Hidden String Order in a Hole Superconductor with Extended Correlated Hopping

Ultracold fermions in one-dimensional, spin-dependent nonoverlapping optical lattices are described by a nonstandard Hubbard model with next-nearest-neighbor correlated hopping. In the limit of a kinetically constraining value of the correlated hopping equal to the normal hopping, we map the invariant subspaces of the Hamiltonian exactly to free spinless fermion chains of varying lengths. As a result, the system exactly manifests spin-charge separation and we obtain the system properties for arbitrary filling: ground state collective order characterized by a spin gap, which can be ascribed to an unconventional critical hole superconductor associated with finite long range nonlocal string order. We study the system numerically away from the integrable point and show the persistence of both long range string order and spin gap for appropriate parameters as well as a transition to a ferromagnetic state.

Figure 1

Examples of the configurational mapping from a $2L=16$-site zigzag lattice (ZL) to the charge lattice (CL) for OBC. $\downarrow$ ($\uparrow$) particles are filled circles on the lower (upper) rail of the ZL. Arrows beyond the edges of the ZL are the auxiliary fixed boundary variables needed for the classification of boundary segments in the mapping. (a) $N_t=6$ particles in the sequence $\mathrm{\Sigma }=(\downarrow \uparrow \downarrow \downarrow \uparrow \uparrow )$. There are $\mathbf{d}=(2,0,2,3,2,1,0)$ holes interjected by particles from the left to right edge on the ZL. So $M_0=2$ and the length $L_{\mathrm{eff}}=10$. From Eq. (2), $\mathbf{h}=(1,0,1,1,1,0,0)$ holes on the CL. (b) $N_t=7$ and $\mathrm{\Sigma }=(\uparrow \uparrow \downarrow \downarrow \uparrow \uparrow \downarrow )$ with $\mathbf{d}=(3,1,0,3,0,1,0,1)$ holes. So $L_{\mathrm{eff}}=9$ ($M_0=5$). Using Eq. (2), $\mathbf{h}=(1,0,0,1,0,0,0,0)$.

Figure 2

(a),(b) In the sequence ${\mathrm{\Sigma }}_0$ hosting the global ground state, nearest neighbor holes are uniquely paired in all configurations (above). Each single particle hop is equivalent to a hole-pair hop—e.g., two single hops lead from (a) to (b). (Below) The bound hole-pairs (green) are fictitious particles (filled squares) that hop on the edges (squares) of the ZL. Unique pairing of holes (above) implies simultaneously occupied nearest neighbor sites on the dual lattice are forbidden, i.e., infinite repulsion $J_z$ of paired holes. (c) A single spin flip in the sequence ${\mathrm{\Sigma }}_0$ breaks a bound hole pair leading to the spin gap.

Figure 3

Exact diagonalization (ED) (a),(b) and TEBD algorithm [40, 41] (c),(d) results. (a) $O$ vs $V$ for $X=J$ where $2N=8$, $2L=20$ (black), $2N=8$, $2L=24$ (red), $2N=6$, $2L=24$ (blue). (b) Extrapolated spin gap ${\mathrm{\Delta }}_S^{(\infty )}$ to thermodynamic limit by fitting finite size gaps with quadratic polynomials in $1/L$ (using sizes up to $2L=30$) for $\rho =1/2$ (red) and $\rho =1/3$ (blue) for $V=0$. (c),(d) Finite size scaling $O_S$ vs distance $x$ under OBC for different densities and $V=0$. Data are for $X=1.0J$ (diamonds), $0.6J$ (circles), $0.3J$ (squares), $0.0J$ (triangles). Compared are lattice lengths $2L=24$ (blue), $2L=48$ (red) and $2L=60$ (green) for every $X$.