Chiral and pair superfluidity in triangular ladder produced by state-dependent Kronig-Penney lattice

Abstract

We propose a concrete realization of a triangular ladder for ultracold atoms, which simultaneously hosts geometric frustration and unusual two-body interactions, and in particular controllable pair hopping and density-induced tunneling. This is done by means of a spin-dependent Kronig-Penney lattice created using a spatially-dependent tripod-type atom-light coupling. We apply density matrix renormalization group (DMRG) calculations to derive the quantum phase diagram. We find that pair tunneling stabilizes a robust pair superfluid, characterized by power-law decay of pair correlations. Additionally, a chiral superfluid arises from frustration induced by competing nearest neighbor (NN) and next-nearest neighbor (NNN) tunnelings. Finally, in the high barrier regime, we map our system onto the XXZ spin model and find the exact phase transition points.

Figures (6)

• Figure 1: (a) Tripod atom-light coupling configuration for ultracold atoms. (b) Spatial dependence of the modulus of the Rabi frequencies $\Omega_{l}$ ($l\in \left\{1,2,3\right\}$) obtained from Eq. \ref{['eq:Tripod-Omegas']} for $\epsilon=0.1$.
• Figure 2: (a) The geometric scalar potential $\phi\approx\hbar^2\theta^{\prime 2}/2m$EdvinasScipost in the units of the recoil energy $E_{\mathrm{R}}=\hbar^2k^2/2m$ for $\epsilon=\Omega_\mathrm{p}/\Omega_\mathrm{c}=0.1$. The potential represents a periodic array of barriers that act in the alternating manner on the symmetric (green) or antisymmetric (yellow) superposition of atomic ground states $|\pm\rangle$. (b) Schematic representation of the corresponding Wannier functions of the lowest (the $s$) Bloch band for atoms in the symmetric (green) or antisymmetric (yellow) superposition states $|\pm\rangle$.
• Figure 3: (a) Triangular ladder with opposite-sign single-particle hoppings $J_1$ and $J_2$, yielding effective geometric frustration. Consequently, each triangular plaquette has a $\pi$-flux. (b) The corresponding linearized 1D chain described by Hamiltonian of Eq. \ref{['eq:WannierBasis']} containing the NN and NNN couplings $J_1$ and $J_2$.
• Figure 4: (a) Ground state phase diagram for the Hamiltonian given by Eqs. \ref{['eq:HMain']}-\ref{['eq:HIntMain']} with $D=0$ (i.e., $g_z=0$), showing the dependence of the phases on parameter ratios $g_x/g_0$ and $J_1/g_0G_{000}$. Solid lines indicate sharp phase transitions, while dotted line specifies a BKT transition. Yellow markers for $J_1=0$ (corresponding to the horizontal $g_x/g_0$ axis) indicate exact transition points determined by the spin mapping, explained in Sec. \ref{['sec:spin-mapping']}. The $g_x$ dependence of: (b) even parity order parameter $O_{\rm even}\left(\Delta j\right)$, defined in Eq. \ref{['eq:even-parity']}; (c) chirality-chirality correlation $\kappa_2 \left(\Delta j\right)$, defined in Eq. \ref{['eq:chirality-chirality-corr']}; (d) the structure factor $S\left(k\right)$ at $k=\pi$, defined in Eq. \ref{['eq:struct-fact']}; (e) pair correlation function $C_2\left(\Delta j\right)$, defined in Eq. \ref{['eq:C2-corr']}; for $J_1/g_0G_{000}=0.075$ and $\Delta j = 159$. Note that $C_1\left(\Delta j\right)$ is non-zero for long ranges in the CSF region, but it is not shown for simplicity.
• Figure 5: Energies per site $e$ vs. the interaction strength ratio $g_{x}/g_{0}$. The Mott insulator energy per site $e_{{\rm MI}}$ is represented by a solid blue line, while AFM energy per site $e_{{\rm AFM}}$ is represented by a dashed red line.