Isospin correlations in two-partite hexagonal optical lattices

Two-component mixtures in optical lattices reveal a rich variety of different phases. We employ an exact diagonalization method to obtain the relevant correlation functions in hexagonal optical lattices which characterize those phases. We relate the occupation difference of the two species to the magnetic polarization. “Iso” -magnetic correlations disclose the nature of the system, which can be of easy-axis type, bearing phase segregation, or of easy-plane type, corresponding to super-counter-fluidity. In the latter case, the correlations reveal easy-plane segregation, involving a highly entangled state. We identify striking correlated supersolid phases appearing within the superfluid limit.

Figure 1

Numerical results for the on-site square of the spin components, easy-axis (a) and easy-plane (b). Insets of (a) depicts the FM $\left(V>U\right)$ and AFM $\left(U>V\right)$ phase, with the broken lines marking the unit cell boundaries.

Figure 2

(a) Numerical results for the particle fluctuation $\langle \Delta {\widehat{n}}_{}\rangle$ as a function of the on-site interactions, $U$ and $V$. A low-mobility MI phase (dark blue) and a high-mobility SF phase (red to cyan) are apparent. The dashed line defines the boundary between both limits (arbitrary). (b) Numerical results of ${\mathcal{G}}_1^z$ as a function of $U$ and $V$. The sign of ${\mathcal{G}}_1^z$ determines the magnetic phases: FM (blue-purple-black) and AFM (red-yellow).

Figure 3

Numerical results for easy-axis (a)–(c) (note the different color scales) and easy-plane (d) and (e) correlation functions for the first [(a) and (d)], second [(b) and (e)], and third [(c) and (f)] NNs, for a unit cell with eight sites.

Figure 4

First (dark, red), second (light gray, green), and third (dark gray, blue) NN (a) easy-plane pair-correlation function, ${\mathcal{G}}_n^{\parallel }$ and (b) doubly occupied flip-flop pair-correlation function, ${\kappa }_n$. Vertical broken (solid) lines: boundaries of SS (SF) phase. All the calculations were performed with $V=50t$. Inset: supersolid configuration. The black broken lines mark the unit cell boundaries.

Figure 5

Expectation value of NN population difference $\langle n_B^i-n_B^{i+1}\rangle$ calculated for ${\epsilon }_A=0.1t$ as a function of $U$ and $V$ (a) and for $V=50t$ as a function of $U$ (b). Inset: $V^{\prime }$ defines two alternating sublattices (dark, red and light gray, blue).

Figure 6

FM phase of an (a) 8- and (b) 10-site unit cell. The former is a stripe phase, whereas the latter presents phase separation. The broken lines depict the boundaries of two unit cells in either case. The numbering relates to the neighbor count in the computational scheme.

Figure 7

Numerical results for easy-axis (a-c) and easy-plane (d-e) correlation functions for the first [(a) and (d)], second [(b) and (e)], and third [(c) and (f)] NN, for a 10-site unit cell. Note the different color scales used in (a)–(c).