Ground state of the one-dimensional Bose gas with Feshbach-resonant interactions above the critical density: zero-temperature Yang-Yang equations with a nonmonotonic kernel

The one-dimensional Bose gas whose interactions are dominated by a narrow Feshbach resonance was solved exactly at zero temperature by Gurarie [Phys. Rev. A 73, 033612 (2006)] using the asymptotic Bethe ansatz, but only below a critical density: there the dressed energy develops a roton-like minimum at the resonant momentum which, at the critical density, dips below zero and destabilizes the standard filled rapidity sea. The correct zero-temperature Yang-Yang equations for the regime above the critical density were formulated (Eqs. (78)-(79) of that paper) but have never been solved, because the resonant scattering kernel $K(\lambda)$ is nonmonotonic (peaked at the resonant momentum transfer): the standard iteration-convergence proof uses monotonicity of $K$ and breaks down, and any naive iterative numerical scheme diverges. The open problem is to determine the true ground state, the occupied-rapidity filling pattern and the excitation branches above the critical density in both the small-$c$ and large-$c$ regimes, and to derive analytic critical criteria at large $c$ analogous to the known small-$c$ formulas. No resolution of this problem is known; it remains open.

Background

The Lieb-Liniger model - bosons on a line interacting through a repulsive delta-function potential - is exactly solved by the Bethe ansatz (Lieb and Liniger, 1963; Lieb, 1963). The Bethe ansatz labels many-body eigenstates by rapidities $\lambda_j$: the momenta the particles carry when far apart. In the thermodynamic limit the ground state occupies rapidities in a single interval $[-\lambda_F,\lambda_F]$; the density of occupied states obeys the Lieb-Liniger integral equation, and particle and hole excitations are governed by a dressed energy $\epsilon(\lambda)$, the energy of a particle added at rapidity $\lambda$ after its interactions with the occupied sea are included. Thermodynamics at finite temperature follows from the nonlinear integral equation of Yang and Yang (1969); at zero temperature it reduces to a self-consistency condition in which exactly the states with $\epsilon(\lambda)<0$ are occupied. All of these constructions rest on a property of the kernel $K(\lambda)=d\,\theta(\lambda/2)/d\lambda$, the derivative of the two-body scattering phase shift $\theta$: for delta-function interactions $K(\lambda)=2c/(c^2+\lambda^2)$ is positive and monotonically decreasing for $\lambda>0$, and this monotonicity underlies the standard proof that the nonlinear integral equations can be solved by iteration (Korepin, Bogoliubov and Izergin, 1993).

Real atomic scattering is not of pure delta type. A Feshbach resonance - the coupling of two colliding atoms to a closed-channel molecular state whose energy (detuning) is tunable by a magnetic field - produces a scattering phase shift with strong energy dependence, i.e. a finite effective range; when the resonance is narrow this structure cannot be absorbed into a contact coupling. Gurarie (2006) studied the one-dimensional two-channel Hamiltonian $H=\int dx\left[\frac{1}{2m_a}\partial_x a^\dagger\partial_x a+\frac{1}{4m_a}\partial_x b^\dagger\partial_x b+\epsilon_0\,b^\dagger b+\frac{c}{2}a^{\dagger 2}a^2+\frac{g}{\sqrt{2}}\left(b\,a^{\dagger 2}+b^\dagger a^2\right)\right],$ describing bosonic atoms $a$ of mass $m_a$ and molecules $b$ of mass $2m_a$ that convert into each other at rate $g$, with detuning $\epsilon_0$, an atom-atom contact repulsion $c$, and restricted to the no-bound-state regime $\epsilon_0-g^2/c>0$ (Eq. (2) of that paper). The model was solved at zero temperature by the asymptotic Bethe ansatz: the many-body wavefunction is built from the exact resonant two-body S-matrix, which is legitimate when the interparticle spacing is large compared to the interaction range so that simultaneous three-particle encounters are rare; the neglected three-body processes were estimated to be small in the studied low-density window (Appendix E there).

The solution showed that at higher density the resonant gas differs qualitatively from the delta-function gas. Three regimes exist depending on $c$ relative to the resonance scale $\sqrt{\epsilon_0/m_a}$. In the regime $c\sim\sqrt{\epsilon_0/m_a}$ the Fermi momentum can never exceed a certain cutoff. In the regimes of small or large $c$, as the density grows the dressed energy $\epsilon(\lambda)$ develops a local (roton-like, recall the famous minimum of the superfluid helium excitation curve) minimum at the resonant momentum, where atom pairs are resonant with the molecular level. At a critical density $D_{\rm crit}$ this minimum touches zero, and beyond it the minimum is negative: the state with only $[-\lambda_F,\lambda_F]$ filled is no longer the ground state, and additional rapidity intervals around the resonant momentum are expected to fill. At small $c$ the critical point is known in closed form, $D_{\rm crit}=\left(\frac{9m_a\epsilon_0^3}{2(c\epsilon_0-g^2)}\right)^{1/3},\qquad \lambda_F^{\rm crit}=\left(6m_a^2(c\epsilon_0-g^2)\right)^{1/3},$ (Eqs. (4)-(5) there), but no analogous analytic result exists at large $c$.

The kernel of the resonant problem, $K(\lambda)=d\,\theta(\lambda/2)/d\lambda$, is no longer monotone: for small $c$ it is a sum of Lorentzians centered at $0$ and $\pm 2\lambda_0$, and for large $c$ two Lorentzians centered at $\pm 2\lambda_1$ (Eqs. (41)-(42) there), where $\lambda_0=\sqrt{2m\epsilon_0}$ and $\lambda_1=\sqrt{2m\epsilon_1}$ with $\epsilon_1=\epsilon_0-g^2/c$ the detuning of the resonant scattering state and $m$ the reduced mass of a colliding pair. For densities above $D_{\rm crit}$ the correct zero-temperature Yang-Yang equation for the dressed energy is $\epsilon(\lambda)-\frac{1}{4\pi}\int_{-\infty}^{\infty}d\mu\;K(\lambda-\mu)\left[\epsilon(\mu)-\left|\epsilon(\mu)\right|\right]=\frac{\lambda^2}{2}-h,$ with chemical potential $h$ (Eq. (78) there), supplemented by the companion density-of-states equation for $\rho(\lambda)$ (Eq. (79) there, understood with the sign bracket evaluated at the integration variable so that the integral is restricted to the occupied region where $\epsilon<0$, as required by its low-density reduction to the Lieb-Liniger equation (47) over the filled interval; the sign function of the outer variable printed in the source is a typographical slip). Because the integrand involves the nonmonotonic kernel, the standard iteration-convergence proof fails and naive iterative schemes were found to diverge; no analytic or controlled numerical solution is known, and the nature of the phase above the critical density, at both small and large $c$, is undetermined.

Problem Statement

Determine the zero-temperature ground state and excitation structure of the one-dimensional Bose gas with Feshbach-resonant interactions - the two-channel Hamiltonian above, in the no-bound-state regime $\epsilon_0-g^2/c>0$ - at densities exceeding the critical density $D_{\rm crit}$ at which the dressed energy of the low-density solution becomes negative near the resonant momentum. Concretely, solve the zero-temperature Yang-Yang equation $\epsilon(\lambda)-\frac{1}{4\pi}\int_{-\infty}^{\infty}d\mu\;K(\lambda-\mu)\left[\epsilon(\mu)-\left|\epsilon(\mu)\right|\right]=\frac{\lambda^2}{2}-h$ together with the companion density-of-states equation $\rho(\lambda)-\frac{1}{4\pi}\int_{-\infty}^{\infty}d\mu\;K(\lambda-\mu)\,\rho(\mu)\left[1-\mathrm{sign}\,\epsilon(\mu)\right]=\frac{1}{2\pi},$ in which the sign bracket, evaluated at the integration variable $\mu$, restricts the integral to the occupied region where $\epsilon(\mu)<0$ (these are Eqs. (78)-(79) of Gurarie, Phys. Rev. A 73, 033612 (2006); the printed Eq. (79) carries the sign function of the outer variable $\lambda$, a typographical slip, since with it the bracket would merely rescale the whole integral instead of selecting the occupied region). The kernel $K$ is the resonant one (Eqs. (40)-(42) there), nonmonotonic on $\lambda>0$. The required results, which jointly characterize the above-critical-density state, are:

1. the filling pattern, i.e. the set of occupied rapidity intervals in the ground state, as a function of the chemical potential $h$ (or density) and the interaction parameters, in both the small-$c$ and large-$c$ regimes, together with the dressed energy $\epsilon(\lambda)$, the density of states $\rho(\lambda)$ and the ground-state energy;

2. the excitation branches above this ground state and their stability, establishing the nature of the phase (or phases) into which the roton-like instability resolves, including the character of the transition at $D_{\rm crit}$;

3. analytic (or controlled asymptotic) criteria for the critical rapidity and critical density at large $c$, analogous to the small-$c$ formulas $D_{\rm crit}=(9m_a\epsilon_0^3/2(c\epsilon_0-g^2))^{1/3}$ and $\lambda_F^{\rm crit}=(6m_a^2(c\epsilon_0-g^2))^{1/3}$.

References

1. V. Gurarie, 'One-dimensional gas of bosons with Feshbach-resonant interactions', Physical Review A 73, 033612 (2006). DOI: 10.1103/PhysRevA.73.033612; arXiv:cond-mat/0511411, https://arxiv.org/abs/cond-mat/0511411
2. E. H. Lieb and W. Liniger, 'Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State', Physical Review 130, 1605 (1963). DOI: 10.1103/PhysRev.130.1605, https://doi.org/10.1103/PhysRev.130.1605
3. E. H. Lieb, 'Exact Analysis of an Interacting Bose Gas. II. The Excitation Spectrum', Physical Review 130, 1616 (1963). DOI: 10.1103/PhysRev.130.1616, https://doi.org/10.1103/PhysRev.130.1616
4. C. N. Yang and C. P. Yang, 'Thermodynamics of a One-Dimensional System of Bosons with Repulsive Delta-Function Interaction', Journal of Mathematical Physics 10, 1115 (1969). DOI: 10.1063/1.1664947, https://doi.org/10.1063/1.1664947
5. V. E. Korepin, N. M. Bogoliubov and A. G. Izergin, 'Quantum Inverse Scattering Method and Correlation Functions', Cambridge University Press (1993). DOI: 10.1017/CBO9780511628832, https://doi.org/10.1017/CBO9780511628832
6. V. Gurarie, 'One dimensional gas of bosons with integrable resonant interactions', arXiv:0912.3204 (2009, unpublished preprint). https://arxiv.org/abs/0912.3204
7. Yu-Wen Lee and Yu-Li Lee, 'Quantum phases of a Feshbach-resonant atomic Bose gas in one dimension', Physical Review A 73, 043606 (2006). DOI: 10.1103/PhysRevA.73.043606; arXiv:cond-mat/0506216, https://arxiv.org/abs/cond-mat/0506216
8. M. A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac and M. Rigol, 'One dimensional bosons: From condensed matter systems to ultracold gases', Reviews of Modern Physics 83, 1405 (2011). DOI: 10.1103/RevModPhys.83.1405; arXiv:1101.5337, https://arxiv.org/abs/1101.5337
9. M. J. Bhaseen, S. Ejima, F. H. L. Essler, H. Fehske, M. Hohenadler and B. D. Simons, 'Discrete symmetry breaking transitions between paired superfluids', Physical Review A 85, 033636 (2012). DOI: 10.1103/PhysRevA.85.033636; arXiv:1111.6778, https://arxiv.org/abs/1111.6778
10. Ran Qi and Xiwen Guan, 'Many-body properties of quasi-one-dimensional boson gas across a narrow CIR', EPL 101, 40002 (2013). DOI: 10.1209/0295-5075/101/40002, https://doi.org/10.1209/0295-5075/101/40002
11. A. Imambekov, A. A. Lukyanov, L. I. Glazman and V. Gritsev, 'Exact Solution for 1D Spin-Polarized Fermions with Resonant Interactions', Physical Review Letters 104, 040402 (2010). DOI: 10.1103/PhysRevLett.104.040402; arXiv:0910.2269, https://arxiv.org/abs/0910.2269
12. Lei Pan, Shu Chen and Xiaoling Cui, 'Many-body stabilization of a resonant p-wave Fermi gas in one dimension', Physical Review A 98, 011603(R) (2018). DOI: 10.1103/PhysRevA.98.011603; arXiv:1801.05902, https://arxiv.org/abs/1801.05902
13. Abhinav Prem and Victor Gurarie, 'One-dimensional model of chiral fermions with Feshbach resonant interactions', Journal of Statistical Mechanics: Theory and Experiment 2018, 023111 (2018). DOI: 10.1088/1742-5468/aaa78d; arXiv:1707.00105, https://arxiv.org/abs/1707.00105
14. D. V. Kurlov, S. I. Matveenko, V. Gritsev and G. V. Shlyapnikov, 'One-dimensional two-component fermions with contact even-wave repulsion and SU(2)-symmetry-breaking near-resonant odd-wave attraction', Physical Review A 99, 043631 (2019). DOI: 10.1103/PhysRevA.99.043631; arXiv:1812.08677, https://arxiv.org/abs/1812.08677
15. Tianhao Ren and Igor Aleiner, 'Bethe-ansatz analysis of near-resonant two-component systems in one dimension', Physical Review A 99, 023611 (2019). DOI: 10.1103/PhysRevA.99.023611, https://doi.org/10.1103/PhysRevA.99.023611
16. Yuta Sekino and Yusuke Nishida, 'Field-theoretical aspects of one-dimensional Bose and Fermi gases with contact interactions', Physical Review A 103, 043307 (2021). DOI: 10.1103/PhysRevA.103.043307; arXiv:2011.11971, https://arxiv.org/abs/2011.11971