Laughlin-like States in Bosonic and Fermionic Atomic Synthetic Ladders

The combination of interactions and static gauge fields plays a pivotal role in our understanding of strongly correlated quantum matter. Cold atomic gases endowed with a synthetic dimension are emerging as an ideal platform to experimentally address this interplay in quasi-one-dimensional systems. A fundamental question is whether these setups can give access to pristine two-dimensional phenomena, such as the fractional quantum Hall effect, and how. We show that unambiguous signatures of bosonic and fermionic Laughlin-like states can be observed and characterized in synthetic ladders. We theoretically diagnose these Laughlin-like states focusing on the chiral current flowing in the ladder, on the central charge of the low-energy theory, and on the properties of the entanglement entropy. Remarkably, Laughlin-like states are separated from conventional liquids by Lifschitz-type transitions, characterized by sharp discontinuities in the current profiles, which we address using extensive simulations based on matrix-product states. Our work provides a qualitative and quantitative guideline towards the observability and understanding of strongly correlated states of matter in synthetic ladders. In particular, we unveil how state-of-the-art experimental settings constitute an ideal starting point to progressively tackle two-dimensional strongly interacting systems from a ladder viewpoint, opening a new perspective for the observation of non-Abelian states of matter.

Popular Summary

The quantum Hall effect is one of the most spectacular phenomena in condensed-matter physics. A two-dimensional gas of electrons in a magnetic field exhibits a perfectly quantized electrical conductance—a manifestation of quantum mechanics at the macroscopic scale. Remarkably, despite being composed of whole particles, the quantization can have a fraction of an electron charge. This is the basis of several proposals for fault-tolerant quantum-computing hardware. There are unprecedented possibilities for future quantum technologies if the fractional quantum Hall effect could be created in an ultracold atomic gas, but such a state has not been observed yet. We theoretically show that the one-dimensional analog of this effect appears in several experimentally relevant setups, and we discuss how this result could be verified in the near future.

Our approach relies on the concept of synthetic dimensions, a recently proposed and experimentally verified scheme that interprets atomic states as positions along a lattice. We focus on “synthetic ladders,” namely, systems consisting of two coupled one-dimensional chains. Using several complementary techniques, ranging from analytical methods to numerical ones, we show unambiguous signatures of the fractional quantum Hall effect in such setups by characterizing the current counterflowing on the two chains and by measuring other more sophisticated observables.

We believe that our work opens a path toward the experimental realization of the fractional quantum Hall effect with ultracold atoms and provides the scientific community with landmark results for this research field.

Figure 1

From Laughlin to Laughlin-like states. (a) A two-dimensional FQH system with counterpropagating fractional edge modes (purple arrows) is split into (b) a long array of one-dimensional wires, where the edge modes localize on the extremal ones. (c) Finally, Laughlin-like physics is studied in the limit of two coupled chains (a ladder).

Figure 2

Scheme of the two-leg ladder (spin projections $m=\pm 1/2$) in a magnetic flux $\mathrm{\Phi }$. Longitudinal and transverse hopping parameters are denoted by $t$ and $t_{\perp }$, respectively. The currents for the two spin components $⟨{\widehat{J}}_{\pm 1/2}⟩$ flow in opposite directions (purple arrows).

Figure 3

Numerical results for $J_c$ (in units of $t$) of the bosonic $\nu =1/2$ Laughlin-like state versus flux. The flux is parametrized as $\mathrm{\Phi }=2\pi N_{\mathrm{\Phi }}/(L+1)$. Simulation parameters are $L=120$, $D_{\max }=180$, $t_{\perp }/t={10}^{-2}$, and $N=30$, 40, 50 for (a)–(c), respectively. Several values of $V_{\perp }/t$ are considered. The resonance $\mathrm{\Phi }=2\pi n$ for the $\nu =1/2$ Laughlin-like state appears at $N_{\mathrm{\Phi }}=N$.

Figure 4

Fit of the current data presented in Fig. 3 for $V_{\perp }/t=30$, $L=120$, $N=40$, $t_{\perp }/t={10}^{-2}$ (blue dots) using Eq. (3), where $\mathrm{\Delta }{J}_{\mathrm{fit}}=[J_c(\mathrm{\Phi })-J_{\mathrm{\Phi }}-C_2({\mathrm{\Phi }}_1-\mathrm{\Phi })]/C_1$ (see text), and $J_{{\mathrm{\Phi }}_1}$ and ${\mathrm{\Phi }}_1$ are taken from the cusp maximum at $N_{\mathrm{\Phi }}=40$. We show the data in log-log scale, and fit for $N_{\mathrm{\Phi }}<N_{{\mathrm{\Phi }}_1}$. In the inset, we plot the data in linear scale with the fitted function superimposed.

Figure 5

Numerical results for the $\nu =1/2$ resonance for HCBs, with $L=120$, $N=50$, $t_{\perp }/t={10}^{-1}$, $V_{\perp }/t=30$, and $D_{\max }=400$. (a) Data for $J_c$. We expect the signal of the $\nu =1/2$ Laughlin-like state to occur at $N_{\mathrm{\Phi }}=50$. In the inset, we show the fit with Eq. (3), for $N_{\mathrm{\Phi }}>N_{{\mathrm{\Phi }}_2}$ (see text). (b) Entanglement entropy before the C-IC transition, for $N_{\mathrm{\Phi }}=45$, 46, 47, 48. (c) Central charge as a function of $N_{\mathrm{\Phi }}$. For $48\lesssim N_{\mathrm{\Phi }}\lesssim 51$, the central charge is equal to one, consistent with the presence of a Laughlin-like state.

Figure 6

Current configuration along the ladder for the data in Fig. 5. (a) Data for $N_{\mathrm{\Phi }}=47$ (just before the beginning of the helical region), (b) for $N_{\mathrm{\Phi }}=48$ (just after the beginning of the helical region), (c) for $N_{\mathrm{\Phi }}=51$ (just before the end of the helical region), and (d) for $N_{\mathrm{\Phi }}=52$ (just after the end of the helical region). Blue dots on the sites of the ladder represent the local densities $⟨{\widehat{n}}_{j,m}⟩$.

Figure 7

Transverse current fluctuations as a function of $N_{\mathrm{\Phi }}$ for the data in Fig. 5. We see that the transverse current fluctuations are minimized inside the Laughlin-like region, in agreement with bosonization predictions.

Figure 8

Numerical results for the $\nu =1/2$ resonance for HCBs, with $L=120$, $N=50$, and $D_{\max }=200$. (a) Data for $J_c$ at $V_{\perp }/t=30$ and for different values of $t_{\perp }/t$. (b) $J_c$ at $t_{\perp }/t=0.1$ for different values of $V_{\perp }/t$. (c) Double-occupation probability $⟨{\widehat{n}}_{-1/2}{\widehat{n}}_{+1/2}⟩$ as a function of $V_{\perp }/t$ for the data in (b) with $N_{\mathrm{\Phi }}=50$.

Figure 9

Chiral current for interacting fermions for different values of $U/t$, and $V/t=0$ in Eq. (6). Here, we use $L=120$, $N=30$, $t_{\perp }/t={10}^{-1}$ and vary $U$ as in the legend. The series with $U=\infty$ has been taken by implementing the hard-core limit as a reduction of the dimension of the local Hilbert space, on each site. The arrows indicate the expected values of $N_{\mathrm{\Phi }}$ at which the $\nu =1$ and $\nu =1/3$ resonances occur (the latter expected only if $V\ne 0$), which are $N_{\mathrm{\Phi }}\simeq 15$ and $N_{\mathrm{\Phi }}\simeq 45$, respectively.

Figure 10

Chiral current for the $\nu =1/3$ Laughlin-like state for the model ${\widehat{H}}_{\mathrm{ex}}$. The actual system is at $\nu =1/3$, with $\xi =3$, $L=225$, $\mathrm{\Phi }\simeq 0.658\pi$. The data are obtained through a simulation of a ${\nu }^{\prime }=1$ phase using $L^{\prime }=78$, $N_{\mathrm{\Phi }}^{\prime }=26$ ($\mathrm{\Phi }\simeq 0.658\pi$), $t_{\perp }/t={10}^{-2}$, and $U/t=10$. In the inset, we highlight the behavior described in Eq. (10).

Figure 11

Band structure $E_{\pm }(k)$ (solid black lines). The Fermi energy is plotted inside the lower gapless region (green solid line, the red points represent filled states), in the middle of the gap (green dash-dotted line), and inside the upper gapless region (green dashed line). The condition for which $k_F=k_{\mathrm{SO}}$ defines the $\nu =1$ phase, when the Fermi energy is at midgap (green dash-dotted horizontal line). The band maximum is denoted by $E_1\equiv E_{-}(0)$.

Figure 12

Numerical results for free fermions ($\nu =1$ state), with $L=240$, $N=120$, and $t_{\perp }/t=2\times {10}^{-1}$. We perform the same analysis as in Fig. 5. The analogy of (a) the chiral current and the fit with Eq. (3) in the inset, (b) the EE, and (c) the central charge through the C-IC transition with the data on the $\nu =1/2$ Laughlin-like state (Fig. 5) clearly emerges.

Figure 13

Band structure $E_{\pm }(k)$ (black lines), and corresponding chiral fields, ${\widehat{\varphi }}_{\mu ,m}$ (see Appendix pp1). Density-density interactions are ${\widehat{\psi }}_{\mu ,m}^{†}{\widehat{\psi }}_{\mu ,m}{\widehat{\psi }}_{{\mu }^{\prime },m^{\prime }}^{†}{\widehat{\psi }}_{{\mu }^{\prime },m^{\prime }}$, where $\mu ,{\mu }^{\prime }=R$, $L$ and $m\ne m^{\prime }$. The pair tunneling process ${\widehat{\psi }}_{L,-1/2}{\widehat{\psi }}_{L,+1/2}^{†}{\widehat{\psi }}_{R,-1/2}^{†}{\widehat{\psi }}_{R,+1/2}$ is represented by red arrows.

Figure 14

Numerical results for free fermions ($\nu =1$ Laughlin-like state). (a) Simulation with $L=240$, $N=120$, and $t_{\perp }/t=2\times {10}^{-1}$, as in Fig. 12, with the inclusion of a finite harmonic trap, for different values of $w$. The harmonic trap is implemented as explained in the text. (b) Simulations at finite temperature. We simulate free fermions with $L=240$, $N=120$, and $t_{\perp }/t=2\times {10}^{-1}$ for different values of $k_{B}T$. (c) Ratio between the peak of $J_c$ at temperature $T$ and the peak at zero temperature. To increase numerical accuracy, we simulate free fermions with $L=480$, $N=240$, and with different values of $t_{\perp }/t$.

Figure 15

(a) Numerical data for $S(\ell )$ for different values of $D_{\max }$, for $\ell \in [7:113]$ and $N_{\mathrm{\Phi }}=40$. Other simulation parameters as in Fig. 5. We see that, for $D_{\max }=50$, 100, the EE in the bulk of the system is flat. We start seeing the behavior predicted by Eq. (4) for larger values of $D_{\max }$. For $D_{\max }=400$, we fit the EE as explained in Sec. 2b, and show $S(\ell )$ from Eq. (4) with $c_{\min }=1.77492$ (blue dashed line), $c_{\max }=2.18872$ (purple dashed line), and with the average value $\overline{c}=(c_{\max }+c_{\min })/2=1.98182$ (green full line). (b) Numerical data for $J_c$ for different values of $D_{\max }$. In contrast to the data on the EE (a), the chiral current is not drastically affected by the finite value of $D_{\max }$. The helical region remains indeed visible, and for $D_{\max }=400$ the square-root behavior emerges, for $\mathrm{\Phi }>{\mathrm{\Phi }}_2$ (see Fig. 5).

Figure 16

Numerical data for $S(\ell )$, with $\ell \in [7:113]$, $D_{\max }=400$, and $N_{\mathrm{\Phi }}=49$. Other simulation parameters as in Fig. 5. We fit the EE as explained in Sec. 2b, and show $S(\ell )$ from Eq. (4), with $c_{\min }=0.986867$ (blue dashed line), $c_{\max }=1.37001$ (purple dashed line), and with the average value $\overline{c}=(c_{\max }+c_{\min })/2=1.178438$ (green full line). The fact that we do not exactly fit $c=1$ can be due to both finite-size effect and nonperfect convergence of the numerical algorithm.

Figure 17

Band structure $E_{\pm }(k)$ (solid black lines). Red points represent filled states, and the Fermi energy is represented by the solid green line. The four points ($h=1$, 2, 3, 4), identifying left edge, left bulk, right bulk, and right edge, respectively, label the four different dispersion relations in Eq. (d1), and $E_1=E_{-}(0)$.