Quantal molecular description and universal aspects of the spectra of bosons and fermions in the lowest Landau level arXiv:1001.1090v1 [cond-mat.quant-gas] 7 Jan 2010 Constantine Yannouleas ∗ and Uzi Landman † School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430 (Dated: 29 November 2009) Through the introduction of a class of trial wave functions portraying combined rotations and vibrations of molecules formed through particle localization in concentric polygonal rings, a cor- related basis is constructed that spans the translationally invariant part of the lower-Landau-level (LLL) spectra. These trial functions, referred to as rovibrational molecular (RVM) functions, gen- eralize our previous work which focused exclusively on electronic cusp states, describing them as pure vibrationless rotations. From a computational viewpoint, the RVM correlated basis enables controlled and systematic improvements of the original strongly-correlated variational wave func- tion. Conceptually, it provides the basis for the development of a quantal molecular description for the full LLL spectra. This quantal molecular description is universal, being valid for both bosons and fermions, for both the yrast and excited states of the LLL spectra, and for both low and high angular momenta. Furthermore, it follows that all other translationally invariant trial functions (e.g., the Jastrow-Laughlin, compact composite-fermion, or Moore-Read functions) are reducible to a description in terms of an excited rotating/vibrating quantal molecule. PACS numbers: 03.75.Hh, 03.75.Lm, 73.43.-f, 73.21.La ticipated that small (and/or mesoscopic) assemblies of ul- tracold bosonic atoms will become technically available in the near future [32–35] and that they will provide an ex- cellent vehicle [27, 28, 31–35] for experimentally reaching exotic phases and for testing the rich variety of proposed LLL trial wave functions. Despite the rich literature and unabated theoretical interest, a unifying physical (as well as mathematical) description of the full LLL spectra (including both yrast [36] and all excited states), however, is still missing. In this paper, a universal theory for the LLL spectra of a finite number of particles valid for both statistics (i.e., for both bosons and fermions) is introduced. The LLL spectra are shown to be associated with fully quantal [37] and strongly correlated ro-vibrational molecular (RVM) states, i.e., with (analytic) trial functions describing vi- brational excitations relative to the set of the special yrast states known as cusp states. The cusp states exhibit enhanced stability and magic angular momenta (see below), and as such they have at- tracted considerable attention. However, the cusp states represent only a small fraction of the LLL spectrum. The molecular trial functions associated with them are purely rotational (i.e., vibrationless) and were introduced for the case of electrons in Ref. [12] under the name rotating electron molecules (REMs). The corresponding purely rotational bosonic analytic trial functions for cusp states [called rotating boson molecules (RBMs)] are introduced in this paper; see Section II A. More importantly, this pa- per shows that the quantal molecular description can be extended to all other LLL states (beyond the special cusp states) by introducing (see Section II D) analytic expres- sions for trial functions representing ro-vibrational exci- tations of both REMs and RBMs. These ro-vibrational trial functions include the REM or RBM expressions as a special case, and they will be referred to in general as More recently, the emerging field of graphene quantum dots [20, 21], and the burgeoning field of rapidly rotat- ing trapped ultracold neutral gases [22–34] have gener- ated significant interest pertaining to strongly correlated states in the lowest Landau level. Furthermore, it is an- Following the discovery [1] of the fractional quantum Hall effect (FQHE) in two-dimensional (2D) semiconduc- tor heterostructures under high magnetic fields ( B ) in the 1980’s, the description of strongly correlated elec- trons in the lowest Landau level (LLL) developed into a major branch of theoretical condensed matter physics [2–19]. Early on, it was realized that the essential many- body physics in the LLL could be captured through trial wave functions. Prominent examples are the Jastrow- type Laughlin (JL) [2], composite fermion (CF) [6], and Moore and Read’s (MR) [7] Pfaffian functions, represent- ing quantum-liquid states [2]. In the last ten years, the field of semiconductor quantum dots [15] helped to focus attention on finite systems with a small number ( N ) of electrons. Theoretical investigations of such finite sys- tems led to the introduction of “crystalline”-type LLL trial functions referred to as rotating electron molecules [12, 15] (REMs). In particular in their intrinsic frame of reference, the REMs describe electrons localized at the apexes of concentric polygonal-ring configurations ( n$_{1}$, n$_{2}$, ..., n$_{r}$ ), where ∑ r q $_{=1}$n$_{q}$ = N and r is the num- ber of concentric rings. I. INTRODUCTION A. Motivation $^{∗}$Electronic address: Constantine.Yannouleas@physics.gatech.edu $^{†}$Electronic address: Uzi.Landman@physics.gatech.edu