An exactly solvable model describing the low density limit of the spin-1 bosons in a onedimensional optical lattice is proposed. The exact Bethe ansatz solution shows that the low energy physics of this system is described by a quantum liquid of spin singlet bound pairs. Motivated by the exact results, a mean-field approach to the corresponding three-dimensional system is carried out. Condensation of singlet pairs and coexistence with ordinary Bose-Einstein condensation are predicted.

This article presents an elementary introduction on various aspects of the prototypical integrable model the Lieb-Liniger Bose gas ranging from the cooperative to the collective features of many-body phenomena. In 1963 Lieb and Liniger first solved this quantum field theory many-body problem using the Bethe's hypothesis, i.e. a particular form of wave function introduced by Bethe in solving the one-dimensional Heisenberg model in 1931. Despite the Lieb-Liniger model is arguably the simplest exactly solvable model, it exhibits rich quantum many-body physics in terms of the aspects of mathematical integrability and physical universality. Moreover, the Yang-Yang grand canonical ensemble description for the model provides us with a deep understanding of quantum statistics, thermodynamics and quantum critical phenomena at the many-body physics level. Recently, such fundamental physics of this exactly solved model has been attracting growing interest in experiments. Since 2004, there have been more than 20 experimental papers that report novel observations of different physical aspects of the Lieb-Liniger model in the lab. So far the observed results to date are seen to be in excellent agreement with results obtained using the analysis of this simplest exactly solved model. Those experimental observations reveal the unique beauty of integrability.

The free fermion nature of interacting spins in one dimensional (1D) spin chains still lacks a rigorous study. In this letter we show that the length-1 spin strings significantly dominate critical properties of spinons, magnons and free fermions in the 1D antiferromagnetic spin-1/2 chain. Using the Bethe ansatz solution we analytically calculate exact scaling functions of thermal and magnetic properties of the model, providing a rigorous understanding of the quantum criticality of spinons. It turns out that the double peaks in specific heat elegantly mark two crossover temperatures fanning out from the critical point, indicating three quantum phases: the Tomonaga-Luttinger liquid (TLL), quantum critical and fully polarized ferromagnetic phases. For the TLL phase, the Wilson ratio RW = 4Ks remains almost temperature-independent, here Ks is the Luttinger parameter. Furthermore, applying our results we precisely determine the quantum scalings and critical exponents of all magnetic properties in the ideal 1D spin-1/2 antiferromagnet Cu(C4H4N2)(NO3)2 recently studied in Phys. Rev. Lett. 114, 037202 (2015)]. We further find that the magnetization peak used in experiments is not a good quantity to map out the finite temperature TLL phase boundary. Of central importance to the study of the 1D spin-1/2 antiferromagnetic Heisenberg chain is the understanding of spin excitations [1,[3][4][5][6][7][8][9][10][11][12][13]. Elementary spin excitations in this model may exhibit quasi-particle behaviour which is described by spinons carrying half a unit of spin. Such fractional quasiparticles are responsible for the TLL in the model [10,14,15].Regarding to the Bethe ansatz solution of the 1D spin-1/2 chain, a significant development is Takahashi's discovery of spin string patterns [2], i.e., magnon bound states with different string lengths. Takahashi's spin strings give one full access to the thermodynamics of the model through Yang and Yang's grand canonical approach [18], namely the so-called thermodynamic Bethe ansatz (TBA) equations [2]. However, the problems of how such spin strings determine the free fermion nature of spinons and how spin strings comprise universal scalings of thermal and magnetic properties still lack a rigorous understanding. In this paper we present a full answer to these questions.Using spin string solutions to the TBA equations, we obtain the following results: I) we obtain exact scaling functions, critical exponents and a benchmark of quantum magnetism for the 1D spin-1/2 Heisenberg chain, revealing the microscopic origin of the quasiparticle spinons, free fermions and magnons that emerge in different physical regimes; II) We find that the Wilson ratio [19,20], the ratio between the susceptibility χ and the specific heat c v divided by the temperature T ,πkB gµB 2 χ/(c v /T ), significantly characterises the TLL of spinons and marks the crossover temperature between the quantum critical phase and the TLL [21], see Fig. 1. When the magnetic field is larger than the saturation field, dilute magnon beh...