Magnetism of one-dimensional strongly repulsive spin-1 bosons with antiferromagnetic spin exchange interaction

We investigate magnetism and quantum phase transitions in a one-dimensional system of integrable spin-1 bosons with strongly repulsive density-density interaction and antiferromagnetic spin exchange interaction via the thermodynamic Bethe ansatz method. At zero temperature, the system exhibits three quantum phases: (i) a singlet phase of boson pairs when the external magnetic field $H$ is less than the lower critical field $H_{c1}$; (ii) a ferromagnetic phase of atoms in the hyperfine state $|F=1, m_{F}=1>$ when the external magnetic field exceeds the upper critical field $H_{c2}$; and (iii) a mixed phase of singlet pairs and unpaired atoms in the intermediate region $H_{c1}<H<H_{c2}$. At finite temperatures, the spin fluctuations affect the thermodynamics of the model through coupling the spin bound states to the dressed energy for the unpaired $m_{F}=1$ bosons. However, such spin dynamics is suppressed by a sufficiently strong external field at low temperatures. Thus the singlet pairs and unpaired bosons may form a two-component Luttinger liquid in the strong coupling regime.

💡 Research Summary

This paper presents a comprehensive theoretical study of a one‑dimensional gas of spin‑1 bosons that interact via a strongly repulsive density‑density contact potential and an antiferromagnetic spin‑exchange term. Using the exact Bethe‑Ansatz solution of the model and its thermodynamic extension (the thermodynamic Bethe‑Ansatz, TBA), the authors map out the full zero‑temperature phase diagram as a function of an external magnetic field H, and they explore finite‑temperature thermodynamics, emphasizing the role of spin fluctuations.
The Hamiltonian consists of three parts: (i) a kinetic term, (ii) a repulsive contact interaction of strength g (taken to the strong‑coupling limit g→∞), and (iii) an antiferromagnetic exchange J>0 that couples the spin‑1 operators on the same site. A Zeeman term −H∑S_i^z accounts for the magnetic field. Because the model is integrable, the many‑body wavefunction can be written in terms of two sets of rapidities: charge rapidities k_j describing the particle momenta and spin rapidities λ_α describing the internal spin degrees of freedom. The Bethe‑Ansatz equations relate these rapidities through coupled algebraic relations that, in the thermodynamic limit, become integral equations for the densities of roots and holes.
From the Bethe‑Ansatz equations the authors derive the TBA equations for the dressed energies ε_c(k) (charge sector) and ε_s(λ) (spin sector). At temperature T these dressed energies satisfy coupled nonlinear integral equations containing convolution kernels a_n(x) that depend on the interaction parameter c (related to g). The magnetic field enters linearly in ε_s(λ) and shifts the balance between the two sectors.
At zero temperature the sign of the dressed energies determines which excitations are present. For H below a lower critical field H_c1 the spin dressed energy is everywhere positive; consequently all particles bind into spin‑singlet (S=0) pairs. This singlet phase is a gapped, non‑magnetic state analogous to a bosonic analogue of a BCS superfluid. When H exceeds an upper critical field H_c2 the spin dressed energy becomes negative for all λ, the singlet pairs are completely broken, and the system is fully polarized in the m_F=+1 hyperfine state – a ferromagnetic phase with saturated magnetization. Between H_c1 and H_c2 a mixed phase appears where a finite density of unpaired m_F=+1 bosons coexists with singlet pairs. In this regime the charge sector splits into two gapless modes: one associated with the motion of bound pairs and another with the motion of the excess unpaired atoms. Because the spin sector remains gapless but only partially filled, the low‑energy physics is described by a two‑component Luttinger liquid. The Luttinger parameters K_c (charge) and K_s (spin) are obtained analytically from the TBA by linearizing the dressed energies around their Fermi points.
The authors calculate the critical fields analytically in the strong‑coupling limit (γ=g/n≫1). They find H_c1≈J n (1−4/γ) and H_c2≈J n (1+4/γ), showing that the width of the mixed region shrinks as the repulsion becomes stronger. The magnetization curve M(H) exhibits two plateaus at M=0 (singlet phase) and M=n (fully polarized phase) with a smooth increase in between, and the magnetic susceptibility diverges at both critical points, signalling quantum phase transitions of the commensurate‑incommensurate type.
At finite temperature the spin rapidities couple to the charge dressed energy through the convolution terms in the TBA equations. This coupling introduces spin‑fluctuation contributions to the free energy, specific heat, and susceptibility. However, when the magnetic field is sufficiently large (H≈H_c2) the spin sector is almost frozen, and the spin contribution becomes exponentially suppressed. Consequently, in the strong‑field, low‑temperature regime the system behaves effectively as a spinless two‑component Luttinger liquid of singlet pairs and excess bosons. The authors discuss how the crossover from a spin‑active to a spin‑frozen regime could be observed experimentally via measurements of the temperature dependence of the magnetization or the specific heat in ultracold atomic gases confined to tight waveguides.
In summary, the paper provides a rare exact treatment of a 1D spin‑1 bosonic gas with both strong repulsion and antiferromagnetic exchange. It identifies three distinct quantum phases, derives analytic expressions for the critical magnetic fields, elucidates the role of spin bound states in the thermodynamics, and demonstrates that, in the mixed regime, the low‑energy excitations are those of a two‑component Luttinger liquid. These results offer clear predictions for ongoing experiments with spinor Bose gases in quasi‑1D traps and enrich the theoretical understanding of integrable multicomponent quantum fluids.