Synthetic-gauge-field-induced resonances and Fulde-Ferrell-Larkin-Ovchinnikov states in a one-dimensional optical lattice

Coherent coupling generated by laser light between the hyperfine states of atoms, loaded in a one-dimensional (1D) optical lattice, gives rise to the “synthetic dimension” system which is equivalent to a Hofstadter model in a finite strip of square lattice. An SU($M$) symmetric attractive interaction in conjunction with the synthetic gauge field present in this system gives rise to unusual effects. We study the two-body problem of the system using the $T$-matrix formalism. We show that the two-body ground states pick up a finite momentum and can transform into two-body resonancelike features in the scattering continuum with a large change in the phase shift. As a result, even for this 1D system, a critical amount of attraction is needed to form bound states. These phenomena have spectacular effects on the many-body physics of the system analyzed using the numerical density matrix renormalization group technique. We show that the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states form in the system even for a “balanced” gas and the FFLO momentum of the pairs scales linearly with flux. Considering suitable measures, we investigate interesting properties of these states. We also discuss a possibility of realization of a generalized interesting topological model, called the Creutz ladder.

Figure 1

Schematic plot of the synthetic dimension system with atoms in the $F=1$ ($M=3$) ground-state manifold in the transformed basis. The “physical sites” of the 1D optical lattice are labeled by $i$ and the hyperfine states are labeled by $\sigma =1$, 2, and 3. Tunneling along the physical direction is with the amplitude $t$ and the phase factor ${\theta }_{\sigma }$ (here, $j=\sqrt{-1}$). The spin flip hopping along the synthetic direction accompanies the amplitude ${\mathrm{\Omega }}_{\sigma }$ in going from $\sigma \to \sigma +1$ at a particular physical site $i$. Thus moving around a plaquette of this synthetic ladder gives rise to the flux $\varphi$. The SU($M$) symmetric two-body interaction is of strength $U$ and is “long-ranged” in the synthetic direction shown by the dashed-dotted curves.

Figure 2

Single-particle dispersion of the first band for the $M=2$ SD system with $\mathrm{\Omega }/t=1$ and different values of $\varphi$.

Figure 3

Two-body bound-state properties of a fermionic SD system with $M=2$. Panels (a) and (b) respectively show the variations of the pairing threshold ($E_{\mathrm{th}}^p$) and the bound-state energy ($E_{\mathrm{bs}}$) as a function of the canonical COM $Q$ with $\mathrm{\Omega }/t=1$. The minima of $E_{\mathrm{th}}^p$ [see panel (a)] and $E_{\mathrm{bs}}$ [see panel (b)] occur at $Q=Q_g$ and it scales as $Q_g\propto \varphi$ [inset of panel (b)]. We note from panel (a) that $E_{\mathrm{bs}}\left(Q\right)$ of $\varphi =0$ (which is with $U/t=4$) can be just above $E_{\mathrm{th}}^p\left(Q\right)$ of $\varphi \ne 0$ in some regime of $Q$ (hatched regimes below the black curve). In this situation, the fate of the bound states is explored in panels (c) and (d) focusing at $Q=0$. Panels (c) and (d) respectively show the total PDOS $\left[g_p\left(\omega \right)\right]$ and the phase shift $\left[\delta \right(\omega \left)\right]$ as a function of energy ($\omega$) with $\mathrm{\Omega }/t=1$. Finally, in panels (e) and (f), we show as a function of $\varphi$ the behaviors of the mass ($m$) and the binding energy ($E_b$), respectively, for the strongest bound state occurring at $Q=Q_g$ with $U/t=5$ (which is larger than $U/t=4$ at which resonances occur). Here, $m_0$ and $E_b^0$ are the values of $m$ and $E_b$, respectively, for a 1D free Fermi gas ($\varphi =\mathrm{\Omega }=0$).

Figure 4

Synthetic-gauge-field-induced FFLO states. Variation of the pair momentum distribution function $\mathcal{X}\left(k_n\right)$ for the $M=2$ (fermionic) case with $\mathrm{\Omega }/t=1,L=100$ (number of physical sites), and total density of particles $n=0.3$, as a function of the pair momentum ($k_n$) for different values of $\varphi$. Panel (a) shows that the peak of $\mathcal{X}\left(k_n\right)$ shifts to a finite value of $k_n=Q_F$ as $\varphi$ becomes nonzero, signaling the emergence of the FFLO states. The FFLO momentum $Q_F\propto \varphi$ [shown in the inset of panel (a)]. This has its origin in the formation of finite-momentum dimers shown in Fig. 3. Panel (b) shows that at a fixed $\varphi$, the FFLO peak disappears continuously as $U$ decreases.

Figure 5

Properties of the FFLO states for $M=2,\mathrm{\Omega }/t=1,n=0.3$, and $L=100$. Panels (a) and (b) respectively show the variations of the peak anomaly ($\mathcal{P}$) and the area ($\mathcal{A}$) under the $\mathcal{X}\left(k_n\right)$ curve as a function of $U$ for different values of $\varphi$. Here, ${\mathcal{A}}_0$ is the value of $\mathcal{A}$ at $U=0$.

Figure 6

Local and nonlocal correlation functions. Parameters are $M=2,\mathrm{\Omega }/t=1,\varphi /\left(2\pi \right)=0.2,n=0.3$, and $L=100$. (a) Variations of the local average density of particles $\langle n_{i,1}\rangle$ in the lowest $\sigma =1$ leg as a function of the site number $i$ are shown for two different values of $U$. In the inset, we show the variation of the difference $\mathrm{\Delta }{n}_i=\langle n_{i,1}\rangle -\langle n_{i,2}\rangle$ for the same $U$ values. (b) Behaviors of the nonlocal correlation functions ${\mathcal{X}}_0,{\mathcal{C}}_1$, and ${\mathcal{C}}_2$ (defined in the text) with respect to the central site at $L/2$ are shown. We note that the FFLO correlation (${\mathcal{X}}_0$) is the slowest to decay.