Unconventional phases of attractive Fermi gases in synthetic Hall ribbons

An innovative way to produce quantum Hall ribbons in a cold atomic system is to use $M$ hyperfine states of atoms in a one-dimensional optical lattice to mimic an additional “synthetic dimension.” A notable aspect here is that the SU($M$) symmetric interaction between atoms manifests as “infinite ranged” along the synthetic dimension. We study the many-body physics of fermions with SU($M$) symmetric attractive interactions in this system using a combination of analytical field theoretic and numerical density-matrix renormalization-group methods. We uncover the rich ground-state phase diagram of the system, including unconventional phases such as squished baryon fluids, shedding light on many-body physics in low dimensions. Remarkably, changing the parameters entails interesting crossovers and transition; e.g., we show that increasing the magnetic field (that produces the Hall effect) converts a “ferrometallic” state at low fields to a “squished baryon superfluid” (with algebraic pairing correlations) at high fields. We also show that this system provides a unique opportunity to study quantum phase separation in a multiflavor ultracold fermionic system.

Figure 1

Kinetic energy: Schematic plot of the $M=2$ SD system with $\text{SU}\left(2\right)$ gauge and Zeeman field. ${\mathbb{A}}_{\zeta ,\zeta }$ are the flavor preserving hoppings and ${\mathbb{A}}_{1,2}$ and ${\mathbb{A}}_{2,1}$ are the flavor-orbital couplings. Note that the $\zeta$ states at any site $i$ are Zeeman split and there is no hopping between them.

Figure 2

Kinetic energy at $\varphi =\pi$: Schematic plot showing the crisscross hopping structure special to the $\varphi =\pi$ case. For $\mathrm{\Omega }\gg {\mathrm{\Omega }}_c$, the low-energy sector is marked by the hatched box. In this limit, an effective theory can be constructed in the low-energy sector by considering two neighboring sites (shown by cyan boxes) to form a unit cell where the states on the odd (even) physical sites can be thought of as spin $\uparrow$ (spin $\downarrow$) states. The onsite energies of different $\zeta$ flavors are ${\omega }_{\zeta }$.

Figure 3

Schematic view of (a) a local $p$-baryon at the site $i$ and (b) a squished $p$-baryon of $\zeta =1$ flavor in the neighboring sites of $i$.

Figure 4

Squished pair-correlation functions: Comparison of DMRG and field theoretic results of the algebraic decay of the nonlocal pair correlation ${\mathrm{\Delta }}_s\left(r\right)$ for $\pi$ flux with $t/U=0.5, \mathrm{\Omega }/U=4, L=160$, and $N=32$. Panels (a), (b), (c), and (d) show the $M=2$, 3, 4, and 5 cases, respectively.

Figure 5

Comparison of the DMRG results of the correlation functions of squished pairs ${\mathrm{\Delta }}_s\left(r\right)$ and usual dimers ${\mathrm{\Delta }}_h\left(r\right)$ of the $M=2$ SD system for $\pi$ flux with $t/U=0.5, \mathrm{\Omega }/U=4, L=160$, and $N=32$ (lines are a guide to the eye).

Figure 6

$M=2$ results: $t/U=0.5, n=0.1$. Phase diagrams in the $\mathrm{\Omega }-\varphi$ plane corresponding to (a) normalized $f_s^{\left(2\right)}$, (b) normalized $f_h^{\left(2\right)}$, and (c) central charge $c$ (DMRG data for $L=80$ sites). $f_s^{\left(2\right)}$ and $f_h^{\left(2\right)}$ are normalized by their maximum values in the $\mathrm{\Omega }-\varphi$ plane shown here ($\mathrm{Max}\left[f_s^{\left(2\right)}\right]=0.0428$ and $\mathrm{Max}\left[f_h^{\left(2\right)}\right]=0.0466$). Symbols depict estimates of transition points by DMRG while the solid lines are guides to the eye. Excitation spectra are shown, along different cut directions of the phase diagrams, first in (d), (e), and (f) vs $\mathrm{\Omega }$ for $\varphi /\pi =0.05$, 0.2, and 1, respectively, and then in (g) vs $\varphi$ for fixed $\mathrm{\Omega }/U=0.8$ ($L=80$). Dashed lines show the transition points between different phases seen in the phase diagrams. In the large $\mathrm{\Omega }>{\mathrm{\Omega }}_c$ limit, variation of the number of Fermi points of the corresponding noninteracting system in the $\mathrm{\Omega }-\varphi$ plane is shown in the left panel of (h). Its right panels show the structure of the Fermi surface with the blue lines showing the chemical potential in the first band $\left[e_1\left(k\right)\right]$ for the chosen filling.

Figure 7

$M=3$ results: $t/U=0.5$ and $n=0.1$. Phase diagrams corresponding to (a) normalized $f_s^{\left(3\right)}$, (b) normalized $f_h^{\left(3\right)}$ and (c) central charge $c$, respectively ($L=40$). $f_s^{\left(3\right)}$ and $f_h^{\left(3\right)}$ are normalized by their maximum values in the $\mathrm{\Omega }-\varphi$ plane shown here. The maximum value of $f_h^{\left(3\right)}$ is $\mathrm{Max}\left[f_h^{\left(3\right)}\right]=0.0566$ and the maximum value of $f_s^{\left(3\right)}$, calculated by “masking” the phase separation (PS) region shown by the hatched lines and setting $f_s^{\left(3\right)}$ to be zero in this region, is $\mathrm{Max}\left[f_s^{\left(3\right)}\right]=0.0226$. Symbols, estimates of transition points by DMRG; solid lines, guides to the eye; dotted lines, numerical estimates of bound-state transitions in the dilute limit; and dashed lines, Lifshitz transitions (also discussed in Appendix pp1). (d)–(f) Excitation spectra along different cuts of the phase diagrams ($L=80$; green dotted lines, extent of the FFLO region). (g) $S_{\mathrm{vN}}$ of a subsystem of size $l$ with fits to the Calabrese-Cardy formula [54] for $L=160$ and $\varphi =\pi$. (h) Variations of onsite populations $\langle n_{i,\zeta }\rangle$ in the phase separation (PS) regime ($L=80$), and clustering of particles near the central site is seen.

Figure 8

Lifshitz transitions occurring in the $\mathrm{\Omega }-\varphi$ plane for the noninteracting $M=3$ SD system with $t=0.5$ and $n=0.1$ are shown here by looking at the number of Fermi points in the noninteracting Fermi surface with the given density.

Figure 9

Population ($M=3$): The variations of the total average populations ($N_{\zeta }$) of different $\zeta$ flavors are shown for the $M=3$ case with $t/U=0.5, n=0.1$, and $L=80$. (a) Variations of $N_{\zeta }$ as a function of $\mathrm{\Omega }$ for the $\varphi =0$ case with the green lines showing the extent of the FFLO region. Similarly, in (b) we show the same for the $\varphi =0.2\pi$ case with the green line showing the transition point from the 3-BF state to the FM state, while in (c) we show the variations of $N_{\zeta }$ as a function of $\varphi$ in the $\mathrm{\Omega }\gg {\mathrm{\Omega }}_c$ with $\mathrm{\Omega }/U=2$. Finally, for the special case $\varphi =\pi$, we show the variations of the effective populations $N_{\zeta }^e$ as a function of $\mathrm{\Omega }$ in (d).

Figure 10

Phase diagram in the dilute limit $n\approx 0$, for (a) $M=2$, (b) $M=3$, and (c) $M=4$. Solid, dotted, and dash-dotted lines indicate various crossings between different few-particle bound-state energies to be the lowest energy for the few-particle system. For the sake of completeness, we add the Lifshitz-transition lines (dashed lines) in the large $\mathrm{\Omega }$ limit.

Figure 11

(a) Entanglement entropy $S_{\mathrm{vN}}$ as a function of the subsystem size $l$ for the $M=2$ case with $\varphi =0.05\pi , t/U=0.5$, and several values of $\mathrm{\Omega }$. The black dotted lines are fits to the Calabrese-Cardy-formula [Eq. (21)] with $c=1$ in the 2-BF ($\mathrm{\Omega }/U=0.05$) and FM ($\mathrm{\Omega }/U=0.28$) phases depending on the data, while the solid yellow line is the same with $c=2$ in the FFLO region ($\mathrm{\Omega }/U=0.14$ and 0.2). The inset shows the average entanglement entropy ${\mathrm{S}}_{\mathrm{vN}}^{\mathrm{av}}$ as a function of $\mathrm{\Omega }/U$ for system sizes $L=40$, 80, and 160 from bottom to top and the green dotted lines denote the extent of the FFLO region as determined by the excitation spectra defined in Eq. (20). (b) The estimated central charge ($c$) using similar procedure as (a) for the $M=3$ case with $\varphi =\pi$ and $t/U=0.5$ as a function of $\mathrm{\Omega }/U$ ($L=80$). Corresponding excitation spectra are shown in Fig. 7. While for the 3-BF ($\mathrm{\Omega }/U\lesssim 0.85$) and the SBF ($\mathrm{\Omega }/U\gtrsim 0.95$) phases we observe $c=1$, the fitting result for the intermediate imbalanced phase may be consistent with $c=2$. The green dotted lines again denote the extent of the imbalanced region as determined by the excitation spectra.

Figure 12

Analysis of the transition from FM to SBF phase as a function of $\varphi$ for $M=2, \mathrm{\Omega }/U=0.8$, and $n=0.1$. (a) Behavior of the fidelity susceptibility ${\chi }_F/L^2$ as a function of $\varphi$ is shown here. The inset shows the linear scaling of the peak height (${\chi }_F^{\max }/L^2$) with the system size. (b) Averaged parity order ${\mathcal{O}}_P^2$ as a function of $\varphi$. The collapse of the data points is shown in the inset (${\varphi }_c\approx 0.55\pi$).

Figure 13

Phase separation ($M=3$): Considering a particular flux $\varphi /\pi =0.6$, we show phase separation in the $M=3$ SD system for $t/U=0.5, n=0.1$, and $L=80$. (a) Behavior of the equation of state of the system, i.e., the density ($n$) as a function of the chemical potential ($\mu$), for different values of $\mathrm{\Omega }$. The chemical potential $\mu$ has been shifted for clarity. The green line shows the density $n=0.1$ under consideration and the $\mathrm{\Delta }N$'s show the steps in the number of particle $N$. (b) Variation of the average occupation at the central site $n_{0,\zeta }$ as a function of $\mathrm{\Omega }$.

Figure 14

Chiral current $J_c$ with $n=0.1$ and $t/U=0.5$ for (a) $M=2$ ($L=80$ rungs) and (b) $M=3$ ($L=40$ rungs). Solid lines are guides to the eye, dotted lines are the numerical estimates of bound-state transitions in the dilute limit, and the dashed lines correspond to the Lifshitz transition. (c) Examples of the local current configurations in the original basis [Eq. (1)] of $M=2$ with $\varphi /\pi =0.05$ and (top to bottom) $\mathrm{\Omega }/U=0.08$, 0.12, 0.16, 0.2, 0.24, 0.28, 0.36, 0.44, and 0.5. The shading marks the current configurations corresponding to the $c=2$ phase.