Superfluidity in the absence of kinetics in spin-orbit-coupled optical lattices

At low temperatures bosons typically condense to minimize their single-particle kinetic energy while interactions stabilize superfluidity. Optical lattices with artificial spin-orbit coupling challenge this paradigm, because here kinetic energy can be quenched in an extreme regime where the single-particle band flattens. To probe the fate of superfluidity in the absence of kinetics we construct and numerically solve interaction-only tight-binding models in flatbands. We find that superfluid states arise entirely from interactions operating in quenched kinetic energy bands, thus revealing a distinct and unexpected condensation mechanism. Our results have important implications for the identification of quantum condensed phases of ultracold bosons beyond conventional paradigms.

Figure 1

Energy versus wave vector for the lowest two energy bands of Eq. (2) with $k_R=2\pi /a,V_{\mathrm{lat}}=E_R$, for (a) the one-dimensional system, with ${\mathrm{\Omega }}^{*}=8.88E_R$, and (b) the two-dimensional system, with ${\mathrm{\Omega }}^{*}=8.31E_R$.

Figure 2

(a) The flatness ratio (defined as the ratio of the gap between the two lowest bands to the width of the lowest band [9]) versus Rabi frequency for $k_R=2\pi /a$ and $V_{\mathrm{lat}}=E_R$ in one dimension. (b) The hopping parameters $t$ and $t^{\prime }$ against the Rabbi frequency. (c) The ratio of interaction parameters $V, P$, and $A$ to $U$. The right column (d)–(f) plots the same quantities for a two-dimensional system (with the same $k_R$ and $V_{\mathrm{lat}}$). Choosing $\mathrm{\Omega }$ to lie at the peak leads to Eq. (3).

Figure 3

The magnitude of the SF order parameter $\left|〈\widehat{a}〉\right|$ against $\mu$ and $A(=V)$ in the model Eq. (3). $z=2(z=4)$ is the coordination number in one (two) dimensions. The MI and CDW phases have $\left|〈\widehat{a}〉\right|=0$ and the SF and SS phases have $\left|〈\widehat{a}〉\right|\ne 0$. The boundary between the SF and the SS phase is set at $\mathrm{var}〈\widehat{a}〉=0.001$. The inset shows the superfluid spin texture (Appendix pp3) in a unit cell for the two-dimensional system in the superfluid phase at its optimal flatness point as in Fig. 1, leading to $V\approx A\approx 0.01U$.

Figure 4

(a) The gapped phase (MI with 1 particle per site) boundary defined by plotting the chemical potential against $A(=V)$ for Eq. (3). (b) The single-particle density matrix $\langle {\widehat{a}}_0^{†}{\widehat{a}}_r\rangle$ versus distance $r$ in the compressible phase [star in (a), with $\mu =U$ and $A=V=0.01U$ ] on a log-log scale. (c) The same as (b) but in the gapped phase [diamond in (a), with $\mu =U$ and $A=V=0.01U$] in the log-linear scale. The lines are a guide to the eye. Note that the chosen values of $A$ in (b) and (c) are consistent with the estimated values for ${}^{87}\mathrm{Rb}$.

Figure 5

The spin textures [Eqs. (C2)] for the Mott state (upper panel) and for the superfluid state (lower panel) in one unit cell at the same parameters as in Fig. 2 in the main text.