wave stiffness coefficient contains contribut
The explicitly soluble Luttinger model is used as a basis for the description of the general interacting Fermi gas in one dimension, which will be called 'Luttinger liquid theory', by analogy with Fermi liquid theory. The excitation spectrum of the Luttinger model is described by density-wave, charge and current excitations; its spectral properties determine a characteristic parameter that controls the correlation function exponents. These relations are shown to survive in non-soluble generalisations of the model with a non-linear fermion dispersion. It is proposed that this low-energy structure is universal to a wide class of 1D systems with conducting or fluid properties, including spin chains.
The explicitly soluble Luttinger model is used as a basis for the description of the general interacting Fermi gas in one dimension, which will be called 'Luttinger liquid theory', by analogy with Fermi liquid theory. The excitation spectrum of the Luttinger model is described by density-wave, charge and current excitations; its spectral properties determine a characteristic parameter that controls the correlation function exponents. These relations are shown to survive in non-soluble generalisations of the model with a non-linear fermion dispersion. It is proposed that this low-energy structure is universal to a wide class of 1D systems with conducting or fluid properties, including spin chains.
We investigate the dynamics of the one-dimensional strongly repulsive spin-1/2 Bose-Hubbard model for filling nu <or= 1. While at nu=1 the system is a Hubbard-Mott insulator exhibiting dynamical properties of the Heisenberg ferromagnet, at nu<1 it is a ferromagnetic liquid with complex spin dynamics. We find that close to the insulator-liquid transition the system admits for a complete separation of spin and density degrees of freedom valid at all energy and momentum scales within the t-J approximation. This allows us to derive the propagator of transverse spin waves and the shape of the magnon peak in the dynamic spin structure factor.