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

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 pseudospin $\mathcal{S}=N/2$, making it possible 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 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 et al. [arXiv:0909.4059 (to be published)].

Figure 1

Sketch of the kinks supported by the U($2N$) repulsive Hubbard model that interpolate between the two degenerate dimerized vacua [the boxes indicate an SU($2N$) singlet made of $2N$ fermions]. Here $N=5$. At large $U$, in the low-energy sector, all sites have exactly $N$ fermions. (a) Neutral kinks, that are the only low-energy excitations at large $U$, and transform in the antisymmetric representation ${\omega }_N$. Note that they are their own antiparticle. (b) Charged kinks, that play a role at small $U$, depicted here with charge $Q=-2$.

Figure 2

Phase diagram obtained by the low-energy approach in the $N=2$ case.

Figure 3

Ratio of the coupling constants, to which the RG flow leads in the IR limit, for $N=3$; $\theta$ spans a ring with fixed radius $R=0.1t$ in the phase diagram: $U=R\cos \theta$, $V=R\sin \theta$. Three different regions can be defined as function of $\theta$.

Figure 4

Values of the coupling constants $g_i$ in the infrared limit, in the vicinity of the transition from CDW phase (II) to BCS critical phase (III), close to the SU(2)${}_c$ symmetric line ($V=NU$). Notations are the same as in Fig. 3.

Figure 5

Values of the coupling constants $g_i$ in the IR limit, in the vicinity of the transition between the BCS (III) and SP (I) phases, displaying the restoration of an SU$\left(2\right)$ symmetry, denoted ${\widetilde{\mathrm{SU}\left(2\right)}}_c$, which is the dual of the lattice SU(2)${}_c$ symmetry ($V=NU$): $g_2=-g_4$ and $g_3=-g_5$. We use the same notations as in Fig. 3.

Figure 6

Phase diagram obtained by the low-energy approach in the $N$ odd case ($N>1$).

Figure 7

Phase diagram obtained by the low-energy approach in the $N$ even case ($N>2$).

Figure 8

Phase diagram of the $N=2$ case. Solid lines are DMRG results (see text for discussion) and dashed lines are results from solving numerically the RG flow in the weak-coupling limit. $c$ stands for the expected central charge on the transition line. Lines with higher symmetries are indicated in black dashed lines. Boundaries of the HI phase obtained by the strong-coupling are shown in dotted lines. As a rule of thumb, DMRG results can be trusted if $\left|U\right|,\left|V\right|\gtrsim t$, while weak-coupling predictions are exact close to the free fermions limit at the origin. (Inset) Zoom on the region of the HI phase where the $x$ axis $(2U-V)/t$ is perpendicular to the $\text{SU}{\left(2\right)}_c\times \text{SO}\left(5\right)$ line.

Figure 9

Behavior of the Haldane gap $\Delta$ along the SU(2)${}_c$ line $V=2U<0$ of Fig. 8. (a) Gaps and finite-size scalings (see text for discussion). (b),(c) The Haldane gap as a function of $U/t$ in units of, respectively, $t$ and the effective antiferromagnetic coupling $J$. The ${\Delta }_{S=1}$ line indicates the value of the gap known for a Heisenberg spin-one chain.

Figure 10

Fitting the entanglement entropy close to the critical lines surrounding the Haldane phase provides central charges $c$ close to the expected values $c=1/2$ (for HI-CDW) and $c=1$ (for HI-RS). The best agreement is found using PBC with DMRG and keeping a large number of kept states $m$.

Figure 11

Scaling of the order parameter ${\mathcal{O}}_{\mathrm{SP}}$ at the RS-SP transition. (a) For $U/t=4$, the exponent is quite close to $1/8$; (b) for $U/t=1$, it is closer to 0.8.

Figure 12

Charge and spin gaps and their ratio along the SO(7) line $V=-2U$ of Fig. 8.

Figure 13

Numerical phase diagram obtained by DMRG in the $N=3$ case.

Figure 14

Pairing and density correlations obtained by DMRG in the $N=3$ case for various interactions corresponding (a) to the exact SU(2)${}_c$ symmetry and (b) to the emergent SU(2) symmetry ${\widetilde{\text{SU}\left(2\right)}}_c$. Note that correlations are measured starting from the middle of the chain.

Figure 15

Local densities obtained by DMRG in the $N=3$ case for $U/t=-2$, $V=3U$, and $L=108$. From bottom to top, data correspond to adding two, four, or six particles to the half-filled system. Data for adding four and six particles are shifted by two and four, respectively for clarity, and in these cases ${\langle n\left(i\right)\rangle }_{N_f=3L+2}$ has been substracted in order to get the bulk contribution.

Figure 16

Pair and density correlations in the $N=3$ case for $V/t=-6$ and $L=72$ and various $U/t$. (Inset) Fitting these data gives an estimate of the Luttinger parameter $K_c$ vs $U/t$ (using a log scale starting at $U/t=-2$).

Figure 17

Bond kinetic energy modulation at the center of a chain of length $L$ for various $V$ at fixed $U/t=3$.

Figure 18

Numerical phase diagram obtained by DMRG in the $N=4$ case with $L=30$.

Figure 19

Pairing correlations for $N=4$ at fixed $V/t=-8$ and $L=64$. (a) Critical correlations are observed close to the HI phase represented by $U/t=-2$ on the SU(2)${}_c$ line. (b) Both in HI and in RS phase, the pairing correlations are short ranged.

Figure 20

(a) Density correlations for $N=4$ at fixed $V/t=-8$ and $L=64$ in the BCS phase. (b) Luttinger liquid parameter $K_c$ vs $U$ for fixed $V/t=-8$. BCS phase is delimited by $K_c\ge 1$.