The low-energy properties of the one-dimensional anyon gas with a -function interaction are discussed in the context of its Bethe ansatz solution. It is found that the anyonic statistical parameter and the dynamical coupling constant induce Haldane exclusion statistics interpolating between bosons and fermions. Moreover, the anyonic parameter may trigger statistics beyond Fermi statistics for which the exclusion parameter is greater than one. The Tonks-Girardeau and the weak coupling limits are discussed in detail. The results support the universal role of in the dispersion relations. Anyons, which are used to describe particles with generalized fractional statistics [1,2], are becoming of increasing importance in condensed matter physics [3] and quantum computation [4]. The concept of anyons provides a successful theory of the fractional quantum Hall (FQH) effect [5]. In particular, the signature of fractional statistics has recently been observed in experiments on the elementary excitations of a two-dimensional electron gas in the FQH regime [3]. These developments are seen as promising opportunities for further insight into the FQH effect, quantum computation, superconductivity, and other fundamental problems in quantum physics.In one dimension, collision is the only way to interchange two particles. Accordingly, interaction and statistics are inextricably related in 1D systems. The 1D Calogero-Sutherland model is seen to obey fractional exclusion statistics [6,7]. In the sense of Haldane exclusion statistics, the 1D interacting Bose gas is equivalent to the ideal gas with generalized fractional statistics [8,9]. We consider an integrable model of anyons with a -function interaction introduced and solved by Kundu [10]. Here we obtain the low-energy properties and Haldane exclusion statistics of this 1D anyon gas. We find that the low energies, dispersion relations, and the generalized exclusion statistics depend on both the anyonic statistical and the dynamical interaction parameters. The anyonic parameter not only interpolates between Bose and Fermi statistics, but can trigger statistics beyond Fermi statistics in a super Tonks-Girardeau (TG) gaslike phase.Bethe ansatz solution.-We consider N anyons with a -function interaction in one dimension with Hamiltonian [10]

Abstract. We provide evidence in support of a recent proposal by Astrakharchik et al for the existence of a super Tonks-Girardeau gas-like state in the attractive interaction regime of quasi-one-dimensional Bose gases. We show that the super TonksGiradeau gas-like state corresponds to a highly-excited Bethe state in the integrable interacting Bose gas for which the bosons acquire hard-core behaviour. The gas-like state properties vary smoothly throughout a wide range from strong repulsion to strong attraction. There is an additional stable gas-like phase in this regime in which the bosons form two-body bound states behaving like hard-core bosons. § Corresponding author (Murray.Batchelor@anu.edu.au)

We extend the exact periodic Bethe Ansatz solution for onedimensional bosons and fermions with δ-interaction and arbitrary internal degrees of freedom to the case of hard wall boundary conditions. We give an analysis of the ground state properties of fermionic systems with two internal degrees of freedom, including expansions of the ground state energy in the weak and strong coupling limits. PACS numbers: 03.75.Ss, 05.30.Fk, 67.60.-g,71.10.Pm IntroductionOne-dimensional quantum gases with two-particle δ-interaction have long been of fascination. The most simple model of δ-interacting spinless bosons in a periodic box was solved in terms of the Bethe Ansatz by Lieb and Liniger [1]. This quantum mechanical model is not only one of the oldest integrable models after the Heisenberg spin chain, but arguably also one of the most important test beds for exploring new ideas and methods, e.g., the Thermodynamic Bethe Ansatz [2] was pioneered for this model. Earlier Girardeau [3] discussed a mapping from strongly repulsive interacting bosons to fermions, corresponding to the limit c → ∞, where c features as the arbitrary interaction strength in the Lieb-Liniger model. Later in seminal work McGuire [4] discussed δ-interaction particles via an optical analogue. Gaudin [5] and Yang [6] then considered spin-1 2 fermions with periodic boundary conditions, the first model with internal states. Sutherland [7] applied the nested Bethe Ansatz, which allowed the treatment of periodic quantum gases with arbitrary spin by repeated application of the Bethe Ansatz, reducing the number of internal states in each step. Gaudin [8] solved the model of spinless bosons with hard wall boundary conditions.The special form of δ-interaction is at the heart of the integrability of the quantum gases. Models tweaking the type of interaction have been considered, but so far are less prominent [9]. Non-integrable models, like the harmonically trapped gas with tunable interaction strength, have to be treated by approximate methods and simulations. One integrable model, the mixture of fermionic and bosonic particles, solved by Lai and Yang [10] has long been dormant in the literature, but is enjoying new interest ‡

Abstract. We consider the integrable one-dimensional δ-function interacting Bose gas in a hard wall box which is exactly solved via the coordinate Bethe Ansatz. The ground state energy, including the surface energy, is derived from the Lieb-Liniger type integral equations. The leading and correction terms are obtained in the weak coupling and strong coupling regimes from both the discrete Bethe equations and the integral equations. This allows the investigation of both finite-size and boundary effects in the integrable model. We also study the Luttinger liquid behaviour by calculating Luttinger parameters and correlations. The hard wall boundary conditions are seen to have a strong effect on the ground state energy and phase correlations in the weak coupling regime. Enhancement of the local two-body correlations is shown by application of the Hellmann-Feynman theorem.

This article considers recent advances in the investigation of the thermal and magnetic properties of integrable spin ladder models and their applicability to the physics of strong coupling ladder compounds. For this class of compounds the rung coupling J ⊥ is much stronger than the coupling J along the ladder legs. The ground state properties of the integrable two-leg spin-1 2 and the mixed spin-( 1 2 , 1) ladder models at zero temperature are analysed by means of the Thermodynamic Bethe Ansatz (TBA). Solving the TBA equations yields exact results for the critical fields and critical behaviour. The thermal and magnetic properties of the models are discussed in terms of the recently introduced High Temperature Expansion (HTE) method, which is reviewed in detail. In the strong coupling region the integrable spin-1 2 ladder model exhibits three quantum phases: (i) a gapped phase in the regime H < H c1 = J ⊥ − 4J , (ii) a fully polarized phase for H > H c2 = J ⊥ + 4J , and (iii) a Luttinger liquid magnetic phase in the regime H c1 < H < H c2 . The critical behaviour in the vicinity of the critical points H c1 and H c2 is of Pokrovsky-Talapov type. The temperature-dependent thermal and magnetic properties are directly evaluated from the exact free energy expression and compared to known experimental results for the strong coupling ladder compounds (5IAP) 2 CuBr 4 · 2H 2 O, Cu 2 (C 5 H 12 N 2 ) 2 Cl 4 , (C 5 H 12 N) 2 CuBr 4 , BIP-BNO and [Cu 2 (C 2 O 2 )(C 10 H 8 N 2 ) 2 )](NO 3 ) 2 . Similar analysis of the mixed spin-( 1 2 , 1) ladder model reveals a rich phase diagram, with a 1 3 and a full saturation magnetization plateau within the strong antiferromagnetic rung coupling regime. For weak rung coupling, the fractional magnetization plateau is diminished and a new quantum phase †