Quantum phases of quadrupolar Fermi gases in coupled one-dimensional systems

Following the recent proposal to create quadrupolar gases [Bhongale et al., Phys. Rev. Lett. 110, 155301 (2013)], we investigate what quantum phases can be created in these systems in one dimension. We consider a geometry of two coupled one-dimensional (1D) systems, and derive the quantum phase diagram of ultracold fermionic atoms interacting via quadrupole-quadrupole interactions within a Tomonaga-Luttinger-liquid framework. We map out the phase diagram as a function of the distance between the two tubes and the angle between the direction of the tubes and the quadrupolar moments. The latter can be controlled by an external field. We show that there are two magic angles ${\theta }_{B,1}^c$ and ${\theta }_{B,2}^c$ between 0 and $\pi /2$, where the intratube quadrupolar interactions vanish and change signs. Adopting a pseudospin language with regard to the two 1D systems, the system undergoes a spin-gap transition and displays a zigzag density pattern, above ${\theta }_{B,2}^c$ and below ${\theta }_{B,1}^c$. Between the two magic angles, we show that polarized triplet superfluidity and a planar spin-density-wave order compete with each other. The latter corresponds to a bond-order solid in higher dimensions. We demonstrate that this order can be further stabilized by applying a commensurate periodic potential along the tubes.

Figure 1

(a) Sketch of an atom with a quadrupole moment and of two coupled one-dimensional (1D) systems. The unit vector $\widehat{B}$ indicates the direction of the external magnetic field, along which the quadrupoles are aligned. We assume that $\widehat{B}$ is in the $y$-$z$ plane. The angle between $\widehat{B}$ and the $z$ axis is ${\theta }_B$. The distance between the two 1D systems is $d$, and $a$ is the length scale of the confinement wave functions. $V_R^0$ and $V_R^1$ denote the intra- and intertube interactions, respectively. In (b), (c), and (d) we sketch the three quantum phases SDW${}_z$, TS${}_{\pm }$, and SDW${}_{x,y}$ that occur in the phase diagram. SDW${}_z$ order corresponds to a spontaneous zigzag density order, TS${}_{\pm }$ corresponds to $p$-wave pairing in each tube. SDW${}_{x,y}$ corresponds to an order parameter that consists of one atom in one tube and one hole in the other. These are depicted by a filled and a transparent symbol, respectively. We elaborate further on these quantum phases in the text.

Figure 2

(a) The intratube interaction $V_0\left(k\right)$ and (b) the intertube interaction $V_1\left(k\right)$ at momenta $k=0$ and $k=2k_F=n\pi$ versus the angle ${\theta }_B$ are shown. We choose $d=4a$, the density $n={10}^{-3}$ nm${}^{-1}$ and $a=50$ nm. We note that there are two magic angles ${\theta }_B^c=0.53$ and $1.22$ where the intratube interaction vanishes.

Figure 3

(a) Quantum phase diagram as a function of the distance apart, $d$, of the two tubes and of the angle ${\theta }_B$. We choose the density $n={10}^{-3}$ nm${}^{-1}$ and $a=50$ nm. The shaded regime indicates a spin-gapped state. (b) The corresponding regimes of ($K_{\rho }<1,K_{\sigma }<1$), ($K_{\rho }<1,K_{\sigma }>1$), and ($K_{\rho }>1,K_{\sigma }>1$).

Figure 4

(a) Quantum phase diagram and (b) the corresponding regimes of ($K_{\rho }<1,K_{\sigma }<1$), ($K_{\rho }<1,K_{\sigma }>1$), and ($K_{\rho }>1,K_{\sigma }>1$) for a lower density than in Fig. 3, specifically $n=5\times {10}^{-4}$ nm${}^{-1}$. We again choose $a=50$ nm.

Figure 5

(a) Quantum phase diagram and (b) the corresponding regimes of ($K_{\rho }<1,K_{\sigma }<1$), ($K_{\rho }<1,K_{\sigma }>1$), and ($K_{\rho }>1,K_{\sigma }>1$) for density $n=2.5\times {10}^{-4}$ nm${}^{-1}$ and $a=50$ nm.

Figure 6

Quantum phase diagram for $n={10}^{-3}$ nm${}^{-1}$ and $a=50$ nm, in the presence of umklapp scattering. For the strength of umklapp scattering we choose $g_u=0.02$ in (a) and $g_u=0.05$ in (b).