# Competing orders in one-dimensional half-filled multicomponent fermionic cold atoms: The Haldane-charge conjecture

### H. Nonne 1, P. Lecheminant 1, Sylvain Capponi 2, G. Roux 3, E. Boulat 4

#### Physical Review B 84 (2011) 125123

We investigate the nature of the Mott-insulating phases of half-filled 2N-component fermionic cold atoms loaded into a one-dimensional optical lattice. By means of conformal field theory techniques and large-scale DMRG calculations, we show that the phase diagram strongly depends on the parity of $N$. First, we single out charged, spin-singlet, degrees of freedom, that carry a pseudo-spin ${\cal S}=N/2$ allowing to formulate a Haldane conjecture: for attractive interactions, we establish the emergence of Haldane insulating phases when $N$ is even, whereas a metallic behavior is found when $N$ is odd. We point out that the $N=1,2$ cases do \emph{not} have the generic properties of each family. The metallic phase for $N$ odd and larger than 1 has a quasi-long range singlet pairing ordering with an interesting edge-state structure. Moreover, the properties of the Haldane insulating phases with even $N$ further depend on the parity of N/2. In this respect, within the low-energy approach, we argue that the Haldane phases with N/2 even are not topologically protected but equivalent to a topologically trivial insulating phase and thus confirm the recent conjecture put forward by Pollmann {\it et al.} [Pollmann {\it et al.}, arXiv:0909.4059 (2009)].

• 1. Laboratoire de Physique Théorique et Modélisation (LPTM),
CNRS : UMR8089 – Université de Cergy Pontoise
• 2. Laboratoire de Physique Théorique - IRSAMC (LPT),
CNRS : UMR5152 – Université Paul Sabatier - Toulouse III
• 3. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
CNRS : UMR8626 – Université Paris XI - Paris Sud
• 4. Matériaux et Phénomènes Quantiques (MPQ),
CNRS : UMR7162 – CNRS : FR2437 – Université Paris VII - Paris Diderot