Generalization of a Fermi Liquid to a Liquid with Fractional Exclusion Statistics in Arbitrary Dimensions: Theory of a Haldane Liquid

Abstract

In 1957 Landau [1] introduced the famous concept of quasiparticles with Fermi-Dirac statistics (FDS) in a Fermi liquid in order to describe electrons in a metal where the interaction between electrons was believed to be very significant, as a generalization of the Sommerfeld theory for a degenerate Fermi gas [2]. This description was later justified by Luttinger [3]. Recently Haldane [4] introduced the concept of fractional exclusion statistics (FES) which quasiparticles in strongly interacting systems in arbitrary dimensions might satisfy as a generalization of Pauli’s exclusion principle. Wu [5] first formulated quantum statistical mechanics (QSM) in the state representation and derived the distribution function for an ideal gas with FES as a generalization of the FD and the Bose-Einstein (BE) distribution functions. [Throughout this Letter the FES is referred to as the Haldane-Wu statistics (HWS) such as FDS and BES, while a gas or liquid with HWS is called a Haldane gas or liquid such as a gas or liquid with FDS (BES) was called a Fermi (Bose) gas or liquid.] This concept has played a very important role to understand strongly interacting system such as the TomonagaLuttinger model (TLM) [6], the Calogero-Sutherland model (CSM) [7], and the Haldane-Shastry model (HSM) [8] in one dimension and the fractional quantum Hall effect in two dimensions [9]. Especially, notable is that Dasnieres de Veigy and Ouvry [10] revealed a deep connection between the FES and the fractional quantum Hall system, while Bernard and Wu [11] and Isakov [12] found the one between the FES and the CSM. More recently, QSM formulation has been developed further by Nayak and Wiczek [13], Isakov, Arovas, Myrheim, and Polychronakos [14], and the author [15]. Here, the QSM formulation allows us to evaluate the equation of state for an ideal gas with HWS in arbitrary dimensions with obtaining all the exact cluster coefficients in the cluster expansion [14,15]. However, interacting quasiparticles with HWS in arbitrary dimensions have never been investigated yet, and therefore the concept of a Haldane liquid is still missing. In this Letter I will explore this concept as a generalization of the Landau’s Fermi liquid theory and generalize the Luttinger’s theorem for a Fermi liquid to that for a Haldane liquid.