Correlations and enlarged superconducting phase of $t-J_{\perp }$ chains of ultracold molecules on optical lattices

We compute physical properties across the phase diagram of the $t-J_{\perp }$ chain with long-range dipolar interactions, which describe ultracold polar molecules on optical lattices. Our results obtained by the density-matrix renormalization group indicate that superconductivity is enhanced when the Ising component $J_z$ of the spin-spin interaction and the charge component $V$ are tuned to zero and even further by the long-range dipolar interactions. At low densities, a substantially larger spin gap is obtained. We provide evidence that long-range interactions lead to algebraically decaying correlation functions despite the presence of a gap. Although this has recently been observed in other long-range interacting spin and fermion models, the correlations in our case have the peculiar property of having a small and continuously varying exponent. We construct simple analytic models and arguments to understand the most salient features.

Figure 1

Phase transition and crossover lines obtained by the DMRG after extrapolating to the thermodynamic limit: (a) $t-J_{\perp }$ chain with NN interactions (4) (also see Ref. [40]); (b) $t-J_{\perp }$ model with NNN interactions (5); and (c) dipolar $t-J_{\perp }$ model (3). The different regions are SDW, the two-channel Luttinger liquid (LL) with dominant spin-density wave (SDW) correlations; SS+SG, the singlet superconductor (SS) with finite spin gap (6) (SG), a one-channel LL phase; PS, phase separation, characterized by a vanishing inverse compressibility (7); and CDW, the one-channel LL phase with dominating charge-density wave (CDW) correlations and finite spin gap. The dotted lines indicate constant values of the Luttinger parameter $K_{\rho }$ and the blue line indicates $K_{\rho }=1$. At the green line, the spin gap becomes larger than $3\times {10}^{-3}t$ (estimated numerical accuracy after extrapolating to the thermodynamic limit) upon increasing $J_{\perp }/t$. The purple line indicates the onset of phase separation. We display only results for $J_{\perp }/t>3$, since for smaller values the systems seem to be in the SDW phase. Note that in (b) and (c) we present numerical results for the phase transition and crossover lines only for densities $n\le 0.7$, as described in the text. Also note that in (a) we do not further distinguish between triplet SC (TS) and SS, since we are neglecting logarithmic corrections, which can make the TS channel dominant [58, 59].

Figure 2

Spin gap (6) as a function of filling $n$ and $J_{\perp }/t$ for the NN model (4). The short-dashed line indicates ${\mathrm{\Delta }}_S/t=3\times {10}^{-3}$, which was used in Fig. 1 as the border of the spin-gapped region. The long-dashed line indicates the onset of the phase separation region, as in Fig. 1. The inset shows typical finite-size scaling behavior.

Figure 3

Luttinger liquid parameter $K_{\sigma }$ for the NN model (4) as a function of the filling $n$ and of $J_{\perp }/t$ obtained from fitting the structure factor of the longitudinal spin correlations (13) as discussed in the text. The short-dashed line indicates ${\mathrm{\Delta }}_S/t=3\times {10}^{-3}$, which was used in Fig. 1 as the border of the spin gapped region. The long-dashed line indicates the onset of phase separation, as in Fig. 1.

Figure 4

Typical fits for obtaining the Luttinger liquid parameters of the NN model (4) from the structure factors of the respective correlation functions. (a) Fit of the slope for $k\to 0$ in (a) the charge structure factor from Eq. (9) and (b) the structure factor of the longitudinal spin correlation function (15). The insets show the extrapolation to the thermodynamic limit: left panel, $K_{\rho ,\sigma }$; right panel, value of $N\left(k\right)$ or $S\left(k\right)$ at $k=0$.

Figure 5

(a) Inverse compressibility ${\kappa }^{-1}$ [Eq. (7)] for different values of the filling $n$ as a function of $J_{\perp }/t$ for the system with nearest-neighbor interactions (4). (b) Line in the $(n,J_{\perp })$ plane at which ${\kappa }^{-1}=0$, indicative of the phase separation region for the chain with NN [Eq. (4), purple squares], NNN [Eq. (5), green circles], and dipolar [Eq. (3), blue triangles] interactions.

Figure 6

Example of a fit of the density-density correlation function (8) using Eq. (24).

Figure 7

Typical fits of the structure factors to obtain (a) $K_{\rho }$ and (b) $K_{\sigma }$ for the system with dipolar interactions. The results look similar for the NNN case. The insets show extrapolations to the thermodynamic limit of the values of the Luttinger liquid parameters as well as of the values of the structure factors at $k=0$.

Figure 8

Typical results for the exponents of the correlation functions of the system with dipolar interactions (3): the dominant exponents are shown at filling (a) $n=0.4$ and (b) $n=0.8$ as a function of $J_{\perp }/t$. The dotted vertical lines in (a) indicate the position of the intermediate CDW phase obtained from the spin gap and the $K_{\rho }=1$ line in Fig. 1.

Figure 9

Spin gap ${\mathrm{\Delta }}_S/t$ at filling $n=0.1$ for the usual $t-J$ chain [Eq. (2), data from Ref. 60], the $t-J_{\perp }$ chain with NN interactions [Eq. (4)], with NNN interactions [Eq. (5)], and with dipolar interactions [Eq. (3)] as a function of $J_{\perp }/t$ ($J/t$ for the standard $t-J$ chain).

Figure 10

The DMRG results for the spin gap (6) for two particles on a lattice with $L=100$ sites for systems with NN interactions: red squares: standard $t-J$ model (2); green down triangles, chain with $J_z=0$ but $V=-J_{\perp }/4$; blue up triangles, $V=0$ but $J_z=J_{\perp }$; and magenta circles, $J_z=V=0$ as in the NN $t-J_{\perp }$ chain (4).

Figure 11

(a) Estimate of the size of the superconducting region (in units in which $t\equiv 1$) at low fillings as a function of $\alpha =J_z/J_{\perp }$ and $\beta =V/J_{\perp }$. The result is obtained from the difference of Eqs. (34) minus (33). In the yellow region, the size is greater than or equal to 25; in the white region, $J_c^{\left(1\right)}>J_c^{\left(2\right)}$, indicating the absence of SC. (b) and (c) Size of the SC region when keeping $\alpha$ or $\beta$ fixed, as indicated. The vertical dashed and dotted lines show the position of the poles of $(J_c^{\left(2\right)}-J_c^{\left(1\right)})/t$ at which the value becomes negative, indicating the absence of the SC phase.

Figure 12

Effect of truncating the range of the dipolar interactions on algebraically decaying correlation functions, here the density-density correlation function for $n=0.2$ and $J_{\perp }/t=6$. The plot displays results for NN interactions, for a truncation in the interaction range after three and ten sites, and results for the full range of the interactions, as indicated.

Figure 13

Effect of long-range ${1/|i-j|}^3$ interactions on the transverse spin correlation functions. The results shown are at $n=0.2$ and $J_{\perp }/t=8$ for dipolar interactions truncated after three and ten lattices sites and for the full range of interactions, as indicated. The black line is a fit of an algebraic function with exponent 8.7 in the long-distance part in the case of full-range interactions.

Figure 14

Results for the exponent of the algebraic tail for the dipolar $t-J_{\perp }$ chain (3) as function of $J_{\perp }/t$ at fillings $n=0.1$ (red squares) and $n=0.2$ (green circles), obtained from fits to the transverse spin correlation function.