Synthetic helical liquids with ultracold atoms in optical lattices

We discuss a platform for the synthetic realization of key physical properties of helical Tomonaga Luttinger liquids (HTLLs) with ultracold fermionic atoms in one-dimensional optical lattices. The HTLL is a strongly correlated metallic state where spin polarization and propagation direction of the itinerant particles are locked to each other. We propose an unconventional one-dimensional Fermi-Hubbard model which, at quarter filling, resembles the HTLL in the long wavelength limit, as we demonstrate with a combination of analytical (bosonization) and numerical (density matrix renormalization group) methods. An experimentally feasible scheme is provided for the realization of this model with ultracold fermionic atoms in optical lattices. Finally, we discuss how the robustness of the HTLL against backscattering and imperfections, well known from its realization at the edge of two-dimensional topological insulators, is reflected in the synthetic one-dimensional scenario proposed here.

Figure 1

(a) Schematic of the two-component model Hamiltonian (3) for $b={\alpha }_R=1$, with hopping $t$ between sites of the same component, and operations $S^{+(-)}$ associated with a change in component and a hopping from left (right) to right (left), respectively. (b) Schematic of a model projected onto the lower band, resulting in an effective single component model. Gray ovals denote operators ${\gamma }_j$ of the effective nearest neighbor hopping model in the lower band [cf. Eq. (9)]. Physical lattice sites are denoted by dashed lines. (c) Band structure of the Hamiltonian (3) for $t=b={\alpha }_R=1$ and chemical potential $\mu$ at quarter filling. The coloring of the plots visualizes the momentum-dependent spin polarization of the fermions, where red denotes spin up and blue spin down.

Figure 2

Proposed setup for the implementation of the spin-dependent hopping in Eq. (3) at the point ${\alpha }_R=b$ using ${}^{173}\mathrm{Yb}$. The two spin states are given by the nuclear Zeeman states of the ${}^1{S}_0$ ground state $|m_f=-\frac{5}{2}\rangle , |m_f^{\prime }=-\frac{1}{2}\rangle$, subject to a Zeeman splitting of $2{\mathrm{\Delta }}_z$. Raman assisted tunneling with lasers with Rabi frequencies ${\mathrm{\Omega }}_{1/2}$, frequencies ${\omega }_{1/2}$, and polarizations ${\sigma }^{+/-}$, respectively, couple nearest neighbor sites via a ${|}^3{P}_1\rangle$ virtual excited state, detuned by an amount $\delta$. The lattice is tilted with equal tilting $\mathrm{\Delta }$ of the two spin states.

Figure 3

The decay exponent $\xi$ extracted from a fit to $|r_1-r_2{|}^{-2\xi }$ as a function of the Hubbard interaction $U$, as extracted from DMRG measurements of the correlation function $\langle S^z\left(r_1\right)S^z\left(r_2\right)\rangle$ (solid line) and $\langle S^y\left(r_1\right)S^y\left(r_2\right)\rangle$ (dashed line), respectively. Here $r_1=3L/4$ and $r_2=L/4$, where the system size $L$ is varied between 40 and 140 sites. Different colors correspond to separation of the bands increasing from $b={\alpha }_R=t$ to $b={\alpha }_R=50t$. Additionally, the solution of the Bethe ansatz is shown for comparison. The value of $\xi =0.5$ indicating the phase transition between the Luttinger liquid and charge-density wave phase is shown with a dashed black line as a guide for the eye. The spread around $\xi =1$ at $U=0$ gives an indication of the magnitude of the numerical error. Inset: Correlation function $\langle S^z\left(r_1\right)S^z\left(r_2\right)\rangle$ as a function of $|r_1-r_2|$, and the best fit from which the exponent $\xi$ can be extracted. Here all data is at ${\alpha }_R=b=10t$, and the colors indicating $U=0$ increasing to $U=4t$.

Figure 4

Upper panel: Decay exponent $\xi$ from fit to $|r_1-r_2{|}^{-2\xi }$ as a function of the Hubbard interaction $U$, as extracted from DMRG measurements of the correlation function parallel $\langle {S\Vert }^{(}{r}_1){S\Vert }^{(}{r}_2)\rangle$ (dashed line) and perpendicular $\langle S^{\perp }\left(r_1\right)S^{\perp }\left(r_2\right)\rangle$ (solid line) to the spin quantization axis. Here $r_1=3L/4$ and $r_2=L/4$, where the system size $L$ is varied between 40 and 140 sites. Different colors correspond to $b=4.2t$ to $b=5t$, with ${\alpha }_R=5t$ fixed for every value of $b$. Lower panel: Band structure of the lower band for the parameters of the upper panel.