Phase diagrams of one-dimensional half-filled two-orbital $\mathrm{SU}\left(N\right)$ cold fermion systems

We investigate possible realizations of exotic $\mathrm{SU}\left(N\right)$ symmetry-protected topological (SPT) phases with alkaline-earth cold fermionic atoms loaded into one-dimensional optical lattices. A thorough study of two-orbital generalizations of the standard $\mathrm{SU}\left(N\right)$ Fermi-Hubbard model, directly relevant to recent experiments, is performed. Using state-of-the-art analytical and numerical techniques, we map out the zero-temperature phase diagrams at half-filling and identify several Mott-insulating phases. While some of them are rather conventional (nondegenerate, charge-density wave, or spin-Peierls-like), we also identify, for even $N$, two distinct types of SPT phases: an orbital Haldane phase, analogous to a spin-$N/2$ Haldane phase, and a topological $\mathrm{SU}\left(N\right)$ phase, which we fully characterize by its entanglement properties. We also propose sets of nonlocal order parameters that characterize the $\mathrm{SU}\left(N\right)$ topological phases found here.

Figure 1

The two-leg ladder representation of the $g\text{−}e$ model (2). Two single-band $\mathrm{SU}\left(N\right)$ Hubbard chains are coupled to each other only by the interchain particle exchange ($V_{\mathrm{ex}}^{g\text{−}e}$) and the interchain density-density interaction ($V$). Note that splitting of a single physical chain into two is fictitious.

Figure 2

The two-leg ladder representation of the $p$-band model (27). On top of the interactions included already in the $g\text{−}e$ model, pair-hopping processes between the two orbitals are allowed.

Figure 3

The ratio $U_1/U_2$ for anharmonic potential (30) obtained by solving the Schrödinger equation (23b) numerically.

Figure 4

Four translationally invariant Mott states for $N=2$: (a) spin Haldane (SH), (b) orbital Haldane (OH), (c) charge Haldane (CH), and (d) rung-singlet (RS) phases (see also Appendix pp4). Singlet bonds formed between spins (orbital pseudospins) are shown by thick solid (dashed) lines [singlet bonds are not shown in (c)]. Dashed ovals (rectangles) denote spin singlets (triplets).

Figure 5

Two density-wave states for $N=2$. In-phase and out-of-phase combinations of two density waves in $g$ and $e$ orbitals, respectively, form (a) CDW and (b) ODW.

Figure 6

The $N$-leg ladder representation of the model (2). $N$ Hubbard-type chains for “spinful” fermions ($g$ and $e$) are coupled to each other by $U$ (interchain density-density interaction among like fermions), $V$ (that between $g$ and $e$), and the interchain Hund couplings ($V_{\mathrm{ex}}^{g\text{−}e}$).

Figure 7

$\mathrm{SU}\left(N\right)$ VBS states are constructed out of a pair of self-conjugate representations at each site. Dashed lines denote maximally entangled pairs. (a) SU(4) with 20-dimensional representation and (b) SU(6) with 175-dimensional representation. (a)${}^{\prime }$ and (b)${}^{\prime }$ are the corresponding matrix-product states.

Figure 8

General phase diagrams for the $N=6$ generalized Hund model (18) obtained by solving numerically the one-loop RG equations (73) with initial conditions (71). $T$-coupling constants $(J_z,J,U)$ are parametrized by $(\theta ,\varphi )$ as Eqs. (118) and the meaning of the extra bold lines is discussed in the text. The signs of $J_z$, $J$, and $U$ in each quadrant are indicated. In the region shown as “no-DSE,” RGE flow does not exhibit dynamical symmetry enlargement. For other abbreviations, see Table 3.

Figure 9

Phase diagram for the $N=6$ generalized Hund model (18) with SU(2)${}_{\mathrm{o}}$ symmetry obtained by solving numerically the one-loop RG equations (73) with initial conditions (71).

Figure 10

Phase diagram for the $N=6$ generalized Hund model (18) obtained by solving numerically the one-loop RG equations (73) with initial conditions (71). From top to bottom, and from right to left, $J_z/t=-0.03$, $J_z/t=0$, and $J_z/t=0.03$.

Figure 11

Phase diagram for the $N=6$ $g\text{−}e$ model (2) obtained by solving numerically the one-loop RG equations (73) with initial conditions (72). From top to bottom, and from right to left: SU(2)${}_{\mathrm{o}}$ symmetry, $V_{\mathrm{ex}}^{g\text{−}e}/t=-0.06$, $V_{\mathrm{ex}}^{g\text{−}e}/t=0$, and $V_{\mathrm{ex}}^{g\text{−}e}/t=0.02$.

Figure 12

Phase diagram for $N=2$ $g\text{−}e$ model (2) obtained by DMRG. Top and bottom panels correspond, respectively, to $V_{\mathrm{ex}}^{g\text{−}e}/t=-1$ and $V_{\mathrm{ex}}^{g\text{−}e}/t=1$. Due to the symmetry Eq. (39) (which exists only for $N=2$), CH and OH, as well as CDW and ODW, appear in a symmetrical way with respect to the symmetry axis $V=V_{\mathrm{ex}}^{g\text{−}e}/2$ indicated with a dashed line.

Figure 13

Phase diagram for $N=2$ $p$-band model (29) obtained by DMRG. Note the mapping (35) $(U_1,U_2)\to (-U_1,-U_2)$ which interchanges spin and charge. The line $U_1=3U_2$ corresponds to the axially symmetric trapping scheme. The two other lines denote the transitions and are compatible with the expected $c=2$ Luttinger-liquid behavior.

Figure 14

Phase diagrams for $N=4$ $g\text{−}e$ model (2) obtained by DMRG. From top to bottom, panels correspond, respectively, to $V_{\mathrm{ex}}^{g\text{−}e}/t=-1$, $V_{\mathrm{ex}}^{g\text{−}e}/t=0$, and $V_{\mathrm{ex}}^{g\text{−}e}/t=1$. Symbols correspond to the numerical data obtained by DMRG with $L=36$ while colored regions and dashed lines indicate the one-loop numerical RG results.

Figure 15

DMRG results for the local fermion densities and kinetic energies for each flavor $\alpha =1,...,4$ in the $N=4$ case. Panel (a) corresponds to the $g\text{−}e$ model with $U/t=8$, $V=0$, and $V_{\mathrm{ex}}/t=1$ on the $L=54$ chain. Panel (b) corresponds to the $p$-band model with $U_1/t=12$ and $U_2/t=4$ on the $L=54$ chain. The presence of localized edge states is clearly visible in both cases.

Figure 16

Phase diagram for the $N=4$ $g\text{−}e$ model (2) with SU(2)${}_{\mathrm{o}}$ symmetry, i.e., $U_{mm}-V=V_{\mathrm{ex}}^{g\text{−}e}$. Symbols correspond to the numerical data obtained by DMRG with $L=36$ while colored regions and dashed lines indicate the one-loop numerical RG results. We also plot the special line $V=U_{mm}/5$ (see text).

Figure 17

Phase diagram for the $N=4$ generalized Hund model (18) with SU(2)${}_{\mathrm{o}}$ symmetry. Symbols correspond to the numerical data obtained by DMRG with $L=36$ while colored regions and dashed lines indicate the one-loop numerical RG results. We also plot the special line $J=4U/3$ (see text).

Figure 18

Phase diagram for the $N=4$ generalized Hund model (18) with $J_z=0$. Symbols correspond to the numerical data obtained by DMRG with $L=36$ while colored regions and dashed lines indicate the one-loop numerical RG results. We also plot the special line $J=U$ where the model can be mapped onto the $N=4$ $p$-band model.

Figure 19

Phase diagram for the $N=4$ generalized Hund model (18) with $J_z/t=4$ (a) and $J_z/t=-4$ (b). Symbols correspond to the numerical data obtained by DMRG with $L=36$. Dashed-dotted lines correspond to SU(2)${}_{\mathrm{o}}$ lines when $J=J_z$. The critical region for $J_z/t=-4$ is almost invisible on this scale (see text).

Figure 20

Absolute values of the transverse and longitudinal pseudospin correlation functions measured from the middle of the chain on a $L=72$ system with parameters $U=0$, $J/t=-4.1$, and $J_z/t=-4$. Both can be fitted with a similar Luttinger parameter $K_{\mathrm{o}}\simeq 1.09>1$ and appropriate functional forms (see text).

Figure 21

Von Neumann entanglement entropy $S_{\mathrm{vN}}$ of a block of $x$ sites (starting from the left open edge) vs conformal distance $d\left(x\right|L)=(L/\pi )sin(\pi x/L)$ for $U=0$ and $J/t=-4$ with varying parameters $J_z$ from the SU(2)${}_{\mathrm{o}}$ point $J_z=J$ with OH phase to the $J_z=0$ RS singlet phase. In the intermediate region, there is an extended critical gapless phase compatible with $c=1$ central charge.

Figure 22

Phase diagram for half-filled $N=4$ $p$-band model (27) obtained by DMRG on $L=32$. Dashed line corresponds to the condition $U_1=3U_2$ satisfied for an axially symmetric trap. $U_2=0$ correspond to two decoupled SU(4) Hubbard chains (see text).

Figure 23

Contour plots of squared wave functions $|{\varphi }_{n_x,n_y}{|}^2$ for three orbitals $(n_x,n_y)=(0,0)$, $(1,0)$, and $(0,1)$.

Figure 24

Phase diagram for $N=2$ $p$-band model (29) obtained by solving numerically the one-loop RG equations (D5) with initial conditions (D4). The line $U_1=3U_2$ corresponds to the axially symmetric trapping scheme.