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arXiv:1811.08804 · Physical Review A

Bethe-ansatz analysis of near-resonant two-component systems in one dimension

Tianhao Ren, I. L. Aleǐner

Published Feb 11, 2019• 3 citations
arXivHTMLPDF

Abstract

We introduce a type of model of two-component systems in one dimension that can be analyzed by the Bethe-ansatz approach. The interspecies interactions are tunable via Feshbach resonant interactions, and the Yang-Baxter equation for exact solvability is fulfilled by fine-tuning the resonant energies. Although the strict exact solvability beyond the level of two-body scatterings would require extra singular counterterms, the models introduced here can still be well described by the Bethe ansatz for relatively small densities. The resulting systems can be described by introducing intraspecies repulsive and interspecies attractive couplings ${c}_{1}$ and ${c}_{2}$. This kind of system admits two types of interesting solutions: In the regime with ${c}_{1}>{c}_{2}$, the ground state is a Fermi sea of two-strings, where the Fermi momentum $Q$ is constrained to be smaller than a certain value ${Q}^{*}$, and it provides an ideal scenario to realize the BCS-BEC crossover (from weakly attractive atoms to weakly repulsive molecules) in one dimension. In the opposite regime with ${c}_{1}<{c}_{2}$, the ground state is a single bright soliton even for fermionic atoms, which reveals itself as an embedded string solution.

DOI
10.1103/physreva.99.023611

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We investigate the ground-state properties of two-component Bose gases confined in one-dimensional harmonic traps in the scheme of density-functional theory. The density-functional calculations employ a Bethe-ansatz-based local-density approximation for the correlation energy, which accounts for the correlation effect properly from the weakly interacting regime to the strongly interacting regime. For the binary Bose mixture with spin-independent interaction, the homogeneous reference system is exactly solvable by the Bethe-ansatz method. Within the local-density approximation, we determine the density distribution of each component and study its evolution from Bose distributions to Fermi-like distribution with the increase in interaction. For the binary mixture of Tonks-Girardeau gases with a tunable interspecies repulsion, with a generalized Bose-Fermi transformation we show that the Bose mixture can be mapped into a two-component Fermi gas, which corresponds to exact soluble Yang-Gaudin model for the homogeneous system. Based on the ground-state energy function of the Yang-Gaudin model, the ground-state density distributions are calculated for various interspecies interactions. It is shown that with the increase in interspecies interaction, the system exhibits composite-fermionization crossover.
Few and many-body systems usually feature interesting and novel behaviors compared with their counterparts in three dimensions. On one hand, low dimensional physics presents challenges due to strong interactions and divergences in the perturbation theory; On the other hand, there exist powerful theoretical tools such as the renormalization group and the Bethe ansatz. In this thesis, I discuss two examples: three interacting bosons in two dimensions and interacting bosons/fermions in one dimension. In both examples, there are intraspecies repulsion as well as interspecies attraction, producing a rich spectrum of phenomena. In the former example, a universal curve of three-body binding energies versus scattering lengths is obtained efficiently by evolving a matrix renormalization group equation. In the latter example, exact solutions for the BCS-BEC crossover are obtained and the unexpected robust features in their excitation spectra are explained by a comprehensive semiclassical analysis.
Exactly solvable models of ultracold Fermi gases are reviewed via their thermodynamic Bethe ansatz solution. Analytical and numerical results are obtained for the thermodynamics and ground state properties of two- and three-component one-dimensional attractive fermions with population imbalance. New results for the universal finite temperature corrections are given for the two-component model. For the three-component model, numerical solution of the dressed energy equations confirms that the analytical expressions for the critical fields and the resulting phase diagrams at zero temperature are highly accurate in the strong coupling regime. The results provide a precise description of the quantum phases and universal thermodynamics which are applicable to experiments with cold fermionic atoms confined to one-dimensional tubes.
Exactly solvable models of ultracold Fermi gases are reviewed via their thermodynamic Bethe Ansatz solution. Analytical and numerical results are obtained for the thermodynamics and ground state properties of two- and three-component one-dimensional attractive fermions with population imbalance. New results for the universal finite temperature corrections are given for the two-component model. For the three-component model, numerical solution of the dressed energy equations confirm that the analytical expressions for the critical fields and the resulting phase diagrams at zero temperature are highly accurate in the strong coupling regime. The results provide a precise description of the quantum phases and universal thermodynamics which are applicable to experiments with cold fermionic atoms confined to one-dimensional tubes.
Using Yang and Yang's particle–hole description, we present a thorough derivation of the thermodynamic Bethe ansatz equations for a general SU (κ) fermionic system in one dimension for both the repulsive and attractive regimes under the presence of an external magnetic field. These equations are derived from Sutherland's Bethe ansatz equations by using the spin–string hypothesis. The Bethe ansatz root patterns for the attractive case are discussed in detail. The relationship between the various phases of the magnetic phase diagrams and the external magnetic fields is given for the attractive case. We also give a quantitative description of the ground-state energies for both strongly repulsive and attractive regimes.
Trapped two-component Fermi gases allow the investigation of the so-called BCS-BEC crossover by tuning the interspecies atom-atom sss-wave scattering length a(aa){a}^{(\mathit{aa})}a(aa) from attractive to repulsive, including vanishing and infinitely large values. Here, we numerically determine the energy spectrum of the equal-mass spin-balanced four-fermion system---the smallest few-particle system that exhibits BCS-BEC crossoverlike behavior---as a function of a(aa){a}^{(\mathit{aa})}a(aa) using the stochastic variational approach. For comparative purposes, we also treat the two- and three-particle systems. States with vanishing and finite total angular momenta as well as with natural and unnatural parities are considered. In addition, the energy spectrum of weakly attractive and weakly repulsive gases is characterized by employing a perturbative framework that utilizes hyperspherical coordinates. The hyperspherical coordinate approach allows the straightforward assignment of quantum numbers and furthermore provides great insight into the strongly interacting unitary regime.